Stochastic models of intracellular transport

Paul C. Bressloff1,2 and Jay M. Newby2 1Department of Mathematics 2Mathematical Institute, University of Utah University of Oxford, 155 South 1400 East 94-96 St. Giles’, Salt Lake City Oxford, UT 84112 OX1 3LS UK

The interior of a living is a crowded, heterogenuous, fluctuating environment. Hence, a major challenge in modeling intracellular transport is to analyze stochastic processes within complex environments. Broadly speaking, there are two basic mechanisms for intracellular transport: passive diffusion and motor-driven active transport. Diffusive transport can be formulated in terms of the motion of an over-damped Brownian parti- cle. On the other hand, active transport requires chemical energy, usually in the form of ATP hydrolysis, and can be direction specific, allowing biomolecules to be transported long distances; this is particularly important in neurons due to their complex geometry. In this review we present a wide range of analytical methods and models of intracellular transport. In the case of diffusive transport, we consider narrow escape problems, dif- fusion to a small target, confined and single-file diffusion, homogenization theory, and fractional diffusion. In the case of active transport, we consider Brownian ratchets, random walk models, exclusion processes, random intermittent search processes, quasi- steady-state reduction methods, and mean field approximations. Applications include receptor trafficking, axonal transport, membrane diffusion, nuclear transport, - DNA interactions, virus trafficking, and the self–organization of subcellular structures.

PACS numbers: 87.10.-e, 87.16.Wd, 05.40.-a

CONTENTS G. Exclusion processes 48 1. Asymmetric exclusion process and the I. Introduction1 hydrodynamic limit 48 2. Method of characteristics and shocks 49 II. Diffusive transport: First-passage problems3 3. mRNA translation by ribosomes 51 A. Derivation of the diffusion equation3 H. Motor transport and random intermittent search 1. Random walks3 processes 52 2. Langevin equation4 1. Mean-field model 54 B. First passage times6 2. Local target signaling 56 C. Narrow escape problems8 I. Active transport on DNA 57 D. Diffusion-limited reaction rates 10 E. Diffusive search for a protein-DNA binding site 13 V. Transport and self-organization in cells 59 A. Axonal elongation and cellular length control 59 III. Diffusive transport: Effects of molecular crowding, traps B. Cooperative transport of in cellular and confinement 16 organelles 61 A. Anomalous diffusion 16 C. Cell polarity 62 B. Molecular crowding 18 C. Diffusion–trapping models 21 VI. Discussion 64 1. Sequence-dependent protein diffusion along DNA 21 2. Diffusion along spiny dendrites 22 Acknowledgements 64 3. Diffusion in the plasma membrane 25 D. Diffusion in confined geometries 27 References 64 1. Fick-Jacobs equation 27 2. in a periodic potential with tilt 29 3. Single-file diffusion 31 E. Nuclear transport 32 I. INTRODUCTION

IV. Active intracellular transport 35 The efficient delivery of proteins and other molecular A. Modeling molecular motors at different scales 35 1. Brownian ratchets 36 products to their correct location within a cell (intracel- 2. PDE models of active transport 38 lular transport) is of fundamental importance to normal B. Tug-of-war model of bidirectional transport 39 cellular function and development (Alberts et al., 2008). C. Quasi–steady–state (QSS) reduction of PDE models of Moreover, the breakdown of intracellular transport is a active transport 41 D. Fast and slow axonal transport 43 major contributing factor to many degenerative diseases. E. Active transport on microtubular networks 44 Broadly speaking, there are two basic mechanisms for in- F. Virus trafficking 47 tracellular transport; passive diffusion within the cytosol 2 or the surrounding plasma membrane of the cell, and ac- processes that are particularly relevant to intracellular tive motor–driven transport along polymerised filaments transport, some of which have not been emphasised in such as and F- that comprise the cy- other reviews. These include the following: toskeleton. Newly synthesised products from the nucleus 1. Since the aqueous environment (cytosol) of a cell is are mainly transported to other intracellular compart- highly viscous at the length scale of macromolecules (low ments or the cell membrane via a microtubular network Reynolds number), a diffusing particle can be treated as that projects radially from organising centres (centro- an overdamped Brownian particle where inertial effects somes). The same network is used to transport degraded are ignored. cell products back to the nucleus. Moreover, various ani- 2. One of the characteristics of diffusive transport inside mal viruses including HIV take advantage of - the cell is that often a particle is confined to a domain based transport in order to reach the nucleus from the cell with small exits on the boundary of the domain. Exam- surface and release their genome through nuclear pores ples include an ion looking for an open ion channel within (Damm and Pelkmans, 2006). The challenges of intra- the cell membrane (Grigoriev et al., 2002), the trans- cellular transport are particularly acute for brain cells port of newly transcribed mRNA from the nucleus to the (neurons), which are amongst the largest and most com- cytoplasm via nuclear pores (Gorski et al., 2006; Mis- plex cells in , in particular, with regards to the telli, 2008), the confinement of neurotransmitter recep- efficient trafficking of newly synthesized proteins from tors within a synapse of a neuron (Holcman and Schuss, the cell body or soma to distant locations on the axon 2004), and the confinement of calcium and other signal- or dendrites. In healthy cells, the regulation of protein ing molecules within sub cellular compartments such as trafficking within a neuron provides an important mech- dendritic spines (Biess et al., 2011). This has led to recent anism for modifying the strength of synaptic connections interest in using Green’s function and asymptotic meth- between neurons (Bredt and Nicoll, 2003; Choquet and ods to solve the so–called narrow escape problem (Beni- Triller, 2003; Newpher and Ehlers, 2008; Triller and Cho- chou and Voituriez, 2008; Grigoriev et al., 2002; Holcman quet, 2005), and synaptic plasticity is generally believed and Schuss, 2004; Pillay et al., 2010; Schuss et al., 2007; to be the cellular substrate of learning and memory. On Singer et al., 2006a,b). the other hand, various types of dysfunction in protein 3. A related class of problems involves the search for trafficking appear to be a major contributory factor to a small target within the interior of a cellular domain. a number of neurodegenerative diseases associated with In this case it is necessary to extend the Smoluchowski memory loss including Alzheimers (de Vos et al., 2008). theory of diffusion limited reaction rates to bounded do- Over the past 20 years, intracellular transport has been mains or to more complex transport processes than sim- a major application area within the statistical physics ple diffusion. One example is the arrival of a receptor at community, and has driven a large number of papers on a localised reaction site on the surface of an immune cell, the stochastic modeling and analysis of molecular motors which is a key step in the signaling cascade responsible and diffusion in complex environments. Many excellent for activating the cell (Coombs et al., 2009). Another reviews have been written on topics relevant to intracel- important example is a promotor protein searching for lular transport including anomalous diffusion (Bouchaud its binding site on DNA, which is facilitated by an in- and Georges, 1990; Metzler and Klafter, 2000, 2004), termittent search process in which the particle switches molecular motors (Julicher et al., 1997; Keller and Bus- between 3D and 1D diffusion (Berg et al., 1981; Coppey tamante, 2000; Kolomeisky and Fisher, 2007; Lipowsky et al., 2004; Halford and Marko, 2004; Kolomeisky, 2011; and Klumpp, 2005; Reimann, 2002), reaction kinetics Mirny et al., 2009; Sheinman et al., 2012). (Hanggi et al., 1990), confined diffusion (Burada et al., 4. The intracellular environment is extremely crowded 2009), random intermittent search processes (Benichou with macromolecules, subcellular compartments and con- et al., 2011b), and exclusion processes (Chou et al., 2011; finement domains, suggesting that anomalous subdiffu- Evans and Hanney, 2005; Schadschneider et al., 2010). sion is likely to occur (Dix and Verkman, 2008). The However, as far as we are aware, there has not been a plasma membrane is also a complex heterogeneous en- substantial review in which intracellular transport itself vironment (Jacobson et al., 2007; Kusumi et al., 2005; is the central topic. Vereb et al., 2003). Thus, many papers model diffusion The overall goal of the current review is to provide an in such environments in terms of continuous time ran- up to date and unified perspective on stochastic models dom walks and fractional Brownian motion. However, of intracellular transport. One of the major aims is to it is still unclear to what extent intracellular diffusion is cover a wide range of models and analytical methods, anomalous in the long–time limit rather than just at in- highlighting links between them wherever possible. Al- termediate times. This motivates studying diffusion in though it is not possible to cover every topic in complete the presence of obstacles and transient traps whereby detail, sufficient details are provided to make the review normal diffusion is recovered asymptotically (Bressloff as self-contained and pedagogical as possible. Another and Earnshaw, 2007; Santamaria et al., 2006; Saxton, aim of the review is to highlight aspects of stochastic 1994, 2007). 3

5. Another common form of diffusion within cells is crucial role in axonal growth and guidance during neuro- the transport of particles through a narrow biological genesis and cortical development (Goldberg, 2003; Gra- pore or channel (Hille, 2001; Roux et al., 2004; Schuss ham et al., 2006; Suter and Miller, 2011). et al., 2001). Restricting the volume of the phase space The structure of the paper is as follows. In Sec. II, dif- available to the diffusing particles by means of confining fusive transport is developed from the perspective of the boundaries or obstacles causes striking entropic effects for an overdamped Brownian particle, (Burada et al., 2009). Moreover, various mechanisms of and various first passage time problems are considered facilitated diffusion can occur through interactions be- (points 1-3). In Sec. III, the anomalous effects of molec- tween a diffusing particle and proteins within the chan- ular crowding, trapping and confinement are discussed nel, as exemplified by nuclear pore complexes, which are (points 4 and 5). The differences in diffusive behavior at the sole mediators of exchange between the nucleus and multiple timescales are highlighted. In Sec. IV, the the- cytoplasm (Fahrenkrog et al., 2004; Macara, 2001; Rout ory of motor-driven active transport is reviewed, empha- et al., 2003; Tran and Wente, 2006). When a channel sizing the connection with random intermittent search becomes sufficiently narrow, particles are no longer able processes (point 6). A method for reducing the complex- to pass each other (single-file diffusion), which imposes ity of models is also described, and used strong constraints on the diffusive motion. In particu- to study the effects of local signaling. Finally, in Sec. V lar, a tagged particle exhibits anomalous subdiffusion on some examples illustrating the role of intracellular trans- long time scales (Barkai and Silbey, 2009; Harris, 1965; port in self-organizing systems are presented (point 7). Lebowitz and Percus, 1967; Levitt, 1973) 6. There have been many stochastic models of motor- driven transport at multiple spatial and temporal scales, II. DIFFUSIVE TRANSPORT: FIRST-PASSAGE PROBLEMS ranging from Brownian ratchet models (Reimann, 2002) to random walk models (Lipowsky and Klumpp, 2005; A. Derivation of the diffusion equation Muller et al., 2008b) to systems of partial differential equations (PDEs) (Reed et al., 1990; Smith and Sim- 1. Random walks mons, 2001). However, many of these treatments neglect the fact that the goal of such transport is to deliver molec- Consider a particle that hops at discrete times between ular cargo to specific sites. This then naturally leads to neighboring sites on a one–dimensional (1D) lattice with a connection with random intermittent search processes unit spacing. At each step, the random walker moves a (Bressloff and Newby, 2009; Loverdo et al., 2008; Newby unit distance to the right with probability p or to the left and Bressloff, 2010a,b). It also raises the important ques- with probability q = 1 − p. Let Pn(r) denote the proba- tion regarding signaling mechanisms responsible for lo- bility that the particle is at site r at the Nth time step. calizing a motor complex at a target. Another issue in The evolution of the probability distribution is described active transport involves exclusion effects due to multi- by the discrete-time master equation ple motors moving along the same filament track (Blythe and Evans, 2007; Chou et al., 2011; Schadschneider et al., PN (r) = pPN 1(r − 1) + qPN 1(r + 1), r ∈ Z. (2.1) − − 2010). 7. Most cellular structures including the If q = p = 1/2 then the random walk is symmetric or and various organelles are highly dynamic, open systems unbiased, whereas for p > q (p < q) it is biased to the that are constantly exchanging energy and molecules right (left). In order to solve this equation, introduce the with the cytosol or other compartments. The formation discrete Laplace-Fourier transform and maintenance of such dynamic structures have prop- X∞ X∞ erties suggestive of far from equilibrium self-organizing Pe(k, z) = zN eikrP (r, N). (2.2) systems (Heinrich and Rapoport, 2005; Mistelli, 2001; N=0 r= −∞ Semplice et al., 2012). One of the challenges in cellular Applying this transform to the master equation and mul- biology is understanding how the coupling of diffusive or tiplying by an extra factor of z, it is straightforward to vesicular transport with chemical reactions and cell sig- show that naling generates self-organizing structures within a cell. One clear example is given by the actin and microtubular X∞ ikr ik ik , which not only provide tracks for intracel- Pe(k, z) − P0(r)e = z(pe + qe− )Pe(k, z). r= lular transport, but also determine cell shape and polar- −∞ ity (Lacayo et al., 2007), drive cell motility (Mogilner and Assuming that the particle starts at the origin so that Edelstein-Keshet, 2002; Rafelski and Theriot, 2004), and P0(r) = δr,0, we have form the during cell division (Eggert et al., 2006; Glotzer, 2009). Self-organization of the cy- 1 ik ik Pe(k, z) = , u(k) = pe + qe− . (2.3) toskeleton and its regulation by cell signaling also plays a 1 − zu(k) 4

Here u(k) is the Fourier transform of the single-step hop- lowing equation for the probability density ρ(x, t): ping probability. The original probability distribution can now be reconstructed using the inverse transform ρ(x, t) = pρ(x − δx, t − δt) + qρ(x + δx, t − δt)  ∂ρ  ∂ρ I Z π ≈ (p + q) ρ(x, t) − δt − (p − q) δx dz dk ikr ∂t ∂x PN (r) = N+1 e− Pe(k, z) (2.4) 2πiz π 2π 2 − (p + q) ∂ ρ 2 + 2 δx , with the z-contour taken around the unit circle. Tay- 2 ∂x lor expanding the solution for Pe(k, z) in terms of z thus where ρ has been Taylor expanded to first order in δt yields and to second order in δx. Note that p + q = 1. Dividing Z π through by δt and taking the continuum limit δx, δt → 0 1 ikr N PN (r) = e− u(k) dk (2.5) such that the quantities V,D are finite, where 2π π − 2 Z π N   δx δx 1 ikr X N m N m ik(N 2m) V = lim (p − q) ,D = lim , = e− p q − e− − dk δx,δt 0 δt δx,δt 0 2δt 2π π m → → − m=0 yields the Fokker–Planck (FP) equation with constant N! (N+r)/2 (N r)/2 = p q − . N + r  N − r  drift ! ! 2 2 ∂ρ(x, t) ∂[ρ(x, t)] ∂2ρ(x, t) = −V + D . (2.7) ∂t ∂x ∂x2 Using Stirling’s approximation for large N, Note that p = 0.5+κδx and q = 0.5−κδx with κ = O(1). 1 log n! ≈ n log n − n + log(2πn), Applying the same continuum limit to the Gaussian dis- 2 tribution (2.6) gives the density and assuming p, q ≈ 1/2, it can be shown that 1 (x V t)2/4Dt ρ(x, t) = √ e− − , (2.8) 4πDt 1 [r N(p q)]2/2N PN (r) ∼ √ e − − . 2πN which is the fundamental solution of equation (2.7) un- der the initial condition ρ(x, 0) = δ(x). Although we will (To be more precise, PN (r) should be multiplied by a fac- mainly consider continuum models of diffusion, it should tor of 2, since N −r must be even). Indeed, the Gaussian be noted that random walk models are particularly help- form of PN (r) in the large time limit arises universally ful in developing theories of diffusion in complex hetero- whenever the mean and variance of the displacement ∆r geneous media and associated phenomena such as anoma- in a single step are finite. This is basically a statement lous diffusion (Bouchaud and Georges, 1990; Hughes, of the central-limit theorem. One way to see this is to 1995; Metzler and Klafter, 2000), see also sectionIIIA. note that when h∆ri and h∆r2i are both finite, u(k) has the small-k series expansion 2. Langevin equation 1 2 2 ik ∆r 1 k2 ∆r2 u(k) = 1 + ikh∆ri − k h∆r i + ... ∼ e h i− 2 h i. 2 Consider a microscopic particle moving in a water solu- tion such as found in the interior of cells (the cytoplasm). Substituting this approximation into the first line of Suppose that it is subject to some external force of size F . equation (2.5) using the fact that the integral is domi- Collisions with fluid molecules have two distinct effects. nated by the behavior in the region around k = 0 when First, they induce an apparent diffusive or Brownian mo- N is large, the resulting Gaussian integral yields the ap- tion of the particle, and second they generate an effective proximation frictional force that opposes motion induced by the ex- ternal force. In the case of microscopic particles, water 1 (r N ∆r )2/2N ∆r2 PN (r) ∼ e − h i h i. (2.6) acts as a highly viscous medium (low Reynolds number) p 2 2πNh∆r i so that any particle quickly approaches terminal veloc- ity and inertial effects can be ignored. The effects of all Having analyzed the discrete random walk, it is now collisions on the motion of the particle can then be repre- possible to take an appropriate continuum limit to obtain sented in terms of the Langevin or stochastic differential a diffusion equation in continuous space and time. First, equation (Gardiner, 2009) introduce infinitesimal step lengths δx and δt for space and time and set PN = ρ(x, t)δx with x = rδx, t = Nδt. dX(t) F (X(t)) = + ξ(t), (2.9) Substituting into the master equation (2.1) gives the fol- dt γ 5 where X(t) is the stochastic position of the particle at where time t, γ is a drag coefficient, and ξ(t) is a Gaussian Z ∞ n noise term with αn(x, t) = w(x, t; u, δt)u du. (2.13) −∞ hξ(t)i = 0, hξ(t)ξ(t )i = 2Dδ(t − t ). (2.10) 0 0 The Langevin Eq. (2.9) can be used to calculate the Suppose, for the moment, that F is a constant. Formally coefficients αn. First, rewrite Eq. (2.9) in the infinitesi- integrating equation (2.9) with X(0) = 0 shows that mal form

Z t X(t + δt) = x + F (x)δt/γ + δtξ(t), X(t) = V t + ξ(t0)dt0 0 given that X(t) = x. This implies that the transition probability w can be written as with V = F/γ the terminal velocity. Averaging with respect to the noise then implies that w(x, t; u, δt) = hδ(x + u − X(t + δt))iξ = hδ(u − F (x)δt/γ − δtξ(t))i . hX(t)i = V t, h(X(t) − V t)2i = 2Dt. ξ Discretizing time in units of δt means that ξ(t) becomes Thus the mean-square displacement about the determin- a Gaussian random variable with zero mean and variance h i istic trajectory x(t) = V t is given by ∆X(t) = 2Dt, 2D/δt. The corresponding probability density is which suggests identifying D as a diffusion coefficient. Moreover, X(t) is itself a Gaussian process whose proba- p ξ2δt/4D p(ξ) = δt/4πDe− . bility density p(x, t) evolves according to the FP equation (2.7). Under the initial condition p(x, 0) = δ(x), this can Hence, averaging with respect to ξ(t), be solved to give the Gaussian distribution (2.8). r We now present a more formal derivation of the FP 1 (u F (x)δt/γ)2/4Dδt w(x, t; u, δt) = e− − . equation applicable to position–dependent forces F (x). 4πDδt Since X(t) is a stochastic variable, each simulation of It follows that the Langevin equation generates one sample out of the set of all possible trajectories. This motivates an alter- 2 α0 = 1, α1 = F (x)δt/γ, α2 = 2Dδt + α , (2.14) native way of thinking about such a stochastic process, 1 2 namely in terms of the conditional probability density and αm = O(δt ) for m > 2. Substituting these results p(x, t|x0, t0) that the particle is at x at time t, given into Eq. (2.12) and taking the limit δt → 0 finally leads that it started at x0 at time t0. Exploiting the fact that to the Fokker–Planck (FP) equation the stochastic process is Markovian, that is, X(t + δt) ∂p(x, t) 1 ∂[F (x)p(x, t)] ∂2p(x, t) only depends on the state at the previous time step − = + D 2 . (2.15) X(t), it follows that p(x, t|x0, t0) satisfies the Chapman– ∂t γ ∂x ∂x Kolmogorov equation Note that it is straightforward to generalize the above Z ∞ analysis to higher dimensions. Assuming for simplicity p(x, t|x0, t0) = p(x, t|x0, t0)p(x0, t0|x0, t0)dx0 (2.11) isotropic diffusion and friction, Eq. (2.9) becomes −∞ dXi Fi(X) for any t0 ∈ [t0, t]. Such an equation is a defining property = + ξi(t), i = 1, . . . , d (2.16) of a Markov process. Consider an infinitesimal version of dt γ this equation by taking t → t + δt, t0 → t and setting with hξi(t)i = 0 and hξi(t)ξj(t0)i = 2Dδi,jδ(t − t0). The w(x, t; u, δt) = p(x + u, t + δt|x, t): corresponding multivariate FP equation is Z ∞ ∂p(x, t) 1 p(x, t + δt) = w(x − u, t; u, δt)p(x − u, t)du, = − ∇ · [F(x)p(x, t)] + D∇2p(x, t). (2.17) ∂t γ −∞ where the initial argument (x0, t0) has been suppressed. The 1D FP Eq. (2.15) can be rewritten as a probability Now suppose that over a sufficiently small time window conservation law according to δt, large jumps u in position are highly unlikely, so that ∂p(x, t) ∂J(x, t) u can be treated as a small variable. Performing a Taylor = − , (2.18) expansion with respect to u gives ∂t ∂x where p(x, t + δt) = α0(x, t)p(x, t) − ∂x[α1(x, t)p(x, t)] 1 1 ∂p(x, t) + ∂2 [α (x, t)p(x, t)] + ... (2.12) J(x, t) = F (x)p(x, t) − D (2.19) 2 xx 2 γ ∂x 6 is the probability flux. An equilibrium steady-state solu- distinguish between the two cases, it is necessary to incor- tion corresponds to the conditions ∂p/∂t = 0 and J ≡ 0. porate details regarding the microscopic dynamics using, This leads to the first-order ODE for the equilibrium den- for example, kinetic theory (Bringuier, 2009; van Milligen 1 sity P (x): DP 0(x) − γ− F (x)P (x) = 0, which has the et al., 2005). From the perspective of the FP equation, a solution space–dependent diffusion coefficient arises when the cor- responding Langevin equation is driven by multiplicative Φ(x)/γD P (x) = N e− . (2.20) (state-dependent) noise, and the position of the diffusion x coefficient will depend on whether the noise is interpreted Here Φ(x) = − R F (y)dy is a potential energy function in the sense of Ito or Statonovich (Gardiner, 2009; Kam- N and is a normalization factor (assuming that it ex- pen, 1992). The situation is even more complicated in ists). Comparison of the equilibrium distribution with anisotropic heterogeneous media, where it is no longer the Boltzmann-Gibbs distribution of statistical mechan- possible to characterize the rate of diffusion in terms of a ics yields the Einstein relation single coefficient. One now needs to consider a diffusion tensor, see Sec. IVE. Dγ = kBT, (2.21) where T is the temperature (in degrees Kelvin) and kB ≈ 23 1 B. First passage times 1.4 × 10− JK− is the Boltzmann constant. This for- mula relates the variance of environmental fluctuations to the strength of dissipative forces and the temperature. In One of most important ways of quantifying the effi- the case of a sphere of radius R moving in a fluid of vis- cacy of diffusive transport is in terms of the first passage cosity η, Stoke’s formula can be used, that is, γ = 6πηR. time to reach a target (Gardiner, 2009; Redner, 2001). 3 1 1 For water at room temperature, η ∼ 10− kg m− s− so In the case of intracellular transport, such a target could 9 that a particle of radius R = 10− m has a diffusion coef- represent a substrate for a subsequent biochemical re- 2 1 ficient D ∼ 100µm s− . action or an exit from some bounded domain such as a So far, we have considered diffusive–like motion from chemical synapse. (Although we will focus on spatially the probabilistic perspective of a single microscopic par- continuous processes, an analogous theory of first pas- ticle moving in a fluid medium. However, it is possi- sage times can be developed for spatially discrete random ble to reinterpret Eq. (2.15) or (2.17) as a deterministic walks (Hughes, 1995; Kampen, 1992)). Consider a parti- advection–diffusion equation for the concentration u(x, t) cle whose position evolves according to the 1D Langevin of many particles. That is, ignoring any interactions or Eq. (2.9) with motion restricted to the bounded domain correlations between the particles, set u(x, t) = Np(x, t) x ∈ [0,L]. Suppose that the corresponding FP Eq. (2.15) where N is the total number of particles (assumed large). has a reflecting boundary condition at x = 0 and an ab- Multiplying both sides of Eq. (2.15) by N then leads to sorbing boundary condition at x = L: the corresponding advection–diffusion (or Smoluchowski) equation for u(x, t) with NJ(x, t) interpreted as the par- J(0, t) = 0, p(L, t) = 0. (2.22) ticle flux arising from a combination of advection and Fickian diffusion. In this review we will often switch be- We would like to determine the stochastic time T (y) for tween the microscopic probabilistic formulation of dif- the particle to exit the right hand boundary given that it ∈ fusion and the macroscopic deterministic formulation, starts at location y [0,L] at time t. As a first step, we which can be viewed as a mean-field limit of the former. introduce the survival probability P(y, t) that the particle However, the relationship between macroscopic and mi- has not yet exited the interval at time t: croscopic formulations is more complicated when chemi- Z L cal reactions are included. Macroscopically, reactions are P(y, t) = p(x, t|y, 0)dx. (2.23) described in terms of the deterministic law of mass action, 0 whereas microscopically they are modeled stochastically It follows that Prob[T (y) ≤ t] = 1 − (y, t) and we can using a chemical master equation. Differences between P define the first passage time (FPT) density according to the two levels of modeling become significant when the number of interacting molecules becomes small (Kam- ∂P(y, t) pen, 1992). f(y, t) = − . (2.24) ∂t Finally, note that another important issue arises in the case of space–dependent diffusion coefficients. From the The FPT density satisfies a backwards FP equation, macroscopic picture of Fickian diffusion, the conserva- which can be derived from the Chapman-Kolmogorov Eq. tion equation ∂tu = −∇ · J can lead to two different (2.11) by differentiating boths sides with respect to the forms of the diffusion equation, depending on whether intermediate time t0 and using the forwards equation. Us- J(x, t) = ∇[D(x)u(x, t)] or J(x, t) = D(x)∇u(x, t). ing the fact that ∂t0 p(x, t|x0, t0) = −∂tp(x, t|x0, t0), which (These are equivalent when D is a constant). In order to follows from time-translation invariance, then yields the 7 backwards FP equation for p: average time it takes is infinite. (This can also be un- derstood in terms of the scaling properties of the FPT 2 ∂tp(x, t|x0, t0) = A(x0)∂x0 p(x, t|x0, t0)+D∂x0x0 p(x, t|x0, t0), density, see below). Now suppose that L is finite and (2.25) the particle starts at the left–hand boundary. The corre- where A(x) = F (x)/γ. Taking x0 → y, t0 = 0 and inte- sponding MFPT is then τ = L2/D. Within the cytosol grating with respect to x shows that P(y, t), and hence of cells, macromolecules such as proteins tend to have 2 1 f(y, t), also satisfy a backwards FP equation: diffusivities D < 1µm s− , which is due to effects such as molecular crowding, see Sec.III. This implies that ∂P(y, t) ∂P(y, t) ∂2P(y, t) = A(y) + D . (2.26) the mean time for a diffusing particle to travel a distance ∂t ∂y ∂y2 100µm is at least 104s (a few hours), whereas to travel a distance 1mm is at least 106s (10 days). Since neu- A quantity of particular interest is the mean first pas- rons, which are the largest cells in humans, have axonal sage time (MFPT) τ(y) defined according to and dendritic protrusions that can extend from 1mm up

Z ∞ to 1m, the mean travel time due to passive diffusion be- τ(y) = hT (y)i ≡ f(y, t)tdt (2.27) comes prohibitively large, and an active form of transport 0 Z Z becomes essential. ∞ ∂P(y, t) ∞ = − t dt = P(y, t)dt, It is also possible to extend the above analysis to the 0 ∂t 0 case where the particle can exit from either end (Gar- after integration by parts. Hence, integrating both sides diner, 2009; Redner, 2001). It is often of interest to keep of Eq. (2.26) shows that the MFPT satisfies the ODE track of which end the particle exits, which leads to the concept of a splitting probability. Let G0(x, t) denote the dτ(y) d2τ(y) probability that the particle exits at x = 0 after time t, A(y) + D = −1. (2.28) dy dy2 having started at the point x. Then Z ∞ Eq. (2.28) is supplemented by reflecting and absorbing G0(x, t) = − J(0, t0|x, 0)dt0 (2.31) boundary conditions for the backwards FP equation: t with τ 0(0) = 0, τ(L) = 0. (2.29) ∂p(y, t|x, 0) J(0, t|x, 0) = A(0)p(0, t|x, 0) − D . It is straightforward to solve Eq. (2.28) by direct inte- ∂y y=0 gration (Gardiner, 2009). First, introduce the integration Differentiating with respect to t and using the backwards factor FP Eq. (2.25) gives  Z y  1 Z − ∂G0(x, t) ∞ ∂J(0, t0|x, 0) ψ(y) = exp A(y0)dy0 = exp ( V (y)/kBT ) , | − D 0 = J(0, t x, 0) = dt0 ∂t t ∂t0 1 1 1 ∂G (x, t) ∂2G (x, t) where D− A(y) = (Dγ)− F (y) = −(kBT )− V 0(y) and = A(x) 0 + D 0 . (2.32) V (y) is a potential energy. Eq. (2.28) becomes ∂x ∂x2 The hitting or splitting probability that the particle ex- d ψ(y) [ψ(y)τ 0(y)] = − its at x = 0 (rather than x = L) is Π0(x) = G0(x, 0). dy D Moreover, the probability that the particle exits at time so that t through x = 0, is Prob(T0(x) > t) = G0(x, t)/G0(x, 0), where T0(x) is the corresponding conditional FPT. Since y 1 Z the conditional MFPT satisfies ψ(y)τ 0(y) = − ψ(y0)dy0, D Z Z 0 ∞ ∂Prob(T0(x) > t) ∞ G0(x, t) τ0(x) = − t dt = dt, ∂t G0(x, 0) where the boundary condition τ 0(0) = 0 has been used. 0 0 Integrating once more with respect to y and using τ(L) = Eq. (2.32) is integrated with respect to t to give 0 then gives 2 ∂Π0(x)τ0(x) ∂ Π0(x)τ0(x) 0 A(x) + D = −Π (x), (2.33) Z L Z y 2 0 dy0 ψ(y00) ∂x ∂x τ(y) = dy00. (2.30) y ψ(y0) 0 D with boundary conditions Π0(0)τ0(0) = Π0(L)τ0(L) = 0. Finally, taking the limit t → 0 in Eq. (2.32) and noting In the case of pure diffusion (A(x) = 0), we have ψ(y) = 1 that J(0, 0|x, 0) = 0 for x =6 0, and τ(y) = (L2 − y2)/2D. It follows that for any finite 2 L−y, τ(y) → ∞ as L → ∞. Thus, although 1D diffusion ∂Π0(x) ∂ Π0(x) A(x) + D = 0, (2.34) is recurrent i.e. the particle surely reaches the origin, the ∂x ∂x2 8 with boundary conditions Π0(0) = 1, Π0(L) = 0. A simi- C. Narrow escape problems lar analysis can be carried out for exit through the other end x = L such that Π0(x) + ΠL(x) = 1. Within the context of intracellular transport, there has The above formalism extends to higher spatial dimen- been a growing interest in a particular class of first pas- sions. In particular, suppose that a particle evolves ac- sage processes, namely, the escape of a freely diffusing cording to the Langevin Eq. (2.16) in a compact domain molecule from a 2D or 3D bounded domain through small Ω with boundary ∂Ω. Suppose that at time t = 0 the absorbing windows on an otherwise reflecting boundary particle is at the point y ∈ Ω and let T (y) denote the (Benichou and Voituriez, 2008; Grigoriev et al., 2002; first passage time to reach any point on the boundary ∂Ω. Holcman and Schuss, 2004; Schuss et al., 2007). Exam- The probability that the particle has not yet reached the ples include the FPT for an ion to find an open ion chan- boundary at time t is then nel situated within the cell membrane or the FPT of a Z protein receptor to locate a particular target binding site. P(y, t) = p(x, t|y, 0)dx, (2.35) Within the context of intracellular transport in neurons, Ω recent applications include analyzing the confinement of where p(x, t|y, 0) is the solution to the multivariate FP neurotransmitter receptors within the synapse of a neu- Eq. (2.17) with an absorbing boundary condition on ∂Ω. ron (Bressloff and Earnshaw, 2009; Holcman and Schuss, The FPT density is again f(y, t) = −dP(y, t)/dt which, 2004; Holcman and Triller, 2006) and the the role of den- on using Eq. (2.17) and the divergence theorem, can be dritic spines in confining signaling molecules such as cal- expressed as cium (Biess et al., 2007; Holcman et al., 2005). (A re- Z lated class of problems is the FPT for a particle to find a f(y, t) = − [−A(x)p(x, t|y, 0) + D∇p(x, t|y, 0)] · dσ ∂Ω small target within the interior of a cellular domain. One example concerns the arrival of receptors at a localized with A = F/γ. Similarly, by constructing the corre- reaction site on the surface of an immune cell, which is a sponding backwards FP equation, it can be shown that key step in the signaling cascade leading to the activation the MFPT satisfies the equation of the cell (Coombs et al., 2009), see Sec. II.D). In order A(y) · ∇τ(y) + D∇2τ(y) = −1 (2.36) to develop the basic theory, we will focus on diffusion in a two–dimensional domain Ω ⊂ 2 whose boundary can be with ∂nτ(y) = 0 for y ∈ ∂Ω. R In the case of 1D diffusion, it is straightforward to decomposed as ∂Ω = ∂Ωr ∪ ∂Ωa, where ∂Ωr represents calculate the FPT density explicitly. In the absence of the reflecting part of the boundary and ∂Ωa the absorb- boundaries we can set the conditional probability density ing part. We then have a narrow escape problem in the limit that the measure of the absorbing set |∂Ωa| = O(ε) p(x, t|x0, 0) = p(x − x0, t). Similarly the FPT density of arriving for the first time at x at time τ starting from is asymptotically small, that is, 0 < ε  1. It follows from the analysis of exit times, see Eq. (2.36), that the x0 can be written as f(x, τ|x0, 0) = f(x − x0, τ). The densities p and f are related according to the integral MFPT to exit the boundary ∂Ωa satisfies the equation equation (in the absence of external forces) Z t 2 1 p(x − x0, t) = p(x − x0, t − τ)f(x0 − x0, τ)dτ. (2.37) ∇ τ(x) = − , x ∈ Ω (2.40) 0 D Taking Laplace transforms, with boundary conditions

p(x − x0, s) = p(x − x0, s)fe(x0 − x0, s). (2.38) N e e τ(x) = 0, x ∈ ∂Ωa = ∪j=1∂Ωj (2.41) Laplace transforming the Gaussian√ distribution (2.8) for 1 x2s/D and V = 0 yields p(x, s) = e− , so that e √4πDs √ ∂nτ(x) = 0, x ∈ ∂Ωr. (2.42) (x x0)s/D fe(x − x0, s) = e− − . The inverse Laplace transform then yields the Levy- The absorbing set is assumed to consist of N small dis- Smirnov distribution joint absorbing windows ∂Ωj centered at xj ∈ ∂Ω. In the 2D case, each window is a small absorbing arc of length r 2 1 (x − x ) 2 0 (x x0) /4Dt |∂Ω | = εl with l = O(1). It is also assumed that the f(x − x0, t) = e− − . (2.39) j j j t 4πDt windows are well separated, that is, |xi − xj| = O(1) for This inverse-Gaussian decays asymptotically as f(x, t) ∼ all i =6 j. An example of a Brownian particle in a 2D 3/2 t− , which immediately establishes that it does not unit disk with small absorbing windows on the circular have a finite first moment, that is, the MFPT from x0 boundary is illustrated in Fig.1. R ∞ to x diverges. On the other hand, 0 f(x − x0, t)dt = 1 Since the MFPT diverges as ε → 0, the calculation so that the diffusing particle will almost surely hit any of τ(x) requires solving a singular perturbation prob- point x during its motion. lem. There have been a number of studies of the narrow 9 √ ε 2 2 l4 where dj = lj/4, |y| = |x − xj|/ε = zb + sb and Aj ∂Ω is an unknown constant (that is determined by matching r x 4 sj with the outer solution). s Ω As far as the outer solution is concerned, each absorb- ing arc shrinks to a point xj ∈ ∂Ω as ε → 0. Each ε z l1 x1 x point xj effectively acts as a point source that generates a logarithmic singularity resulting from the asymptotic matching of the outer solution to the far–field behavior x3 of the inner solution. Thus the outer solution satisfies x2 εl 1 3 ∇2 x − x ∈ ε τ( ) = , Ω, (2.46) l2 D with reflecting boundary condition FIG. 1 Example trajectory of a Brownian particle moving in a 2D unit disk with small absorbing windows on an otherwise ∂ τ = 0 for x ∈ ∂Ω\{x ,..., x } (2.47) reflecting circular boundary. Inset: a local coordinate system n 1 N around the jth arc. and

