Revista Mexicana de Física ISSN: 0035-001X [email protected] Sociedad Mexicana de Física A.C. México

Ambía, F.; Híjar, H. dynamics of a Brownian motor based on morphological changes Revista Mexicana de Física, vol. 63, núm. 4, julio-agosto, 2017, pp. 314-327 Sociedad Mexicana de Física A.C. Distrito Federal, México

Available in: http://www.redalyc.org/articulo.oa?id=57050930003

How to cite Complete issue Scientific Information System More information about this article Network of Scientific Journals from Latin America, the Caribbean, Spain and Portugal Journal's homepage in redalyc.org Non-profit academic project, developed under the open access initiative RESEARCH Revista Mexicana de F´ısica 63 (2017) 314-327 JULY-AUGUST 2017

Stochastic dynamics of a Brownian motor based on morphological changes

F. Amb´ıa and H. H´ıjar Grupo de Sistemas Inteligentes, Facultad de Ingenier´ıa, Universidad La Salle, Benjam´ın Franklin 47, 06140, Ciudad de Mexico.´ Received 16 February 2017; accepted 18 April 2017

We introduce a simplified model for a microscopic system that performs directed Brownian due to coordinated morphological adap- tations. This system consists of two spherical with adaptable size, that interact through elastic and repulsive . We propose an algorithm to control the time dependence of the system’s shape that turns it into a Brownian motor, whose stochastic dynamics is analyzed by means of a Langevin model. We restrict ourselves to the simplified case of motors with small shape asymmetries and slow morphological changes, and calculate the average speed at which they should move. We compare the theoretical predictions with the results from Brownian Dynamics simulations and find that they are in very good quantitative agreement. We carry out a comparison of the proposed rectifying algorithm with a classical one based on a ratchet potential and show that in some cases morphological adaptations could produce larger . We thus propose the locomotion mechanism based on controlled structural changes as a novel alternative method from which Brownian motors could operate autonomously, i.e., requiring neither a substrate nor a ratchet field.

Keywords: Brownian motor; Langevin dynamics; rectified .