Aj τ(x) ∼ + Aj log |x − xj| as x → xj, j = 1,...,N, escape problem using a combination of singular pertur- µj bation theory and Green’s function methods for a vari- (2.48) ety of geometries in two and three dimensions (Benichou where and Voituriez, 2008; Chevalier et al., 2011; Cheviakov 1 et al., 2010; Holcman and Schuss, 2004; Pillay et al., 2010; µj ≡ − . (2.49) Schuss et al., 2007; Singer et al., 2006a,b). Here we will log(εdj) review the particular approach of Ward and collabora- This can be solved in terms of the Neumann Green’s tors (Pillay et al., 2010; Ward, 2000). The basic idea is function, defined as the unique solution of to construct the asymptotic solution for the MFPT in the limit ε → 0 using the method of matched asymptotic 2 1 ∇ G(x, x0) = − δ(x − x0), x ∈ Ω (2.50a) expansions. That is, an inner or local solution valid in a |Ω| O(ε) neighborhood of each absorbing arc is constructed 1 G(x, xj) ∼ − log |x − xj| + R(xj, xj) as x → xj ∈ ∂Ω and then these are matched to an outer or global solution π that is valid away from each neighborhood. (2.50b) In order to construct an inner solution near the jth Z absorbing arc, Eq. (2.40) is rewritten in terms of a local ∂nG(x, x0) = 0, x ∈ ∂Ω, G(x, xj)dx = 0, (2.50c) Ω orthogonal coordinate system (z, s), in which s denotes arc length along ∂Ω and z is the minimal distance from where R(x, x0) is the regular part of G(x, x0). It follows ∂Ω to an interior point x ∈ Ω, as shown in the inset of that the outer solution can be expressed in terms of the Fig.1. Now introduce stretched coordinates zb = z/ε Green’s function G and an unknown constant χ, and s = (s − sj)/ε, and write the solution to the inner b N problem as τ(x) = w(z, s). Neglecting terms of O(ε), it X b b τ(x) = −π A G(x, x ) + χ. (2.51) can be shown that w satisfies the homogeneous equation i j j=1 (Pillay et al., 2010) Integrating both sides of Eq. (2.51) shows that χ is the ∂2w ∂2w ∞ −∞ ∞ MFPT averaged over all possible starting positions: 2 + 2 = 0, 0 < zb < , < sb < (2.43) ∂ zb ∂ sb 1 Z ≡ x x with the following boundary conditions on zb = 0: χ = τ τ( )d . (2.52) |Ω| Ω ∂w = 0 for |s| > lj/2, w = 0 for |s| < lj/2. (2.44) The problem has reduced to solving N + 1 linear equa- ∂z b b b tions for N + 1 unknowns Ai, χ. The first N equations The resulting boundary value problem can be solved by are obtained by matching the near-field behavior of the introducing elliptic cylinder coordinates. However, in or- outer solution as x → xj with the far–field behavior of der to match the outer solution we need only specify the the corresponding inner solution (2.45). After cancella- far-field behavior of the inner solution, which takes the tion of the logarithmic terms, we have form X Aj −πAjRj − π AiGji + χ = , (2.53) µj w(x) ∼ Aj [log |y| − log dj + o(1)] as |y| → ∞, (2.45) i=j 6 10 for j = 1,...,N, where Gji ≡ G(xj, xi) and Rj ≡ speed of reaction k is limited by their encounter rate R(xj, xj). The remaining equation is obtained by noting via diffusion. (Also note that k has units of volume 2 PN 2 s 1). One can then formulate the problem as an idealized that ∇ τ(x) = −π j=1 Aj∇ G(x, xj) and, hence − first passage process, in which one molecule is fixed and N treated as the target, whilst the other reactants diffuse 1 X 1 π|Ω|− Aj = . (2.54) and are absorbed if they hit the target. It is assumed that D j 1 − the density of the particles is sufficiently small, so that re- actions amongst the diffusing particles can be neglected, In the case of a single absorbing window of arc length that is, a reaction only occurs if one of the background 2ε (d = 1/2), Eqs. (2.53) and (2.54) are easily solved to diffusing particles comes within the reaction radius of the give A = Ω|/πD and 1 target molecule. The steady–state flux to the target (if |Ω|  1  it exists) is then identified as the reaction rate k. Let τ(x) ∼ − log(ε/2) + R(x , x ) − G(x, x ) , D π 1 1 1 Ω denote the target domain (which is often treated as a sphere of radius a) and ∂Ω its absorbing boundary. We (2.55) then need to solve the diffusion equation for the concen- |Ω|  1  tration c(x, t) of background molecules exterior to the τ ∼ − log(ε/2) + R(x , x ) . D π 1 1 domain Ω:

All that remains is to calculate the regular part of the ∂c(x, t) = D∇2c(x, t), c(x ∈ ∂Ω, t) = 0, c(x, 0) = 1, Neumann Green’s function R(x, xj), which will depend ∂t on the geometry of the domain Ω. In certain cases such as (2.59) the unit disk or a rectangular domain, explicit formulae subject to the far–field boundary condition c(x, t) = 1 for R can be obtained, otherwise numerical methods are for x → ∞. The flux through the target boundary is required (Holcman and Schuss, 2004; Pillay et al., 2010; Z Singer et al., 2006a,b). The Green’s function for a unit J = D ∇c · dS. (2.60) disk when the source xj is on the unit circle has the well ∂Ω known formula Note the sign, which is due to the fact that the flux is 1 |x|2 1 from the exterior to interior of the target. G(x, xj) = − log |x − xj| + − . (2.56) π 4π 8π Let d denote the spatial dimension of the target. For d > 2, a diffusing particle is transient, which means that x x It immediately follows that R( j, j) = 1/8π (since there is a non-zero probability of never reaching the tar- |x |2 j = 1) and get. Hence, the loss of reactants by target absorption is 1 balanced by their resupply from infinity. It follows that τ = [− log(ε) + log 2 + 1/8] . (2.57) D there exists a steady state in which the reaction rate is finite. On the other hand, for d ≤ 2, reactants are sure to For a rectangular domain of width L2 and height L1, hit the target (recurrent diffusion) and a depletion zone the Green’s function can be solved using separation of continuously develops around the target so that the flux variables and expanding the result in terms of logarithms, and reaction rate decay monotonically to zero with re- see (Bressloff and Newby, 2011; Pillay et al., 2010). spect to time. Although a reaction rate does not strictly exist, it is still useful to consider the time-dependent flux as a time–dependent reaction rate. The two–dimensional D. Diffusion-limited reaction rates case is particularly important when considering interac- tions of molecules embedded in the plasma membrane Another important type of first passage process arises of a cell or the surrounding an intracellular in Smoluchowski rate theory for diffusion–controlled re- compartment. actions (Collins and Kimball, 1949; Keizer, 1982; Red- First consider the case of a spherical target of radius a ner, 2001; Rice, 1985; Smoluchowski, 1917). The simplest (d = 3). Exploiting the radial symmetry of the problem, version of the theory concerns the bimolecular reaction it is possible to set u(r, t) = rc(r, t) such that the 3D A+B → AB for which the concentrations evolve accord- diffusion equation for c reduces to a 1D diffusion equation ing to the following law of mass action: for u (Redner, 2001): d[AB] = k[A][b]. (2.58) ∂u(r, t) ∂2u(r, t) dt = D (2.61) ∂t ∂r2 We assume that an A molecule and a B molecule re- act immediately to form the complex AB when they en- with u(r, 0) = r, u(a, t) = 0 and u(r, t) = r as r → ∞. counter each other within a reaction radius, so that the Laplace transforming this equation gives sue(r, s) − r = 11

Due00(r, s), which has the solution Over the years there have been various generalizations √ of Smoluchowski’s rate theory. For example, Collins and 1 h (r a) s/Di u(r, s) = r − ae− − . Kimball(1949) considered the case where molecules in e s √ proximity to each other do not react immediately. Thus, 1 r s/D Since the inverse Laplace transform√ of s− [1 − e− ] the target is assumed to act like an imperfect absorber, is the error function erf(r/ 4Dt), one finds that which can be taken into account by modifying the bound-  a a  r − a  ary condition at the surface of the (spherical) target: c(r, t) = 1 − + erf √ . (2.62) r r 4Dt 2 ∂c(r, t) 4πa D = k0c(a, t), (2.66) It follows that the time–dependent flux is ∂r r=a

∂c J(t) = 4πa2D (2.63) where k0 is the intrinsic reaction rate. Incorporating this ∂r r=a modified boundary condition into Smoluchowki’s theory  a  leads to a new expression for the diffusion–controlled re- = 4πaD 1 + √ → 4πaD. t action rate of the form πDt →∞ Hence, we obtain the Smoluchowski reaction rate k = 4πDak0 k = . (2.67) 4πaD. As highlighted by Redner(2001), it is straightfor- 4πDa + k0 ward to generalize the steady–state result to other three– Another important extension was developed by Keizer dimensional targets by making a connection with electro- (1982), who used the theory of nonequilibrium pair cor- statics. That is, setting φ(x) = 1 − c(x) in steady–state, relation functions to include many body effects that be- it follows that φ satisfies Laplace’s equation with φ = 1 come important at higher concentrations of reactants. on the target boundary and φ = 0 at infinity, so that So far it has been assumed that the diffusion of the φ is equivalent to the electrostatic potential generated background reactants occurs in an unbounded domain by a perfectly conducting object Ω held at unit poten- with a uniform concentration at infinity. The analysis tial. Moreover, the steady–state reaction rate k = 4πDQ becomes considerably more involved when the boundary where Q is the total charge on the surface of the conduc- of the domain is taken into account. Recently, however, tor, which for a unit potential is equal to the capacitance, Straube et al. (2007) have shown how methods similar to Q = C. Thus, determining the reaction rate for a gen- the analysis of the narrow escape problem, Sec. II.C, can eral 3D target is equivalent to finding the capacitance of be used to determine the reaction rate in the asymptotic a perfect conductor with the same shape; see also (Chevi- limit that the target is much smaller than the domain akov et al., 2010). size. Here we sketch the basic steps of their analysis. Although it is possible to calculate the exact time– Consider a target disk Ω of radius ε  1 and center x dependent flux for d ≤ 2, a much simpler method is to use ε 0 that is located in the interior of a rectangular domain a quasi–static approximation (Redner, 2001). Consider, Ω of size O(1). The calculation of the reaction rate can for example a target disk of radius r = a. The region ex- be formulated in terms of the solution to the following terior to the disk is divided into a near zone that extends √ diffusion equation: a distance Dt from the surface and a complementary far zone. In the near zone, it is assumed that diffusing ∂c(x, t) 2 = D∇ c(x, t), x ∈ Ω\Ωε (2.68) particles have sufficient time to explore the domain before ∂t being absorbed by the target so that the concentration in with ∂ c = 0 on the exterior boundary ∂Ω and c = 0 on the near zone can be treated as almost steady or quasi– n the interior boundary ∂Ω . The initial condition is taken static. Conversely, it is assumed that the probability of a ε to be c(x, 0) = 1. Following Straube et al. (2007), we seek particle being absorbed by the target is negligible in the a solution in the form of an eigenfunction expansion, far zone, since√ a particle is unlikely to diffuse more than a distance Dt over a time interval of length t. Thus, ∞ √ X λj Dt c(r) ≈ 1 for r > Dt + a. The near zone concentration c(x, t) = cjφj(x)e− (2.69) is taken to be a radially symmetric solution of Laplace’s j=0 equation, which for d = 2 is c(r) = A + B log r. Match- where the eigenfunctions φj(x) satisfy the Helmholtz ing√ the solution to the boundary√ conditions c(a) = 0 and equation c( Dt) = 1 then gives (for Dt  a) 2 log(r/a) ∇ φj + λjφj = 0, x ∈ Ω\Ωε (2.70) c(r, t) ≈ √ . (2.64) log( Dt/a) subject to the same boundary conditions as c(r, t). The The corresponding time–dependent flux is eigenfunctions are orthogonalized as 2πD Z J(t) ≈ √ . (2.65) φi(x)φj(x)dx = δi,j. (2.71) log( Dt/a) Ω Ωε \ 12

The initial condition then implies that Matching the near–field behavior of the outer solution Z with the far–field behavior of the inner solution then cj = φj(x)dx. (2.72) yields a transcendental equation for the principal eigen- Ω Ωε \ value: Taking the limit ε → 0 results in an eigenvalue problem 1 R(x , x ; λ ) = − . (2.79) in a rectangular domain without a hole. It is well known 0 0 0 2πν that the eigenvalues are ordered as λ = 0 < λ ≤ λ ≤ 0 1 2 Finally, the normalization condition for φ determines .... This ordering will persist when 0 < ε  1 so that 0 the amplitude A according to in the long–time limit, the solution will be dominated by Z the eigenmode with the smallest eigenvalue: 2 2 2 4π A G(x, x0; λ0) dx = 1. (2.80) λ0t Ω c(x, t) ∼ c0φ0(x)e− . (2.73) The Helmholtz Green’s function and its regular part It can also be shown that the principal eigenvalue has an can be calculated along similar lines to Sec. II.C. Here infinite logarithmic expansion (Ward et al., 1993): it is sufficient to note that, since 0 < λ0  1 for a small 1 target, the Green’s function has the expansion λ = νΛ + ν2Λ + . . . , ν ≡ − . (2.74) 0 1 2 log ε 1 2 G(x, x0; λ0) = − +G1(x, x0)+λ0G2(x, x0)+O(λ0) Moreover, the eigenfunction φ0(x) will develop a bound- λ0|Ω0| (2.81) ary layer in a neighborhood of the target, where it R changes rapidly from zero on the boundary ∂Ωε to a with Ω Gj(x, x0)dx = 0. The regular part R(x, x0; λ0) value of O(1) away from the target. This suggests di- can be expanded in an identical fashion. Hence, neglect- viding the domain into inner/outer regions, and using ing terms of O(λ0) and higher, substitute R(x, x0; λ0) ≈ 1 matched asymptotics along analogous lines to the study −(λ0|Ω0|)− + R1(x, x0) into the transcendental Eq. of the narrow escape problem. (2.79). This yields a linear equation for λ0 such that Therefore, introduce stretched coordinates y = (x − 2πν 1 x0)/ε and write the inner solution of the principal eigen- λ0 ≈ . (2.82) |Ω0| 1 + 2πνR1(x0, x0) function as ϕ(y) = φ0(εy). Using the logarithmic expan- sion of λ0 shows that the righthand side of the rescaled Substituting the expansion (2.81) into Eq. (2.80) shows 2 2 k eigenvalue equation is of O(ε ν ) = o(ν ) for all k ≥ 0. that to leading order in λ0, 2 Thus to logarithmic accuracy, it follows that ∇ ϕ(y) = 0 p 2 1 1 |Ω |λ on R \S where S is the unit circle centered about the A ≈ 0 0 . (2.83) origin, and ϕ = 0 on |y| = 1. Hence, ϕ(y) = A log |y| 2π and the inner solution has the far–field behavior R Moreover, using Eqs. (2.78), (2.80) and Ω Gj(x)dx = 0, the coefficient c is ϕ ∼ A log(|x − x0|/ε). (2.75) 0 Z 2πA The outer solution satisfies the equation c0 = −2πA G(x, x0; λ0)dx = . (2.84) Ω λ0 ∇2φ + λ φ = 0, x ∈ Ω\{x }, 0 0 0 0 We now have all the components necessary to determine φ0 ∼ A log(|x − x0|/ε), x → x0, (2.76) the time–dependent reaction rate k(t). That is, using the Z 2 inner solution ϕ(x) = A log(r/ε), r = |x − x0|, we com- φ0(x)dx = 1. Ω bine Eqs. (2.73), (2.83) and (2.84) to obtain the result (Straube et al., 2007) Following the analysis in Sec. II.C, the outer problem can be solved in terms of the Neumann Green’s function Z 2π   λ0t ∂ϕ for the Helmholtz equation: J(t) = Dc0e− r dθ 0 ∂r r=ε 2 λ0t λ0Dt ∇ G(x, x0; λ0) + λ0G(x, x0; λ0) = −δ(x − x0), x ∈ Ω = 2πDc0e− A ≈ c0|Ω0|λ0e− . (2.85) (2.77a) Note that Straube et al. (2007) applied the above analysis ∂nG(x, x0; λ0) = 0, x ∈ ∂Ω (2.77b) to the particular problem of protein receptor clustering 1 on a cylindrical surface membrane. The only modifica- G(x, x0; λ0) ∼ − log |x − x0| + R(x0, x0; λ0), x → x0. 2π tion to the rectangular domain Ω is that the left and right (2.77c) side boundaries are identified by replacing the reflecting That is, boundary conditions with periodic boundary conditions. This generates the topology of a cylinder and modifies φ0(x) = −2πAG(x, x0; λ0). (2.78) the form of the Helmholtz Green’s function accordingly. 13

E. Diffusive search for a protein-DNA binding site (a)

sliding A wide range of cellular processes are initiated by a 3D diffusion single protein binding a specific target sequence of base DNA pairs (target site) on a long DNA molecule. The pre- cise mechanism whereby a protein finds its DNA binding protein site remains unclear. However, it has been observed ex- target perimentally that reactions occur at very high rates, of 10 1 1 around k = 10 M − s− (Richter and Eigen, 1974; Riggs et al., 1974). This is around 100 times faster than the rate based on the Smoluchowski theory of diffusion–limited re- action rates (Sec. II.D), and 1000 times higher than most known protein-protein association rates. (Note, how- ever, that some protein-protein association rates are also (b) much larger than the predictions of Smoluchowski the- ory (Schreiber et al., 2009)). This apparent discrepancy in reaction rates, suggests that some form of facilitated x=-M x=0 x=L diffusion occurs. The best known theoretical model of facilitated diffusion for proteins searching for DNA tar- FIG. 2 (a) Mechanism of facilitated diffusion involving alter- gets was originally developed by Berg, Winter and von nating phases of 3D diffusion and 1 D diffusion (sliding along Hippel (BHW) (Berg and von Hippel, 1985; Berg et al., the DNA). (b) 1D representation of facilitated diffusion. 1981; Winter et al., 1981), and subsequently extended by a number of groups (Coppey et al., 2004; Halford and ability of finding the specific promoter site is p = n/N. Marko, 2004; Hu et al., 2006; Hu and Shklovskii, 2006; It follows that the probability of finding the site after R Mirny et al., 2009; Slutsky and Mirny, 2004). The ba- rounds is (1−p)R 1p. Hence, the mean number of rounds sic idea of the BHW model is to assume that the pro- − is r = 1/p = N/n. Assuming that 1D sliding occurs via tein randomly switches between two distinct phases of √ normal diffusion, then nb = 2 D τ where D is the 1D motion, 3D diffusion in solution and 1D diffusion along 1 1 1 diffusion coefficient, and we have (Mirny et al., 2009) DNA (sliding), see Fig.2. Such a mechanism is one ex- ample of a random intermittent search process, see Sec. N p τ = (τ + τ ), n = 2 D τ /b. (2.86) IVB. The BHW model assumes that there are no correla- n 1 3 1 1 tions between the two transport phases, so that the main factor in speeding up the search is an effective reduction Since τ3 depends primarily on the cellular environment in the dimensionality of the protein motion. However, and is thus unlikely to vary significantly between pro- as recently reviewed by Kolomeisky(2011), there are a teins, it is reasonable to minimize the mean search number of discrepancies between the BHW model and time with respect to τ1 while τ3 is kept fixed. Setting dτ/dτ = 0 implies that the optimal search time occurs experimental data, which has led to several alternative 1 p theoretical approaches to facilitated diffusion. We first when τ1 = τ3 with τopt = 2Nτ3/n = Nb τ3/D1. Com- review the BHW model and then briefly discuss these paring with the expected search time for pure 3D diffu- alternative models. sion by setting τ1 = 0, n = 1 gives τ3D = Nτ3. Thus A simple method for estimating the effective reaction facilitated diffusion is faster by a factor n/2. Further in- rate of facilitated diffusion in the BHW model is as fol- sights to facilitated diffusion may be obtained by using lows (Mirny et al., 2009; Slutsky and Mirny, 2004). Con- the Smoluchowski formula for the rate at which a diffus- ing protein can find any one of N binding sites of size N, sider a single protein searching for a single binding site 1 on a long DNA strand of N base pairs, each of which has namely, τ3− = 4πD3Nb. Using this to eliminate N shows length b. Suppose that on a given search, there are R that the effective reaction rate of facilitated diffusion is rounds labeled i = 1,...,R. In the ith round the pro- (Mirny et al., 2009) tein spends a time T3,i diffusing in the cytosol followed   1 τ3 by a period T1,i sliding along the DNA. The total search k ≡ τ − = 4πD3 nb. (2.87) PR τ1 + τ3 time is thus T = i=1(T3,i + T1,i), and the mean search time is τ = r(τ3 + τ1). Here r is the mean number of This equation identifies two competing mechanisms in rounds and τ3, τ1 are the mean durations of each phase facilitated diffusion. First, sliding diffusion effectively in- of 3D and 1D diffusion. Let n denote the mean number creases the reaction cross-section from 1 to n base pairs, of sites scanned during each sliding phase with n  N. thus accelerating the search process compared to stan- If the binding site of DNA following a 3D diffusion phase dard Smoluchowski theory. This is also known as the is distributed uniformly along the DNA, then the prob- antenna effect (Hu et al., 2006). However, the search is 14 also slowed down by a factor τ3/(τ1 + τ3), which is the where P1(t) = hP1(x, t)i,Q1(t) = hQ1(x, t)i and fraction of the time the protein spends in solution. That 1 R L hg(x, t)i ≡ (L+M)− M g(x, t)dx for an arbitrary func- is, a certain amount of time is lost by binding to non- tion g. In order to determine− the FPT density F (x, t) for specific sites that are far from the target. finding the target, it is necessary to sum over all pos- A more complicated analysis is needed in order to sible numbers of excursions and intervals of time, given take into account the effects of boundaries, for example. Pn Pn 1 the constraint t = i=1 ti + i=1− τn. Thus, setting Here we review the particular formulation of Coppey et. F (t) = hF (x, t)i, one finds that al., which generalizes the original analysis of Berg et al. Z (1981). Suppose that DNA is treated as a finite track of X∞ ∞ F (t) = dt . . . dtndτ . . . dτn (2.92) length l = L + M with reflecting boundaries at x = −M 1 1 n=1 0 and x = +L and a point–like target at x = 0, see Fig.2. n n 1 ! n 1 n 1 Rather than modeling 3D diffusion explicitly, each time X X− Y− Y− δ ti + τn − t Q1(tn) P3(τi) P1(ti). the protein disassociates from DNA it simply rebinds at i=1 i=1 i=1 i=1 a random site at a time t later that is generated from an exponential waiting time density. This is based on the Finally, Laplace transforming this equation gives assumption that 3D excursions are uncorrelated in space. (Coppey et al., 2004) 1 It might be expected that excursions would be correlated " #− 1 − fe(λ1 + s) due to the geometric configuration of DNA. However, in Fe(s) = fe(λ1 +s) 1 − , (2.93) solution DNA is a random coil so that even short 3D trips (1 + s/λ1)(1 + s/λ3) can generate long displacements relative to the linear po- R st ∞ h i sition of the protein along the DNA strand, resulting in with fe(s) = 0 e− f(x, t) dt. Given Fe(s), the MFPT to find the target (averaged over the starting position x) decorrelation of excursions. If P3(t) denotes the proba- bility density that the protein in solution at time t = 0 is then binds to the DNA at time t at a random position, then Z ∞ dFe(s) τ = tF (t)dt = − . (2.94) λ3t P3(t) = λ3e− , (2.88) 0 ds s=0 where τ3 = 1/λ3 is again the mean time spent in so- which can be evaluated to give lution. Next, let P1(t, x) be the conditional probability density that the protein disassociates from the DNA at 1 − fe(λ1) 1 1 τ = (λ1− + λ3− ). (2.95) time t without finding the target, given that it is at linear fe(λ1) position x along the DNA at time t = 0: All that remains is to determine fe(x, s) averaged with λ1t P1(x, t) = λ1e− P(x, t) (2.89) respect to x. If x < 0 (x > 0), then one simply needs to where τ1 = 1/λ1 is the mean time of each sliding phase, determine the FPT density for a 1D Brownian particle and P(t, x) is the conditional probability density that the on the interval [−M, 0] ([0,L]) with a reflecting bound- protein starting at x has not met the target at time t. ary at x = −M (x = L) and an absorbing boundary at Finally, let Q1(x, t) be the conditional probability density x = 0. Recall from Sec. (II.B) that f(x, t) satisfies the that the protein starting at x finds the target at time t: backwards FP equation 2 λ1t ∂f(x, t) ∂ f(x, t) Q1(x, t) = e− f(x, t), (2.90) = D , (2.96) ∂t 1 ∂x2 where f(x, t) = −dP(x, t)/dt is the FPT density asso- ciated with diffusion along the DNA strand. That is with f(x, 0) = 0, f(0, t) = δ(t) and ∂xf(L, t) = 0 or f(x, t)dt is the probability that starting at x at t = 0, ∂xf(−M, t) = 0. Taking Laplace transforms, the protein finds the target during a single phase of slid- ∂2fe(x, s) ing diffusion in the time interval [t, t+dt]. (Protein-DNA sfe(x, s) = D1 2 , (2.97) binding is assumed to be diffusion-limited so that as soon ∂x as the protein reaches the target site it reacts). with fe(0, s) = 1, ∂xfe(L, s) = 0 or √∂xfe(−M, s) =√ 0. The Suppose that in a given trial, a protein starting at x s/D1x s/D1x general solution is fe(x, s) = Ae− + Be− at time t = 0 executes n−1 excursions before finding the with the coefficients A, B determined by the boundary target with t1, . . . , tn the residence times on DNA and conditions. Solving for A, B separately when x < 0 and τ1, . . . , τn 1 the excursion times. The probability density − Pn Pn 1 x > 0 and averaging with respect to x finally gives for such a sequence of events with t = t + − τ i=1 i i=1 n r is 1 D1 fe(s) = L + M λ1 Pn(x, {ti, τi}) = Q1(tn)P3(τn 1)P1(tn 1) ... − − h p p i × tanh(L λ /D ) + tanh(M λ /D ) . × P1(t2)P3(τ1)P1(x, t1), (2.91) 1 1 1 1 15

Thus, setting τi = 1/λi, i = 1, 3, Electrostatic interactions. One alternative hypothesis is  √  that the observed fast association rates are due to electro- (L + M)/ τ1D1 τ = √ √ − 1 (τ1 + τ3) , static interactions between oppositely charged molecules, tanh(L/ τ1D1) + tanh(M/ τ1D1) and thus do not violate the 3D diffusion limit (Halford, 2009). This is motivated by the theoretical result that the which recovers the original result√ of Berg et al.√ (1981). It also recovers Eq. (2.86) when L/ τ1D1,M/ τ1D1  1. maximal association rate in Smoluchowski theory when There have been a number of extensions of the BHW there are long-range interactions between the reacting model that incorporate various biophysical effects. For molecules is example, sequence-dependent protein-DNA interactions Z ∞ dr generate a rugged energy landscape during sliding mo- k = 4πDa/β, β = eU(r)/kB T , (2.98) r2 tion of the protein (Hu and Shklovskii, 2006; Mirny et al., a 2009; Slutsky and Mirny, 2004) - see also Sec. III.C. This where U(r) is the interaction potential. The standard re- observation then leads to an interesting speed-stability sult is recovered when U(r) = 0 for r > a, see Eq. (2.63). paradox (Mirny et al., 2009; Sheinman et al., 2012). On It follows that long–range attractive interactions can sig- 2 the one hand, fast 1D search requires that the variance σ nificantly increase diffusion–limited reaction rates. It has of the protein-DNA binding energy be sufficiently small, been further argued that in vitro experiments tend to be that is, σ ∼ kBT , whereas stability of the protein at performed at low salt concentrations so that the effects the DNA target site requires σ ∼ 5kBT . One suggested of screening could be small. However, experimentally- resolution of this paradox is to assume that a protein- based estimates of the Debye length, which specifies the DNA complex has two conformational states: a recog- size of the region where electrostatic forces are impor- nition state with large σ and a search state with small tant, indicate that it is comparable to the size of the σ (Mirny et al., 2009; Slutsky and Mirny, 2004). If the target sequence. Hence, electrostatic forces are unlikely transitions between the states are sufficiently fast then to account for facilitated diffusion. target stability and fast search can be reconciled. (For a recent review of the speed-stability paradox and its Colocalization Another proposed mechanism is based on implications for search mechanisms see Sheinman et al. the observation that in bacteria, genes responsible for (2012)). Other effects include changes in the conforma- producing specific proteins are located close to the bind- tional state of DNA and the possibility of correlated as- ing sites of these proteins. This colocalization of proteins sociation/disassociation of the protein (Benichou et al., and binding sites could significantly speed up the search 2011a; Hu et al., 2006), and molecular crowding along process by requiring only a small number of alternating DNA (Li et al., 2009) or within the cytoplasm (Isaacson 3D and 1D phases (Mirny et al., 2009). However, such et al., 2011). Molecular crowding will be considered in a mechanism might not be effective in cells, Sec. IIIB. where transcription and translation tend to be spatially The BHW model and its extensions provide a plausible and temporally well separated. Moreover, colocalization mechanism for facilitated diffusion that has some support breaks down in cases where proteins have multiple tar- from experimental studies, which demonstrate that pro- gets on DNA. teins do indeed slide along DNA (Gorman and Greene, 2008; Gowers et al., 2005; Li et al., 2009; Tafvizi et al., 2008; Winter et al., 1981). In particular, recent advances Correlations. Yet another theoretical mechanism in- in single–molecule spectroscopy means that the motion volves taking into account correlations between 1D slid- of flourescently labeled proteins along DNA chains can ing and 3D bulk diffusion. These correlations reflect the be quantified with high precision, although it should be fact that attractive interactions between a protein and noted that most of these studies have been performed non-specific binding sites means that there is a tendency in vitro. A quantitative comparison of the BHW model for a protein to return back to a neighborhood of the with experimental data leads to a number of discrepan- DNA site from which it recently disassociated (Cherstvy cies, however. For example, it is usually assumed that et al., 2008; Zhou, 2005). Although such interactions D1 ≈ D3 in order to obtain a sufficient level of facilita- tend to slow down proteins moving along DNA, they also tion. On the other hand, single–molecule measurements increase the local concentration of proteins absorbed to indicate that D1  D3 (Elf et al., 2007; Yang et al., DNA. This suggests that facilitated diffusion can occur 2006). Such experiments have also shown that τ1  τ3, at intermediate levels of protein concentration and inter- which is significantly different from the optimal condition mediate ranges of protein–DNA interactions. τ1 = τ3. Hence the intermittent search process could ac- tually result in a slowing down compared to pure 3D dif- fusion (Hu et al., 2006). The RHW model also exhibits unphysical behavior in certain limits. These issues have motivated a number of alternative models of facilitated diffusion, as recently highlighted by Kolomeisky(2011). 16

III. DIFFUSIVE TRANSPORT: EFFECTS OF generating long-time correlations in the particle’s trajec- MOLECULAR CROWDING, TRAPS AND tory. This memory effect leads to subdiffusive behavior CONFINEMENT that can be modeled in terms of fractional Brownian mo- tion (FBM) or the fractional Langevin equation (FLE) A. Anomalous diffusion (Burov et al., 2011; Mandelbrot and Ness, 1968). In con- trast to CTRW and diffusion on fractals, the probabil- In normal (unobstructed) diffusion in d dimensions, the ity density for unconfined subdiffusion in FBM/FLE is a mean-square displacement (MSD) of a Brownian particle Gaussian (with a time–dependent diffusivity). Moreover, is proportional to time, hR2i = 2dDt, which is a con- FBM/FLE are ergodic systems, although under confine- sequence of the central limit theorem. A general sig- ment time-averaged quantities behave differently from nature of anomalous diffusion is the power law behavior their ensemble-averaged counterparts (Jeon and Metzler, (Bouchaud and Georges, 1990; Metzler and Klafter, 2000) 2012). hR2i = 2dDtα, (3.1) Determining which type of model best fits experimen- tal data is a nontrivial task, particularly since CTRW, corresponding to either subdiffusion (α < 1) or superdif- OD and FBM/FLE generate similar scaling laws for fusion (α > 1). Due to recent advances in single-particle ensemble-averaged behavior in the long-time limit. Thus tracking methods, subdiffusive behavior has been ob- other measures such as ergodicity are being used to help served for a variety of biomolecules and tracers within identify which model provides the best characterization living cells. Examples include messenger RNA molecules for anomalous diffusion in living cells. A number of re- (Golding and Cox, 2006) and chromosomal loci (Weber cent studies provide examples where FBM/FLE appears et al., 2010) moving within the cytoplasm of bacteria, to give a better fit to the data than CTRW (Magdziarz lipid granule motion in yeast cells (Jeon et al., 2011), et al., 2009; Szymanski and Weiss, 2009; Weber et al., viruses (Seisenberger et al., 2001), telemores in cell nu- 2010). However, other studies suggest that both ergodic clei (Bronstein et al., 2009), and protein channels moving (OD or FBM/FLE) and nonergodic processes (CTRW) within the plasma membrane (Weigel et al., 2011). can coexist (Jeon et al., 2011; Weigel et al., 2011). There are currently three subcellular mechansims In this review we are going to focus on biophysical thought to generate subdiffusive motion of particles in models of the cellular environment and how it effects dif- cells, each with its own distinct type of physical model fusive transport via molecular crowding, trapping and (Weber et al., 2010). The first mechanism, which is typ- confinement, rather than on generic models of anomalous ically modeled using the continuous time random walk transport such as CTRW and FBM/FLE. The reasons (CTRW) (Hughes, 1995; Scher and Montroll, 1975), in- are two-fold: (i) there are already a number of compre- volves transient immobilization or trapping, see Sec. hensive reviews of such models (Kou, 2008; Mandelbrot IIIC. That is, if a diffusing particle encounters a binding and Ness, 1968; Metzler and Klafter, 2000; Scher and site then it will pause for a while before disassociating Montroll, 1975; Tothova et al., 2011), and (ii) it is still un- and diffusing away. Multiple binding events with a range clear to what extent intracellular diffusion is anomalous of rate constants can generate long tails in the waiting in the long–time limit rather than just at intermediate time distribution leading to subdiffusive behavior (Sax- times. This motivates studying diffusion in the presence ton, 1996, 2007). In addition to having a heavy-tailed of obstacles and transient traps whereby normal diffu- waiting time distribution, the CTRW is weakly noner- sion is recovered asymptotically. However, before pro- godic; the temporal average of a long particle trajectory ceeding, we briefly sketch the basic structure of CTRW differs from the ensemble average over many diffusing and FBM/FLE models. particles (He et al., 2008; Jeon and Metzler, 2012; Jeon The CTRW considers a particle performing random et al., 2011; Weigel et al., 2011). The second mechanism jumps whose step length is generated by a probabil- for subdiffusion in cells is obstructed diffusion (OD) due ity density with finite second moments. However, the to molecular crowding or cytoskeletal networks that im- waiting times between jumps are assumed to be dis- pose obstacles around which diffusing molecules have to tributed according to a power law (rather than an ex- navigate, see Sec. IIIB. If the concentration of obsta- ponential waiting time density characteristic of Marko- cles is sufficiently high, then subdiffusive behavior oc- vian random walks). The resulting heavy tailed wait- curs, in which the domain of free diffusion develops a ing times generate subdiffusive behavior. Consider, for fractal-like structure (Saxton, 1994). Diffusion on a frac- example, a 1D CTRW generated by a sequence of in- tal is a stationary process and is thus ergodic. The final dependent identically distributed (iid) positive random mechanism involves the viscoelastic properties of the cy- waiting times T1,T2,...,Tn, each having the same prob- toplasm due to the combined effects of macromolecular ability density function φ(t), and a corresponding se- crowding and the presence of elastic elements such as nu- quence of (iid) random jumps X1,X2,... ∈ R, each hav- cleic acids and cytoskeletal filaments. As a particle moves ing the same probability density w(x). Setting t0 = 0 through the cytoplasm, the latter “pushes back”, thus and tn = T1 + T2 + ... + Tm for positive integers n, 17 the random walker makes a jump of length Xn at time with 0 < α < 1 for subdiffusive behavior and C(−t) = tn. Hence, its position is x0 = 0 for 0 ≤ t < T1 and C(t). Using a generalized fluctuation-dissipation theo- xn = X1 + X2 + ... + Xn for tn ≤ t < tn+1. It follows rem, the functions φ(t),C(t) are related according to that the probability density p(x, t) that the particle is at position x at time t satisfies the integral equation C(t) = kBT φ(t), t ≥ 0. (3.6)

p(x, t) = δ(x)Ψ(t) (3.2) We now show how to derive a FP equation for the t   generalized Langevin equation following Wang (Wang, Z Z ∞ + φ(t − t0) w(x − x0)p(x0, t0)dx0 dt0, 1992). Since the fluctuating force is described by a Gaus- 0 −∞ sian process then so is the random variable X(t). Let R p(x, t) denote the probability density for X(t) = x and where Ψ(t) = ∞ φ(t )dt is the survival probability that t 0 0 introduce the characteristic function or Fourier transform at time t the particle has not yet moved from its initial Z position at x = 0. In the special case of an exponential ikX(t) ∞ ikx 1 t/τ Γ(k, t) ≡ he i = p(x, t)e dx. (3.7) waiting time density φ(t) = τ − e− , we can differenti- ate both sides of Eq. (3.2) to obtain −∞ It follows that Γ is given by the Gaussian   ∂p(x, t) Z ∞ τ = −p(x, t0) + w(x − x0)p(x0, t0)dx0 , ikX(t) k2σ2(t)/2 ∂t Γ(k, t) = e − , (3.8) −∞ Setting w(x) = 0.5δ(x − δx) + 0.5δ(x + δx) then yields where a spatially discrete version of the diffusion equation. On 2 2 the other hand, anomalous subdiffusion occurs if φ(t) ∼ X(t) = hX(t)i, σ (t) = h[X(t) − X(t)] i. (3.9) t α 1 with 0 < α < 1 (Metzler and Klafter, 2000). − − The mean and variance can be determined from Laplace In order to motivate the FLE, recall that a molecule transforming the generalized Langevin equation (3.3): moving through a fluid is subject to a frictional force and a random fluctuating force originating from random colli- sXe(s) − X(0) = Ve(s), (3.10a) sions between the Brownian molecule and particles of the 2 surrounding fluid. In the Langevin Eq. (2.9), the random s Xe(s) − V (0) − sX(0) + φe(s)V (s) = Fe(s). (3.10b) fluctuations are represented by a zero-mean Gaussian Rearranging (3.10b) gives noise term with a two-point correlation function taken to be a Dirac function. Moreover, the amplitude-squared of 1 Xe(s) − s− X(0) − He(s)V (0) = He(s)Fe(s), the noise (diffusion coefficient) is related to the friction coefficient via the Einstein relation, which is an exam- where ple of the fluctuation-dissipation theorem. It has been 1 suggested that the Langevin equation can be generalized He(s) = . (3.11) to the case of diffusion in complex heterogeneous media s[s + φe(s)] such as the cytoplasm or plasma membrane by consid- ering a Brownian particle subject to frictional and fluc- Inverting the equation for Xe(s) thus yields tuating forces with long–time correlations that exhibit Z t power-law behavior (Lutz, 2001; Porra et al., 1996; Wang, X(t) − X(0) − V (0)H(t) = H(t − t1)F (t1)dt1 (3.12) 1992; Wang and Lung, 1990). For example, consider the 0 following generalized Langevin equation for a Brownian It immediately follows from equation (3.12) that particle with unit mass moving in a 1D medium (Wang and Lung, 1990): X(t) = X(0) + V (0)H(t). (3.13)

X˙ (t) = V (t), (3.3a) Calculation of the variance is a little more involved. Z t First, using equations (3.12) and (3.13), X¨(t) + φ(t − t )V (t )dt = F (t), (3.3b) 1 1 1 Z t Z t 0 2 σ (t) = dt1 dt2H(t1)H(t2)hF (t − t1)F (t − t2)i where φ(t) represents a friction memory kernel with 0 0 φ(t) = 0 for t < 0. The Gaussian noise term F (t) satisfies Z t Z t = dt1 dt2H(t1)H(t2)C(t1 − t2) hF (t)i = 0, hF (0)F (t)i = C(t). (3.4) 0 0 Z t Z t1 = 2 dt1 dt2H(t1)H(t2)C(t1 − t2) Following Wang(1992); Wang and Lung(1990), the cor- 0 0 relation function C(t) is governed by a power law, Z t Z t1 = 2k T dt dt H(t )H(t )φ(t − t ) α B 1 2 1 2 1 2 C(t) = F0(α)t− , t > 0, (3.5) 0 0 18

The last two lines follow from the fact that C(t) is an Finally, note that one can construct a FBM model that even function and the generalized fluctuation-dissipation exhibits the same Gaussian behavior as the FLE in the theorem, respectively. The final step is to note that long-time limit (Lutz, 2001). The former considers a par- ticle evolving according to a Langevin equation of the Z t1 form A(t1) ≡ H(t2)φ(t1 − t2)dt2 0 dX 1 1 = −kX + ξ(t), (3.19) −→L Ae(s) = He(s)φe(s) = − dt s s + φe(s) −1 dH(t1) which is driven by fractional Gaussian noise of zero mean −−−→L A(t1) = 1 − , dt1 and a slowly decaying, power-law autocorrelation func- tion (t =6 t0) where L denotes the Laplace transform operator. Sub- 2 α 2 stituting back into the expression for σ gives (Wang, hξ(t)ξ(t0)i ∼ α(α − 1)K|t − t0| − . (3.20) 1992), In contrast to FLE, this process is driven by external  Z t  2 2 σ (t) = kBT 2 H(τ)dτ − H(t) . (3.14) noise without considering the fluctuation-dissipation the- 0 orem. Note that the Gaussian solution of FBM breaks down in the case of confined motion. The final step in the analysis is to substitute equations (3.13) and (3.14) into equation (3.8) and to differentiate the resulting expression for Γ(k, t) with respect to time B. Molecular crowding t. This gives

∂Γ(k, t)  2  One of the characteristic features of the interior aque- = ikV (0)h(t) − k kBTH(t)(1 − h(t)) Γ(k, t), ∂t ous environment of cells (cytoplasm) and intracellular (3.15) compartments such as the and mi- where h(t) = dH(t)/dt. Finally, carrying out the inverse tochondria is that they are crowded with macromolecules Fourier transform gives the FPE and skeletal proteins, which occupy 10%−50% of the vol- ume (Dix and Verkman, 2008; Fulton, 1982; Luby-Phelps, ∂p(x, t) ∂p(x, t) = −h(t)V (0) 2000). Cell membranes are also crowded environments ∂t ∂x containing lipids (molecules consisting of nonpolar, hy- ∂2p(x, t) + kBTH(t)(1 − h(t)) . (3.16) drophobic hydrocarbon chains that end in a polar hy- ∂x2 drophylic head), which are often organized into raft struc- For a given choice of correlation function C(t) and the tures, and various mobile and immobile proteins (Kusumi asymptotic properties of Laplace transforms in the small et al., 2005). One consequence of molecular crowding, s limit, one can determine the large t behavior of the which we will not consider further here, is that it can α drastically alter biochemical reactions in cells (Schnell FP equation. For example, if C(t) = F0(α)t− , it can be shown that when t → ∞ and 0 < α < 1, the FP equation and Turner, 2004; Zhou et al., 2008). That is, volume has the asymptotic form (Wang, 1992) or area exclusion effects increase the effective solute con- centration, thus increasing the chemical potential of the ∂p(x, t) α 2 ∂p(x, t) solute. Another consequence of molecular crowding is = −a (α)V (0)t − V (0) ∂t 1 ∂x that it hinders diffusion, although there is an ongoing de- 2 α 1 ∂ p(x, t) bate regarding to what extent this results in anomalous b (α)kBT t − , (3.17) 1 ∂x2 diffusion rather than a simple reduction in the normal diffusion coefficient (Banks and Fradin, 2005; Dix and for α–dependent coefficients a1, b1. Note in particular Verkman, 2008; Weiss et al., 2004). that the diffusion coefficient is time–dependent and there One of the difficulties in experimentally establishing 6 is a drift term when V (0) = 0. Both of these reflect the the existence of anomalous diffusion is that the behavior non–Markovian nature of the process. Given the initial of hR2i can depend on the spatial or temporal scale over condition p(x, 0) = δ(x), the asymptotic FP equation has which observations are made. Consider, for example, the the solution effects of obstacles on protein diffusion (Saxton, 1994; 1 Sung and Yethiraj, 2008). The presence of obstacles re- √ − α 1 2 α  p(x, t) = α exp (x + at − ) /(4D0t ) , 4πD0t duces the space available for diffusion, and consequently (3.18) decreases the effective diffusion coefficient. As the vol- where D0 = b1kBT/α, a = a1V (0)/(α − 1). It can also ume or area fraction of obstacles φ is increased, there is be shown that the mean-square displacement h[X(t) − a fragmentation of the available space in the sense that 2 α X] i = 2D0t , thus signifying anomalous subdiffusion. many paths taken by a diffusing protein terminate in a 19 dead end and thus do not contribute to diffusive trans- (a) port. The region of free diffusion develops a fractal–like structure resulting in anomalous diffusion at intermedi- 2 α √ate times, hR i ∼ t , α < 1. (For sufficiently small times Dt  ξ, where ξ is the mean distance between ob- stacles, so that diffusion is normal). However, assuming ν that the volume or area fraction is below the percolation ν' threshold, diffusion is expected to be normal on suffi- ciently long time-scales, hR2i ∼ t. On the other hand, above the percolation threshold, proteins are confined 10 (b) 1 (c) ] 2 2 1 0.8 and hR i saturates as t → ∞. The time it takes to m µ

[ III > cross over from anomalous to normal diffusion increases 0 0.6 2 0.1 /D R

with the volume or area fraction φ, and diverges at the < II eff 0.4

2 α D percolation threshold φ where hR i ∼ t for all times. 0.01 c 0.2 Another difficulty in interpreting experimental data is I 0.001 0 0.001 0.01 0.1 1 10 0 0.2 0.4 0.6 0.8 1 that there are certain practical limitations of current φ t [sec] methods (Dix and Verkman, 2008). The most effec- tive method for describing membrane diffusion is single- FIG. 3 (a) Diffusion of a finite size particle (tracer) be- particle tracking (SPT). This involves the selective label- tween obstacles of volume ν (left) can be modeled as diffusion ing of proteins or lipids with fluorophores such as quan- of a point particle between effective obstacles of volume ν0 tum dots, green fluorescent protein (GFP) or organic (right). Effective obstacles can partially overlap. (b) Sketch 2 dyes so that continuous high resolution tracking of in- of MSD hR i against time t, illustrating three different dif- dividual molecules can be carried out. SPT can yield fusion regimes: unobstructed diffusion (I), anomalous inter- mediate diffusion (II), and normal effective diffusion (III). (c) nanometer spatial resolution and submillisecond tempo- Illustrative plot of the normalized effective diffusion coeffi- ral resolution of individual trajectories. However, it is cient Deff (φ)/D0 for random spheres. The scale of the curves not currently suitable for measuring diffusion in three in (b,c) are based on the results of Novak et al. (2009). dimensions due to the relatively rapid speed of 3D dif- fusion and the problems of imaging in depth. Hence, in the case of diffusion within the cytosol, it is necessary to ple argument for this (Novak et al., 2009) is to consider use a method such as fluorescence recovery after photo- a set of N identical overlapping objects placed in a box bleaching (FRAP). Here fluorescently labeled molecules of total volume |Ω|. Let ν denote the volume of each are introduced into the cell and those in some specified obstacle. The probability P (x) that a randomly selected volume are bleached by a brief intense laser pulse. The point x ∈ Ω is outside any given obstacle is 1 − ν/|Ω|. diffusion of unbleached molecules into the bleached vol- Hence, the probability for that point to be outside all ume is then measured. FRAP is limited because it only obstacles is P (x)N = (1 − ν/|Ω|)N . The volume fraction provides ensemble averaged information of many fluores- of accessible space at fixed number density n = N/|Ω| is cent particles, and it also has a restricted measurement then time, making it difficult to capture long-tail phenomena expected in anomalous subdiffusion. N N nν V lim (1 − ν/|Ω|) = lim (1 − νn/N) = e− = e− N N Recently, homogenization theory has been used to de- →∞ →∞ velop a fast numerical scheme to calculate the effects of excluded volume due to molecular crowding on diffusion and the result follows. The mean distance between ob- in the cytoplasm (Novak et al., 2009). The basic idea is stacles can then be determined in terms of φ and the to model the heterogeneous environment in terms of ran- geometry of each obstacle. domly positioned overlapping obstacles. (Note, however, As previously discussed, three regimes of diffusion are that this is an oversimplification, since single-particle expected below the percolation threshold, φ < φc, as il- tracking experiments indicate that the cytoplasm is more lustrated in Fig.3(b). For sufficiently short times there properly treated as a dynamic, viscoelastic environment). is unobstructed diffusion, for intermediate times there is Although obstacles don’t overlap physically, when the fi- anomalous diffusion, and for long times there is normal nite size of a diffusing molecule (tracer) is taken into effective diffusion. Novak et al. (2009) use homogeniza- account, the effective volume excluded by an obstacle tion theory to estimate the effective diffusion coefficient increases so that this can result in at least partially over- Deff in the last regime. The starting point for their anal- lapping exclusion domains, see Fig.3(a). In the absence ysis is to consider a periodic arrangement of identical of any restrictions on the degree of overlap, the fraction obstacles in a large rectangular box of volume Ω with V of inaccessible volume is φ = 1 − e− , where V is the accessible volume Ω1 and φ = 1 − |Ω1|/|Ω|. The spatial sum of the individual obstacles per unit volume. A sim- periods of the arrangement in Cartesian coordinates are 20 aj, j = 1, 2, 3 such that the ratio Finally, averaging both sides of Eq. (3.25c) with respect 3 q to y over a unit volume |ω0|/ of the periodic structure, 2 2 2 p3  = a1 + a2 + a3/ |Ω|  1. (3.21) using the divergence theorem, and expressing u1 in terms of u0 yields the homogenized diffusion equation The heterogeneous diffusion coefficient is 3 2  X ∂ u0(x) D0 if x ∈ Ω1 Deeff,ij = 0, (3.28) D(x) = (3.22) ∂xi∂xj 0, otherwise. i,j=1 Inhomogeneous Dirichlet conditions are imposed on the with the anisotropic diffusion tensor boundaries of the box in order to maintain a steady–state 3 Z    ∂wi(y) diffusive flux. In the case of a heterogeneous diffusion Deeff,ij = D(y)δi,j + D(y) dy. |ω | ∂y coefficient, the flux is determined by the steady–state 0 ω0 j (3.29) diffusion equation for the tracer distribution u(x): Finally, rewriting the diffusion tensor in a more symmet- ∇ · (D(x)∇u(x)) = 0. (3.23) ric form using Eq. (3.27) and integration by parts gives (Novak et al., 2009) The basic idea of the homogenization method is to rep- resent the diffusive behavior of a tracer on two differ- Z 3     D0 X ∂wbi(x) ∂wbj(x) ent spatial scales (Pavliotis and Stuart, 2008; Torquato, Deeff,ij = δi,k + δj,k + dx, |ω0| ω ∂xk ∂xk 2002): one involving a macroscopic slow variable x and 1 k=1 the other a microscopic fast variable y ≡ x/ so that u where ω1 is the accessible region of the fundamental do- is periodic with respect to y. Thus, we write main ω0. The function w(x) has been rescaled according 1 to w(x) = w(x/) so that u = u(x, y), ∇u = ∇xu(x, y) + − ∇yu(x, y). b 3   x ≡ x y ∂D(x) X ∂ ∂wbi(x) Also D( ) D( /) = D( ) with D and u having the + D(x) = 0 (3.30) ∂xi ∂xj ∂xj same periodicity in y. j=1 A solution to Eq. (3.23) is then constructed in terms of the asymptotic expansion over a unit cell with periodic boundary conditions. Note that the concentration u0(x) is only defined in free 2 u = u0(x, y) + u1(x, y) +  u2(x, y) + .... (3.24) space so that the macroscopic concentration is actually u(x) = (1 − φ)u (x) and the macroscopic diffusion ten- Collecting terms of the same order in  then yields a 0 − hierarchy of equations, which up to O(1) are as follows: sor is Deff,ij = Deeff,ij/(1 φ). In the case of isotropic periodic structures Deff,ij = Deff δi,j. ∇y · [D(y)∇yu0(x, y)] = 0 (3.25a) Novak et al. (2009) numerically extended the homog- enization method to a random arrangement of obstacles ∇y · [D(y)∇yu1(x, y)] = −∇y · [D(y)∇xu0(x, y)] by approximating the disordered medium with a periodic − ∇ · [D(y)∇ u (x, y)] x y 0 one, in which the unit cell consists of N randomly placed (3.25b) obstacles. N is taken to be sufficiently large so that for a ∇y · [D(y)∇yu2(x, y)] = −∇x · [D(y)∇xu0(x, y)] given density of obstacles, one obtains a statistically sta-

− ∇y · [D(y)∇xu1(x, y)] tionary Deff . Comparing the homogenized diffusion coef- ficient with that obtained from Monte Carlo simulations, − ∇x · [D(y)∇yu1(x, y)] . (3.25c) Novak et. al. showed that the numerical homogenization method yielded reasonable agreement for N = O(100). Eq. (3.25a) and periodicity with respect to y establishes One of the interesting results of their study was that the that u0(x, y) ≡ u0(x), that is, u0 corresponds to an ho- variation of Deff with the excluded volume fraction φ can mogenized solution. It follows from Eq. (3.25b) that be approximated by the power law µ ∇ D(y) · ∇ u (x) + ∇ · [D(y)∇ u (x, y)] = 0, (1 − φ/φc) y x 0 y y 1 D (φ) = D , (3.31) eff 0 1 − φ which has the solution 3 where the parameters φc, µ depend on the geometry X ∂u0(x) of the obstacles. For example, for randomly arranged u1(x, y) = wi(y), (3.26) ∂xi ≈ ≈ i=1 spheres, φc 0.96 and µ 1.5. A typical plot of Deff (φ) is shown in Fig.3(c). Previously, the above with wi(y) a periodic function satisfying power law behavior had been predicted close to the per- 3   colation threshold (Bouchaud and Georges, 1990), but ∂D(y) X ∂ ∂wi(y) + D(y) = 0. (3.27) these results suggest it also holds for a wider range of ∂yi ∂yj ∂yj j=1 volume fractions. 21

C. Diffusion–trapping models base pair protein In the Smoluchowski theory of reaction kinetics, it is assumed that when a diffusing particle reacts with the target it disappears, that is, we have the trapping reac- DNA tion A+B → B where A denotes a diffusing particle and B denotes an immobile trap. However, within the con- text of intracellular transport, there are many examples target sequence where there is transient trapping of diffusing particles, FIG. 4 Schematic illustration of Barbi et al. (2004a,b) ran- resulting in anomalous diffusion on intermediate time- dom walk model of sequence-dependent protein diffusion scales and normal diffusion on long time-scales. This has along DNA. Each lattice site corresponds to a base pair with been elucidated by Saxton(1996, 2007), who carried out four binding sites that can potentially make hydrogen bonds Monte–Carlo simulations of random walks on a 2D lat- with the diffusing protein: acceptor sites (black dots), donor tice with a finite hierarchy of binding sites, that is bind- sites (gray dots) and missing sites (white dots). In this exam- ing sites with a finite set of energy levels. This means ple, the protein interacts with a base pair sequence of length r = 2. The energy of protein-DNA interactions determines that there are no traps that have an infinite escape time the transition rates to nearest neighbor lattice sites. so that diffusing particles ultimately equilibrate with the traps and diffusion becomes normal. On the other hand, in the case of infinite hierarchies, arbitrarily deep traps ten mediated by hydrogen bonds to a set of four specific exist but are very rare, resulting in a nonequilibrium sys- binding sites on each base pair. Some binding sites form a tem in which anomalous subdiffusion occurs at all times hydrogen bond as an acceptor, some as a donor and some (Bouchaud and Georges, 1990). The latter process can do not to form a bond. Barbi et. al. assume that at each be modeled in terms of a continuous-time random walk, site n of DNA, the protein attempts to form hydrogen see Sec. III.A. Here we will consider some examples of bonds with the local sequence of r base pairs. Hence, diffusive transport in the presence of transient immo- each potential binding site n is represented as a sequence bile traps. Note that a related application is calcium of r-vectors {bn, bn+1,..., bn+r 1}, one for each bp in buffering, where freely diffusing calcium molecules bind the sequence, according to the rule− to large proteins in the cytosol that can either be mobile or immobile. In particular, mobile buffering can lead  (1, −1, 1, 0)T for AT (0, 1, −1, 1)T for TA b = , to a nonlinear advection-diffusion equation that changes n (1, 1, −1, 0)T for GC (0, −1, 1, 1)T for CG many properties of the basic diffusion model, as detailed in Keener and Sneyd(2009). where +1, −1, 0 denote, respectively, an acceptor, a donor and a missing hydrogen bond on a given bp. The protein is then represented by a so–called (r ×4) recogni- 1. Sequence-dependent protein diffusion along DNA tion matrix R describing the pattern of hydrogen bonds formed by the protein and DNA at the target site where We begin by considering a 1D random walk model there is optimal matching. The protein-DNA interaction used to study sequence-dependent protein diffusion along energy is then defined by counting the matching and un- DNA (Barbi et al., 2004a,b). This concerns the impor- matching bonds between the recognition matrix and the tant problem of how a site–specific DNA binding protein DNA sequence at site n: locates its target binding site on DNA. As discussed in Sec. II.E, such a search process is thought to involve a E(n) = −Tr[R · B], (3.32) combination of mechanisms, including one–dimensional sliding along the DNA and uncorrelated 3D diffusion where B is the matrix whose r columns are given by et al. et al. the vectors {bn, bn+1,..., bn+r 1}, and  denotes each (Berg , 1981; Halford and Marko, 2004). Barbi − (2004a,b) model the sliding phase of protein movement in hydrogen bond energy. terms of a 1D random walk, in which the step probabil- Given the above energy landscape, Barbi et. al. model ity to neighboring sites depends on an energy landscape the dynamics of protein sliding motion along DNA as a that reflects sequence dependent protein-DNA interac- 1D random walk, in which the protein is represented as tions, see Fig.4. This is motivated by the idea that the a particle hopping to its nearest neighboring lattice sites protein needs to “read” the underlying sequence of base with rates pairs (bps) as it slides along the DNA in order to be able 1 ∆E 0 /kB T to detect the target site. Thus each nonspecific site on rn n0 = e− n→n , n0 = n ± 1, (3.33) → 2τ DNA acts as a potential trap for the sliding protein. The sequence of the target site usually consists of a where ∆En n0 is the effective energy barrier between → few (r) consecutive bps, and sequence recognition is of- neighboring sites, kB is the Boltzmann constant, and T is 22 the temperature. Various models can be considered relat- drite, the spines act as transient traps as illustrated in ing the barrier energy to the site-dependent energy E(n) Fig.7(a). Following along similar arguments to the case (Barbi et al., 2004b). The simplest is to take ∆En n0 = of diffusion in the presence of obstacles, normal diffusion → max[E(n0)−E(n), 0]. Monte Carlo simulations may then is expected at short and long times and anomalous sub- be used to study the dynamics of the resulting random diffusion at intermediate times. Anomalous subdiffusion walk with transient traps (Barbi et al., 2004a,b). In the was indeed observed by Santamaria et al. (2006), such 2 2/β large time limit, a population of non-interacting random that the mean square displacement hx (t)i ∼ D0t at walkers will reach a stationary Boltzmann distribution of intermediate times with β > 2 and D0 the free diffusion E(n)/kB T the form ρ(n) ∼ e− and the associated dynamics coefficient. As might be expected, β increases (slower will exhibit normal diffusion. However, for large enough diffusion) with increasing spine density. β also increases values of /kBT , some sites along the DNA could trap when the volume of the spine head is increased relative a protein for significant times, suggesting that anoma- to the spine neck, reflecting the fact there is an enhanced lous behavior could be observed at intermediate times. bottleneck. Note that anomalous diffusion can occur at This is indeed found to be the case. Defining the MSD all times if the reactions within each spine are taken to 2 1 PN 2 according to h∆n i = N − i=1(ni(t) − ni(0)) where have a non-exponential waiting time density (Fedotov N is the number of proteins in the population, it was et al., 2010), see also III.A. found numerically that h∆n2i ∼ tα, α < 1 at intermedi- A related problem is the diffusive transport of neu- ate times, with a crossover to normal diffusion (α = 1) at rotransmitter protein receptors within the plasma mem- large times. Moreover, using experimentally based model brane of a dendrite, with each spine acting as a transient parameters, Barbi et. al. showed that the crossover time trap that localizes the receptors at a synapse. The ma- was sufficiently large that anomalous diffusion occurred on time-scales comparable to the typical sliding phase of target search (Barbi et al., 2004a,b). This suggests that anomalous diffusion is likely to dominate. j dendrite

2. Diffusion along spiny dendrites i u(t) Another recent example of anomalous diffusion in the presence of transient traps has been considered by San- soma V(t) wij tamaria et al. (2006). These authors used a combination of experimental and computational modeling to study synapse how the presence of dendritic spines affects the 3D dif- AP fusion of signaling molecules along the dendrites of neu- rons. Neurons are amongst the largest and most complex axon cells in biology. Their intricate geometry presents many FIG. 5 Basic structure of a neuron. [Inset shows a synaptic challenges for cell function, in particular with regards to connection of strength wij from an upstream or presynaptic the efficient delivery of newly synthesized proteins from neuron labeled j and a downstream or postsynaptic neuron the cell body or soma to distant locations on the axon labeled i]. Neurons in the brain communicate with each other or dendrites. The axon contains ion channels for action by transmitting electrical spikes (action potentials). An ac- potential propagation and presynaptic active zones for tion potential (AP) propagates along the axon of a neuron neurotransmitter release, whereas each dendrite contains until it reaches a terminal that forms the upstream or presy- naptic component of the synaptic connection to a downstream postsynaptic domains (or densities) where receptors that or postsynaptic neuron. The arrival of the action potential in- bind neurotransmitter tend to cluster, see Fig.5. At duces the release of chemical transmitters into the synapse. most excitatory synapses in the brain, the postsynaptic These subsequently bind to protein receptors in the postsy- density (PSD) is located within a dendritic spine, which naptic membrane resulting in the opening of various ion chan- is a small, sub-micrometer membranous extrusion that nels. This generates a synaptic current that flows along the protrudes from a dendrite (Sorra and Harris, 2000), see dendritic tree of the postsynaptic neuron and combines with Fig.6. Typically spines have a bulbous head that is con- currents from other activated synapses. If the total synaptic current u(t) forces the membrane potential V (t) at a certain nected to the parent dendrite through a thin spine neck, location within the cell body to cross some threshold, then the and there can exist thousands of spines distributed along postsynaptic neuron fires an action potential and the process a single dendrite. It is widely thought that spines act to continues. One can thus view the brain as a vast collection compartmentalize chemical signals generated by synap- of synaptically–coupled networks of spiking neurons. More- tic activity, thus impeding their diffusion into dendrites over, the strength of synaptic connections within and between (Sabatini et al., 2001; Yuste et al., 2000). Conversely, in networks are modifiable by experience (synaptic plasticity). the case of signaling molecules diffusing along the den- 23

which respond to the neurotransmitter glutamate. There is now a large body of experimental evidence that the fast trafficking of AMPA receptors into and out of spines is a major contributor to activity-dependent, long-lasting changes in synaptic strength (Bredt and Nicoll, 2003; Collinridge et al., 2004; Henley et al., 2011; Shepherd and Huganir, 2007). Single–particle tracking experiments suggest that surface AMPA receptors diffuse freely within the dendritic membrane until they enter a spine, where they are temporarily confined by the geometry of the spine and through interactions with scaffolding proteins and cytoskeletal elements (Choquet and Triller, 2003; FIG. 6 An example of a piece of spine studded dendritic tissue Ehlers et al., 2007; Gerrow and Triller, 2010; Groc et al., (from rat hippocampal region CA1 stratum radiatum). The 2004; Newpher and Ehlers, 2008; Triller and Choquet, dendrite on the right-hand side is ∼ 5µm in length. Taken 2005). A surface receptor may also be internalized via en- with permission from SynapseWeb, Kristen M. Harris, PI, docytosis and stored within an intracellular pool, where http://synapses.clm.utexas.edu/ it is either recycled to the surface via exocytosis or de- graded (Ehlers, 2000), see Fig.7(b). is the physical process whereby vesicles are formed within the plasma membrane and then internalized, and exocyto- jority of fast excitatory synaptic transmission in the cen- sis is the complementary process in which intracellular tral nervous system is mediated by AMPA (α-amino-3- vesicles fuse with the plasma membrane and release their hydroxy-5-methyl-4-isoxazole-propionic acid) receptors, contents (Doherty and McMahon, 2009). Molecular mo- tors transport internalized vesicles to intracellular com- (a) partments that either recycle vesicles to the cell surface dendrite (early and recycling endosomes) or sort them normal for degradation (late endosomes and lyososomes) (Max- diffusion field and McGraw, 2004; Soldati, 2006). A number of single spine models have explored the combined effects of diffusion, trapping, receptor clustering and recycling on dendritic spines the number of synaptic AMPA receptors (Burlakov et al., 2012; Czondora et al., 2012; Earnshaw and Bressloff, anomalous 2006; Holcman and Triller, 2006; Shouval, 2005). In such diffusion models, the synapse is treated as a self-organizing com- partment in which the number of AMPA receptors is a dynamic steady-state that determines the strength of the synapse; activity-dependent changes in the strength of (b) the synapse then correspond to shifts in the dynamical AMPA receptor set-point. When receptor-receptor interactions are in- lateral scaffolding protein cluded a synapse can exhibit bistability between a non- diffusion clustered and clustered state (Shouval, 2005), which can recycling be understood in terms of a liquid-vapor phase transition (Burlakov et al., 2012). It is also possible to develop a diffusion–trapping model degradation of receptor trafficking at multiple spines by considering a 2D version of the Santamaria et al. (2006) model, in FIG. 7 (a) Schematic illustration of the anomalous diffusion which receptors diffuse on the surface of a cylindrical model of Santamaria et al. (2006), who carried out detailed dendrite containing multiple disk-like traps; when a re- 3D simulations of diffusion in a spiny dendrite treated as ceptor transiently enters a trap it can undergo various a system of connected cylinders with the following baseline parameter values: spine neck diameter 0.2µm, neck length reactions corresponding to processes within a spine such 0.6µm, head length and diameter 0.6µm , dendrite diame- as receptor recycling and binding to anchoring proteins. ter 1µm and a spine density of 15 spines/µm. The dendritic Using asymptotic methods similar to those of Straube spines act as transient traps for a diffusing particle within the et al. (2007), see Sec. II.D, one can show that the 2D dendrite, which leads to anomalous diffusion on intermediate model is well approximated by a reduced 1D cable model time scales. (b) Schematic illustration of various pathways of in which dendritic spines are treated as point–like sources AMPA receptor trafficking at a dendritic spine. or sinks (Bressloff et al., 2008). The advantage of the 1D 24 model is that the associated 1D Green’s function is non– (3.34a), (3.34c) and using the initial conditions, gives singular. Therefore, consider a population of N identi- 2 NX cal spines distributed along a uniform dendritic cable of ∂ pe X 1 −zp + D0 = Ω[pj − Rej/A]δ(x − xj) − l− δ(x), length L and circumference l, with xj, j = 1,...,N, the e ∂x2 e position (axial coordinate) of the jth spine. Let p(x, t) j=1 (3.37a) denote the probability density (per unit area) that a sur- face receptor is located within the dendritic membrane zRej = Ω[pej − Rej/A] − kRej + σSej, (3.37b) at position x at time t. Similarly, let Rj(t),Sj(t) de- zSej = −σSej + kRej, (3.37c) note the probability that the receptor is trapped at the surface of the jth spine or within an associated intra- where pej(z) = pe(xj, z). In the limit z → 0, Eqs. (3.37b)– cellular pool, respectively. A simple version of the 1D (3.37c) imply that Apej(0) = Rej(0) = σSej(0)/k, and Eq. diffusion–trapping model of AMPA receptor trafficking (3.37a) becomes takes the form (Bressloff and Earnshaw, 2007; Earnshaw and Bressloff, 2008) ∂2p(x, 0) lD e = −δ(x). (3.38) 0 ∂x2 N ∂p ∂2p X = D − Ω[pj − Rj/A]δ(x − xj), (3.34a) Imposing the boundary conditions at x = 0,X gives ∂t 0 ∂x2 j=1 lpe(x, 0) = (X − x)/D0. Combining these results, dR j NX = Ω[pj − Rj/A] − kRj + σSj, (3.34b) X2 η X dt τ(X) = + (X − xj), (3.39) 2D D dSj 0 0 j=1 = −σSj + kRj, (3.34c) dt where η = A[1+k/σ]/l. The first term on the right–hand where D0 is the surface diffusivity and pj(t) = p(xj, t). side of this equation is the MFPT in the absence of any The term Ω(pj − Rj/A) with A the surface area of a spines, whereas the remaining terms take into account spine represents the probability flux into the jth spine the effects of being temporarily trapped at a spine. with Ω an effective hopping rate. (This rate depends In order to calculate an effective diffusivity, consider on the detailed geometry of the dendritic spine (Ashby the simple example of identical spines distributing uni- et al., 2006)). It is assumed that surface receptors within formly along the cable with spacing d. That is, xj = jd, the jth spine can be recycled with respect to the intra- j = 1,...,N such that Nd = L and NX = X/d for cellular pool with k, σ the rates of endocytosis and ex- X  d. Eq. (3.39) then becomes (for NX  1) ocytosis, respectively. Eq. (3.34a) is supplemented by N reflecting boundary conditions at the ends of the cable: X2 X2 η XX τ(X) ≡ = + (X − jd). D∂xp(0, t) = 0 and D∂xp(L, t) = 0. 2D 2D D 0 0 j=1 The effective diffusivity of a receptor in the long-time limit, which takes into account the effects of trapping Using the approximation at spines, can be determined by calculating the MFPT NX 2 τ(X) to travel a distance X from the soma. Introducing X (NX + 1)NX d X (X − jd) = NX X − ≈ an absorbing boundary condition at x = X, the function 2 2d j=1