PACS: 05.40.Jc; 05.40.-a; 05.70.Ln

1. Introduction case, detailed balance is lost as a consequence of temperature variations and directed motion is achieved having no other In accordance with the second law of thermodynamics, it source than thermal fluctuations [18]. In a very similar fash- is not possible to obtain a net current of Brownian parti- ion, Brownian motion can be rectified by producing cyclic cles when they are subjected only to equilibrium thermal variations of the amplitude of the substrate potential [6]. The forces [1–3]. However, in the presence of ratchets or intel- so-called Stokes’ drift, a deterministic mechanism that drags ligent control systems, e.g., a Maxwell’s demon, Brownian along particles suspended in a viscous medium traversed by a motion can be rectified and a current of Brownian particles longitudinal wave [19], can be also used together with time- can be generated which is maintained by the unbiased fluctu- dependent unbiased forces to achieve the Brownian motor be- ating forces exerted by the thermal bath [4]. Understanding havior [20]. In addition to these examples, a great variety of the precise mechanisms that allow a microscopic system to non-equilibrium perturbations exist that can be used for pro- perform directed Brownian motion in an isotropic noisy en- pelling a Brownian motor, in which a ratchet potential is not vironment, is a problem that has received considerable atten- involved. This is the case discussed in the excellent review tion during the last decades [5–7]. The interest on this sub- in Ref. 7, of using stochastic non-equilibrium perturbations ject has increased since the discovery that processes are with vanishing mean but higher order odd moments different conducted by Brownian motors or molecular machines, that from zero. perform multiple specialized tasks in the cells of living or- ganisms [8]. Examples of such molecular motors include: ki- Thanks to the theoretical analysis of these mechanisms, it nesins [9,10], ATP synthases [11–13], and helicases [14,15]. has been possible to formulate a general definition of a Brow- It is now known that these machines are sophisticated molec- nian motor based on the role that the breakdown of symme- ular structures that use the spontaneous fluctuations occur- tries and fluctuations play in its operation [6, 7]. According ring in their surroundings together with ratchet mechanisms, to this definition, a genuine Brownian motor is any physi- in order to break the spatial symmetry of their dynamics [16]. cal system supplemented with a rectification mechanism that is critically affected by the spatio-temporal periodicity, on From the theoretical point of view, many models have which all the forces and gradients average zero, and which been formulated through the years that predict the emergence is kept away from thermodynamic equilibrium by the rupture of directed Brownian motion from unbiased noise, most of of the detailed balance symmetry. Most importantly, in order which are based on the use of a ratchet potential, i.e., a pe- to constitute an authentic Brownian motor, the random forces riodic potential that lacks reflection symmetry. Even in ab- acting on the system, having thermal or non-thermal origin, sence of spatial symmetry, a net drift of a Brownian must assume a principal role. is forbidden by the second law. However, if the particle is in the presence of an additional having time correlations, Models of directed Brownian motion could lead in the fu- detailed balance is broken and a non-vanishing drift is pro- ture to the fabrication of micro and nanomachines aimed at duced [3]. A resembling situation occurs when a Brownian functions such as: targeted drug delivery [21,22]; stirring and particle in a ratchet potential is immersed in a bath with a pumping in microfluidic devices and biological systems [22]; temperature that changes in time periodically [17]. In this and separation of colloids, macromolecules, chromosomes or STOCHASTIC DYNAMICS OF A BROWNIAN MOTOR BASED ON MORPHOLOGICAL CHANGES 315 viruses [23–25]. Although, up to now, diverse artificial au- responding quantities of the second one. The particles inter- 2 tonomous micromotors have been realized, some of which act through the elastic potential, U = k (x2 − x1 − l) /2, are powered by catalytic reactions [26], Janus particles [27], where k is the restitution coefficient and l is an equilibrium or micro-electromechanical technologies [28], the operation distance, representing the average elongation of the complete of these nanoscale devices is controlled mainly by determin- system. As it is usual, the dynamics of the system is de- istic mechanical or chemical forces, and thermal noise does scribed using the low Reynolds number approximation, since not necessarily play a major role [7]. The case is the same viscous forces are expected to be much larger than inertial for some theoretical models of microsystems that perform ef- contributions. Under such conditions, the mass is not rel- ficient locomotion in viscous fluids [29–31]. What is worse, evant on the dynamics of the particles and their stochastic thermal fluctuations are even considered as problematic when equations of motion can be cast in the form such microdevices are designed, since molecular collisions µ ¶ µ ¶ might alter their desired trajectories [22]. We will show here dxc.m. k 1 1 1 1 1 =− − (xr−l)+ A1+ A2 , (1) that, in contrast to this point of view, thermal forces could be dt 2 β2 β1 2 β1 β2 used as a source to propel a theoretical elastic motor that ad- vances by adapting its shape according to the configuration µ ¶ dxr 1 1 1 1 that the same thermal forces produce. = −k + (xr − l) + A2 − A1. (2) dt β β β β Specifically, in this paper we will analyze in detail the 1 2 2 1 fluctuating dynamics of a theoretical Brownian motor that, In the first place, notice that the previous equations have as far as we can tell, has only been studied by ourselves in been written in terms of the the center of mass position, terms of a very basic formulation [32]. This model consists xc.m. = (x1 + x2) /2, and the current extension of the motor, of an elastic system with adaptable shape that can be pro- xr = x2 − x1. In addition, βµ = 6πηRµ, (throughout this pelled by thermal noise and coordinated unbiased morpho- paper Greek indices will run over the values 1 and 2), rep- logical changes, without needing for a ratchet potential. We resent the drag coefficients on the spheres assuming no-slip will study the statistical properties of the system in terms of a boundary conditions at their surfaces. Moreover, in Eqs. (1) Langevin model in Sec. 2 restricting ourselves to the case of and (2), Aµ, denotes the stochastic force acting on the µth motors with purely elastic interactions and having small and particle, which satisfies hAµi = 0, and the usual fluctuation- slow morphological modifications. From this model we will dissipation relation be able to obtain an explicit expression for the average speed of the motor. In Sec. 3 we will implement Brownian Dynam- 0 0 hAµ (t ) Aν (t)i = 2kBT βµδ (t − t) δµν . (3) ics simulations that corroborate the theoretical predictions within the limit corresponding to the approximations of the where kB is the Boltzmann constant; and no summation over model. Afterwards in Sec. 4, we will extend our model and repeated indices is implied. simulation procedure to supplement them with features that In the model described in Ref. 32, it is assumed that the are commonly found in microscopic systems, namely: inter- configuration of the motor is maintained constant over an en- molecular repulsive interactions and finite nonlinear elastic tire interval of time, say from time 0 to time t. In this interval, forces. This has the purpose of showing that it is plausible to the formal solution of Eqs. (1) and (2) is given by think on an actual physical realization of our proposal. For this extended case, we will be able to give again an expres- ¡ ¢ x = xo +η ˜ 1 − e−αt (xo − l) sion for the average motor’s , which is found to be in c.m. c.m. r very good quantitative agreement with the results of the nu- Zt · merical implementation. Finally, in Sec. 6 we will summarize (1) + dξ ψc.m. (t − ξ) A1 (ξ) our main conclusions. 0 ¸ (2) 2. Directed Brownian motion from morpho- + ψc.m. (t − ξ) A2 (ξ) , (4) logical adaptations Zt · µ ¶ o −αt (1) 2.1. Basic algorithm for directed motion xr = l + (xr − l) e + dξ ψr t − ξ A1 (ξ) 0 We will review first the basic idea for the operation of our ¸ Brownian motor based on morphological changes, as it was (2) + ψ (t − ξ) A2 (ξ) , (5) introduced in Ref. 32. The model is constituted by two r spherical particles, with identical mass m, that perform one- o o dimensional Brownian motion along the x direction, in a fluid where xc.m. and xr are the¡ initial values¢ of xc.m. and x , respectively; α=k β−1+β−1 , and η˜ = with η and temperature T . The symbols x1 and r 1 2 (µ) R1 denote, respectively, the position of the center and the (R2 − R1) /2 (R1 + R2); and the auxiliary functions ψc.m., (µ) radius of the first sphere, while x2 and R2 indicate the cor- ψr , are given by