N Z X XX finally gives (Bressloff and Earnshaw, 2007) P(X, t) ≡ l p(x, t)dx + [Rj(t) + Sj(t)] (3.35) 0 j 1 =1  A  k − D = D 1 + 1 + . (3.40) 0 ld σ is then the probability that t < τ(X); i.e., the probability that a receptor which was initially at the origin has not As expected, the presence of traps reduces the effective yet reached the point x = X in a time t. Here NX is diffusivity of a receptor. In particular, the diffusivity is the number of spines in the interval [0,X). The MFPT reduced by increasing the ratio k/σ of the rates of endo- R ∞ is then τ(X) = 0 P(X, t)dt. It follows that the MFPT cytosis and exocytosis or by increasing the surface area can be expressed in terms of Laplace transforms; A of a spine relative to the product of the spine spac- ing d and circumference of the cable l. Interestingly, D N Z X XX h i does not depend on the hopping rate Ω. Taking typi- 2 1 τ(X) = pe(x, 0)dx + Rej(0) + Sej(0) (3.36) cal measured values of the diffusivity (D = 0.1µm s− ) 0 j=1 (Ashby et al., 2006; Groc et al., 2004), the area of a spine (A = 1µm2), the spacing between spines (d = 1µm) and R ∞ zt where fe(z) ≡ 0 e− f(t)dt. Laplace transforming Eqs. the circumference of a dendrite (l = 1µm)(Sorra and 25

membrane skeleton anchored proteins Harris, 2000), it follows that Deff = 0.5D when k = σ, whereas D  D0 when k  σ. There is experimental evidence that the rates of exo/endocytsosis are activity– dependent (Ehlers et al., 2007) so that the ratio k/σ, and hence D, may be modifiable by experience. There have been various generalizations of the above model to include the effects of binding/unbinding to cy- toskeletal proteins (Earnshaw and Bressloff, 2008) and dendritic branching (Bressloff, 2009). In the latter case, homogenization theory can be used to replace the discrete Membrane-skeleton (fence) Anchored-protein (picket) distribution of spines by a continuum density. One ma- jor simplification of the diffusing–trapping model is that FIG. 8 Picket-fence model of membrane diffusion. The it neglects the detailed structure of a spine and the as- plasma membrane is parceled up into compartments whereby sociated PSD. A more comprehensive model would need both transmembrane proteins and lipids undergo short-term to take into the complex organization of the PSD, in- confined diffusion within a compartment and long-term hop diffusion between compartments. This corralling is assumed teractions with scaffolding proteins, and the geometry of to occur by two mechanisms. (a) The membrane-cytoskeleton the spine (Freche et al., 2011; Kerr and Blanpied, 2012; (fence) model: transmembrane proteins are confined within MacGillavry et al., 2011; Sekimoto and Triller, 2009). Fi- the mesh of the actin-based membrane skeleton. (b) The nally, note that the coupling between exocytosis and en- anchored-protein (picket) model: transmembrane proteins, docytosis during AMPA receptor recycling is one exam- anchored to the actin-based cytoskeleton, effectively act as ple of a more general transport mechanism that occurs rows of pickets along the actin fences. in neurons (and other secretory cells) via the so–called endocytic pathway (Gundelfinger et al., 2003). Other examples include the insertion and removal of membrane cobson, 1997). This has led to a modification of the proteins during axonal elongation and guidance (Bloom original fluid mosaic model, whereby lipids and trans- and Morgan, 2011), see also Sec. V.A, and the stimulus- membrane proteins undergo confined diffusion within, induced release of secretory molecules (neurotransmit- and hopping between, membrane microdomains or cor- ters) at the presynaptic terminal of a synapse (see Fig. rals (Kusumi et al., 2005, 2010; Vereb et al., 2003); the 5). The latter is regulated by the exocytosis of synaptic corraling could be due to “fencing” by the actin cytoskele- vesicles; endocytic processes then have to be coordinated ton or confinement by anchored protein “pickets”, see so that there is an efficient reuptake of vesicles in order Fig.8. These microdomains could also be associated to restore functionality of the synapse. For a detailed with lipid rafts (Jacobson et al., 2007; Kusumi et al., discussion of whole cell kinetic models of receptor recy- 2010). cling and its role in chemical signaling see Lauffenburger Partitioning the membrane into a set of corrals im- (1996); Wiley et al. (2003). plies that anomalous diffusion of proteins will be ob- served on intermediate timescales, due to the combined 3. Diffusion in the plasma membrane effects of confinement and binding to the actin cytoskele- ton. However, on time-scales over which multiple hop- At the simplest level, the plasma membrane can be ping events occur, normal diffusion will be recovered. treated as a 2D lipid sheet into which proteins are em- A rough estimate of the corresponding diffusion coeffi- bedded. In the fluid mosaic model of Singer and Nicol- cient is D ∼ L2/τ, where L is the average size of a mi- son(1972), the membrane lipids are treated as the sol- crodomain and τ is the mean hopping rate between mi- vent (water concentrations are very low within the mem- crodomains. A typical range of values for various types of brane) into which proteins are dissolved. One of the con- mammalian cell are L ∼ 30 − 240 nm and τ ∼ 1 − 20 ms. sequences of the fluid mosiac model is that protein clus- In the case of confinement by anchored protein pickets, tering, which alters the effective size of a diffusing par- τ can be estimated by treating each corral as a domain ticle, has only a weak effect on diffusion in the plasma with a set of small holes (gaps) between anchored pro- membrane. This follows from the hydrodynamic mem- teins, and solving the narrow escape problem (Holcman brane diffusion model of Saffman and Delbruck (Saffman and Schuss, 2004; Holcman and Triller, 2006). (Another and Delbruck, 1975), which implies that the diffusion co- approach to estimating τ has been developed by Kalay efficient for a cylinder of radius r in a 2D membrane et al. (2008); Kenkre et al. (2008), based on a random varies as log r. Although the diffusion of lipids appears walker moving on a 1D lattice with either periodically or to be Brownian in pure lipid bilayers, single-particle randomly distributed semipermeable barriers). On the tracking experiments indicate that lipids and proteins other hand, the membrane cytoskeleton surrounding a undergo anomalous diffusion in the plasma membrane corral is usually modeled as an effective energy barrier (Feder et al., 1996; Kusumi et al., 2005; Saxton and Ja- over which a diffusing protein must escape. For exam- 26 ple, Saxton(1995) carried out a computational study of a multi–state version of the stochastic gate, which cor- a particle diffusing inside a corral surrounded by a static responds to the random opening and closing of multiple energy barrier. It was assumed that when the particle small holes within the PSD boundary. The master Eq. hit the barrier it had a fixed probability of escape. The (3.41) is solved for a single realization of the stochastic MFPT out of the corral was numerically determined for process described by Eq. (3.42). As a consequence, dif- a wide range of corral sizes, shapes and escape probabil- ferent realizations of µ(t) will yield different probability ities. In earlier work, Saxton(1989, 1990) considered a distributions Pn. static fence model in which a protein could only move In order to analyze the above model, introduce the from one corral to another if the particular barrier sepa- generating function rating the two corrals was dissociated. In this particular model, large-scale diffusion only occurs if there exists a X∞ n G(u, t) = u Pn(t), (3.43) percolation network. However, estimates of the density n=0 of the actin cytoskeleton in red blood cells (erythrocytes), for example, suggests that the fraction of disassociated It follows from the master Eq. (3.41) that G satisfies the cytoskeleton is below the percolation threshold. Hence, first-order linear partial differential equation it is necessary to modify the percolation model by con- ∂G ∂G + µ(t)(u − 1) = σ(t)(u − 1)G (3.44) sidering time-dependent, fluctuating energy barriers. ∂t ∂u Consider, for example, the spatially homogeneous n0 stochastic gating model of Brown et al. (2000); Leitner with initial condition G(u, v, 0) = u . Eq. (3.44) can be solved using the method of characteristics (Kampen, et al. (2000). Let Pn(t) denote the probability that there are n free particles within the corral at time t. Denote 1992): the time-dependent rates of protein influx and loss by n0 C(1 (t))(u 1) G(u, t) = [1 + N (t)(u − 1)] e −N − , (3.45) λ(t) and µ(t), respectively. The probability distribution is then taken to evolve according to the master equation where  Z t  dPn = σ(t)Pn 1 + µ(t)(n + 1)Pn+1(t) − [σ(t) + µ(t)n]Pn N (t) = exp − µ(t0)dt0 . (3.46) dt − 0 (3.41) Given G(u, t), the mean and variance of n can be cal- with n ≥ 0 and P 1(t) ≡ 0. The positive terms on the culated according to the formulae right-hand side represent− the various transitions into the ∂G ∂2G state (n) whereas the negative terms represent the vari- 2 − Eµ(n) = , Eµ(n n) = 2 , ous transitions from the state (n). The initial condition ∂u u=v=1 ∂u u=v=1 is P (0) = δ ; i.e., at time t = 0 there are n free n n,n0 0 where the subscript µ indicates that these means are cal- particles within the corral. In the model of Brown et al. culated with respect to a single realization of the ran- (2000); Leitner et al. (2000), the escape of a protein from dom variable µ only, and may therefore take on different the corral is controlled by a stochastic gate that can be values for different realizations of µ. Calculating these in two states, an open state for which µ(t) = µ > 0 and o derivatives yields a closed state for which µ(t) = µc = 0. The opening 2 and closing of the stochastic gate is governed by the rate Eµ(n) = (n0 − C)N (t) + C, Varµ(n) = Eµ(n) − n0N (t) . equations A more useful characterization of the means and vari- o dP o c ances can be obtained by averaging N (t) with respect to = −γ P + γ+P , dt − all possible stochastic realizations of the gate, which is c dP o c denoted by hN i. This can be performed using a method = γ P − γ+P , (3.42) dt − originally developed by Kubo(1962) in the study of spec- where Po(t)(Pc(t)) is the probability that the gate is tral line broadening in a quantum system, and subse- open (closed) at time t, and γ are the transition rates quently extended to chemical rate processes with dynam- between the two states. Thus the± time-dependent escape ical disorder by Zwanzig(1990). One thus finds that the rate µ(t) describes a dichotomous noise process. From µ-averaged mean and variance are detailed balance, the rate at which receptors enter the E(n) = (n0 − C)hwi + C, E(N) = E(n) + L, PSD is then taken to be σ(t) = Cµ(t) with C fixed. (At 2 2 2 2 equilibrium C can be identiffied with the mean number Var(n) = E(n) − n0hw i + (n0 − C) hw i − hwi , of particles in the corral). One possible interpretation where of the stochastic gate is that it represents the random  T      opening and closing of a small hole within the bound- j 1 jµo + γ −γ+ Πo w(t) = exp −t − ary of the PSD. This suggests that one could consider 1 −γ γ+ Πc − 27 for j = 1, 2. Here Πl, l = o, c, are the stationary proba- 1. Fick-Jacobs equation bility distributions for the dichotomous noise process of Eq. (3.42): We begin by deriving the Fick-Jacobs equation for a Brownian particle diffusing in a 2D channel as shown in γ+ γ Fig.9(a). We follow the particular derivation of Zwanzig Πo = , Πc = − . γ+ + γ γ+ + γ (1992), see also (Reguera and Rubi, 2001). It is assumed − − that the channel walls at y = ±w(x) confine the mo- The averages hwji, j = 1, 2, approach zero as time in- tion of the particle but do not exchange energy with it. creases, hence the steady-state means and variances are Thus the probability flux normal to the boundary is zero. E (n) = Var (n) = C. There have been a number of This condition can be imposed by introducing a confin- ∞ ∞ extensions of the stochastic gating model. For example, ing potential U(x, y) such that U(x, y) = 0 for |y| < w(x) Bressloff and Earnshaw(2009) have considered the ef- and U(x, y) = ∞ for |y| ≥ w(x). Let p(x, y, t) denote fects of proteins binding to scaffolding proteins within a the probability that the particle is located at position corral, whereas Reingruber and Holcman(2010) have an- x = (x, y) at time t with periodic boundary conditions in alyzed the narrow escape problem for a particle that can the longtitudinal direction, p(x + L, y, t) = p(x, y, t). For switch between different conformational states and can a general potential U(x, y), the 2D FP equation takes the only exit a domain in one of these states. form ∂p 1 ∂[F p] ∂[F p]  ∂2p ∂2p  = − x + y + D + , ∂t γ ∂x ∂y 0 ∂x2 ∂y2 D. Diffusion in confined geometries

where Fx = −∂xU, Fy = −∂yU. Using the Einstein rela- 1 Another common form of diffusion within cells is the tions D0γ = kBT = β− , the FP equation can be rewrit- transport of particles through a narrow biological pore or ten as channel. Examples include membrane transport through ∂p ∂ βU(x,y) ∂ βU(x,y) ion channels and pumps (Hille, 2001), and the translo- = D0 e− e p(x, y, t) cation of structured polynucleotides through nanopores ∂t ∂x ∂x et al. ∂ βU(x,y) ∂ βU(x,y) (Peskin , 1993), which is an important technique for + D0 e− e p(x, y, t). (3.47) investigating the translocation dynamics of biologically ∂y ∂y et al. relevant macromolecules (Gerland , 2004; Keyser In order to reduce to a 1D equation, first integrate both et al. , 2006). In such examples, changes in the motion of sides of the FP equation with respect to the transverse a particle occur mainly in the axial direction along the coordinate y: channel, whereas local equilibrium is rapidly reached in the transverse directions. Thus transport is quasi one– Z w(x) ∂P (x, t) ∂ βU(x,y) ∂ βU(x,y) dimensional and the effects of the boundaries of the chan- = D0 e− e p(x, y, t)dy, ∂t ∂x w(x) ∂x nel can be incorporated by introducing an entropic bar- − rier into the dynamics of a Brownian particle, leading to where P (x, t) is the reduced probability density the so–called Fick–Jacobs equation (Burada et al., 2009, 2007; Jacobs, 1967; Kalinay and Percus, 2006; Reguera Z w(x) and Rubi, 2001; Rubi and Reguera, 2010; Zwanzig, 1992). P (x, t) = p(x, y, t)dy. (3.48) w(x) Typically a 3D narrow channel is represented by a cylin- − der that extends axially in the x–direction and has a pe- riodically varying cross section that is rotationally sym- (a) metric about the x-axis, see Fig.9(a). Denoting the w(x) space–dependent radius by w(x), the cross-section varies as A(x) = πw(x)2. In the case of a corresponding 2D channel, w(x) represents the half–width of the channel. An extreme version of confined diffusion along a channel is single–file diffusion, in which the channel is so narrow x that particles cannot pass each other. In other words, (b) the longtitudinal motion of each particle is hindered by the presence of its neighbors, which act as moving ob- stacles, see Fig.9(b). Hence, interparticle interactions can suppress Brownian motion and lead to subdiffusive behavior (Barkai and Silbey, 2009; Jepsen, 1965; Levitt, FIG. 9 Confined diffusion in a narrow cylindrical channel with 1973; Percus, 1974; Rodenbeck et al., 1998; Taloni and a periodically modulated boundary w(x) in the axial direc- Marchesoni, 2006). tion. (a) Small diffusing particle. (b) Single-file diffusion. 28

The major step in the reduction is to assume that the the x direction, then Eq. (3.51) still holds except that probability density reaches equilibrium in the transverse F(x) = −F0x − kBT log σ(x). direction. That is, p(x, y, t) is assumed to factorize as Given an external force F0 and the periodic entropic follows: barrier potential V(x), it remains to determine the mean and variance of the particle position in the long time p(x, y, t) ≈ P (x, t)ρ(x, y), (3.49) limit, which naturally leads to the following definitions of the drift mobility and diffusion coefficient of the particle: where ρ(x, y) is a normalized Boltzmann-Gibbs probabil- ity density: hX˙ i hX(t)i µ(F0) ≡ , hX˙ i = lim , (3.54) t βU(x,y) Z w(x) F0 →∞ t e− β (x) 1 βU(x,y) ρ(x, y) = β (x) , e− F = e− dy, A0e− F A0 w(x) and − (3.50) h 2i − h i2 R L X(t) X(t) where A = 2 w(x)dx and F(x) interpreted as an ef- D(F0) = lim . (3.55) 0 0 t 2t fective x–dependent free energy. Under this factorization →∞ the averaged FP equation becomes Note that the relationship between hX˙ i and the long time limit of hX(t)i/t is a consequence of ergodicity (Reimann, ∂P (x, t) ∂ β (x) ∂ β (x) ≈ D0 e− F e F P (x, t). (3.51) 2002). In order to determine µ and D, it is necessary to ∂t ∂x ∂x extend the classical problem of Brownian motion in a This holds for a general potential energy function U(x, y) periodic potential with tilt (Burada et al., 2007; Hanggi (Reguera and Rubi, 2001). et al., 1990; Reimann et al., 2002; Stratonovich, 1958), If U is now taken to be the confining potential of the see also III.D.2. The force–dependence of the mobility β (x) and diffusion coefficient have been studied both analyti- channel boundary, then e− F = 2w(x)/A0 ≡ σ(x) and we obtain the Fick–Jacobs equation cally and numerically in the case of a sinusoidal boundary function (Burada et al., 2007; Reguera et al., 2006) ∂P (x, t) ∂ ∂ P (x, t) = D σ(x) . (3.52) ∂t ∂x 0 ∂x σ(x) w(x) = a[sin(2πx/L) + b], a > 0, b > 1. (3.56)

The same equation is obtained in 3D with σ(x) = The basic results are sketched in Fig. 10. A number of 2 A(x)/A0 with A(x) = πw(x) and A0 the mean cross- interesting observations emerge from this study. First, sectional area. The Fick–Jacobs equation is valid pro- the mobility only depends on the temperature via the di- vided that |w0(x)|  1. However, it has been shown that mensionless parameter F0L/kBT . Hence, increasing the the introduction of an x–dependent diffusion coefficient temperature reduces the mobility. Second, as the force is into the Fick-Jacobs equation can considerably increase increased the effective diffusion coefficient D(F0) exceeds the accuracy of the reduced FP equation and thus ex- the free diffusion coefficient D0. Using scaling arguments, tend the domain of validity (Kalinay and Percus, 2006; it can also be shown that the analysis based on the Fick- Reguera and Rubi, 2001; Zwanzig, 1992): Jacobs equation begins to break down at a critical force F0,c where (Burada et al., 2009) D0 D(x) = α , (3.53)  2 [1 + w (x)2] F0,cL 1 L 0 ≈ 2 . (3.57) kBT 2(1 + b) a with α = 1/3, 1/2 for 2D and 3D respectively. Note that in the absence of any external forces, the effective free The Fick-Jacobs equation represents diffusion through energy F(x) = V(x), where V(x) ≡ −kBT log[A(x)/A0] a narrow channel in terms of a 1D overdamped Brown- reflects the existence of an entropic barrier to diffusion ian particle moving in an effective potential U(x) that (Reguera and Rubi, 2001). That is, using the standard arises from entropic effects. Such a 1D model has also thermodynamic definition of free energy F = E − TS, been the starting point for a series of studies of channel- where E is internal energy and S is the entropy, it facilitated membrane transport, where now U(x) reflects follows that S(x) ∼ log A(x) where A(x) is the cross- the constructive role of attractive interactions between sectional area of the channel at x. This is consistent permeating particles and proteins forming the channel with the microcanonical ensemble definition of entropy. pore (Berezhkovskii and Bezrukov, 2005; Berezhkovskii That is, in equilibrium there is a uniform probability et al., 2003, 2002; Bezrukov et al., 2000). In these stud- density ρ0 in the channel, so that the equilibrium x– ies, mixed boundary conditions are assumed at the ends dependent density Peq(x) = ρ0A(x)/A0 and the number x = 0,L of the channel: J(0, t) = −κ0P (0, t) and of microstates available to a diffusing particle at location J(L, t) = −κLP (L, t). The probability of crossing the x is proportional to the area of the channel. It also fol- channel and the mean time in the channel can then be cal- low that when there is a constant external force F0 in culated using the standard theory of first passage times 29

1 1.5 where V (x) is an L–periodic potential, V (x + L) = V (x) 0.75 for all x, and F0 is a constant external force, see Fig. 11. For the moment, we will take the diffusion coef-

0 1 0.5 γµ ficient to be constant. Within the context of motion D/D through a narrow channel V (x) can be identified with 0.25 0.5 the entropic potential V(x) = −kBT log[A(x)/A ]. How- (a) (b) 0 0 0 ever, there are many other important applications where 15 30 45 60 75 20 60 40 80 a periodic potential arises, including Brownian ratchet F0L/kBT F0L/kBT models of molecular motors (Reimann, 2002), see Sec. FIG. 10 Illustrative sketches of how mobility and diffusivity IV.A.1. Again we will focus on spatially continuous pro- vary with non-dimensionalized applied force F0L/kB T in the cesses. Note, however, that an alternative approach has case of a 2D channel with a sinusoidally varying half-width been developed in a seminal paper by Derrida(1983), (3.56). (a) Effective mobility µ in units of γ. In the limit which is concerned with calculating the effective diffu- −1 F0 → ∞, µ → γ . (b) Diffusion coefficient D in units of sion and velocity of particles on a discrete lattice, and free diffusivity D0. In the limit F0 → ∞, D → D0. Sketches is a spatilly discrete version of the processes considered are based on numerical results of (Reguera et al., 2006) for a = L/2π and b = 1.02. here. The approach of Derrida has many applications, in- cluding motor protein modeling (Kolomeisky and Fisher, 2007). and splitting probabilities, see Sec. II.B. It can be shown that there is an optimal form of the interaction poten- V(x) F = 0 tial that maximizes the flux through the channel, and 0 involves a play off between increasing the translocation x probability through the channel and decreasing the av- erage time particles spend in the channel (Berezhkovskii and Bezrukov, 2005). For a complementary approach to studying channel-facilitated transport that is based V(x)−F0x on spatially discrete stochastic site-binding models, see (Chou, 1998; Kolomeisky, 2007). Finally, note that stochastic models of confined diffusion through channels have also been developed for charged particles flowing F0 > 0 through a nanopore connected to two large reservoirs of electrolyte solutions (Nadler et al., 2004; Schuss et al., FIG. 11 Brownian particle moving in a periodic potential V (x). In the absence of tilt (F = 0) the mean velocity in the 2001). The motion of the ions is sensitive to the specific 0 long time limit is zero. On the other hand, in the presence of nanoscale geometry and the charge distribution around a tilt (F0 6= 0) the net motion of the particle is in the direction the channel so that standard continuum mean-field mod- of the force. els breakdown. Two of the most common mean-field models are the equilibrium Poisson-Boltzmann equation We begin by considering the standard Stratonovich- and the non-equilibrium Poisson-Nernst-Planck equation based calculation of the mean velocity (Hanggi et al., (Roux et al., 2004). These equations assume a constitu- 1990; Stratonovich, 1958). Introduce the effective poten- tive relation between the average ion flux and an effective tial or free energy F(x) = V (x) − F0x and note that mean field potential that satisfies Poisson’s equation for F 0(x) is periodic even though F is not. Next consider the average charge concentrations. We will not consider the reduced probability density and currents ion transport further in this review. See for example Keener and Sneyd(2009) for a detailed discussion of the X∞ X∞ role of ion transport in cell physiology. pb(x, t) = p(x+nL, t), Jb(x, t) = J(x+nL, t) n= n= −∞ −∞ (3.59) with 2. Brownian motion in a periodic potential with tilt  1 ∂p  J(x, t) = −D0 F 0(x)p + . Motivated by the problem of a particle diffusing in kBT ∂x a narrow channel with a periodically varying boundary, consider the 1D FP equation It immediately follows that

∂p  1 ∂[V (x) − F ]p ∂2p  Z L = D 0 0 + , (3.58) 0 2 pb(x + L, t) = pb(x, t), pb(x, t)dx = 1. (3.60) ∂t kBT ∂x ∂x 0 30

Moreover, multiplying both sides of the FP equation by determing D (as well as the mobility µ) is to exploit a x and integrating with respect to x gives well-known recursion relation for the moments of the first passage time (Burada et al., 2007; Reguera et al., 2006; Z Z L dhX(t)i ∞ Reimann et al., 2002). Let T (x → b) denote the first = J(x, t)dx = Jb(x, t)dx. (3.61) 0 dt 0 passage time for the diffusing particle to reach the point b −∞ given that it started at x0 with x0 < b. The nth moment ˙ n (Using the identity J(x, t) = hX(t)δ(x − X(t))i and in- of the first passage time is τn(x0 → b) = hT (x0 → b)i. tegrating both sides with respect to x, it follows that It can then be shown that for the stochastic process de- hX˙ (t)i = dhX(t)i/dt (Reimann, 2002)). The periodicity scribed by the FP Eq. (3.58), the moments satisfy the of F 0(x) implies that if p(x, t) is a solution of the FP recursion relation (Hanggi et al., 1990) equation, then so is p(x + nL, t). The principle of super- position for a linear PDE then shows that p satisfies the n Z b Z x b τ (x → b) = dx e (x)/kB T dy e (y)/kB T FP equation n 0 F F D0 x0 −∞ × τn 1(y → b), n = 1, 2 ..., (3.69) ∂p(x, t) ∂Jb(x, t) − b + = 0, (3.62) ∂t ∂x with τ0(y → b) = 1. Note for n = 1, x0 = y, b = L with we recover Eq. (2.30) (after taking the left hand bound- ary to −∞). The basic derivation proceeds as follows.   1 ∂pb First, it can be shown that T (x0 → x0 + lL), integer l Jb(x, t) = −D0 F 0(x)pb+ (3.63) kBT ∂x is statistically equivalent to a sum of (iid) random vari- ables T (x0 → x0 + L),T (x0 + L → x0 + 2L),...,T (x0 + and periodic boundary conditions at x = 0,L. There (l − 1)L → x0 + lL). Hence, for large l, the central exists a stationary solution pb0 of the reduced FP equation limit theorem (Gardiner, 2009) implies that the FPT with constant flux Jb0 such that T (x0 → x0 +lL) approaches a Gaussian distribution with   mean lτ1(x0 → x0 + L) and variance l∆τ2(x0 → x0 + L) d (x)/kB T Jb0 (x)/kB T 2 eF p (x) = − eF . (3.64) where ∆τ = τ − τ . Second, since µ and D are de- dx b0 D 2 2 1 0 fined in the large t limit, the evaluation of hx(t)i and Integrating this equation from x to x + L and using pe- hx2(t)i can be partitioned into a set of large but finite riodicity yields the stationary solution steps over which the statistics of the corresponding FPT is well approximated by a Gaussian distribution. This Jb0N (x) has the important implication that if any two stochas- p0(x) = , (3.65) b  F0L/kB T  1 − e− tic processes described by an FP equation of the form (3.58) have the same mean τ1(x0 → x0 +L) and variance where ∆τ2(x0 → x0 + L), then they have the same hX˙ i and D. Z x+L It finally follows that (Reimann et al., 2002) 1 (x)/kB T (y)/kB T N (x) = e−F eF dy. (3.66) D L L2 ∆τ (x → x + L) 0 x h ˙ i 2 0 0 X = ,D = 3 . τ1(x0 → x0 + L) 2 [τ1(x0 → x0 + L)] Finally, Jb0 is determined by imposing the normalization ˙ condition on pb0. Since hX(t)i = LJb0 for constant current, The proof of the last step simply consists of verifying that these formulae hold when V (x) = 0 (no periodic poten- F0L/kB T ˙ ˙ 1 − e− tial), for which hXi = F/γ and D = kBT/γ. Having ob- hX(t)i = L L . (3.67) R N (x)dx tained D in terms of the first and second order moments 0 of the FPT, Eq. (3.69) can now be used to calculate D. It can be seen that there is no net motion in a purely After some algebra (see appendix A of (Reimann et al., periodic potential, since the numerator vanishes when 2002)), one finds that F0 = 0. Moreover the net direction of motion for F0 =6 0 Z x0+L is in the direction of the applied force. Note that in the N (x)2N (x)dx/L case of a space–dependent diffusion coefficient D(x), the D = D x0 , (3.70) N 0 " #3 above analysis is easily extended with (x) now given Z x0+L by (Burada et al., 2007) N (x)dx/L x0 Z x+L (x)/kB T 1 (y)/kB T N (x) = e−F eF dy. (3.68) with N (x) given by Eq. (3.66) and x D(y) Z x 1 (x)/kB T (y)/kB T The calculation of the diffusion coefficient is consid- N (x) = eF e−F dy. (3.71) D0 x L erably more involved. However, an elegant method for − 31

3. Single-file diffusion reflection symmetry ensures that hX(t)i = 0, where X(t) is the stochastic position of the tagged particle at time When a pore or channel becomes sufficiently narrow, t. The main underlying idea is to map the many-body particles are no longer able to pass each other, which im- problem to a non-interacting one by allowing particles to poses strong constraints on the diffusive motion. An ide- pass through each other and keeping track of the particle alized model of single–file diffusion considers a 1D collec- label, see Fig. 12(b). That is, assuming that collisions tion of diffusing particles with hard-core repulsion. The are elastic and neglecting n-body interactions for n > 2, many–body problem of single–file diffusion was originally it follows that when two particles collide they exchange tackled by relating the dynamics of the interacting sys- momenta and this is represented as an exchange of parti- tem with the effective motion of a free particle (Harris, cle labels. The probability density for the tagged particle 1965; Jepsen, 1965; Lebowitz and Percus, 1967; Levitt, to be at X(t) = XT at time t then reduces to the problem 1973; Percus, 1974). In particular, in the case of an infi- of finding the probability that the number of free parti- nite system and a uniform initial particle density, it was cle trajectories that started at x0 < 0 and are now to the shown that a tagged particle exhibits anomalous subdif- right of XT is balanced by the number of free particle 2 1/2 fusion on long time scales, hX (t)i ∼ t . (On the other trajectories that started at x0 > 0 and are now to the hand, the center of mass of the system of particles ex- left of XT . j j hibits normal diffusion). More recently, a variety of com- Thus, let PLL(x0− )(PLR(x0− )) denote the probability plementary approaches to analyzing single-file diffusion j that the jth free particle trajectory starting from x0− < 0 have been developed (Barkai and Silbey, 2009; Centres at t = 0 is to the left (right) of XT at time t. Similarly, let and Bustingorry, 2010; Lizana and Ambjornsson, 2008; j j PRR(x0)(PRL(x0)) denote the probability that the jth Rodenbeck et al., 1998; Taloni and Lomholt, 2008). Here j free particle trajectory starting from x0 > 0 at t = 0 is to we review the particular formulation of Barkai and Silbey the right (left) of XT at time t. Let α be the net number (Barkai and Silbey, 2009), which develops the analysis of free particle trajectories that are on the opposite side of of a tagged particle in terms of classical reflection and XT at time t compared to their starting point (with left transmission coefficients. to right taken as positive). The associated probability (a) distribution for α given 2N untagged particles is (Barkai and Silbey, 2009)

Z π N 1 Y j j iαφ PN (α) = Γ(φ, x− , x )e dφ, (3.72) 2π 0 0 π j=1 1 − (b) where 1 1 j j iφ j j j j 1 Γ(φ, x0− , x0) = e PLR(x0− )PRR(x0) + PLL(x0− )PRR(x0) j j iφ j j 2 + PLR(x0− )PRL(x0) + e− PLL(x0− )PRL(x0). (3.73) x 2 2 2 t The integration with respect to φ ensures that the net R π iφn number of crossings is α, that is, π e = δn,0. Since FIG. 12 (a) Single-file diffusion of a tagged particle (red) the trajectories are independent and− the initial conditions surrounded by other impenetrable particles. (b) Equivalent are (iid) random variables, P (α) can be averaged with noninteracting picture, in which each trajectory is treated as N a non-interacting Brownian particle by keeping track of the respect to the initial conditions to give exchange of particle label. Z π 1 N iαφ hPN (α)i = hΓ(φ)i e dφ, (3.74) 2π π Suppose that the tagged particle is initially at the ori- − gin with N particles to its left and N particles to its right, where see Fig. 12(a). The motion of each particle in the ab- iφ  iφ  sence of hard core interactions is taken to be overdamped hΓ(φ)i = hPRRi + e− hPRL hPLLi + e hPLRi . Brownian motion as described by the Langevin Eq. (2.9) (3.75) or the corresponding FP Eq. (2.15). As a further simplifi- The averages hPLRi etc. can be calculated using the R x cation, the potential energy function V (x) = F (x0)dx0 Green’s function G(x, x0, t) of the corresponding FP Eq. is taken to be symmetric, V (x) = V (−x), as is the initial (2.15) with G(x, x0, 0) = δ(x − x0). For example, distribution of particles. That is, if the initial position x of a particle is drawn from f (x ) for x > 0 and Z 0 Z l 0 R 0 0 hP i = f (x ) G(x, x , t)dx dx , (3.76) − LR L 0 0 0 from fL(x0) for x0 < 0, then fR(x0) = fL( x0). This l XT − 32 where 2l is the length of the 1D domain, which can be Taking a uniform initial distribution f(x0) = 1/l with taken to be infinity. l → ∞ and fixed particle density ρ = N/l, one finds Eq. (3.74) takes the form of the generating function for anomalous subdiffusion for large times t (Harris, 1965): a discrete random walk of N steps and a net displacement √ 2 Dt of α. Hence, for large N, application of the central limit hX(t)2i ∼ √ . (3.86) theorem leads to the Gaussian approximation π ρ

1 2 2 On the other hand, for particles initially centered at the PN (0) ∼ √ exp(−Nµ1/2σ ), (3.77) 2πNσ2 origin, f(x0) = δ(x0), diffusion is normal 2 2 πDt where σ = µ2 − µ1 and µ1, µ2 are the first two moments hX(t)2i ∼ . (3.87) of the structure function: 2N 1 In the case of a bounded domain or a Gaussian initial con- hΓ(φ)i = 1 + iµ φ − µ φ2 + O(φ3). (3.78) 1 2 2 dition, anomalous diffusion occurs at intermediate times only (Barkai and Silbey, 2009). Hence,

µ = hP i − hP i, σ2 = hP ihP i + hP ihP i. 1 LR RL RR RL LL LR E. Nuclear transport (3.79) Since hX(t)i = 0 and N is assumed to be large, µ and 1 The nucleus of is surrounded by a protective σ2 can be Taylor expanded with respect to X about T nuclear envelope (NE) within which are embedded nu- X = 0. Reflection symmetry then implies that T clear pore complexes (NPCs). The NPCs are the sole me- diators of exchange between the nucleus and cytoplasm. hPLLi|X =0 = hPRRi|X =0 ≡ R, T T In general small molecules of diameter ∼ 5nm can diffuse through the NPCs unhindered, whereas larger molecules hPLRi| = hPRLi| ≡ T = 1 − R, XT =0 XT =0 up to around 40nm in diameter are excluded unless they are bound to a family of soluble protein receptors known h i| − h i| ≡ J ∂XT PLR XT =0 = ∂XT PRL XT =0 . as karyopherins (kaps), see the reviews (Fahrenkrog et al., 2004; Macara, 2001; Rout et al., 2003; Tran and Wente, R T The time–dependent functions and may be inter- 2006). Within the cytoplasm kap receptors bind cargo to preted as reflection and transmission coefficients deter- be imported via a nuclear localization signal (NLS) that mining whether or not a free particle trajectory crosses results in the formation of a kap-cargo complex. This XT = 0. The resulting mean and variance are complex can then pass through an NPC to enter the nu- 2 2 cleus. A small enzyme RanGTP then binds to the kap, µ1 = −2J XT + O(XT ), σ = 2R(1 − R) + O(XT ). (3.80) causing a conformational change that releases the cargo. The sequence of events underlying the import of cargo is Thus, hPN (α)i for α = 0 reduces to a Gaussian distribu- shown in Fig. 13(a). In the case of cargo export from tion for the position X(t) = XT : the nucleus, kaps bind to cargo with a nuclear export sig-  2  1 XT nal (NES) in the presence of RanGTP, and the resulting P (XT , t) = p exp − , (3.81) 2πhX(t)2i 2hX(t)2i complex passes through the NPC. Once in the cytoplasm, RanGTP undergoes hydrolysis to form RanGDP, result- with ing in the release of the cargo. The export process is illus- R(1 − R) trated in Fig. 13(b). Finally, RanGDP is recycled to the hX(t)2i = . (3.82) 2NJ 2 nucleus by another molecule NFT2 and is reloaded with GTP to begin another import/export cycle. This cycle Finally, using Eq. (3.76), allows a single NPC to support a very high rate of trans- Z l Z l port on the order of 1000 translocations/sec (Ribbeck R = f(x0) G(x, x0, t)dxdx0 (3.83) and Gorlich, 2001). Since the transportation cycle is di- 0 0 rectional and accumulates cargo against a concentration gradient, an energy source combined with a directional Z l cue is required. Both of these are provided by the hydrol- J = f(x0)G(0, x0, t)dx0. (3.84) ysis of RanGTP and the maintenance of a concentration 0 gradient of RanGTP across the NPC. The RanGTP gra- In the special case of zero external forces, the free par- dient is continuously regenerated by GTP hydrolysis in ticle Green’s function is the cytoplasm, translocation of RanGTD into the nucleus

2 1 (x x0) /4Dt by NFT2, and replacement of GDP by GTP in the nu- G(x, x0, t) = √ e− − . (3.85) 4πDt cleus. It is important to note that the energy generated 33 from RanGTP hydrolysis is ultimately used to create a 2011; Moussavi-Baygi et al., 2011). Here we will review concentration gradient of kap-cargo complexes between the first two types of model in a little more detail. the nucleus and cytoplasm, so that the actual transloca- tion across the NPC occurs purely via diffusion. Entropic gate model. Recall from Sec. III.D that a macromolecule diffusing in a confined geometry (such as CYTOPLASM NUCLEUS a nuclear pore) experiences an entropic barrier due to nuclear envelope (a) importing excluded volume effects. Within the NPC this would be karyopherin RanGTP enhanced by the densely packed FG-repeats. One way to counteract the effects of the entropic barrier is for the NPC kaps to have an affinity for and bind to the FG-repeat regions (Rout et al., 2003; Zilman et al., 2007), thus low- kap-Cargo complex ering the effective Gibbs free energy of the cargo-complex NLS-cargo within the NPC. The degree of affinity has to be suffi- ciently high to overcome the entropic barrier but not too

exporting high otherwise the complex can be trapped within the (b) RanGTD karyopherin NPC and the rate of translocation would be too small. One possible solution, is to have a large number of low– NPC affinity binding sites within the nuclear pore. Recently, hydrolysis a mathematical model for the effects of binding on dif- kap-Cargo complex fusion within the NPC has been developed by Zilman NES-cargo et al. (2007), based on diffusion through an effective en- FIG. 13 Schematic illustration of the (a) import and (b) ex- ergy landscape. This is based on the assumption that port process underlying the karyopherin-mediated transporta- the binding/unbinding rates are relatively fast compared tion of cargo between the nucleus and cytoplasm via a nuclear to the diffusion rate. The simplest version of the model pore complex (NPC). See text for details. is illustrated in Fig. 14 for the case of nuclear import. The effective potential energy U(X) is taken to be a flat Although the above basic picture is now reasonably potential well of depth E along an NPC, and zero outside well accepted, the detailed mechanism underlying facil- the NPC. Absorbing boundary conditions are placed at itated diffusion of kap-cargo complexes within the NPC the points x = 0,L a distance R from either side of the is still not understood. The NPC is composed of about NPC, which has length L − 2R. The left-hand boundary 30 distinct proteins known collectively as nucleoporins takes into account the fact that not all complexes enter- (nups). It has emerged in recent years that individual ing the NPC will reach the nucleus, that is, some will nups are directly related to a number of human diseases eventually return to the cytoplasm. Diffusion within the including influenza and cancers such as leukemia (Cron- NPC is described by a standard Smoluchowski equation shaw and Matunis, 2004), as well as playing an impor- for the density of cargo-complexes ρ(x), x = [0,L], see tant role in viral infections by providing docking sites for Sec. II.A.2: viral capsids (Whittaker et al., 2000). Associated with ∂ρ ∂J ∂ρ ∂U many of the nups are natively unfolded phenylalanine- = − ,J = −D − Dρ , (3.88) ∂t ∂x ∂x ∂x glycine (FG) repeats, known collectively as FG-repeats. The FG-repeats set up a barrier to diffusion for large with U measured in units of kBT . This equation is sup- molecules so that the key ingredient in facilitated dif- plemented by the boundary conditions ρ(0) = ρ(L) = 0. fusion through the NPC is the interaction between kap absorbing boundaryabsorbing receptors with the FG-repeats. In essence, the major boundaryabsorbing difference between the various theoretical models of NPC potential U(x) transport concerns the built in assumptions regarding the E J properties and spatial arrangements of FG-repeats within JS e the NPC, and the nature of interactions with kaps dur- _ J ing translocation through the NPC (Becskei and Mat- 0 J JL taj, 2005). Current models include the Entropic Gate complex model (Rout et al., 2003; Zilman et al., 2007), Selective Phase models (Bickel and Bruinsma, 2002; Kustanovich x=0 NPC x=L and Rabin, 2004; Ribbeck and Gorlich, 2001, 2002), Di- R R mensionality Reduction model (Peters, 2005), and the Polymeric Brush model (Lim et al., 2007, 2006). A num- FIG. 14 Model of Zilman et al. (2007). Transport of cargo- ber of computational models and molecular-based simu- complex through the NPC is modeled as diffusion in an energy lations are also being developed (Grunwald and Singer, landscape. See text for details. 34

The steady-state solution is obtained by assuming that gesting that they act approximately like a weak reversible there are constant fluxes J0 in [0,R], JL in [L−R,L] and gel. Particles smaller than the mesh size of the network J in [R,L − R] with J0 < 0. These fluxes are related ac- can diffuse freely through the NPC, whereas non-selective cording to JS = J − |J0| and J = JL + Je, where JS is macromolecules larger than the mesh size cannot. On the the total flux of complexes injected into the NPC from other hand, kap-cargo complexes can ’dissolve’ in the gel the cytoplasm, which is proportional to density of com- due to the presence of hydrophobic domains on the sur- plexes in the cytoplasm, and Je denotes the flux due to face of the kap receptors, and then diffuse through the active removal of complexes from the nucleus end of the pore by breaking the weak bonds of the reversible gel NPC by RanGTP. The latter depends on the number (Ribbeck and Gorlich, 2001, 2002). However, as pointed of complexes at the nuclear exit, the rate Jran at which out by Bickel and Bruinsma(2002), the observed high RanGTP molecules hit the exit: Je = Jranρ(L − R)R. permeability of the NPC with respect to the transport of The steady-state rate of transport J can now be deter- kap-cargo complexes is inconsistent with the basic theory mined by solving for ρ(x) in terms of J0,JL, J in each of of reversible gels. The argument proceeds as follows (see the three domains and imposing continuity of the density appendix of Bickel and Bruinsma(2002)). For a homoge- at x = R and x = R − L. The result is that the fraction neous pore filled with reversible gel, the flux of particles of complexes reaching the nucleus is given by (Zilman through the pore is given by J = D∆φ/L, where L is the et al., 2007) length of the pore, D is the diffusivity of dissolved com- plexes, and ∆φ = φL − φR is the difference in concentra- " # 1 Z L R − tions of dissolved complexes at the ends of the pore. The J K 1 − U(x) P = = 2 − + e dx , (3.89) permeability Π, however, is defined in terms of the dif- JS 1 + K R R ference in concentrations of exterior particle reservoirs on either side of the pore, ∆c = c − c . That is, J = Π∆c. with K = J R2/D. It follows that for a sufficiently L R ran In order to relate ∆c and ∆φ, it is necessary to consider deep well (large E), where the integral term is negligible, the fluxes entering and exiting the pore. Equating the and for sufficiently large K (large J ), the probability of ran fluxes at the left and right ends gives translocation is P ≈ 1. On the other hand, if K is small so that RanGTP does not facilitate entry of complexes J = kincL − koutφL = koutφR − kincR (3.90) into the nucleus then Pmax = 0.5. As previously indi- cated, it is not possible to arbitrarily increase the affinity This allows one to express ∆φ in terms of ∆c such that of binding sites and thus the well depth E, since this will lead to trapping of the complexes so that they accumu- kin Π = . (3.91) late within the NPC, resulting in molecular crowding and 2 + koutL/D an unrealistically long time for an individual molecule to pass through the NPC. Thus there is some optimal well From detailed balance the ratio of the rates is kin/kout = β∆F depth that balances an increase of transport probability e where ∆F is the free energy gain (assuming ∆F > P with increased time spent in the NPC (Zilman et al., 0) of entering the gel. Ignoring other contributions to 2007). Finally, note that the model is robust with re- the free energy, ∆F = n, where n is the number of gards the particular shape of the potential well. For ex- interactions of strength  between a kap receptor and the ample, one could represent transport through the NPC gel. It can also be shown that the diffusivity of a spherical as diffusion in an array of overlapping potential wells that kap-cargo complex moving through a reversible gel is represent flexible FG-repeat regions. The shape of each D = D (1 + e n) 1 ≈ D e βn. (3.92) well will depend on the number and affinity of binding 0 − − 0 − sites on each FG-repeat, and the degree of flexibility of Suppose that in the high affinity regime kout is given the polymers which will determine the entropic costs of β∆F by the Arrhenius law kout ∼ (D/δ)e− , where ∆ is bending and stretching the FG-repeats. It can be shown taken to be the size of the boundary layer at either end that for relatively fast binding/unbinding, the multi-well of the pore. Combining all of these results then shows potential can be replaced by a single well along the lines that (Bickel and Bruinsma, 2002) of Fig. 14. D0 Selective phase models. The basic assumption of these Π = . (3.93) L + 2δeβn models is that the NPC can be treated as a channel filled with a hydrophobic medium consisting of a concentrated Thus the permeability decreases with the number of sites polymer solution; the latter is composed of the natively n, implying that increasing the affinity of the complex unfolded, flexible protein domains of FG-repeats (Bickel moving in a reversible gel should decrease the perme- and Bruinsma, 2002; Kustanovich and Rabin, 2004; Re- ability; this contradicts the high permeabilities seen ex- ichenbach et al., 2006; Ribbeck and Gorlich, 2001, 2002). perimentally (Ribbeck and Gorlich, 2002). One sugges- The FG-repeats form weak bonds with each other sug- tion for modifying the original selective phase model is 35

IV. ACTIVE INTRACELLULAR TRANSPORT

A. Modeling molecular motors at different scales

Diffusion inside the cytosol or along the plasma mem- NPC brane of a cell is a means by which dissolved macro- molecules can be passively transported without any in- put of energy. However, there are two main limitations of Cytoplasm Nucleus passive diffusion as a mechanism for intracellular trans- port. First, it can take far too long to travel the long distances necessary to reach targets within a cell, which FIG. 15 Selective phase model (Bickel and Bruinsma, 2002; is particularly acute in the case of the axons and den- Ribbeck and Gorlich, 2001, 2002), in which the FG repeats drites of neurons. Second, diffusive transport tends to within a NPC are treated as a reversible polymer gel. See be unbiased, making it difficult to sort resources to spe- text for details. cific areas within a cell. Active intracellular transport can overcome these difficulties so that movement is both faster and direct specific, but does so at a price. Ac- to assume that the polymer gel is under tension due to tive transport cannot occur under thermodynamic equi- pinning of the polymers to the container walls (Bickel librium, which means that energy must be consumed and Bruinsma, 2002). Thermal fluctuations would then by this process, typically via the hydrolysis of adeno- lead to local rearrangements of the stretched polymer sine triphosphate (ATP). The main type of active intra- network, resulting in rearrangements of the associated cellular transport involves the molecular motors tension field. This, in turn, could generate force fluc- and carrying resources along microtubular fila- tuations on a dissolved macromolecule that could en- ment tracks. Microtubules are polarized polymeric fila- hance its effective diffusivity. Such a mechanism has ments with biophysically distinct (+) and (−) ends, and not yet been confirmed experimentally. However, single this polarity determines the preferred direction in which particle tracking of individual complexes moving through an individual molecular motor moves. In particular, ki- the NPC shows that cargo bound to more kap receptors nesin moves towards the (+) end whereas dynein moves diffuse more freely (Lowe et al., 2010). This is consis- towards the (−) end (Howard, 2001). Each motor protein tent with the tensile gel model but inconsistent with the undergoes a sequence of conformational changes after re- entropic gate model, for which greater affinity implies acting with one or more ATP molecules, causing it to slower transport of individual complexes. step forward along the microtubule in its preferred direc- tion. Thus, ATP provides the energy necessary for the Chaperone–assisted translocation of polymers through molecular motor to do work in the form of pulling its nanopores. Finally, note that here and in Sec. IIID we cargo along the microtubule in a biased direction. have considered diffusion of particles that are smaller The movement of molecular motors and motor/cargo than or comparable in size to the diameter of the chan- complexes occur over several length and time scales nel. However, there are many examples where it is nec- (Julicher et al., 1997; Keller and Bustamante, 2000; essary to consider translocation of an unfolded (or par- Kolomeisky and Fisher, 2007; Lipowsky and Klumpp, tially folded) polymer through a nanopore, including the 2005). In the case of a single motor there are at least translocation of RNA through the nuclear pore mem- three regimes: (i) the mechanicochemical energy trans- brane, as well as newly synthesized proteins into the en- duction process that generates a single step of the mo- doplasmic reticulum (see also Sec. VB). Polymer translo- tor; (ii) the effective biased random walk along a fila- cation through a nanopore involves a large entropic bar- ment during a single run; (iii) the alternating periods rier due to the loss of many conformational states, so that of directed motion along the filament and diffusive or some form of driving force is required. One suggested stationary motion when the motor is unbound from the driving mechanism involves the binding of chaperone pro- filament. A popular model for the stochastic dynamics teins to the translocating polymer on the far side of the of a single motor step in regime (i) is the so–called Brow- nanopore, which prohibits the polymer diffusing back- nian ratchet (Reimann, 2002), which extends the theory ward through the pore, thus speeding up translocation of overdamped Brownian motion in periodic potentials (Matlack et al., 1999). This rectified stochastic motion that was reviewed in Sec. III.D.2. In the case of dimeric was orginally analyzed in terms of a Brownian ratchet or double–headed kinesin, a single step is of length 8µm by Peskin et al. (1993), and has subsequently been devel- and the total conformational cycle takes around 10ms. oped in a number of studies (Ambjornsson and Metzler, In the second regime (ii), multiple steps are taken before 2004; D’Orsogna et al., 2007; Elston, 2000b; Krapivsky a motor disassociates from the filament. For example, and Mallick, 2010). kinesin takes around 100 steps in a single run, covering 36 a distance of around 1µm. Walking distances can be V2(x) substantially increased if several molecular motors pull the cargo. In the third motility regime (iii), molecular ω ω motors repeatedly unbind and rebind to filaments. In 1 2 the unbound state a motor diffuses in the surrounding V1(x) aqueous solution with a diffusion coefficient of the order 2 1 1µm s− . However, molecular crowding tends to con- fine the motor so that it stays close to its detachment x point. At these longer length and time scales, the mo- tion of the motor can be represented in terms of a system FIG. 16 Brownian ratchet model of a molecular motor that can exist in two internal states with associated l–periodic po- of PDEs. This combines a discrete Markov process for tentials V (x) and V (x). State transition rates are denoted the transitions between bound and unbound states with 1 2 by ω1 and ω2. an FP equation for the advective or diffusive motion in the different states (Newby and Bressloff, 2010b; Reed et al., 1990; Smith and Simmons, 2001). In bidirectional where ωij(x) is the rate at which the motor switches from transport, there may be more than one type of bound state j to state i. state. In order to develop the basic theory, consider the sim- ple case of two internal states N = 2 following along the lines of Julicher et al. (1997); Parmeggiani et al. (1999); 1. Brownian ratchets Prost et al. (1994). Then

In performing a single step along a filament track, a ∂p (x, t) ∂J (x, t) 1 + 1 = −ω (x)p (x, t) + ω (x)p (x, t) molecular motor cycles through a sequence of conforma- ∂t ∂x 1 1 2 2 tional states before returning to its initial state (modulo (4.5a) the change in spatial location). Suppose that there is ∂p (x, t) ∂J (x, t) 2 + 2 = ω (x)p (x, t) − ω (x)p (x, t). a total of M conformational states in a single cycle la- ∂t ∂x 1 1 2 2 beled i = 1,...,M. Given a particular state i, the motor (4.5b) is modeled as an overdamped, driven Brownian particle moving in a periodic potential Vi(x). The Langevin equa- Note that adding the pair of equations together and set- tion for the location of the particle X(t) assuming that ting p = p1 + p2, J = J1 + J2 leads to the conserva- it remains in the given state is tion equations ∂tp + ∂xJ = 0. An example of l–periodic ratchet (asymmetric) potentials V1(x),V2(x) is shown in dX V (X) = − i0 dt + ξ(t), (4.1) Fig. 16, with l the basic step length of a cycle along dt γ the filament track. The analysis of the two–state model proceeds along similar lines to the one state model con- with hξ(t)i = 0 and hξ(t)ξ(t0)i = 2Dtδ(t − t0). The cor- sidered in Sec. III.D.2. That is, set responding FP equation is X∞ X∞ ∂pi(x, t) ∂Ji(x, t) p (x, t) = p (x+nl, t), J (x, t) = J (x+nl, t). = − , (4.2) bj j bj j ∂t ∂x n= n= −∞ −∞ (4.6) where pi(x, t) is the probability density that the motor The total probability flux can then be written as particle is in internal state i and at location x at time t, and Ji(x, t) is the probability flux Jb(x, t) (4.7)     1 ∂pb(x, t) 1 ∂ = − V10(x)p1(x, t) + V20(x)p2(x, t) + kBT . Ji(x, t) = −V 0(x) − kBT pi(x, t). (4.3) γ b b ∂x γ i ∂x Consider the steady-state solution for which there is a If the state transitions between the conformational states constant total flux Jb0 so that are now introduced according to a discrete Markov pro- cess, then it is necessary to add source terms to the FP ∂pb(x) V 0(x)p1(x) + V 0(x)p2(x) + kBT = −Jb0γ. equation: 1 b 2 b ∂x ∂p (x, t) ∂J (x, t) i = − i (4.4) Defining λ(x) = pb1(x)/pb(x), this equation can be rewrit- ∂t ∂x ten as X + [ωij(x)pj(x, t) − ωji(x)pi(x, t)] , ∂pb(x) j=i V 0 (x)p(x) + kBT = −Jb0γ, (4.8) 6 eff b ∂x 37 where Clearly detailed balance no longer holds. In general, it is Z x now necessary to determine the steady–state solution of Veff (x) = [λ(y)V10(y) + (1 − λ(y))V20(y)] dy. (4.9) the pair of Eqs. (4.5) numerically. Given such a solution, 0 the efficiency of the motor doing work against a load F Suppose that the system is in thermodynamic equilib- may be determined as follows. First the flux (4.3) has an rium so that detailed balance holds. That is, the state additional term of the form F pi(x, t)/γ. The mechanical transition rates and steady-state probabilities satisfy work done per unit time against the external force is then W˙ = F v where v = lJb0 is the velocity of the motor. On the other hand, the chemical energy consumed per unit ω1(x) [V1(x) V2(x)]/kB T p2(x) = e − = . (4.10) time is Q˙ = r∆µ, where r is the steady–state rate of ATP ω2(x) p1(x) consumption: Therefore, Z l 1 r = [(α1(x) − γ1(x))pb1(x) − (α2(x) − γ2(x))pb2(x)]dx. λ(x) = , (4.11) 0 [V1(x) V2(x)]/kB T 1 + e− − The efficiency of the motor is then defined to be (Julicher and, in particular, λ(x) reduces to an l–periodic function. F v et al., 1997) η = r∆µ . It follows that Veff (x) in Eq. (4.8) is also an l-periodic potential and hence there is no net motion in a particular direction (in the absence of an external force or tilt), see 3 3 3 Sec. III.D.2. In conclusion, in order for a molecular mo- tor to sustain directed motion that can pull against an applied load, we require a net positive supply of chemi- 2 2 2 cal energy that maintains the state transition rates away from detailed balance - this is the role played by ATP. Therefore, consider the situation in which transitions 1 1 1 between the two states occur as a result of chemical re- k-1 k k+1 actions involving ATP hydrolysis. Denoting the two con- ∆ formational states of the motor by M1,M2, the scheme x is taken to be (Parmeggiani et al., 1999) FIG. 17 State transition diagram for a discrete Brownian α2 ratchet that cycles through M = 3 internal states and makes ATP + M1 M2 + ADP + P α1 a single step of length ∆x

γ2 ADP + P + M1 M2 + ATP γ1 It is often convenient to simplify the generalized ratchet model further by taking the transition rate func- β2 M1 M2 tions to be localized at the discrete set of positions β1 x = xk, k = 1,...,K, and to replace the continuum dif- with αj, γj, βj x–dependent. The first reaction pathway fusion and drift terms by hopping rates between nearest involves ATP hydrolysis with chemical free energy gain lattice sites (Liepelt and Lipowsky, 2007; Lipowsky and ∆µ and a corresponding transition from state 1 to state 2, Klumpp, 2005). The resulting discrete Brownian ratchet the second involves hydrolysis in the opposite direction, model can be mapped on to a stochastic network of KM while the third involves thermal state transitions without states as shown in Fig. 17; (see (Kolomeisky and Fisher, any change in chemical potential. Basic chemical kinetics 2007) for an alternative approach to stochastic network implies that models of molecular motors). The stochastic dynamics is now described by a master equation, an example of α1 (V1 V2+∆µ)/kB T γ1 (V1 V2 ∆µ)/kB T = e − , = e − − , which is α2 γ2 β dPkm(t) X 1 (V1 V2)/kB T = [Pkn(t)Wkm kn − Pkm(t)Wkn km] (4.15) = e − . (4.12) dt ; ; β2 n=m 6 + Pk+1,1(t)WkM;k+1,1 + Pk 1,M (t)Wk1;k 1,M It follows that the net transition rates between the two − − conformational states are − Pk,1(t)Wk 1,M;k,1 − Pk,M (t)Wk+1,1;k,M , −

(V1 V2+∆µ)/kB T (V1 V2 ∆µ)/kB T ω1 = α2e − + γ2e − − where Pkm(t) = pm(xk, t) and for “vertical” transitions

(V1 V2)/kB T Wkm;kn = ωmn(xk). In this example steps along the fil- + β e − (4.13) 2 ament (power strokes) only occur between states m = 1 ω2 = α2 + γ2 + β2. (4.14) and m = M. Observing a molecular motor’s motion 38 along a filament on longer time-scales (several cycles) motor complex. One possibility is that the motors inter- suggests that its macroscopic dynamics can be approxi- act through a tug–of–war competition, where individual mated by diffusion with constant drift (Peskin and Oster, motors influence each other through the force they ex- 1995; Wang et al., 1998). That is, in the long-time limit, ert on the cargo (Hendricks et al., 2010; Kural et al., PM 2005; Muller et al., 2008a,b; Soppina et al., 2009; Welte, the probability density ρ(x = k∆x, t) = m=1 Pkm(t) satisfies the FP equation 2004), see Sec. IV.B. When a force is exerted on a mo- tor opposite to its preferred direction, it is more likely ∂ρ ∂ρ ∂2ρ = −V + D . (4.16) to detach from its microtubule. Ultimately the motion ∂t ∂x ∂x2 of the cargo is determined by the random attachments Such an equation can be derived analytically (Elston, and force–dependent detachments from the microtubule 2000a) by considering the vector F (z, t) with compo- of each motor in the motor complex. (An alternative P k coordination mechanism could involve a fast molecular nents Fm(z, t) = k∞= Pkm(t)z . This is related to the −∞ M switch that alternatively turns off kinesin and dynein). probability generating function F (z, t) = P F (z, t) m=1 m When considering the active transport of intracellular for the moments of the discrete location K(t). The cargo over relatively long distances, it is often convenient vector F (z, t) satisfies a matrix equation of the form to ignore the microscopic details of how a motor performs ∂ F (z, t) = A(z)F (z, t) such that the long time behavior t a single step (as described by Brownian ratchet models), of F (z, t) is dominated by the leading eigenvalue λ (z) 0 and to focus instead on the transitions between different of the matrix A(z). From this it can be shown that to types of motion (eg. anterograde vs. retrograde active leading order the drift V = λ (1)∆x and the diffusion 0 transport, diffusion vs. active transport). This has moti- coefficient D = ∆x2(λ (1) + λ (1))/2 (Elston, 2000a). 000 0 vated a class of mesoscopic models that take the form of a system of PDEs (Bressloff and Newby, 2009; Friedman 2. PDE models of active transport and Craciun, 2006; Jung and Brown, 2009; Kuznetsov and Avramenko, 2008; Loverdo et al., 2008; Newby and Over much longer time-scales a molecular motor alter- Bressloff, 2010b; Reed et al., 1990; Smith and Simmons, nates between phases where it is bound to a filament and 2001). For the sake of illustration, consider a simple 3- undergoing several cycles of mechanicochemical trans- state model of a particle moving on a 1D track of length duction, and phases where it is unbound and diffusing L. Such a track could represent a single microtubular fil- in the cytosol. As a further level of complexity, sev- ament. Within the interior of the track, 0 < x < L, the eral molecular motors may be attached to a vesicular particle is taken to be in one of three states labeled by cargo at the same time, which means that the active n = 0, ±: unbound from the track and diffusing (n = 0), transport of the motor/cargo complex will exhibit dif- bound to the track and moving to the right (anterograde) ferent velocity states depending on which combination of with speed v+ (n = +), or bound to the track and mov- motors are currently bound to the filament (Mallik and ing to the left (retrograde) with speed −v (n = −). − Gross, 2004; Welte, 2004). Indeed, experimental observa- For simplicity, take v = v > 0.Transitions between the ± tions of the dynamic behavior of motor/cargo complexes three states are governed by a discrete Markov process. transported along microtubules reveal intermittent be- Let Z(t) and N(t) denote the random position and state havior with constant velocity movement in both direc- of the particle at time t and define P(x, n, t | y, m, 0)dx tions along the microtubule (bidirectional transport), in- as the joint probability that x ≤ Z(t) < x + dx and terrupted by brief pauses or fast oscillatory movements N(t) = n given that initially the particle was at position that may precede localization at specific targets (Bannai Z(0) = y and was in state N(0) = m. Setting et al., 2004; Dynes and Steward, 2007; Gennerich and X ≡ | Schild, 2006; Knowles et al., 1996; Rook et al., 2000). pn(x, t) P(x, t, n 0, 0, m)σm (4.17) In the axon and distal end of dendrites one finds that m Pn microtubule filaments all have the same polarity, with with initial condition pn(x, 0) = δ(x)σn, m=1 σm = the (+) end oriented away from the cell body (Goldstein 1, the evolution of the probability is described by the and Yang, 2000). This suggests a model of bidirectional following system of PDEs for t > 0: transport in which kinesin and dynein motors transport ∂p+ a cargo in opposite directions along a single track. On = −v∂xp − β p + αp (4.18a) ∂t + + + 0 the other hand, dendritic microtubules located close to ∂p the cell body tend to have mixed polarities (Baas et al., − = v∂xp − β p + αp0 (4.18b) 1988), suggesting a model in which motors of the same ∂t − − − ∂p ∂2p directional preference are distributed among two parallel 0 − 0 = β+p+ + β p 2αp0 + D0 2 . (4.18c) microtubules of opposite polarity. In both of the above ∂t − − ∂x scenarios, there has to be some mechanism for coordinat- Here α, β are the transition rates between the station- ing the action of the various motors as part of a larger ary and mobile± states. Eq. (4.18) is supplemented by 39 appropriate boundary condition at x = 0,L. For exam- are two distinct mechanisms whereby such bidirectional ple, a reflecting boundary at x = 0 and an absorbing transport could be implemented (Muller et al., 2008a). boundary at x = L means that First, the track could consist of a single polarized micro- tubule filament (or a chain of such filaments) on which p (0, t) = p+(0, t), p (L, t) = 0. (4.19) − − up to N+ kinesin motors and N dynein motors can at- tach, see Fig. 18. Since individual− kinesin and dynein mo- In the general case that the velocities v in the two di- tors have different biophysical properties, with the former rections are different, the transport will± be biased in the tending to exert more force on a load, it follows that even anterograde (retrograde) direction if v+/β+ > v /β − − when N+ = N the motion will be biased in the antero- (v /β < v /β ). − + + grade direction. Hence, this version is referred to as an It is straightforward− − to generalize the 3–state model to asymmetric tug–of–war model. Alternatively, the track the case of n distinct velocity states as found for example could consist of two parallel microtubule filaments of op- in the tug–of-war model, see IV.B. Introducing the n– posite polarity such that N kinesin motors can attach component probability density vector p(x, t) ∈ n the + R to one filament and N to the other. In the latter case, corresponding system of PDEs takes the form − if N+ = N then the resulting bidirectional transport is − ∂p unbiased resulting in a symmetric tug–of–war model. = Ap + L(p), (4.20) ∂t - + n n where A ∈ R × specifies the transition rates between each of the n internal motor states and the differential operator L has the structure   L1 0 ··· 0 - +  0 L2 0 ··· 0     . . .  L =  . .. .  , (4.21)    Ln 1 0  − FIG. 18 Schematic diagram of an asymmetric tug–of–war 0 ··· 0 Ln model. Two kinesin and two dynein motors transport a cargo where the scalar operators are given by in opposite directions along a single polarized microtubule track. Transitions between two possible motor states are 2 shown. Lj = [−vj∂x + D0,j∂x]. (4.22)

Here vj is the velocity of internal state j and D0,j is the When bound to a microtubule, the velocity of a single corresponding diffusivity. molecular motor decreases approximately linearly with force applied against the movement of the motor (Viss- cher et al., 1999). Thus, each motor is assumed to satisfy B. Tug-of-war model of bidirectional transport the linear force-velocity relation  v (1 − F/F ) for F ≤ F Suppose that a certain vesicular cargo is transported v(F ) = f s s (4.23) along a one–dimensional track via N+ right–moving (an- vb(1 − F/Fs) for F ≥ Fs terograde) motors and N left–moving (retrograde mo- − where F is the applied force, F is the stall force satis- tors). At a given time t, the internal state of the cargo– s fying v(F ) = 0, v is the forward motor velocity in the motor complex is fully characterized by the numbers n s f + absence of an applied force in the preferred direction of and n of anterograde and retrograde motors that are − the particular motor, and v is the backward motor ve- bound to a microtubule and thus actively pulling on b locity when the applied force exceeds the stall force. The the cargo. Assume that over the time–scales of inter- original tug–of–war model assumes the binding rate is est all motors are permanently bound to the cargo so independent of the applied force, whereas the unbinding that 0 ≤ n ≤ N . The tug–of–war model of Muller ± ± rate is taken to be an exponential function of the applied et. al. (Muller et al., 2008a,b) assumes that the motors force: act independently other than exerting a load on motors F with the opposite directional preference. (However, some F π(F ) = π0, γ(F ) = γ0e d , (4.24) experimental work suggests that this is an oversimplifica- tion, that is, there is some direct coupling between mo- where Fd is the experimentally measured force scale on tors (Driver et al., 2010)). Thus the properties of the which unbinding occurs. The force dependence of the motor complex can be determined from the correspond- unbinding rate is based on measurements of the walking ing properties of the individual motors together with a distance of a single motor as a function of load (Schnitzer specification of the effective load on each motor. There et al., 2000), in agreement with Kramers rate theory 40

(Hanggi et al., 1990). Let Fc denote the net load on the of behavior depend on motor strength, which primar- set of anterograde motors, which is taken to be positive ily depends upon the stall force. More recently, Newby when pointing in the retrograde direction. Suppose that and Bressloff(2010a,b) have constructed a system of the molecular motors are not directly coupled to each PDEs describing the evolution of the probability density other so that they act independently and share the load; p(n+, n , x, t) that the motor complex is in the inter- − however, see (Driver et al., 2010). It follows that a sin- nal state (n+, n ) and has position x at time t. This − gle anterograde motor feels the force Fc/n , whereas a version of the tug–of–war model simultaneously keeps − single retrograde motor feels the opposing force −Fc/n+. track of the internal state of the motor complex and Eq. (4.24) implies that the binding and unbinding rates its location along a 1D track. In order to write the for both types of motor take the form model in the general form (4.20), it is convenient to in- troduce the label i(n+, n ) = (N+ + 1)n + (n+ + 1) γ(n, Fc) = nγ(Fc/n), π(n) = (N − n)π0. (4.25) − − b b and set p(n+, n , x, t) = pi(n ,n )(x, t). This then gives − + − The parameters associated with kinesin and dynein mo- an n–component probability density vector p ∈ Rn with tors will be different, so that it is necessary to add the n = (N+ + 1)(N + 1), which satisfies Eq. (4.20). − subscript ± to these parameters. The cargo force Fc is The internal velocity of internal state j = j(n+, n ) is − determined by the condition that all the motors move vj = vc(n+, n ), and the diffusivities Dj are taken to be − with the same cargo velocity vc. Suppose that N+ ≥ N zero unless all motors are detached from the microtubule: − so that the net motion is in the anterograde direction, Di = D0δi,1. The components ai,j, i, j = 1, . . . , n, of the which is taken to be positive. In this case, the forward state transition matrix A are given by the corresponding motors are stronger than the backward motors so that binding/unbinding rates of Eqs. (4.25). That is, setting n+Fs+ > n Fs . Eq. (4.23) implies that i = i(n+, n ), the non–zero off–diagonal terms are − − −

vc = vf+(1 − Fc/(n+Fs+)) = −vb (1 − Fc/(n Fs )). ai,j = π+(n+ − 1) for j = i(n+ − 1, n ), (4.30a) − − − − (4.26) ai,j = π (n − 1), for j = i(n+, n − 1), (4.30b) − − − This generates a unique solution for the load Fc and cargo ai,j = γ+(n+ + 1,Fc), for j = i(n+ + 1, n ), (4.30c) velocity vc: − ai,j = γ (n + 1,Fc), for j = i(n+, n + 1). (4.30d) − − − Fc(n+, n ) = (Fn+Fs+ + (1 − F)n Fs ), (4.27) P − − − The diagonal terms are then given by ai,i = − j=i aj,i. 6 where One of the useful features of the tug-of-war model is that it allows various biophysical processes to be incor- n Fs vf F = − − − , (4.28) porated into the model (Newby and Bressloff, 2010a,b; n Fs vf + n+Fs+vb − − − − Posta et al., 2009), see also Sec. IV.H.2. For example, a and convenient experimental method for changing the stalling force (and hence the mode of motor behavior) is to vary n+Fs+ − n Fs vc(n+, n ) = − − . (4.29) the level of ATP available to the motor complex. At − n+Fs+/vf+ + n Fs /vb − − − low [ATP] the motor has little fuel and is weaker, result- The corresponding expressions when the backward mo- ing in mode (i) behavior; then, as [ATP] increases and more fuel is available, mode (ii) behavior is seen until tors are stronger, n+Fs+ < n Fs , are found by inter- − − changing vf and vb. the stall force saturates at high values of [ATP] where The original study of Muller et al. (2008a,b) consid- mode (iii) behavior takes over. Thus, [ATP] provides a ered the stochastic dynamics associated with transitions single control parameter that tunes the level of intermit- tent behavior exhibited by a motor complex. There are a between different internal states (n+, n ) of the motor complex, without specifying the spatial− position of the number of models of the [ATP] and force-dependent mo- complex along a 1D track. This defines a Markov process tor parameters that closely match experiments for both with a corresponding master equation for the time evo- kinesin (Fisher and Kolomeisky, 2001; Mogilner et al., 2001; Schnitzer et al., 2000; Visscher et al., 1999) and lution of the probability distribution P (n+, n , t). They determined the steady-state probability distribution− of dynein (Gao, 2006; King and Schroer, 2000). It is found internal states and found that the motor complex exhib- that [ATP] primarily affects the stall force, forward mo- ited at least three different modes of behavior: (i) the tor velocity, and unbinding rate. For example, based on motor complex spends most of its time in states with experimental data, the forward velocity may be modeled approximately zero velocity; (ii) the motor complex ex- using Michaelis-Menten kinetics hibits fast backward and forward movement interrupted max vf [ATP] by stationary pauses, which is consistent with experi- vf ([ATP]) = , (4.31) [ATP] + K mental studies of bidirectional transport; (iii) the mo- v max tor complex alternates between fast backward and for- where vf = 1µm/s, Kv = 79.23µM for kinesin max ward movements. The transitions between these modes and vf = 0.7µm/s, Kv = 38µM for dynein. (The 41 backward velocity of both kinesin and dynein is small, active transport (Brooks, 1999; Friedman and Craciun, vb ≈ ±0.006µm/s, so that the [ATP] dependence can 2006; Friedman and Hu, 2007). More recently, Newby be neglected). The binding rate is determined by the and Bressloff(2010a,b) have carried out a QSS reduction time necessary for an unbound motor to diffuse within of the tug-of-war model and used this to study the effi- range of the microtubule and bind to it, which is as- ciency of intracellular active transport within the context sumed to be independent of [ATP]. The unbinding rate of random intermittent search processes, see also IV.H. of a single motor under zero load can be determined us- For a more general discussion of the QSS and projec- ing the [ATP] dependent average run length Lk([ATP]) = tion methods for reducing the dimensionality of stochas- max Lk /([ATP] + Ku). The mean time to detach from the tic models, also referred to as adiabatic reduction, see microtubule is vf ([ATP])/Lk([ATP]) so that Gardiner(2009). The first step in the QSS reduction is to fix the units of max vf ([ATP] + Ku) space and time by setting l = 1 and l/v = 1, where v = γ0([ATP]) = max , (4.32) n Lk ([ATP] + Kv) maxi=1 vi. This corresponds to non–dimensionalizing Eq. (4.18) by performing the rescalings x → x/l and t → max where Lk = 0.86µm, Ku = 3.13µM for kinesin and tv/l. Furthermore, suppose that for the given choice of max Lk = 1.5µm, Ku = 1.5µM for dynein. Finally, a units, aij = O(1/ε) whereas Lij = O(1) for some small model for the [ATP]-dependent stall force of kinesin is 1 parameter ε  1, and set A = ε− Ab. Eq. (4.18) can then max 0 be rewritten in the dimensionless form (after dropping 0 (Fs − Fs )[ATP] Fs([ATP]) = Fs + , (4.33) hat on Ab) Ks + [ATP] ∂p 1 0 max = Ap + L(p), (4.34) where Fs = 5.5pN, Fs = 8pN, Ks = 100µM for ki- 0 max ∂t  nesin and Fs = 0.22pN, Fs = 1.24pN, Ks = 480µM for dynein. The transition matrix A is assumed to be irreducible and conservative so that ψ = (1, 1, ··· , 1)T is in the nullspace N (AT ). Moreover, A has one zero eigenvalue and the C. Quasi–steady–state (QSS) reduction of PDE models of remaining eigenvalues have negative real part. Let pss ∈ active transport N (A) and choose pss so that ψT pss = 1. The next step is to introduce the decomposition p = upss + w where Two important quantities characterizing the effective- u ≡ ψT p and ψT w = 0. Thus u is the component of p ness of motor-driven active transport are the hitting in the left nullspace of A, whereas w is in the orthogonal probability and MFPT to deliver cargo to a specific complement. Multiplying both sides of (4.34) by ψT target within a cell, see Sec. IVH. In the case of the yields the equation 3-state model (4.18), in which the number of internal T ss states of the motor-cargo complex is sufficiently small, it ∂tu = ψ L(up + w). (4.35) is possible to derive exact analytical expressions for the Substituting p = upss + w into (4.34) gives MFPT and hitting probability using Laplace transforms or solving a corresponding system of backwards equa- ss 1 ss ss ∂tw + (∂tu)p = A(w + up ) + L(w + up ). tions (Benichou et al., 2005; Bressloff and Newby, 2009).  However, if the number of internal velocity states be- Using equation (4.35) and the fact that pss is in the right comes large, as in the tug-of-war model, then some form null space of A, shows that of approximation is needed. This also holds for active transport in higher spatial dimensions where the direc- 1 ss T ss ∂tw = Aw + (In − p ψ )L(w + up ), (4.36) tion of motion is random, see Sec. IVE. In this section,  we review a quasi–steady–state (QSS) method for reduc- where In is the n × n identity matrix. ing equations of the general form (4.20) to a scalar FP Now introduce an asymptotic expansion for w: equation under the assumption that the state transition 2 rates are faster than the velocity of motile states on an w ∼ w0 + w1 +  w2 + .... (4.37) appropriate length scale (Newby and Bressloff, 2010a,b). A number of authors have analyzed such equations in Substituting this expansion into (4.36) and collecting 1 this regime. For example, Reed et al. (1990) used singu- O(− ) terms gives Aw0 = 0. Since w is in the orthogo- lar perturbation theory to show that the transport of a nal complement of the left nullspace of A, it follows that chemical along an axon can be analyzed in terms of an w0 = 0. Now collecting terms of O(1) yields the equation approximate traveling-wave solution of a scalar advec- ss T ss Aw1 = −(In − p ψ )L(up ), (4.38) tion diffusion equation for the chemical concentration. Subsequent work rigorously established the validity of where In is the n × n identity matrix. The orthogonal ss T this scalar reduction for a wide range of PDE models of projection In − p ψ ensures that the right–hand side 42 of the above equation is in the range of A, and a unique The resulting 2-rank system of equations for the 3-vector T solution for w1 is obtained by requiring that ψ w1 = 0. w1, Eq. (4.38), can be solved up to the arbitrary element It is convenient to define the mean of an n-vector z with [w1]0 using Gaussian elimination: respect to the stationary distribution pss according to hzi ≡ zT pss Substituting w ∼ εw into Eq. (4.35) then 1 ∓ hvi αΩ 1 [w1] = ∓ ∂xu + , [w1]0 = Ω (4.45) yields ± β2 γ β ± ± n where 2 X ∂tu = −hvi∂xu + hD0i∂xu + ε Ljw1,j. (4.39) v  1 1  j=1 hvi = − . γ β+ β From Eq. (4.38) it can be seen that the components of − 2 T w1 are linear combinations of ∂xu and ∂xu so that Ω is determined by imposing the condition ψ w = 0: 2   w1,j = −θj∂xu + qj∂ u (4.40) 1 1 − hvi 1 + hvi x − αΩ = 2 2 2 ∂xu (4.46) γ β+ β where θj and qj, j = 1, . . . , n, are u–independent. Col- − lecting ∂xu terms in Eq. (4.38) yields an equation for Substituting equations (4.45) and (4.46) into (4.35) yields T θ = (θ1, ..., θn) , the FP Eq. (4.42) with (to O(2)) θ − h i − ss ··· h i − ss T A = (( v v1)p1 , , ( v vn)pn ) . (4.41) αD (v − hvi)2 (v + hvi)2  D = 0 +  + (4.47) T T 2 2 The condition ψ w1 = 0 implies that ψ θ = 0 and γ γβ+ γβ − hence there exists a unique solution for θ. Likewise the equation for q is given by QSS reduction of tug-of-war model. The reduction of the ss ss T Aq = −((hD0i − D0,1)p1 , ··· , (hD0i − D0,n)pn ) . tug–of–war model presented in Sec. IV.B is more in- volved (Newby and Bressloff, 2010a,b). In particular, it is necessary to compute the vector θ by solving equation Finally, assuming that the diffusivity D = O(ε) and 0,j (4.41), which has the general form Aθ = b. The stan- keeping only lowest order terms, leads to the scalar FP dard numerical method for solving a rank deficient linear equation (Newby and Bressloff, 2010a,b) system using singular valued decomposition (SVD) must ∂u ∂u ∂2u be modified slightly. The Fredholm-Alternative theorem = −V + D (4.42) ∂t ∂x ∂x2 implies that a solution to equation (4.41) exists but is not unique. In the case of a standard least squares so- with lution, uniqueness is obtained by requiring the solution 2 T 2 V = hvi + O(ε ),D = hD0i + εv θ + O(ε ).(4.43) to be orthogonal to the nullspace of A. However, in this case a unique solution must be obtained by requiring the In order to compute the O(ε) contribution to D, the rank solution be orthogonal to the nullspace of AT . The fol- deficient Eqs. (4.41) can be solved numerically using the lowing procedure may be used. Let UΣHT = A be a full singular value decomposition of the matrix A, see full singular value decomposition of A. Let z = U T b and below. The probability density function u is the total T y = H θ so that Σy = z. It follows that yi = zi/σi, probability of being in any motor state at position x and i = 1, ..., n − 1, where σi are the non-zero singular values time t. Suppose that the particle was initially injected on ss of A. The last component yn is arbitrary since σn = 0. to the track at x = 0 and σm = pm in equation (4.17) so The standard least squares solution is obtained by set- that the initial state lies in the slow manifold. The initial ting yn = 0. To determine yn here, one requires that condition for u is then u(x, 0) = δ(x). Similarly, typical Pn θi = 0. Since θ = Hy, boundary conditions for u on a finite track of length L i=1 will be V u − D∂ u | = 0 (reflecting) and u(L, t) = 0 n 1 x x=0 X− (absorbing). These boundary conditions follow from sub- θi = hijyj + hinyn. (4.48) ss stituting p = up +εw1 into the corresponding boundary j=1 conditions of Eq. (4.34). where hij are the components of the matrix H. Since Pn QSS reduction of 3–state model. In the case of the 3–state i=1 θi = 0, it follows that model given by Eq. (4.18), the steady–state probability Pn Pn 1 distribution is i=1 j=1− hijyj yn = − Pn . (4.49)  1  i=1 hin 1 β+ 1 1 1 pss = 1 , γ = + + . (4.44)  β−  The QSS reduction determines the generic parameters γ 1 β+ β α α − V,D of the scalar FP equation (4.42) as functions of the 43 various biophysical parameters of the tug–of–war model. where p1 represents the concentration of moving neurofil- These include the stall force Fs, the detachment force Fd, ament proteins, and pi, i > 1 represent the concentrations the maximum forward and backward velocities vf , vb, and in n − 1 distinct stationary states. Conservation of mass P the single motor binding/unbinding rates γ0, π0. These, implies that Ajj = − i=j Aij. The initial condition is 6 in turn will depend on environmental factors such as the pi(x, 0) = 0 for all 1 ≤ i ≤ n, 0 < x < ∞. Moreover concentration of [ATP] and various signaling molecules, p1(0, t) = 1 for t > 0. Reed et al. (1990) carried out an see also Sec. IV.H.2. asymptotic analysis of Eqs. (4.50) that is related to the QSS reduction method of Sec. IV.C. Suppose that p1 is written in the form D. Fast and slow axonal transport x − ut  p1(x, t) = Q √ , t , Axons of neurons can extend up to 1m in large or-  ganisms but synthesis of many of its components oc- ss Pn ss ss where u is the effective speed, u = vp1 / j=1 pj , and p cur in the cell body. Axonal transport is typically di- is the steady-state solution for which Apss = 0. They vided into three main categories based upon the observed then showed that Q(s, t) → Q0(s, t) as  → 0, where Q0 speed (Brown, 2003; Sheetz et al., 1998): fast transport is a solution to the diffusion equation (1 − 9µm/s) of organelles and vesicles and slow trans- 2 port (0.004 − 0.6µm/s) of soluble proteins and cytoskele- ∂Q0 ∂ Q0 = D ,Q0(s, 0) = H(−s) tal elements. Slow transport is further divided into two ∂t ∂x2 groups; actin and actin-bound proteins are transported with H the Heaviside function. The diffusivity D can in slow component A while cytoskeletal polymers such as be calculated in terms of v and the transition matrix A. microtubules and neurofilaments are transported in slow Hence the propagating and spreading waves observed in component B. It had originally been assumed that the experiments could be interpreted as solutions to an effec- differences between fast and slow components were due tive advection-diffusion equation. More recently, Fried- to differences in transport mechanisms, but direct exper- man and Craciun(2005, 2006) have developed a more imental observations now indicate that they all involve rigorous analysis of spreading waves. fast motors but differ in how the motors are regulated. Membranous organelles, which function primarily to de- stationary liver membrane and protein components to sites along the axon and at the axon tip, move rapidly in a unidi- π γ ρ a a0 r0 r on track rectional manner, pausing only briefly. In other words, ρ γ π they have a high duty ratio – the proportion of time a k off k off cargo complex is actually moving. On the other hand, k on k on cytoskeletal polymers and mitochondria move in an in- a rp off track termittent and bidirectional manner, pausing more often p and for longer time intervals, as well as sometimes revers- ing direction. Such transport has a low duty ratio. FIG. 19 Transition diagram of ‘stop-and-go’ moldel for the Neurofilaments are space-filling cytoskeletal polymers slow axonal transport of neurofilaments. See text for defini- tion of different states. that increase the cross-sectional area of axons, which then increases the propagation speed of action poten- In contrast to the above population models, direct ob- tials. Radioisotopic pulse labeling experiments provide servations of neurofilaments in axons of cultured neurons information about the transport of neurofilaments at the using fluorescence microscopy has demonstrated that in- population level, which takes the form of a slowly mov- dividual neurofilaments are actually transported by fast ing Gaussian-like wave that spreads out as it propagates motors but in an intermittent fashion (Wang and Brown, distally. A number of authors have modeled this slow 2001; Wang et al., 2000). Hence, it has been proposed transport in terms of a unidirectional mode similar to that the slow rate of movement of a population is an the 3–state model of Eqs. (4.18)(Blum and Reed, 1989; average of rapid bidirectional movements interrupted by Craciun et al., 2005; Reed et al., 1990). For example, prolonged pauses, the so-called stop-and-go hypothesis Blum and Reed(1989) considered the following system (Brown, 2000; Jung and Brown, 2009; Li et al., 2012). on the semi-infinite domain 0 ≤ x < ∞: Computational simulations of an associated system of   n PDEs shows how fast intermittent transport can account ∂p1 ∂p1 X  − v = A1jpj (4.50a) for the slowly spreading wave seen at the population level. ∂t ∂x j=1 One version of the model assumes that the neurofilaments n can be in one of six states (Brown, 2000; Li et al., 2012): ∂pi X  = Aijpj, 1 < i ≤ N, (4.50b) ∂t anterograde moving on track (state a), anterograde paus- j=1 ing on track (a0 state), anterograde pausing off track 44

(a) (b) (state ap), retrograde pausing on track (state r0), retro- grade pausing off track (state rp), and retrograde moving on track (state r). The state transition diagram is shown in Fig. 19.

E. Active transport on microtubular networks

In the case of axonal or dendritic transport in neu- rons, the microtubles tend to be aligned in parallel (Gold- FIG. 20 Active transport on a disordered microtubular net- stein and Yang, 2000) so that one can treat the trans- work. (a) Random orientational arrangement of microtubles. port process as effectively 1D. On the other hand, intra- (b) Effective 2D random intermittent search in which a par- cellular transport within the soma of neurons and most ticle switches between diffusion and ballistic motion in a ran- non–polarized animal cells occurs along a microtubular dom direction. network that projects radially from an organizing center (centrosome) with outward polarity (Alberts et al., 2008). This allows the delivery of cargo to and from the nucleus. network as illustrated in Fig. 20. (The extension to Moreover, various animal viruses including HIV take ad- 3D networks is relatively straightforward). Suppose that vantage of microtubule-based transport in order to reach after homogenization, a molecular motor at any point the nucleus from the cell surface and release their genome r = (x, y) in the plane can bind to a microtubule with any through nuclear pores (Damm and Pelkmans, 2006; La- orientation θ, resulting in ballistic motion with velocity gache et al., 2009a). In contrast, the delivery of cargo v(θ) = v(cos θ, sin θ) and θ ∈ [0, 2π). If the motor is un- from the cell membrane or nucleus to other localized cel- bound then it acts as a Brownian particle with diffusion lular compartments requires a non–radial path involving coefficient D0. Transitions between the diffusing state several tracks. It has also been found that microtubules and a ballistic state are governed by a discrete Markov bend due to large internal stresses, resulting in a locally process. The transition rate β from a ballistic state with disordered network. This suggests that in vivo transport velocity v(θ) to the diffusive state is taken to be inde- on relatively short length scales may be similar to trans- pendent of θ, whereas the reverse transition rate is taken R 2π port observed in vitro, where microtubular networks are to be of the form αQ(θ) with 0 Q(θ)dθ = 1. Suppose not grown from centrosomes, and thus exhibit orienta- that at time t the motor is undergoing ballistic motion. tional and polarity disorder (Kahana et al., 2008; Salman Let (X(t),Y (t)) be the current position of the searcher et al., 2005). Another example where a disordered mi- and let Θ(t) denote the corresponding velocity direction. crotubular network exists is within the Drosophila oocyte Introduce the conditional probability density p(x, y, θ, t) (Becalska and Gavis, 2009). Kinesin and dynein motor- such that p(x, y, θ, t)dxdydθ is the joint probability that driven transport along this network is thought to be one (x, y, θ) < (X(t),Y (t), Θ(t)) < (x + dx, y + dy, θ + dθ) of the mechanisms for establishing the asymmetric local- given that the particle is in the ballistic phase. Similarly, ization of four maternal mRNAs - gurken, oskar, bicoid take p0(x, y, t) to be the corresponding conditional prob- and nanos- which are essential for the development of the ability density if the particle is in the diffusive phase. embryonic body axes. (For the moment the initial conditions are left unspeci- A detailed microscopic model of intracellular transport fied). The evolution of the probability densities for t > 0 within the cell would need to specify the spatial distribu- can then be described in terms of the following 2D system tion of microtubular orientations and polarity, in order of PDEs (Bressloff and Newby, 2011): to specify which velocity states are available to a motor- ∂p β αQ(θ) cargo complex at a particular spatial location. However, = −∇ · (v(θ)p) − p(r, θ, t) + p0(r, t) a simplified model can be obtained under the “homog- ∂t   enization” assumption that the network is sufficiently (4.51a) dense so that the set of velocity states (and associated ∂p β Z 2π α 0 ∇2 r − r state transitions) available to a motor complex is inde- = D0 p0 + p( , θ0, t)dθ0 p0( , t). ∂t  0  pendent of position. In that case, one can effectively (4.51b) represent the active transport and delivery of cargo to an unknown target within the cell in terms of a two– or In the case of a uniform density, Q(θ) = 1/(2π), Eqs. three-dimensional model of active transport (Benichou (4.51a) and (4.51b) reduce to the 2D model considered by et al., 2007, 2011b; Loverdo et al., 2008). This also pro- Benichou et. al. (Benichou et al., 2007, 2011b; Loverdo vides motivation for extending the QSS analysis of Sec. et al., 2008). In order to carry out a quasi steady–state IV.C to higher–dimensions (Bressloff and Newby, 2011). (QSS) reduction of Eqs. (4.51a) and (4.51b), see Sec. For simplicity, consider a disordered 2D microtubular IV.C, we have fixed the units of space and time according 45 to l = 1 and l/v = 1, where l is again a typical run and (4.51b) yields length. Furthermore, for the given choice of units, we ∂u ∂w have assumed that there exists a small parameter   1 aQ(θ) +  = −v(θ) · ∇ [aQ(θ)u + w] − βw 1 ∂t ∂t such that all transition rates are O(− ), the diffusivity is O() and all velocities are O(1). + αQ(θ)w0. (4.60) In the limit  → 0, the system rapidly converges to and the space-clamped (i.e. ∇p = ∇p0 = 0) steady–state ss ss ∂u ∂w0 distributions (p (θ), p0 ) where b +  = D ∇2 (bu + w ) (4.61) ∂t ∂t 0 0 β αQ(θ) +β hwi − αw . pss = ≡ b, pss(θ) = ≡ aQ(θ). (4.52) 0 0 α + β α + β Now substitute (4.59) into (4.60) and (4.61). Collecting The QSS approximation is based on the assumption that terms to leading order in  and using Eq. (4.56) then for 0 <   1, solutions remain close to the steady–state gives solution. Hence, ab w0(r, t) ∼ [hvi · ∇u] , (4.62) p(r, θ, t) = u(r, t)pss(θ) + w(r, θ, t) (4.53) α + β ss p0(r, t) = u(r, t)p0 + w0(r, t), (4.54) and Q(θ) where w(r, θ, t) ∼ a2(1 + b) hvi − av(θ) · ∇u. (4.63) β Z 2π u(r, t) ≡ p(r, θ, t)dθ + p0(r, t), (4.55) Finally, substituting equations (4.63) and (4.62) into 0 (4.59) yields to O() the FP equation and ∂u 2 Z 2π = −∇ · (Vu) + bD ∇ u + ∇ · (D∇u). (4.64) ∂t 0 w(r, θ, t)dθ + w0(r, t) = 0. (4.56) 0 The diffusion tensor D has components Dkl, k = x, y, l = Furthermore, the initial conditions are taken to be x, y

a 2  u(r, 0) = δ(r − r ), w(r, 0) = w (r, 0) = 0, (4.57) Dkl ∼ hvkvli − hvki hvli + b hvki hvli , (4.65) 0 0 β which are equivalent to the following initial conditions to lowest order in , whilst the effective drift velocity is for the full probability densities: given by V ∼ a hvi. In the case of a uniform direction distribution Q(θ) = p(r, θ, 0) = δ(r−r )pss(θ), p (r, 0) = δ(r−r )pss. (4.58) 0 0 0 0 1/(2π), the diffusion tensor reduces to a scalar. This Thus, the initial internal state of the motor (diffusive or follows from the fact that vx = v cos θ, vy = v sin θ so ballistic with velocity v(θ)) is generated according to the hvxi = hvyi = hvxvyi = 0 and to leading order steady–state distributions pss(θ) and pss. In other words, 0 av2 the motor starts on the slow manifold of the underlying Dxx = = Dyy,Dxy = 0. (4.66) 2β dynamics. If this were not the case, then one would need to carry out a multiscale analysis in order to take into More generally, assuming that Q(θ) is sufficiently account the initial transient dynamics transverse to the smooth, we can expand it as a Fourier series, slow manifold (Gardiner, 2009). Perturbation and projection methods can now be used 1 1 X∞ Q(θ) = + (ωn cos(nθ) +ω ˆn sin(nθ)). (4.67) to derive a closed equation for the scalar component 2π π n=1 u(r, t)(Bressloff and Newby, 2011). First, integrating equation (4.51a) with respect to θ and adding equation Assume further that ω1 =ω ˆ1 = 0 so there is no velocity (4.51b) yields bias i.e. hvxi = hvyi = 0. Then

2 Z 2π 2 ∂u 2 av 2 av = D0∇ p0 − hhv · ∇pii (4.59) Dxx = cos (θ)Q(θ)dθ = (1 + ω2) ∂t β 0 2β 2 2 = bD0∇ u − a hvi · ∇u −  hhv · ∇wii + O( ), 2 Z 2π 2 av 2 av Dyy = sin (θ)Q(θ)dθ = (1 − ω2) , (4.68) π π β 2β where hfi = R 2 Q(θ)f(θ)dθ and hhfii = R 2 f(θ)dθ for 0 0 0 2 Z 2π 2 any function or vector component f(θ). Next, substi- av av Dxy = sin(θ) cos(θ)Q(θ)dθ = ωˆ2. tuting equations (4.53) and (4.54) into equations (4.51a) β 0 2β 46

-v where

Z ∞ Z ∞ hhv[y(t1)]v[y(t2)]ii = dy1 dy2hv(y1)v(y2)ic −∞ −∞ y+ ξ × p(y2, t2)p(y1 − y2, t1 − t2). ξ y (4.71) v y- ξ Using Laplace transforms and the velocity correlation hopping function, 2 2v ξ Z ∞ hhXe 2(s)ii = p(0, s) p(y, s)dy, (4.72) s e e −∞ FIG. 21 Random velocity model of a microtubular network with with quenched polarity disorder. Particles move ballistically √ along parallel tracks in a direction determined by the polarity 1 y s/D p(y, s) = √ e−| | . of the given track. They also hop between tracks according e 4Ds to an unbiased random walk. Performing the integration with respect to y thus shows 2 2 1/2 5/2 that hhXf (s)ii = v ξD− s− , which on inverting the Laplace transform gives It follows that only the second terms in the Fourier series 2 expansion contribute to the diffusion tensor. 4v ξ / hhX2(t)ii = √ t3 2. (4.73) An alternative formulation of transport on disordered 3 πD microtubular networks has been developed by Kahana An equivalent formulation of the problem is to treat et al. (2008) in terms of random velocity fields (Ajdari, hhX2(t)ii as the solution to the differential equation (Ka- 1995; Redner, 1997; Zumofen et al., 1990). In order to hana et al., 2008) describe the basic idea, consider the simplified model an- d2 alyzed by Zumofen et al. (1990). The latter model con- hhX2(t)ii = 2v2ξyp(0, t), (4.74) sists of a set of equally spaced parallel tracks along the dt2 x-axis, say, see Fig. 21. The tracks are assigned random where ξp(0, t) is the probability of turn to the origin at polarities ±1 with equal probabilities corresponding to time√ t within a single lattice spacing ξ, and p(0, t) = quenched polarity disorder. A particle undergoes a ran- 1/ 4πDt. In conclusion, the random velocity model sup- dom walk in the y–direction, whereas when a particle ports anomalous superdiffusion in the x direction. attaches to a certain track it moves ballistically with ve- Kahana et al. (2008) extended the above construc- locity ±1 according to the track’s polarity. It is assumed tion to 2D (and 3D) disordered networks where there that when a particle hops to a neighboring track it binds are parallel tracks in the x and y directions. The distri- immediately. Let X(t) denote the displacement of a ran- bution of polarities are unbiased in both directions. A dom walker in the longtitudinal direction at time t: self-consistent description of the dynamics is obtained by taking Z t 2 2 d 2 2 d 2 2 X(t) = v[y(t0)]dt0. (4.69) hhX (t)ii = 2v ξpy(0, t), hhY (t)ii = 2v ξpx(0, t), 0 dt2 dt2 (4.75) Taking the continuum limit in the y direction means that where px and py are the probability densities of the x and y coordinates. From the symmetry of the network, px(0, t) = py(0, t). Hence, assuming that px(0, t) = 1 y2/4Dt 2 1/2 p(y, t) = √ e− , ChhX (t)ii− for some constant C, and setting φ(t) = 4πDt hhX2(t)ii gives d2 where D is the diffusion coefficient, and the velocity φ1/2 φ = 2Cv2ξ. (4.76) 2 2 field is δ–correlated hv(y)v(y0)ic = v ξδ(y − y0). Here dt averaging is taken with respect to the quenched polar- It follows that φ(t) ∼ t4/3 so that the diffusion is less ity disorder and ξ is the infinitesimal spacing between enhanced than in the case of parallel tracks in one di- tracks. Now consider the second moment hhX2(t)ii of the rection. Finally, note that active transport on the ran- stochastic process averaged with respect to the quenched domly oriented network of Fig. 20 exhibits normal rather disorder and realizations of the random walk: than anomalous diffusion. A major difference from the random velocity model is that the latter has quenched Z t Z t1 2 polarity disorder, whereas the former has dynamical po- hhX (t)ii = 2 dt1 dt2hhv[y(t1)]v[y(t2)]ii, (4.70) 0 0 larity disorder. 47

F. Virus trafficking cell membrane

An interesting example of active transport in 2D or 3D is given by virus trafficking. An animal virus typically Brownian invades a mammalian cell by first undergoing membrane motion nuclear endocytosis from the exterior to the interior of the cell. pore It then has to navigate the crowded cytoplasm without nucleus being degraded in order to reach a nuclear pore and de- liver its DNA to the cell nucleus (Damm and Pelkmans, motor transport 2006). Single particle tracking has established that virus trajectories within the cytoplasm consist of a succession of free or confined diffusion and ballistic periods involv- ing active transport along microtubules or actin networks microtubule (Brandenburg and Zhuang, 2007). A macroscopic com- putational model of the trafficking of a population of FIG. 22 Model of Lagache and Holcman(2008). Diagram of viruses has been developed based on the law of mass a 2D radially symmetric cell with radially equidistant micro- action, which takes into account cell geometry but ne- tubles. A virus trajectory is shown that alternates between ballistic motion along a microtubule and diffusion the cyto- et al. glects stochastic effects (Dinh , 2007, 2005). More plasm. Trajectory starts at the cell membrane and ends at a recently, in a series of papers, Holcman and collabora- nuclear pore. tors (Holcman, 2007; Lagache et al., 2009a,b; Lagache and Holcman, 2008) have developed a stochastic model of a single virus trafficking inside a cell, which involves gache and Holcman, 2008) is to assume that on a coarse- reducing an intermittent search model (see Sec. IV.H) grained time-scale the random intermittent motion of the to an effective Langevin equation, and using the latter virus can be approximated by a Langevin equation with to calculate the mean time to reach a nuclear pore based a radial drift vector: on a narrow escape problem (see Sec. II.C). The basic dr r √ structure of a 2D version of the latter model is shown in = b(r) + 2Ddξdt. (4.77) |r| Fig. 22. dt Following (Lagache and Holcman, 2008), the cell is In order to estimate the drift function b(r), the MFPT treated as a radially symmetric disc consisting of an an- τ(r0) for the effective Langevin particle to start at r0 nular region of cytoplasm of outer radius R and inner b and end at r1 is calculated using the standard theory of radius δ, surrounding a central nuclear disc. N micro- first passage times (see Sec. II.B and (Redner, 2001)), tubules radiate outwards from the nucleus to the cell and then compared to τ(r0). First, τb(r0) satisfies the membrane, and are assumed to be distributed uniformly equation so that the angle between two neighboring microtubules 2 is Θ = 2π/N. (A two-dimensional description of a cell D∇ τb − b(r)∇τb = −1, would be reasonable in the case of cultured cells that are flattened due to adhesion to the substrate). The motion with boundary conditions of a virus particle alternates between diffusive motion dτb within a wedge region Ωb subtending an angle Θ at the (r) = 0, τ(r1) = 0. dr b origin, and binding to one of the two microtubules at the boundary of the wedge. Suppose that a virus par- As a further simplification, it is assumed that b(r) varies ticle starts at some radius r0 < R and arbitrary angle slowly with r so that b(r) ≈ b(r0), leading to the solution within such a wedge. Let τ(r0) denote the MFPT for the Z r0 Z R b(r0)[u v]/D particle to bind to a microtubule, and let ρ(r0) be the ue− − τb(r0) = dudv. mean radial position on the microtubule. Suppose that r1 v Dv the particle moves with a fixed speed v for a time T to- wards the nucleus before being released to a new position Assuming that D  1 the Laplace method can be used to evaluate the integral with respect to u, giving τ(r0) ≈ with radius r1 and arbitrary angle within another wedge. b (r0 − r1)/b(r0). Finally, setting τ(r0) = τ(r0) + T yields It follows that r1 = ρ(r0) − vT . Treating the domain Ωb b as an open wedge by ignoring the reflecting boundary at r − r d − r Θ2/12 r = R, it can be shown that if Θ  1 then (Lagache b(r ) = 0 1 = 0 . (4.78) 0 τ(r ) + T T + r2Θ2/12D et al., 2009a) 0 0

2 2 2 A more detailed calculation of the effective drift function τ(r0) ≈ r Θ /12D, ρ(r0) ≈ r0(1 + Θ /12). 0 b(r) under less restrictive assumptions can be found in The reduction method of (Lagache et al., 2009a; La- (Lagache et al., 2009a). 48

Having reduced the effective motion of the virus to a 1. Asymmetric exclusion process and the hydrodynamic limit Langevin equation, the probability that the virus arrives at a nuclear pore before being degraded at a rate k0 can Let us consider in more detail the system shown in now be calculated by solving a narrow escape problem. Fig. 23, which consists of a finite 1D lattice of N sites The assocated FP equation takes the form labeled i = 1,...,N. The microscopic state of the sys- tem is given by the configuration C that specifies the ∂p = D∇2p(r, t) − ∇ · b(r)p(r, t) − k p(r, t) (4.79) distribution of identical particles on the lattice. That is, ∂t 0 C = {n1, . . . , nN } where each occupation number ni = 1 on the annular region Ω of Fig. 22, together with the if the i-th site is occupied by a single particle and ni = 0 boundary conditions if the site is vacant. Exclusion effects preclude more than one particle at any site. Thus, the state space consists p(r, t) = 0, r ∈ ∂N ,J(r, t) · n = 0, r ∈ ∂Ω − ∂N . a a of 2N configurations. Let P(C, t) demote the probability The boundary ∂Ω of the annulus is taken to be reflect- of finding a particular configuration C at time t. The ing everywhere except for the surface ∂Na of the nucleus evolution of this probability distribution is described by occupied by nuclear pores, which are taken to be perfect a master equation: absorbers. Asymptotic analysis then shows that the hit- dP(C, t) X ting probability P and conditional MFPT T are (Lagache = [W 0 P(C0, t) − W 0 P(C, t)] . dt C →C C→C et al., 2009a; Lagache and Holcman, 2008) 0= C 6 C (4.81) b(δ) 2δν P = , T = , (4.80) The transition rate W 0 from configuration C to C0 is b(δ) + 2δk0ν 2δk0ν + b(δ) determined from the followingC→C set of rules (Parmeggiani where ν = log(1/) with  the fraction of the nucleus et al., 2003): covered by nuclear pores. (a) at sites i = 1,...,N −1, a particle can jump to site i + 1 at a unit rate if the latter is unoccupied; (b) at site i = 1 (i = N) a particle can enter (exit) the G. Exclusion processes lattice at a rate α (β) provided that the site is unoccupied (occupied); So far we have considered a single molecular motor (c) in the bulk of the lattice, a particle can detach from or motor/cargo complex moving along a filament track. a site at a rate ωD and attach to an unoccupied site at a However, in practice there could be many active particles rate ωA. moving along the same track, which could interact with Rules (a) and (b) constitute a TASEP with open each other and exhibit some form of collective behavior. boundary conditions, whereas rule (c) describes Lang- This has motivated a number of studies that model the muir kinetics. It follows that the evolution of the particle movement of multiple motor particles as an asymmetric densities hnii away from the boundaries is given by the et al. exclusion process (ASEP) (Evans , 2003; Klumpp exact equation and Lipowsky, 2003; Kolomeisky, 1998; Lipowsky et al.,

2001; Nowak et al., 2007a; Parmeggiani et al., 2003, 2004; dhnii et al. = hni 1(1−ni)i−hni(1−ni+1)i+ωAh1−nii−ωDhnii. Popkov , 2003; Pronina and Kolomeisky, 2007). In dt − the simplest version of such models, each particle hops (4.82) P unidirectionally at a uniform rate along a 1D lattice; Here hni(t)i = niP(C, t) etc. Similarly, at the bound- the only interaction between particles is a hard-core re- aries C pulsion that prevents more than one particle occupying the same lattice site at the same time. This so–called dhn1i = −hn1(1 − n2)i + αh1 − n1i − ωDhn1i, (4.83a) totally asymmetric exclusion process (TASEP) is com- dt bined with absorption/desorption (Langmuir) kinetics, dhnN i = hnN 1(1 − nN )i + ωAh1 − nN i − βhnN i. in which individual particles can bind to or unbind from dt − the track, see Fig. 23. The TASEP has become the (4.83b) paradigmatic model of non-equilibrium stochastic pro- cesses, and a variety of analytical methods have been developed to generate exact solutions for the stationary ωA β ωD state, see (Blythe and Evans, 2007; Chou et al., 2011; Schadschneider et al., 2010) and references therein. How- α ever, when Langmuir kinetics or other biologically moti- vated extensions of TASEP are included, it is no longer FIG. 23 Schematic diagram of TASEP with Langmuir kinet- possible to obtain exact solutions so that some form of ics, in which particles can spontaneously detach and attach mean-field approximation is required. at rates ωD and ωA, respectively. 49

Note that in the absence of any exclusion constraints, Eq. TASEP, the phase diagram can be calculated explicitly (4.82) reduces to a spatially discrete version of the uni- (Blythe and Evans, 2007; Krug, 1991). Set ΩA = ΩD = 0 directional PDE Eq. (4.18a), with p+(ni∆x, t) = hnii, in Eq. (4.84), and consider constant current solutions β+ = ωD, and p0α = ωA and v+/∆x = 1. The goal is J(x, t) = J. Rewrite Eq. (4.85) in the form to find a non- equilibrium stationary state for which the current flux J along the lattice is a constant. It then ∂ρ 2 = − (ρ − r+)(ρ − r ), (4.86) follows that J has the exact form ∂x  − with J = αh1 − n1i = hni(1 − ni+1)i = βhnN i, i = 1,N − 1. 1 h p i Eqs. (4.82) and (4.83) constitute a non-trivial many- r = 1 ± 1 − 4J0 , (4.87) ± 2 body problem, since in order to calculate the time evolu- tion of hnii it is necessary to know the two-point corre- Eq. (4.86) is easily integrated from the left-hand bound- lations hni 1(1 − ni)i. The latter obey dynamical equa- ary, say, to give tions involving− three–point and four-point correlations. − − Thus, there is an infinite hierarchy of equations of mo- (ρ(x) r+)(ρ(0) r ) 2(r+ r−)x/ − = e− − . (4.88) tion. However, progress can be made by using a mean- (ρ(x) − r )(ρ(0) − r+) field approximation and a continuum limit in order to − derive a PDE for the density of particles (Evans et al., with ρ(0) = α. The unknown current J is obtained 2003; Parmeggiani et al., 2004). The mean–field approx- by setting x = 1 and using the boundary condition imation consists of replacing two-point correlations by ρ(1) = 1 − β. The stationary density profile can then products of single–site averages: be constructed explicitly by carrying out an asymptotic expansion with respect to the small parameter . First, hninji = hniihnji. suppose that J < 1/4 so r are real. Denote the O(1) ± approximation of the current by J0. If r+ ≈ 1 − β so Next introduce the infinitesimal lattice spacing  and set that J0 = β(1 − β) and r ≈ β, then the bulk of the x = k, ρ(x, t) = ρk(t) ≡ hnk(t)i. The continuum limit − domain is in a high density (HD) phase. Since r+ > r is then defined according to N → ∞ and  → 0 such it follows that β < 1/2. Eq. (4.86) implies that the den-− that the length of the track L = N is fixed. (Fix length sity profile is flat except for a boundary layer close to scales by setting L = 1). Expanding ρk 1(t) = ρ(x ± , t) x = 0. Similarly, there exists a low density (LD) phase in powers of  gives ± when r ≈ α, for which J0 = α(1 − α), ρ+ ≈ (1 − α) and α < 1/−2; there is now a boundary layer at x = 1. Fi- 1 2 3 ρ(x ± , t) = ρ(x) ± ∂xρ(x, t) +  ∂xxρ(x, t) + O( ). nally, in the case J > 1/4 (so r are complex) one finds 2 that the density profile consists± of a flat region at the Finally, rescaling the absorption/desorption rates accord- centre of the domain where ρ ≈ 1/2 and J0 = 1/4 with ing to ωA = ΩA, ωD = ΩD, and rescaling time τ = t, boundary layers now at both ends. Moreover,√ writing Eq. (4.82) becomes to O() J = 1/4 + ∆J it can be seen that r+ − r = i ∆J. In order to avoid fast spatial oscillations in the− profile (4.88), 2 ∂ρ  ∂ ρ ∂ρ one requires ∆J = O(2). Carrying out a perturbation = − (1 − 2ρ) + ΩA(1 − ρ) − ΩDρ. (4.84) ∂τ 2 ∂x2 ∂x expansion of Eq. (4.85) in powers of  then establishes that ρ(x) − 1/2 varies as 1/x as one moves away from Similarly, Eq. (4.83) reduces to the boundary conditions the left-hand boundary. A schematic illustration of the ρ(0) = α, ρ(1) = 1 − β. In the continuum limit the flux phase diagram for pure TASEP is shown in Fig. 24. takes the form  ∂ρ J(x, t) = − + ρ(1 − ρ). (4.85) 2 ∂x 2. Method of characteristics and shocks Note that it is also possible to extend the mean-field ap- Eq. (4.84) is mathematically similar in form to the proximation to ASEP with slowly spatially varying hop- viscous Burger’s equation with additional source terms ping rates, although now the effective diffusivity in the (Ockendon et al., 2003). Thus, one expects singularities hydrodynamic limit now depends on the density (Lakatos such as shocks in the density ρ to develop in the inviscid et al., 2006). or non-dissipative limit  → 0+. This can be investigated The next step is to find a stationary non equilibrium more directly by setting  = 0 in Eq. (4.84), which then state for which the current J is constant, and to deter- takes the form of a standard quasilinear PDE mine the corresponding stationary density profile. This then generates a phase diagram with respect to the pa- ∂ρ ∂ρ + (1 − 2ρ) = ΩA(1 − ρ) − ΩDρ. (4.89) rameters α, β and fixed ΩA, ΩD. In the case of a pure ∂τ ∂x 50

1 In order to illustrate the basic method of analysis, let us return to a pure TASEP where ΩA = ΩD = 0. Eq. (4.89) then simplifies to the kinematic wave equation LD MC ∂ρ ∂J0(ρ) J =α(1−α) J =1/4 + = 0,J (ρ) = ρ(1 − ρ) (4.91) 0 0 ∂τ ∂x 0 β 1/2 and the characteristic become straight lines along which ρ is constant. Ignoring boundary effects for the moment, HD the density profile at time t is determine by the propaga- J = tion of the initial density ρ(x, 0) along characteristics as 0 β(1−β) illustrated in Fig. 25. Since higher densities propagate more slowly than lower densities, an initial linear density 0 1/2 1 profile steepens until a shock is formed at the points of in- α tersection where pairs of characteristics meet. In general, a shock propagates with a speed v determined by the FIG. 24 Mean-field phase diagram for the TASEP showing S the regions of α, β parameter space where the low-density so called Rankine-Hugonoit condition (Ockendon et al., (LD), high-density (HD) and maximal-current (MC) phases 2003): exist. Schematic illustrations of the density profiles in the various regions are shown in red. J0(ρ2) − J0(ρ1) vS = = 1 − ρ1 − ρ2 (4.92) ρ2 − ρ1

A well known method for studying such equations is to where ρ1, ρ2 are the densities on either side of the shock. construct characteristic curves x = x(τ) along which For the particular initial density profile shown in Fig. ρ(τ) ≡ ρ(x(τ), τ) satisfies 25, ρ1 = 0 and ρ2 = 1 so that the shock is stationary (vS = 0). The possibility of stationary shocks reflects the dρ ∂ρ dx ∂ρ fact that the current J (ρ) = ρ(1 − ρ) has a maximum, = + . 0 dτ ∂τ dτ ∂τ which means that two different densities can have the same current on either side of the shock. Comparison with Eq.n (4.89) leads to the characteristic The method of characteristics and kinematic wave the- equations (Evans et al., 2003; Kolomeisky et al., 1998) ory yields insights into the dynamics underlying the for- dx dρ mation of the various stationary phases shown in Fig. 24 = 1 − 2ρ, = ΩA(1 − ρ) − ΩDρ. (4.90) (Blythe and Evans, 2007; Kolomeisky et al., 1998; Krug, dτ dτ 1991). The basic idea is to consider kinematic waves These equations can be interpreted as kinematic waves propagating from the left-hand and right-hand bound- that propagate changes in density that move at a variable aries, respectively, which act as particle reservoirs with speed 1 − 2ρ. corresponding densities ρ(0) = α and ρ(1) = 1 − β. A kinematic wave propagates from the left-hand (right- hand) boundary with speed 1 − 2α (2β − 1). Hence, if density profile ρ(0) 1 α < 1/2, β < 1/2, both waves propagate into the inte- density profile ρ(t) rior of the domain and meet somewhere in the middle to form a shock that propagates with speed vS = β − α. If β > α then the shock moves to the right-hand bound- 0 ary and the bulk of the domain is in a low-density (LD) state with ρ ≈ α < 1/2. On the other hand, If β < α shock t then the shock moves to the left-hand boundary and the bulk of the domain is in a high-density (HD) state with 0.5 ρ ≈ 1 − β > 1/2. Note that the line separating the HD and LD phases, α = β < 1/2, is a coexistence line. The ρ = 1 system consists of a low density region separated from ρ = 0 a high density region by a shock. Once higher order x 0 0.5 1 dissipative effects are included, this shock diffuses freely between the ends of the domain, so that the average den- FIG. 25 Formation of a shock for Eq. (4.91). The charac- sity profile is linear. In the case α > 1/2 or β > 1/2, teristics are straight lines of speed 1 − 2ρ with ρ constant the kinematic wave associated with that boundary does along a characteristic. The initial density profile evolves into not propagate into the interior so that the density as- a stationary shock solution. sociated with the other boundary dominates. Finally, if 51 both α > 1/2 and β > 1/2 then the steady-state bulk solution has the maximal current density ρm = 1/2. In order to show this, and to determine how bulk solutions match the boundary conditions, it is necessary to include dissipation effects as in the previous section. The above analytical arguments can be extended to the full molecular motor model that combines TASEP with Langmuir kinetics (Evans et al., 2003; Parmeggiani et al., 2003, 2004). When ΩA, ΩD =6 0 the characteristics are curves in the x − t plane. For example, consider the propagation of density fluctuations along a characteris- tic starting at the left boundary with ρ = α < 1/2 and α < K/(K + 1), where K = ΩA/ΩD. It follows from FIG. 26 Diagram showing how the translation of the mRNA Eq. (4.90) that initially the fluctuation propagates along and the synthesis of proteins is made by ribosomes. [By La- the characteristic with decreasing speed and increasing dyofHats (Public domain), via Wikimedia Commons] density. If K/(1 + K) < 1/2 then ρ will approach the constant value ρ = K/(K + 1) and the speed approaches a constant value. However, if K/(1 + K) > 1/2 then the translation of messenger RNA (mRNA) by ribosomes after a finite time the density reaches ρ = 1/2 and prop- during protein synthesis (MacDonald and Gibbs., 1969; agation ceases. A similar analysis holds for characteris- MacDonald et al., 1968). Proteins are macromolecules tics propagating from the right boundary. Furthermore, formed from chains of amino acids, and the blueprint for characteristics propagating from opposite boundaries can how these proteins are synthesized is contained in the again intersect, implying multivalued densities and thus DNA of the cell nucleus. Protein synthesis involves two a breakdown of the quasilinear equation. The resulting stages: transcription of genetic information from DNA shock has the same wave speed as pure TASEP. Of par- to messenger RNA (mRNA) by RNA polymerase, and ticular interest are stationary solutions for which the cur- translation from mRNA to proteins through ribosome rent J = ρ(1 − ρ) is constant so that any shock solution translocation. The mRNA carries genetic information, is stationary (vS = 0). To a first approximation, these encoded as triplets of nucleotides known as codons. Since can be obtained by finding steady–state solutions of the there are four nucleotides (A, U,C, G), there are 64 dis- mean-field Eq. (4.89): tinct codons, e.g., AUG, CGG, most of which code for a single amino acid. The process of translation consists of ∂ρ (1 − 2ρ) − ΩD[K − (1 + K)ρ] = 0. (4.93) ribosomes moving along the mRNA without backtrack- ∂x ing (from one end to the other, technically known as the The occurrence of stationary shocks is consistent with the 5’ end to the 3’ end) and is conceptually divided into observation that this is a first–order ODE but there are three major stages: initiation, elongation and termina- two boundary conditions. One thus proceeds by integrat- tion. Each elongation step invokes translating or ‘read- ing from the left boundary where ρ(0) = α to obtain a ing’ of a codon and the binding of a freely diffusing trans- density profile ρL(x) and then integrating from the right fer RNA (tRNA) molecule that carries the specific amino boundary where ρ(1) = 1 − β to obtain a second density acid corresponding to that codon. The basic translation profile ρR(x). The full solution is constructed by match- machinery is illustrated in Fig. 26. ing the two profiles at a shock whose position also has to The simplest model of translation is a pure TASEP. be determined. If the shock lies outside the unit inter- However, as originally highlighted by MacDonald and val, then it is necessary to include at least one boundary Gibbs.(1969); MacDonald et al. (1968) and recently re- layer. A detailed analysis of the steady-state solutions viewed by Chou et al. (2011); Zia et al. (2011), this con- with coexisting low and high density phases, and the siderably oversimplifies the biology. For example, ribo- corresponding phase-diagram with respect to the param- somes are large molecules so that they extend over several eters (α, β, ΩD, ΩA) can be found in (Evans et al., 2003; codons or lattice sites (around l = 12). In order to extend Parmeggiani et al., 2004). If the effects of dissipation TASEP to multi-site particles, it is first necessary to spec- are also taken into account then the sharp interfaces and boundary layers become smooth fronts of size O(1/).

β α 3. mRNA translation by ribosomes

One of the first examples of a TASEP model in biology was proposed by Gibbs and collaborators in their study of FIG. 27 A TASEP with extended particles of size l = 3. 52 ify the rules for entry and exit of a ribosome. One possi- tosol reacts with an adaptor protein that binds a cargo bility is ’complete entry, incremental exit’, which assumes to the motor, causing the cargo to be released. How- that a ribosome enters completely provided the first l lat- ever, when a cargo is pulled at a relatively high velocity tice sites are vacant, whereas it exits one step at a time it doesn’t have much time to explore local space and is (Chou and Lakatos, 2003). Inclusion of extended ob- therefore much less likely to participate in such a reac- jects considerably complicates the analysis even though tion. Therefore, one possible interpretation of the fre- the basic structure of the phase diagram is preserved quent pauses observed during motor transport is that it (Chou and Lakatos, 2003; Shaw et al., 2003). In con- provides a mechanism to improve the reaction kinetics trast to pure TASEP, there does not currently exist an required to localize the cargo to its target by giving it exact solution, although mean field approximations do more time to explore local space. This then leads to a provide useful insights. A second biologically motivated simple model of cargo delivery in which there are transi- modification of TASEP is to include site–dependent hop- tions of the internal state of the motor complex between ping rates (Chou and Lakatos, 2004; Dong et al., 2007; directed movement states and stationary or slowly dif- Foulaadvand et al., 2008; Kolomeisky, 1998). This is mo- fusing searching states. If the transitions between these tivated by the fact that the local hopping rate depends states are governed by chemical reactions under the in- on the relative abundance of specific amino-acid carrying fluence of thermodynamic fluctuations, then the model tRNA. Using a combination of Monte Carlo simulations becomes a random intermittent search processs. and mean field theory it can be shown, for example, that Random search has recently been used to model a wide two defects (regions of slow hopping rates) decrease the range of problems (Benichou et al., 2011b), including the steady-state current more when they are close to each behavior of foraging animals (Bell, 1991; Benichou et al., other. Finally, note that a number of researchers have de- 2005; Viswanathan et al., 1999; Viswanathan et al., 2011) veloped models that take into account intermediate steps and the active transport of reactive chemicals in cells in the translocation of a ribosome along the mRNA, in- (Bressloff and Newby, 2009; Loverdo et al., 2008). The cluding the binding of tRNA to the ribosome and hydrol- facilitated diffusion of protein-DNA interactions can also ysis (Basu and Chowdhury, 2007; Ciandrini et al., 2010; be thought of as a random intermittent search process, Garai et al., 2009; Reichenbach et al., 2006). see Sec. II.E. Random intermittent search falls within a class of random processes characterized by a particle with both an ‘internal’ and ‘external’ state. The external state H. Motor transport and random intermittent search typically represents the spatial location or position of the processes particle, and one or more boundary conditions may ap- ply to the process that represent the physical domain in So far we have considered models of motor-driven in- which it moves. The motion of the particle depends on its tracellular transport without any reference to the process internal state, which can be continuous but is usually dis- whereby the cargo or load carried by one or more motors crete. For example, the tug-of-war model from Sec. IV.B is delivered to the correct location within a cell. It is is a process where the position, or external state, changes unlikely that a target is selected according to where a deterministically at a constant velocity, while the velocity mictrotubular track terminates, since a cargo could be depends upon the discrete internal state, the randomly released from its associated motors at any point along changing number of motors bound to the microtubule. the track. Moreover, targeted delivery probably uses In general, the external state need not change determin- the same microtubular “highway” as more general, non– istically, but could also fluctuate. For example, one could targeted intracellular trafficking. It would also be dif- include diffusion of the cargo and add a continuous noise ficult to establish global chemical concentration gradi- term whose amplitude depends on the internal state. ents aimed at guiding a motor–driven cargo to a spe- In order to formulate motor-cargo transport as a ran- cific target, as such signals would be drowned out by the dom intermittent search process, consider a single parti- many other signals originating from additional intracel- cle moving on a 1D track of length L as shown in Fig. lular targets. Therefore, instead of thinking of motor- 28. In contrast to the models considered in Sec. IV.A- driven cargo transport as a direct path to a given target, IV.G, we now assume that there exists a hidden target the random intermittent motion of motor–driven cargo of width 2l centered at X within the interior of the do- observed in experiments suggests that the cell maintains main. We also assume that if the particle is within range a distribution of mobile cargo throughout its interior, and of the target and is in a slowly moving search state then that delivery of a cargo to a specific target is a stochas- it can ‘find’ the target at a rate k. Within the context tic process (Bressloff and Newby, 2009; Loverdo et al., of motor-driven cargo transport, it is assumed that when 2008; Newby and Bressloff, 2010b). Some of the molec- the target is found the particle is immediately removed ular mechanisms that cause a cargo to attach or detach from the system, that is, the cargo (with or without its from a molecular motor have been identified (Goldstein associated set of molecular motors) is delivered to the tar- et al., 2008). In most cases, a protein dissolved in the cy- get. Hence, the target is treated as a partially absorbing 53 trap. One possible application of a 1D model would be to states is more complicated. The simplest scenario is that vesicular transport along the axons and dendrites of neu- the cargo locates its target after it becomes fully de- rons, where microtubules are aligned in parallel with the tached from the microtubule and diffuses within distance (+) end oriented away from the cell body. (In the case of its target, where it binds to scaffolding proteins and is of dendritic domains close to the cell body, the polari- separated from its molecular motors. However, if many ties may be randomly distributed). The hidden target molecular motors are bound to the cargo, the waiting could then be a synapse, an intracellular compartment time between diffusive searching events can be too large such as an , or the growth cone of an elongat- to reliably deliver the cargo. Moreover, if the cargo is ing axon, see Sec. IV.D. In general, the transport process large so that its diffusivity is low or the cargo is moving is expected to be biased, with newly synthesized macro- through a crowded and confined domain, diffusive motion molecules transported away from the cell body (antero- may be restricted, preventing the cargo from reaching the grade transport) whilst products requiring degradation target. Another possibility is that subcellular machinery are transported back to the cell body (retrograde trans- is present to detach the cargo from its motors or inhibit port). the activity of the motors so that scaffolding proteins can It is straightforward to incorporate a hidden target into bind to and sequester the cargo. Delivery then changes the general n-state PDE model (4.20) of a motor complex from a diffusion limited reaction to a waiting time that moving in a 1D domain by taking depends on a reaction occurring between the motor-cargo complex and biomolecules (either freely diffusing or an- ∂p = Ap + L(p), (4.94) chored) local to the target while its moving along the ∂t microtubule. If details of the localization mechanism are n n unknown then the simplest model is to assume that this where A ∈ R × again specifies the transition rates be- tween each of the n internal motor complex states, and waiting time is approximately exponential and to asso- the diagonal operator L now has the modified form ciate a target detection rate kj with each motor state. The model can be simplified further by assuming that 2 Lj = [−vj∂x + D0,j∂x + kjχ(x − X)]. (4.95) detection is unlikely while only one species of motors is engaged and pulling the cargo at its maximum (forward As before, vj is the velocity of internal state j and D0,j is or backward) velocity. This suggests assigning a single the corresponding diffusivity. The additional term takes target detection rate k to those states that have suffi- into account the probability flux into the target with kj ciently low speeds (Newby and Bressloff, 2010c). Thus, the rate for internal state j and χ is the indicator function k(n ,n ) = kΘ(vh −v(n+, n )), where v(n+, n ) denotes + − − −  1, if |x| < l the velocity when n+ kinesin and n dynein motors are χ(x) = (4.96) − 0, otherwise. attached to the track and vh is a velocity threshold. The efficiency of a given search process can be char- In general kj will only be non-zero for a subset of states. acterized in terms of two important quantities. The first For example, in the 3-state model of Eqs. (4.18), kj = is the hitting probability Π that a particle starting at x0 kδj,0, that is, the unbound stationary or diffusing state at time t = 0 finds the target, that is, the particle is ab- is identified as the search state. sorbed somewhere within the domain X − l ≤ x ≤ X + l. In the case of the more biophysically realistic tug-of- The second is the conditional MFPT T for the particle war model, see Sec. IV.B, the identification of the search to find the target given that it is eventually absorbed by the target. Let J(t) denote the probability flux due to absorption by the target at X, motor-cargo l F complex target n Z X+l v X microtubule k J(t) = kj pj(x, t)dx, (4.97) + X l j=1 − x=0 where we have suppressed the initial conditions. It fol- FIG. 28 Diagram depicting a motor-cargo complex perform- lows that ing a random intermittent search for a hidden target on a R Z ∞ ∞ tJ(t)dt one–dimensional track of length L. The motor-cargo com- Π = J(t)dt, T = 0 . (4.98) R ∞ plex is transported along a microtubule by two populations 0 0 J(t)dt of molecular motors with opposing directional preferences. When equal numbers of each type of motor are bound to In the case of the 3-state model, it is possible to calcu- the microtubule, the motor-cargo complex is in a slowly mov- late these quantities directly from the system of PDEs ing search state and can find the target provided that it lies (4.94)(Benichou et al., 2005, 2007, 2011b; Bressloff and within the target domain of width 2l centered at x = X. Newby, 2009). For more complicated models, the QSS Target detection rate is k. reduction technique presented in Sec. IV.C can be used to reduce the system of PDEs to a scalar FP equation. 54

The target detection terms then lead to an inhomoge- that the FPT distribution for a population of identical neous term in the reduced FP equation: searchers is

(N) µ(t) ∂u ∂ ∂  ∂u F (t) ≡ lim F (t) = 1 − e− , (4.103) = − (V p) + D − λχ(x − X)u, (4.99) ∞ N ∂t ∂x ∂x ∂x →∞ with n with λ = P k pss, and the vector pss is the space- n j=1 j j j Z t X Z t Z X+l clamped steady-state distribution (see Sec. IV.C). For µ(t) = J0(s)ds = κj pj(x, s)dxds. 0 0 X l the three-state model, j=1 − (4.104)   ss 1/α The corresponding hitting probability that at least one λ = kp0 = k . (4.100) 1/β+ + 1/β + 1/α particle finds the target in the mean-field limit is − µ( ) Π = lim F (t) = 1 − e− ∞ . (4.105) There are then three effective parameters that describe t ∞ the random search process: the drift V , the diffusivity D, →∞ ∞ ∞ and the target detection rate λ. Each of these parameters Thus, Π < 1 if µ( ) < . The corresponding condi- are themselves functions of the various cargo velocities, tional MFPT is R µ t transition rates, and target detection rates contained in ∞ tµ (t)e ( ) T 0 0 − the full model. The hitting probability and MFPT are = µ( ) . (4.106) 1 − e− ∞ still given by Eqs. (4.98) except that now the flux is Combining the mean-field approximation with the QSS Z X+l reduction leads to the FPT distribution (4.103) with J(t) = λ u(x, t)dx. (4.101) X l Z t Z X+l − µ(t) = λb u(x, s)dxds (4.107) 0 X l − 1. Mean-field model Pn ss and λb = j=1 κjpj . Here u(x, t) is the solution to the FP Eq. (4.99) with λ = 0. On a semi-infinite domain Further simplification can be obtained by considering with a reflecting boundary at the origin, the method of a population of searchers and taking a mean-field limit images can then be used to obtain the explicit solution (Bressloff and Newby, 2012). That is, consider N in- dependent, identical searchers that all start at the ori- 1 (x V t)2/(4Dt) gin at time t = 0. Each searcher evolves according to u(x, t) = √ e− − πDt the system of PDEs (4.94) or the corresponding FP Eq. V x + V t (4.99). Denote the FPT to find the target of the jth − exV/Derfc( √ ). (4.108) 2D 2 Dt searcher by Tj, j = 1,...,N, with each Tj an inde- pendent, identically distributed random variable drawn Unfortunately, it is not possible to derive an explicit ana- from the single searcher first passage time distribution lytical solution for µ(t), although the integral expressions (1) R t F (t) = 0 J(s)ds. The random time T to fill the trap can be evaluated numerically. Nevertheless, it is possible is then given by T = min(T1,T2, ..., TN ), and the distri- to determine the hitting probability Π and the large-time bution for T is behavior of the waiting time density f (t) = dF (t)/dt under the approximation l  X for which∞ ∞ F (N)(t) = Prob(T < t) = 1 − Prob(T > t) Z t = 1 − Prob(T1 > t, T2 > t, ..., TN > t) µ(t) = c u(X, s)ds, c = 2lλ.b (4.109) = 1 − (1 − F (1)(t))N . 0 First, taking the Laplace transform of u(x, t) gives Furthermore, suppose that the rate of detection for a (Bressloff and Newby, 2012) single searcher in internal state j scales as k = κ /N j j [Γ(s) V/2D]x   with κ independent of j. Set J(t) = J (t)/N with J(t) e− − V 1 j 0 ue(x, s) = √ 2 − given by Eq. (4.94). It follows that V 2 + 4sD D Γ(s) + V/2D

N with  κ Z t  F (N)(t) = 1 − 1 − J (s)ds . (4.102) 1p N 0 Γ(s) = (V/D)2 + 4s/D. (4.110) 0 2

In the limit N → ∞, the detection rates kj → 0 so that Assuming that l  X, it follows that the probability density functions pj(x, t) decouple from c µ(∞) ≈ 2lλb lim u(X, s) = . (4.111) the target. Moreover, taking the large N limit shows s 0 e → V 55

Thus the corresponding hitting probability Π < 1 for 4000 200 3500 V > 0 and Π = 1 for V = 0 (pure diffusion). Second, τ = α 160 the large-time behavior of the waiting time density can 3000 be obtained by using the following asymptotic expansion T T 120 2500 τ = β of the complementary error function: 80 2000 2 (a) (d) e z  1  1500 40 erfc(z) = √− 1 − ... . 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 πz 2z2 1/τ Π f 0.03 √ (b) (c) 10-4 Applying√ √ this to Eq. (4.108) with z = (x+V t)/(2 Dt) ≈ 5 x V t/(2 D) for large t and V > 0, leads to the approxi- 4 x10-4 0.02 mation 3 x10-4 √ 2 x10-4 0.01 2c D V 2t/(4D) u(X, t) ∼ √ e− , (4.112) 10 -4 2 3 V πt 0 0.00 0 10000 20000 200 240 280 320 which is independent of target location X. Substituting time t time t this expression into Eq. (4.109) gives FIG. 29 (a) Unbiased (β+ = β− = β) random intermittent  3/2 search in the mean–field population model. The MFPT vs (i) 2c 4D V 2t/(4D) µ(t) ∼ µ(∞) − √ e− ,(4.113) the average search time 1/α with β = 1/ and (ii) the average V π tV 2 duration 1/β of the moving (forward and backward) state with α = 2/. Each curve has a minimum MFPT for a given On the other hand, for V = 0, value of α and β. Parameter values used are  = 0.1, k = 5/N, t X = 50, and l = 0.25. (b) First passage time density for an Z 1 2 p µ(t) = c √ e−X /(4Dt) ∼ 2c t/πD, (4.114) unbiased search with same parameters as (a). The solid curve 0 πDt shows the analytical density function in the mean field limit and the remaining curves are histograms obtained from 104 2 assuming that e−X /(4Dt) ∼ 1 for large t. The large- Monte-Carlo simulations for different numbers of searchers time asymptotic approximation for µ(t) determines how N. ( c) Corresponding first passage time density for a biased search with α = 1/, β = 1/, β = 2/,  = 0.1, k = 5/N, the waiting time density, f (t), scales with time. For + − ∞ X = 50, and l = 0.25. (d) The MFPT vs hitting probability V = 0, for population model. Each curve is parameterized by 0 ≤ β+ ≤ β−. Analytical results are shown as lines, with the 1/2 cˆ√t f (t) ∝ t− e− , (4.115) single searcher in grey and the mean field limit (N = ∞) ∞ √ in black. Averaged Monte Carlo simulations (104 simulation withc ˆ = 2c/ πD, and for V > 0 each) are shown as symbols, sets ranging from grey to black, with a different value of N used in each set. From grey to 1/2 V 2t/(4D) f (t) ∝ t− e− . (4.116) black there are six sets of simulations with N = 1, 2, 3, 4, 5, 25, ∞ respectively. Parameter values are the same as c). For the sake of illustration, consider the 3-state model of Eqs. (4.18), for which V, D, λ are given by Eqs. (4.47) and (4.100). In the case of a single random intermit- average duration of the search phase, 1/α, and (ii) the tent searcher on a finite track of length L with reflect- average duration of the ballistic phase, 1/β. In both cases ing boundary conditions at both ends x = 0,L (so that there exists a minimum MFPT for a particular choice of Π = 1) and unbiased transport (β+ = β = β), it can α, β, consistent with the single-searcher regime. be shown that there exists an optimal search− strategy in Next, consider how the search process changes as more the sense that there exists a unique set of transition rates searchers are added. In particular, the first passage time α, β for which the MFPT is minimized (Benichou et al., density is approximated by Monte-Carlo simulations for 2005, 2007, 2011b). On the other hand, for directed in- different values of N, and the results are compared to termittent search (β+ > β ) on a semi–infinite domain the analytical mean field results. This illustrates how the or a finite domain with an− absorbing boundary at x = L single-searcher process (N = 1) is related to the mean- (so that Π < 1), a unique optimal strategy no longer field population search process (N → ∞). In Fig. 29(b) exists (Bressloff and Newby, 2009; Newby and Bressloff, the unbiased case is shown. The most significant dif- 2010b). One finds that a similar situation holds if there is ference is found in the large time behavior, with power- 3/2 a population of N independent searchers (Bressloff and law scaling t− for the single search and the so-called cˆ√t Newby, 2012). First, consider an unbiased random in- stretched exponential scaling e− (see Eqn. (4.115)) for termittent search process in the mean–field population the mean field N → ∞ limit. A similar plot showing the model, for which N → ∞ and β+ = β = β (V = 0). In first passage time density for a biased search (β+ < β Fig. 29(a) the MFPT is plotted as a− function of (i) the so that V > 0) is shown in Fig. 29c). In this case, adding− 56 more searchers has little qualitative effect on the first a 0.12 b 1 ) 0.1

passage time density, each case having the same exponen- ) 0.5 1 tial large time scaling (see Eqn. (4.115)). In both cases, 0.08 sec − / the results, show that adding more searchers decreases 0.06 0 m sec 0.04 µ the mean search time and the variance. The analysis of (

( −0.5 λ 0.02

the mean-field model showed that the hitting probabil- V 0 −1 ity is less than unity when the velocity bias is positive 0.15 0.2 0.25 0.15 0.2 0.25 (i.e. when β+ < β so that V > 0). In Fig. 29(d), τ τ − the MFPT is plotted against the hitting probability for c 0.3 ) different values of N. Each curve is parameterized by 0.25 sec β , the rate of leaving the forward-moving state, with 0.2 (N+, N ) = (2, 1)

+ / − 2 0 < β+ < β . By changing the value of β+, any hitting 0.15 (N+, N ) = (3, 2) − − m probability can be achieved. As β → β the searcher’s 0.1 (N , N ) = (4, 3) + µ + −

− ( motion becomes unbiased, and the hitting probability in- 0.05 D 0 creases to unity. However, as the searchers become more 0.15 0.2 0.25 unbiased the MFPT also increases, in other words, an op- τ timal search strategy no longer exists. Analytical results FIG. 30 Effects of tau concentration on the tug-of-war model for the single searcher case (N = 1) and the mean-field with N+ kinesin motors and N− dynein motors. The stall limit (N = ∞) are shown as solid curves (black and grey, force FS , forward velocity vf and unbinding rate γ0 are given respectively) and to connect the two, averaged Monte- by Eqs. (4.31)-(4.33) with [ATP] = 103µM. The other single Carlo simulations are shown (as dots) for different values motor parameters are (Muller et al., 2008b): Fd = 3 pN, −1 −1 of N = 1, 2, 3, 4, 25 (each dot is colored in greyscale from γ0 = 1 sec , π0 = 5 sec , and vb = 0.006 µm/sec. The N = 1 in black to N = 25 in light grey). Ten different corresponding parameters of the FP Eq. are obtained using a QSS reduction and plotted as a function of τ. (a) Effective sets of Monte-Carlo simulations are run corresponding capture rate λ. (b) Drift velocity V . (c) Diffusivity D. to ten different values of β+, and in each set the hitting probability decreases and the MFPT increases as more searchers are added. et al., 1986). Another important MAP, called MAP2, is similar in structure and function to tau, but is present in dendrites (Shaft-Zagardo and Kalcheva, 1998); MAP2 2. Local target signaling has been shown to affect dendritic cargo transport (Maas et al., 2009). Experiments have shown that the presence In the case of directed intermittent search there is a of tau or MAP2 on the microtubule can significantly al- playoff between minimizing the MFPT and maximizing ter the dynamics of kinesin; specifically, by reducing the the hitting probability. One way to enhance the effi- rate at which kinesin binds to the microtubule (Vershinin ciency of the search process would be for the target to et al., 2007). Moreover, the tau- and MAP2-dependent generate a local chemical signal that increases the prob- kinesin binding rate have the same form (Seitz et al., ability of finding the target without a significant increase 2002). It has also been shown that, at the low tau concen- in the MFPT. This issue has recently been explored using trations affecting kinesin, dynein is relatively unaffected the QSS reduction of the tug-of-war model (Newby and by tau (Dixit et al., 2008). Bressloff, 2010b,c). Two potential signaling mechanisms Newby and Bressloff(2010c) modeled the effects of tau were considered by the authors, the second of which we by introducing into the tug-of-war model, Sec. IV.B, the will review in more detail here. The first was based on tau concentration-dependent kinesin binding rate the observation that the stall force and other single mo- max π0 tor parameters are strongly dependent on the level of π0(τ) = , (4.117) γ(τ0 τ) [ATP], see Sec. IV.B. Since ATP concentration ([ATP]) 1 + e− − is heavily buffered, a small region of intense ATP phos- where τ is the dimensionless ratio of tau per microtubule max 1 phorylation around a target could create a sharp, local- dimer and π0 = 5s− . The remaining parameters are ized [ATP] gradient, which would significantly slow down found by fitting the above function to experimental data a nearby motor complex, thus increasing the chances of (Vershinin et al., 2007), so that τ0 = 0.19 and γ = 100. target detection. The second signaling mechanism in- Carrying out the QSS reduction of the tug-war-model volved microtubule associated proteins (MAPs). These then leads to the FP Eq. (4.99) with τ-dependent drift V , molecules bind to microtubules and effectively modify the diffusivity D and capture rate λ as illustrated in Fig. 30. free energy landscape of motor-microtubule interactions The most significant alteration in the behavior of the (Telley et al., 2009; Tokuraku et al., 2007). For exam- motor-complex is the change in the drift velocity V as ple, tau is a MAP found in the axon of neurons and is a function of τ. The drift velocity switches sign when τ known to be a key player in Alzheimer’s disease (Kosik is increased past a critical point. That is, by reducing 57 the binding rate of kinesin, the dynein motors become dominant, causing the motor-complex to move in the op- 60 posite direction. The effects of local changes in τ con- 40 centration on the efficiency of random search can now be t Ψ V1 determined by assuming that within range of the target, V2 20 V1 |x − X| < l, τ = τ1 > τ0, whereas τ = τ0 outside the 0 target, |x − X| > l. Carrying out the QSS reduction of X-l X+l 0 5 10 the tug-war-model then leads to the FP Eq. (4.99) with x x x-dependent drift and diffusivity: FIG. 32 Diagram showing the effective potential well created by a region of tau coating a MT, and a representative trajec- V (x) = V0 + ∆V χ(x),D(x) = D0 + ∆Dχ(x), (4.118) tory showing random oscillations. where χ(x) is the indicator function defined in (4.96), − V0 = V (τ0),D0 = D(τ0), ∆V = V (τ1) V0, and granules move rapidly until encountering a fixed location − ∆D = D(τ1) D0. Solving the piecewise-continuous along the dendrite where they slightly overshoot then FP equation then determines the hitting probability Π stop, move backwards, and begin to randomly oscillate T and MFPT as functions of τ1 for fixed τ0. In Fig. 31, back and forth. After a period of time, lasting on the T the hitting probability Π and the MFPT are plotted order of minutes, the motor-driven mRNA stops oscil- as a function of τ1. As τ1 is increased above the critical lating and resumes fast ballistic motion. The duration level τ0 = 0.19, there is a sharp increase in Π but a rela- of the oscillations can be estimated by noting that the tively small increase in the MFPT, confirming that τ can FP equation obtained by carrying out the QSS reduction improve the efficacy of the search process. describes a Brownian particle moving in an effective po- Another interesting effect of a local increase in τ is R x tential well Ψ(x) = X l V (x0)dx0, where the drift V (x) that it can generate stochastic oscillations in the motion of Eq. (4.118) is reinterpreted− as a piecewise constant of the motor-complex (Newby and Bressloff, 2010c). As a force, see Fig. 32. Depending on the length of the re- kinesin driven cargo encounters the tau-coated trapping gion influenced by the trap, and the magnitude of the region the motors unbind at their usual rate and can’t drift velocities, the time spent in the potential well can rebind. Once the dynein motors are strong enough to be quite long. Suppose that a Brownian particle starts pull the remaining kinesin motors off the microtubule, at the bottom of the potential well. The corresponding − the motor-complex quickly transitions to ( ) end di- mean exit time (MET) (see (2.30)) is rected transport. After the dynein-driven cargo leaves the tau-coated region, kinesin motors can then reestab-  Ψ(z)  Z X+l   Z y exp lish (+) end directed transport until the motor-complex Ψ(y) D(z) MB = exp − dy dz returns to the tau-coated region. This process repeats X l D(y) D(z) − −∞   2 until the motor-complex is able to move forward past the 2lD2  ν  1 2l 4l tau-coated region. Interestingly, particle tracking exper- = e D2 − 1 + − , ν V1 ν ν iments have observed oscillatory behavior during mRNA transport in dendrites (Dynes and Steward, 2007; Rook where ν = −V22l is the depth of the well. In general, the et al., 2000). In these experiments, motor-driven mRNA MET will be an exponentially increasing function of the depth of the well. This means that any error generated

1 50 by the QSS approximation will also grow exponentially. ( N + , N ) = (2 , 1) 45 − ( N + , N ) = (3 , 2) This is typical of large-deviation behavior in a stochastic 0.8 − 40 ( N + , N ) = (4 , 3) − process where it is well known that diffusion approxima- 0.6 35 tions break down. One can still obtain an approximation 30 Π (sec) 0.4 25 of the MET using perturbation methods, but they must T 20 be applied to the full CK equation (Keener and Newby, 0.2 15 2011; Newby and Keener, 2011). (a) (b) 0 10 0.15 0.16 0.17 0.18 0.19 0.2 0.21 0.15 0.16 0.17 0.18 0.19 0.2 0.21 τ τ I. Active transport on DNA FIG. 31 Effect of adding tau to the target on the capture probability Π and MFPT T using parameters from Fig. 30. a) The analytical approximation P (solid line) and results from So far we have considered the case where ATP hydroly- Monte-Carlo simulation. b) The analytical approximation T sis by a molecular motor facilitates active transport along along with averaged Monte-Carlo simulations. The synaptic a fixed track. However, it is also possible for ATP hy- trap is located at X = 10µm, the trapping region has radius drolysis by a track to facilitate active transport. Active l = 2µm, and the MT track has length L = 20µm. The transport occurs when waves of ATP hydrolysis along −1 capture rate is taken to be k0 = 0.5s . the track push a passive element much like ocean waves PHYSICAL REVIEW LETTERS week ending PRL 99, 248302 (2007) 14 DECEMBER 2007

Brownian Ratchets Driven by Asymmetric Nucleation of Hydrolysis Waves

Amit Lakhanpal and Tom Chou Department of Biomathematics & Department of Mathematics, UCLA, Los Angeles, California 90095-1766, USA (Received 20 May 2007; published 14 December 2007) We propose a stochastic process wherein molecular transport is mediated by asymmetric nucleation of domains on a one-dimensional substrate, in contrast with molecular motors that hydrolyze nucleotide triphosphates and undergo conformational change. We show that asymmetric nucleation of hydrolysis waves on a track can also result in directed motion of an attached particle. Asymmetrically cooperative kinetics between hydrolyzed and unhydrolyzed states on each lattice site generate moving domain walls that push a particle sitting on the track. We use a novel fluctuating-frame, finite-segment mean field theory to accurately compute steady-state velocities of the driven particle and to discover parameter regimes yielding maximal domain wall flux, leading to optimal particle drift.

DOI: 10.1103/PhysRevLett.99.248302 PACS numbers: 82.39. k, 05.40. a, 87.16.Ac ÿ ÿ

Molecular motors such as , , helicases, by the substrate, rather than directly by a molecular motor, and polymerases convert part of the free energy of, e.g., to perform mechanical work. In this Letter, we develop a ATP hydrolysis to a conformational change [1]. The result- general stochastic theory of track-driven, hydrolysis wave- ing molecular deformation leads to motion of the motor mediated transport. In addition to analyzing our model against a load on a track. Although the literature on such using Monte Carlo (MC) simulations, we also formulate molecular motors is vast, much less attention has been paid a moving-frame mean field theory (MFT) that accurately to the theory of molecular motions that exploit the dynam- predicts novel features of the transport. ics of the track on which translation occurs. Such loads are As in models of the ATP cycle of motors, in propelled by the track, which itself is undergoing catalyzed RecA hydrolysis wave-mediated transport the RecA sub- state changes by, e.g., hydrolysis. units can exist in a number of substates corresponding to Two biological strategies involving track-propelled par- sites that have bound ATP, ADP Pi, ADP, or are empty. ticles are collagenase catalysis and Holliday junction trans- To simplify our model, we will assume‡ that each site of the port. Collagenase MMP-1, an enzyme that associates with substrate lattice exists in one of only two possible states, collagen, cleaves its track asymmetrically as it diffuses, ‘‘hydrolyzed’’ ( 1) and ‘‘unhydrolyzed’’ ( 0). Any thus biasing its motion [2]. The dynamics of this track- lattice site i can transitionˆ from state  1 to stateˆ  i ˆ i ˆ driven propulsion has been modeled using burnt bridge 0 with rate k0. The reverse process, physically correspond- models [3–5]. ing to ‘‘hydrolysis’’ or ‘‘nucleation’’, fills an empty site. In Another system where substrate reactions may lead to our model, an asymmetry arises in the nucleation transi- biased motion is the translocation of Holliday junctions tions. If site i 1 is also in the state i 1 0, then the ÿ ÿ ˆ [6]. The Holliday junction at which two double-stranded transition i 0 i 1 occurs with rate k . However, ˆ ! ˆ ÿ DNA molecules exchange one of their strands may be if i 1 1, then the transition i 0 i 1 occurs moved by the dynamics of hydrolysis states of the DNA withÿ rateˆk . If k Þ k , the processˆ is! asymmetricˆ and binding protein RecA. RecA polymerizes on at least one of can lead to‡ a net steady-state‡ ÿ current of domain walls. If a the strand-exchanging dsDNA molecules, assembling into particle is associated with the lattice, it will be pushed each a long nucleoprotein filament. The RecA monomers hydro- 58 lyze ATP and can exist in different states, much like the pushintermediate a surfer toward hydrolysis the shore. states Several of myosin studies motors. have The ex- dy- aminednamics track-induced of the interconversion transport (Antal among and these Krapivsky hydrolysis, 1 2005;states Artyomov may provideet al., the2008 force; Morozov necessary, 2007 to; rotateSaffarian DNA et al.strands, 2006). about Here, each we discuss other during directed Holliday transport junction of a Hol- trans- location. An especially promising model of this process kkk liday junction along DNA (Lakhanpal and Chou, 2007). k0 + k0 Aexploits Holliday asymmetric junction is cooperativity a site where in the two hydrolysis segments of of the doubleATP stranded cofactors DNA associated (dsDNA) with bind each RecAand exchange monomer. one This of theirintrinsic strands asymmetry in what of is the known filament as genetic gives rise recombina- to ‘‘waves’’ of hydrolyzed monomers with a preferred direction, tion. Strands with complimentary base pairs bind and thereby moving the junction by virtue of its preferential form a junction. The junction then moves along both FIG. 1 (color online). Schematic of the asymmetric nucleation attachment to the hydrolyzed segment of the RecA fila- FIG. 33 Diagram describing stochastic hydrolysis dynamics. segments until it reaches a termination point. It is well process. An intrinsic asymmetry in the lattice sites gives rise to ment [6,7]. These examples constitute only two of many asymmetric cooperativity and nucleation. The transported parti- knownmechanisms that translocation by which chemical of this energyjunction may is be an harnessed active cle is represented by a triangle. transport process, requiring energy from ATP hydrol- ysis. A protein called RecA forms a helical polymer 0031-9007=07=99(24)=248302(4) 248302-1steady state properties 2007 of The the American hydrolysis Physical waves Society on the wrapped around a segment of DNA, and interactions be- track. However, analytical solutions are only possible in tween the RecA polymer and the dsDNA drive junction thermodynamic equilibrium, where directed transport is translocation (Klapstein and Bruinsma, 2000). When not possible. Therefore an approximation must be found, RecA monomers hydrolyze ATP, the nucleoprotein fil- and one approach is to use a mean field approximation. ament shifts into a different conformational state where From the Master equation governing the process, one the two strands rotate around each other. We refer to can derive a hierarchy of equations for the moments of this as the activated state. Sj. This system of equation is not closed so that the One possibility for how the Holliday junction moves equation for the mean, σj ≡ hSji, depends upon terms along dsDNA is by hydrolysis waves along the nucleopro- such as hSj 1Sji. The mean field approximation assumes tein filament. This reaction proceeds in waves because − that hSj 1Sji ≈ σj 1σj, and with this assumption, one activated RecA preferentially catalyzes ATP hydrolysis − − obtains a system of n equations for each σj. However, by the adjacent monomer on the 30 side. That is, a since we must consider the n → ∞ limit, solving the monomer hydrolyses ATP much more rapidly when its system of nonlinear differential equations is impractical. neighboring monomer (on the 50 side) is activated. This The simplest mean-field approximation is to assume that asymmetry creates waves of hydrolysis, with a resulting σj 1 ≈ σj = σ, which yields mechanical effect, from the 50 to the 30 end (hereafter − specified as the forward direction) of the DNA strand. It is the mechanical stretching and unwinding of the DNA p 2 k+ − 2k − k0 + (k+ − k0) + 4k k0 induced by the hydrolysis wave that propels the junction σ = − − . (4.120) 2(k+ − k ) forward. Consider a model of Holliday junction transport − along an infinite lattice of RecA monomers surrounding a dsDNA. Each lattice site can be in one of two states: When compared to Monte-Carlo simulation results, this activated or unactivated. Let S be a random variable as- j approximation is only in qualitative agreement, which sociated with the jth lattice site. In the activated state suggests that accounting for correlations between adja- S = 1, and in the unactivated state S = 0. Transi- j j cent sites is necessary to correctly capture the steady- tions between these two states are governed by a Markov state behavior. switching process,

α(Sj−1) It is a reasonable assumption that the correlations be- (Sj = 0) −→ (Sj = 1). (4.119) tween sites is short ranged. This motivates an approach ←− k0 called the finite segment mean field theory (FSMFT) ap- The rate of transitioning to the activated state depends proximation (Nowak et al., 2007b). Consider a small on the state of the lattice site to the left, that is, α(0) = segment within the track containing m sites so that the m k and α(1) = k+, with k+ > k (see Fig. 33). Be- number of states in the system is M = 2 . Clearly, It is cause− of the coupling between neighboring− lattice sites, much simpler to enumerate the Master equation for this one must consider transitions between states of the full segment than for an infinite lattice. Define the proba- T system. If the track contains a finite number of n sites bility density vector p = (p0, ··· , pM 1) where each pj then the total number of states in the system is 2n, with corresponds to a different state of the− track segment. The each possible state given by the different sequences of 0s master equation is dp/dt = Ap. The simplest mapping and 1s. In other words, the state space corresponds to of the binary state to the index j is the binary represen- all of the possible binary words of length n. tation of base-ten numbers. For example, if m = 2 then As a first step, before including the motion of the junc- j = 0 corresponds to the state (0 · 0), j = 1 corresponds tion on the track, one would like to characterize the to (0 · 1), j = 2 ↔ (1 · 0), and j = 3 ↔ (1 · 1). In this 59 example, the matrix of transition rates is As long as m is chosen large enough so that σ is close to the far-field bulk value, the FSMFT approximation   −2k − σ∆ k0 k0 0 accurately captures the correlations near the junction; in −  k −k0 − k − σ∆ 0 k0  practice the authors find the m = 5 is a good choice. One A =  − −  ,  k + σ∆ 0 −k+ − k0 k0  expects the velocity of the junction to depend strongly − 0 k σ∆ k+ −2k0 upon the catalyzed hydrolysis rate, k+. As this rate is in- − creased, the asymmetry should increase the speed of the where ∆ = k+ − k . Recall that the rate of activation forward hydrolysis waves. While this is true for small for each site depends− upon the state of its neighbor to values of k+, the velocity begins to decrease after a crit- the left. The approximation made by the FSMFT is to ical value of k+ where the velocity is maximal. This is set the activation rate of the leftmost site in the finite due to an increase in the total fraction of activated sites. segment to k (1 − σ) + k+σ, where 0 ≤ σ ≤ 1 is the − A high fraction of activated sites means that the junc- mean activation level of the site immediately to the left tion is more likely to be trapped within a long segment of the segment. One way to determine σ is to set it equal of purely activated sites where it does not move. to the mean activation level of the rightmost site in the segment. This results in the auxiliary equation, V. TRANSPORT AND SELF-ORGANIZATION IN CELLS M 1 X− hSmi = pj = σ. (4.121) A. Axonal elongation and cellular length control j=0 j odd During neural development, the formation of synapses To find the steady state hydrolysis activity, one first involves the elongation of an axon of one cell to locate solves the linear system of equations for the nullspace the dendrites of another cell. Axon elongation is a con- of A using standard methods. Then, combining the re- sequence of the interplay between force generation at the sult with (4.121) results in a nonlinear equation to solve growth cone that pulls the axon forward, pushing forces for σ. Notice that the simplest mean field approximation due to microtubule and actin polymerization and depoly- (4.120) is recovered by setting m = 1. merization, the rate of protein synthesis at the cell body, The movement of the Holliday junction can be incor- and the action of cytoskeletal motors (Baas and Ahmad, porated into the model by considering a segment (with 2001; Goldberg, 2003; Lamoureux et al., 1989; Mitchi- m odd), centered on the position of the junction, that son and Kirschner, 1988; O’Toole et al., 2008; Suter and moves with the junction. For simplicity take m = 3. The Miller, 2011). Several models of axonal elongation have frame shifts whenever a wall passes through the junction. focused on the sequence of processes based on the pro- When the frame shifts, the state of the new site is deter- duction of dimers at the cell body, the active mined by the far field mean activity, σ. This requires transport of these proteins to the the tip of the grow- modification of transition from states (1 · 0 · 0), (1 · 0 · 1), ing axon, and microtubule extension at the growth cone and (0 · 0 · 1) to appropriate frame-shifted states after (Graham et al., 2006; Kiddie et al., 2005; McLean and activation of the center cite. Let the master equation be Graham, 2004; Miller and Samulels, 1997; van Veen and ˆ dq/dt = Aq, where q(t) is the probability distribution van Pelt, 1994). One motivation for identifying the poly- for the activation state of the sites surrounding the junc- merization of microtubules as a rate limiting step is that tion, with indexing the same as for p. Assume that the axonal growth occurs at a similar rate to the slow axonal only way to reach the (1 · 1 · 1) state, where the center transport of tubulin, namely, around 1mm per day. (It cite is activated, is from (1·0·1); this transition does not is possible that short, freshly nucleated microtubles are result in movement of the junction. Note that transitions also actively transported into axons (Baas and Buster, where the shifted segment has the same state as the orig- 2004)). For the sake of illustration, consider a continuum inal do not apear in the modified transition-rate matrix. model of the active transport of tubulin (Graham et al., For example, suppose a wall moves the junction to the 2006; McLean and Graham, 2004). Let c(x, t) denote the right via a transition from (1 · 0 · 0) when the center cite concentration of tubulin at position x along the axon at is activated. The shifted segment can be either (1 · 0 · 1) time t. Suppose that at time t the axon has length l(t) or (1 · 0 · 0). Transition to state (1 · 0 · 1) occurs at a rate so that x ∈ [0, l(t)]. The transport of tubulin is modeled k+σ, and transition to (1 · 0 · 0) at a rate k+(1 − σ), but macroscopically in terms of an advection-diffusion equa- the transition to (1 · 0 · 0) does not apear in the modified tion with an additional decay term representing degra- transition rate matrix because it has the same activation dation at a rate g: state as the pre-shifted segment. ∂c ∂2c ∂c Let q+ correspond to state (1 · 0 · 0) and q to state − − − = D 2 V gc. (5.1) (0·0·1) (or the sum of qj over appropriate states if m > 3). ∂t ∂x ∂x Then, the steady state velocity and effective diffusivity of Such a model can be derived from a more detailed the junction is V = k+q+ − k q and D = k+q+ + k q . stochastic model of active transport as detailed in Sec. − − − − 60

(IV.C), with V the effective drift due to motor-driven the growth cone due to interactions with the actin cy- transport and D the effective diffusivity. It is assumed toskeleton are neglected (Suter and Miller, 2011). Third, that there is a constant flux of newly synthesized tubulin there are a number of other processes that could act as from the cell body at x = 0 so that rate limiting steps in axonal growth, namely, the recy- cling of lipid membrane and the maintenance of the en- ∂c ergy needs at the growth tip via the transport of mi- = −0c0 at x = 0. (5.2) ∂x tochondria (Hollenbeck and Saxton, 2005; Miller and et al. The flux at the growing end x = l(t) is equal to the Sheetz, 2004; Morris and Hollenbeck, 1993; O’Toole , difference between the fluxes associated with microtubule 2008). A recent stochastic model incorporates a number et al. assembly and disassembly: of these features (Atanasova , 2009). First, the rate of growth of the axon tip is determined by the rates at ∂c which newly delivered membrane proteins are inserted = −lc + γl at x = l(t). (5.3) ∂x into the tip via exocytosis and are removed via endocy- tosis. Meanwhile, microtubules grow via polymerization Finally, the rate of growth is also taken to be proportional until they reach the axon tip, where they are stabilized to the difference between these two fluxes according to by interactions with the actin cytoskeleton. This in turn reduces the rate of endocytosis of membrane vesicles. dl = Ω [lc − γl] , x = l(t). (5.4) Axonal length control is one example of how cells reg- dt ulate the size of their organelles and internal structures. The constant Ω depends on the size of each tubulin dimer, Size control mechanisms, which are critical for proper the number of microtubules at the tip and the cross- cell function, can be distinguished according to whether sectional area of the axon. the underlying structure is static (remains intact once It is straightforward to determine the steady-state assembled (Katsura, 1987; Keener, 2005)) or dynamic. length L of the axon (McLean and Graham, 2004). First, Dynamic structures are constantly turning over so that dl/dt = 0 implies that c(L) = cL ≡ γl/l and and in order for them to maintain a fixed size, there must be ∂c/∂x = 0 at x = L. The steady-state concentra- a balance between the rates of assembly and disassem- λ x λ x tion profile takes the√ form c(x) = Ae + + Be − with bly. If these rates depend on the size in an appropriate λ = (V/2D) 1 ± 1 + 4h and h = Dg/V 2. The co- way then there will be a unique balance point that stabi- efficients± A, B are determined from the boundary condi- lizes the size of the organelle. Recent experimental work tions at x = L. Finally, a transcendental equation for suggests that such a dynamic mechanism may also oc- the steady-state length L is obtained by imposing the cur in eukaryotic flagella (Ishikawa and Marshall, 2011; boundary condition (5.2): Marshall et al., 2005; Marshall and Rosemblum, 2001). These are microtubule–based structures that extend to D c λ−L λ+L 0 0 about 10 µm from the cell and are surrounded by an ex- F (L) ≡ e− − e− = (λ+ − λ ), (5.5) g cL − tension of the plasma membrane. They are at least an order of magnitude longer than bacterial flagella. Flagel- having used λ+λ = −g/D. For small L, the expo- − lar length control is a particularly convenient system for nentials can be Taylor expanded to give L ≈ D0 c0 , g cL studying organelle size regulation, since a flagellum can whereas for large L the first exponential is dominant and   be treated as 1D structure whose size is characterized by L = V log D0 c0 . The last equation follows from tak- g g cL a single length variable. The length of a eukaryotic flag- ing h  1 so that λ+ ≈ V/D and λ ≈ −g/V . In the ellum is important for proper cell motility, and a number first regime diffusion is dominant, whereas− in the other of human diseases appear to be correlated with abnormal active transport is dominant. Numerical simulations of length flagella (Gerdes and Kasanis, 2005). the full time-dependent model show that these steady- Radioactive pulse–labeling has been used to measure states are stable and in both regimes the approach to protein turnover in the flagella of Chlamydomonas, a uni- steady-state is overdamped. On the other hand, for inter- cellular green alga with genetics similar to budding yeast mediate values of L damped oscillations occur resulting (Marshall and Rosemblum, 2001). Such measurements in overshoot (Graham et al., 2006). have suggested that turnover of tubulin occurs at the There are a number of simplifications assumed in the distal + end of flagellar microtubules, and that the as- above model (Goldberg, 2003; Vitriol and Zheng, 2012). sembly part of the turnover is mediated by intraflagellar First, the rate of elongation is based on the average rate transport (IFT). This is a motor-assisted motility within of assembly and disassembly of a bundle of microtubules, flagella in which large protein complexes move from one which neglects the stochastic switching between periods end of the flagellum to the other (Kozminski et al., 1993; of of elongation and rapid contraction exhibited by in- Scholey, 2003). Particles of various size travel to the dividual microtubules (Mitchison and Kirschner, 1984). flagellar tip (anterograde transport) at 2.0 µm/s, and Second, tensile forces acting on the microtubules within smaller particles return from the tip (retrograde trans- 61

l(t) v degradation flagellum V C C + 1 2

IFT particle v+ cargo insertion

FIG. 34 Schematic diagram of intraflagellar transport (IFT), in which IFT particles travel with speed v± to the ± end C C of a flagellum. When an IFT particle reaches the + end it 1 2 releases its cargo of protein precursors that contribute to the assembly of the flagellum. Disassembly occurs independently of IFT transport at a speed V . FIG. 35 LEFT: Diagram of secretory pathway including nu- cleus, ER and . 1. Nuclear membrane; 2. Nuclear pore; 3. RER; 4. SER; 5. Ribosome; 6. Protein; 7. Transport vesicles; 8. Golgi apparatus; 9. Cis face of port) at 3.5 µm/s after dropping off their cargo of as- Golgi apparatus; Trans face of Golgi apparatus ; 11 Cister- nae of Golgi apparatus; 12. Secretory vesicle; 13. Plasma sembly proteins at the + end. A schematic diagram of membrane; 14. Exocytosis; 15 Cytoplasm; 16. Extracellu- IFT transport is shown in Fig. 34. Immunoflourescence lar domain. [Public domain image from WikiMedia Com- analysis indicates that the number of IFT particles (esti- mons]. RIGHT: When two compartments continually ex- mated to be in the range 1–10) is independent of length change products via vesicular transport, a symmetry breaking (Marshall et al., 2005; Marshall and Rosemblum, 2001). mechanisms is needed to maintain nonidentical compartments If a fixed number of transport complexes M move at a (C1 6= C2). fixed mean speedv ¯, then the rate of transport and assem- bly should decrease inversely with the flagellar length L. On the other hand, measurements of the rate of flagellar proper 3D structure. The ER can be partitioned into shrinkage when IFT is blocked indicate that the rate of the rough ER (RER), which is rich in ribosomes, and disassembly is length–independent. This has motivated the smooth ER (SER), which has only a few sparse ribo- the following simple deterministic model for length con- somes and tends to form a tubular structure. In neurons, trol (Marshall and Rosemblum, 2001): the RER is present in the soma and proximal dendritic compartments, whereas the SER is distributed in distal dL avM¯ = − V, (5.6) dendrites (including some dendritic spines) and axons. dt 2L The diffusivity of proteins within the tubular-like SER is where a is the size of the precursor protein transported 3-6 times smaller than within the cytoplasm. However, by each IFT particle and V is the speed of disassem- the ER is constantly being remodeld by motor-driven bly. Eq. (5.6) has a unique stable equilibrium given by sliding along microtubules, for example, which could add an active component to protein transport (Ramirez and L∗ = avM/¯ 2V . Using the experimentally based values Couve, 2011; Valenzuela et al., 2011). Moreover, the thin M = 10,v ¯ = 2.5 µm/s, L∗ = 10 µm and V = 0.01 µm/s, the effective precursor protein size is estimated to be tubular structure of the SER reduces the effective spa- a ≈ 10 nm. A stochastic version of a model for flagellar tial dimension of diffusion, thus enhancing progression length control has also been developed using the theory along a dendrite. Another important aspect of the secre- of continuous time random walks (Bressloff, 2006) tory pathway is that it is tightly regulated (Lippincott- Schwartz et al., 2000). Proteins accumulate at specific exit sites and leave the ER in vesicles that transfer the B. Cooperative transport of proteins in cellular organelles cargo to organelles forming the Golgi network where final packaging and sorting for target delivery is carried out. The extensive secretory pathway of eukaryotic cells In most eucaryotic cells the Golgi network is confined provides an alternative system for transporting newly to a region around the nucleus known as the Golgi ap- synthesized lipids and proteins along axons and dendrites paratus, whereas in neurons there are Golgi “outposts” (Kennedy and Ehlers, 2006; Ramirez and Couve, 2011; distributed throught the dendrite. Thus it is possible Valenzuela et al., 2011). One major organelle of the that some proteins travel long distances within the SER secretory pathway is the endoplasmic reticulum (ER), (rather than via active transport along microtubules) be- which tends to be dispersed throughout the cytoplasm fore being sorted for local delivery at a synapse. of a cell (Lippincott-Schwartz et al., 2000), see Fig. 35. One of the significant features of the secretory pathway Proteins and lipids destined for the plasma membrane is that there is a constant active exchange of molecules enter the ER from the nucleus as they are translated between organelles such as the ER and Golgi apparatus, by ER-associated ribosomes, where they fold into their which have different lipid and protein compositions. Such 62 an exchange is mediated by motor-driven vesicular trans- ing molecules along the cytoskeleton to specific locations port. Vesicles bud from one compartment or organelle, on the plasma membrane (Altschuler et al., 2008; Layton carrying various lipids and proteins, and fuse with an- et al., 2011; Marco et al., 2007), whereas the other in- other compartment. Transport in the anterograde direc- volves the coupling of membrane diffusion with bistable tion has to be counterbalanced by retrograde transport enzymatic dynamics (Gamba et al., 2009; Mori et al., in order to maintain the size of the compartments and 2008; Semplice et al., 2012). to reuse components of the transport machinery. Since One example of the first class of model is shown in bidirectional transport would be expected to equalize Fig. 36. Here the asymmetric distribution of a signaling the composition of both compartments, there has been molecule within the plasma membrane ∂Ω and the orien- considerable interest in understanding the self-organizing tation of actin filaments are mutually enhanced through mechanisms that allow such organelles to maintain their a positive feedback loop (Marco et al., 2007). Let u(r, t) distinct identities while constantly exchanging material denote the concentration of signaling molecules within (Mistelli, 2001), see Fig. 35. One model for generating the plasma membrane. Then u depends on six physically stable, non-identical compartments has been proposed interpretable quantities: (i) the membrane diffusivity D; by Heinrich and Rapoport(2005), (see also Binder et al. (ii) the index function χ(r) indicating the region of the (2009); Dmitrieff and Sens(2011); Gong et al. (2010); plasma membrane to which cytoskeletal tracks are at- Klann et al. (2012)), based on the observation that vesic- tached, that is, the cluster within which u is high; (iii) ular transport involves a complex network of molecular the total amount of signaling molecule Ntot; (iv) the rate interactions between vesicles, transported molecules and of directed transport h; (v) the endocytosis rate k within recipient organelles (Barlowe, 2000; Lippincott-Schwartz the cluster; and (vi) the endocytosis rate K outside the and Phair, 2010; Pelham, 2001). That is, the rates of cluster with K < k. The density u evolves according to vesicle exchange between compartments are influenced the macroscopic equation (Marco et al., 2007) by their composition. An inutitive understanding of the ∂u χ basic mechanism can be obtained by considering the ex- = D∆u − [kχ + Kχ¯] u + hN , (5.7) ∂t cyt R χdr change of four types of protein (SNARES), X, U, Y, V ∂Ω say, between two compartments. Suppose that in steady- where ∆ is the Laplace-Beltrami operator for diffusion state many vesicles with a low content of X,U move in in the membrane,χ ¯(r) = 1 − χ(r), and Ncyt is the total one direction (from the first to the second compartment), amount number of signaling molecules within the cyto- whereas a few vesicles with a large content of X,U move R plasm: Ncyt = Ntot − ∂Ω udr. The cytoplasmic pool is in the opposite direction so that the total protein fluxes assumed to be homogeneous due to the fast dispersion are balanced. This would reflect differences in composi- of vesicles in the cytosol. Numerical simulations show tion of X,U in the two compartments. However, lipid that a stable spatially asymmetric distribution of sig- balance would not be maintained because there would naling molecules within the plasma membrane can be be a net flux of vesicles in one direction. However a maintained. Moreover, the degree of polarization can be balance of lipid fluxes can also be achieved by having optimized by varying the rates of endocytosis. One lim- a complemetary transport of Y,V molecules in the op- itation of the model, however, is that the packaging of posite direction. The asymmetric states are stablilized signaling molecules into discrete vesicles is ignored, that by taking the rates of budding and fusion to depend on is, the model treats transport as a continuous flux of interactions between vesicles and compartments medi- proteins. As highlighted by Layton et al. (2011), incor- ated by the protein pairs (X,U) and (Y,V )(Heinrich porating vesicular transport into the model makes cell and Rapoport, 2005). polarization more difficult to sustain. A simple argu- ment for this proceeds as follows. Exocytic vesicles need to have higher concentrations of the signaling molecule C. Cell polarity than the polarization site in order to enhance the concen- tration. A dynamic equlibrium of recycling can only be Many cellular processes depend critically on the stable maintained if endocytic vesicles also have an enhanced maintenance of polarized distributions of signaling pro- concentration of signaling molecules. This appears to teins on the plasma membrane. These include cell motil- put unrealistically strong constaints on the mechanisms ity, epitheleal morphogenesis, embryogenesis, and stem for loading vesicles with cargo prior to transport. cell differentiation. In many cases cell polarity can occur The second basic mechanism for establishing cell polar- spontaneously in the absence of pre-existing spatial cues. ity does not depend on active transport, and can be mod- Various experimental studies suggest that there are at eled in terms of a reaction–diffusion system. One exam- least two independent but coordinated positive feedback ple of such a model is described in Fig. 37(Gamba et al., mechanisms that can establish cell polarity (Wedlich- 2009; Semplice et al., 2012). Consider a macroscopic ver- Soldner et al., 2004). One involves the reinforcement of sion of the model, in which φ± denote the concentration spatial asymmetries by the directed transport of signal- of activated and inactivated signaling molecules within 63

membrane diffusion S + (a) extracellular (b) Φ - rich domain

X signaling cytosol molecule − + Φ Φ membrane − endocytosis Φ - rich domain Y cytosol actin filaments recycling FIG. 37 Model of Semplice et al. (2012). (a) A set of receptors transduce an external distribution of chemotactic cues into FIG. 36 Model of Marco et al. (2007). Signaling molecules an internal distribution of activated enzymes X that catalyze the switch of a signaling molecule from an unactivated state can attach and orient actin filaments that deliver vesicles car- − + rying the signaling molecule from the cytoplasm to the plasma Φ to an activated state Φ . A counteracting enzyme Y transforms Φ+ back to Φ−. Amplifying feedback loops, in membrane. The additional signaling molecules orient more + − actin filaments that transport more molecules in a positive which Φ activates X and Φ activates Y , result in chemical feedback loop. The local orientation of actin filaments also bistability. The signaling molecules are permanently bound increases the rate of endocytosis within the cluster. to the plasma membrane, where they exhibit lateral diffusion, whilst the enzymes are free to move between the membrane and the cytosol. (b) Cell polarization occurs when there is phase separation into two stable chemical states. the plasma membrane. Let xc, yc be the concentration of the counteracting enzymes in the cytosol (which is assumed to be homogeneous), let x , y be the concentra- 0 Under the adiabatic approximation, the dynamics can be tions of membrane associated enzymes activated by Φ+ written in the variational form and Φ−, respectively, and let x00 denote the concentration of membrane associated enzyme activated by a distribu- ∂φ δF[φ] (r, t) = − (5.12) tion of receptors s. The model equations take the form ∂t δφ(r, t) (Semplice et al., 2012) + with F an effective energy functional ∂φ±/∂t = D∆φ± ± g(φ , φ−, x0, x00, y) (5.8a) − Z ∂x0/∂t = ka0 sxc0 kd0 x0 (5.8b) F[φ] = D(∇φ)2 + V (φ) dS. (5.13) + ∂Ω ∂x00/∂t = k00aφ xc − kd0 x00, (5.8c) ∂y/∂t = kaφ−yc − kdy. (5.8d) Here integration is with respect to the membrane sur- face. Stable homogeneous solutions correspond to min- Here ∆ is the Laplace-Beltrami operator, ka0 , ka00, ka and ima of the potential V (φ) for which V 0(φ) = 0. One finds kd0 , kd00, kd are the forward and backward reaction rates of the signaling and feedback pathways, and that for a range of parameter values, the system exhibits bistability. That is, there exist two stable equilibria ϕ + ± x0φ− x00φ− yφ corresponding to phases enriched in Φ± separated by an g = kc0 + kc00 − kc (5.9) K0 + φ− K00 + φ− K + φ− unstable equilibrium. + The polarization of the cell membrane can now be un- is the enzymatic conversion rate of Φ to Φ−. The total + + derstood in terms of the theory of phase separation kinet- amount of Φ and Φ is conserved, φ + φ− = c, as are the total amounts of− each enzyme X, Y . Using a time- ics familiar from the study of condensed matter systems scale separation in which the equilibria for the concen- (Gamba et al., 2009; Semplice et al., 2012). A polar- ized state will exist when the cell membrane is divided trations x0, x00, y, xc, yc are reached much faster than the into two complementary regions, see Fig. 37(b), that equilibria for the surface distributions φ±, the dynamics + correspond to two distinct stable chemical phases, sepa- for the concentration difference φ = φ − φ− reduces to the system (Semplice et al., 2012) rated by a thin diffusive interface. Such a spatially in- homogeneous solution has to minimize both terms in the ∂φ functional (5.13). One condition for stability is phase = D∆φ + V 0(φ), (5.10) ∂t coexistence, that is, where Z ϕ+ 2 2 ∆V = V (ϕ+) − V (ϕ ) = V 0(φ)dφ = 0. (5.14) V 0(φ) = (c − φ )[Γ0(φ) + Γ00(φ) + Γ(φ)], (5.11) − ϕ− with A second condition is that the diffusive “energy” associ- 2(kc0 ka0 /kd0 )xcs 2(kc00ka00/kd00)xc ated with the interface is minimized. Even when a stable Γ0 = , Γ00 = (2K0 + c − φ)(c + φ) 2K00 + c − φ polarized state exists, the evolution to such a state in- 2(k k /k )y volves a complex process of nucleation and competitive Γ = c a d c . 2K + c − φ growth of heterogeneous patches. Suppose, for example, 64 that the membrane is initially in a metastable state con- ACKNOWLEDGEMENTS siting of the ϕ phase. External stimulation may make − the ϕ+ phase energetically more favorable but there is This publication was based on work supported in part an energy barrier to overcome, which blocks a continu- by the National Science Foundation (DMS-1120327) and ous transition to the ϕ+ phase. 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