Rev. Mex. Fis. 63 (2017) 314-327 316 F. AMBIA´ AND H. HIJAR´

· −αt −αt ¸ (µ) e 1 − e with the temperature, the difference ∆R ≡ Rmax − Rmin, and ψc.m. (t) = + , 2βµ β1 + β2 the elastic coefficient of the motor, while it decreases with the viscosity of the surrounding medium, see Eq. (9) in Ref. 32. e−αt and ψ(µ) (t) = (−1)µ . (6) In addition it is found that the standard deviation of the dis- r β µ tribution function for the position of an ensemble of motors Observe from Eq. (4) that an average displacement of increases linearly with time. o xc.m. will occur whenever η˜ 6= 0 and xr 6= l. That is, when the particles have different sizes, the bigger one experiences 2.2. Morphological changes at finite times and hydrody- a larger viscous drag and plays the role of anchor for the mo- namic effects tor, while the smaller one is able to move and, by the effect The model discussed so far, is not completely satisfactory of the elastic interaction, produces the displacement of xc.m.. from the physical point of view for two principal reasons. Such displacement will tend to occur in the direction dictated First, it requires for instantaneous changes in the shape of the by the sign of η˜(xr − l). This result suggests that thermal motor. Second, it does not take into account the flow pro- forces and morphological changes can be combined in favor duced around the spheres as a consequence of these changes. of producing directed motion. The main idea is to use the fact In favor of analyzing a more consistent situation, we will as- that stochastic forces will drive xr away from l persistently, sume now that the update of the radii Rµ between the values and to supplement the motor with an intelligent control sys- Rmin and Rmax, occurs at finite time intervals of characteristic tem that breaks detailed balance by measuring xr and produc- duration τR. An appealing way to model such changes is to ing a variation of R1 and R2 in such a way that the product assume that, when the condition for a modification in shape η˜(xr − l) will have a preferred sign during the whole evolu- is met, R1 and R2 go from their current values to R or tion of the system. min Rmax exponentially. Accordingly, we propose to update the In Ref. 32, it was assumed that R1 and R2 can be sizes of the spheres 1 and 2 as follows: switched between two prescribed values Rmax, and Rmin (Rmax > Rmin), according to the rule if xr > l then ³ ´ R = R , and R = R , if x < l; −ζ(t−tp) −ζ(t−tp) 1 max 2 min r (7) R1 (t) = R1 (tp) e + Rmin 1 − e , R1 = Rmin, and R2 = Rmax, if xr > l. ³ ´ −ζ(t−tp) −ζ(t−tp) R2 (t) = R2 (tp) e + Rmax 1 − e ; which guarantees that the tendency of xc.m. will be to move along the positive direction, as it is illustrated in Fig. 1. if xr < l then One of the main results in Ref. 1 is that the basic rectifi- ³ ´ −ζ(t−tp) −ζ(t−tp) cation algorithm described by Eq. (1) makes the motor able R1 (t) = R1 (tp) e + Rmax 1 − e , to move at a constant average speed. Such speed increases ³ ´ −ζ(t−tp) −ζ(t−tp) R2 (t) = R2 (tp) e + Rmin 1 − e ; (8)

−1 in which ζ = τR , tp denotes the instant at which the pth structural change takes place, and tp ≤ t ≤ tp+1. On the other hand, hydrodynamic effects can be incorpo- rated in a similar way as it is done in the case of microswim- mer models [30, 31]. More precisely, if we limit ourselves to study very elongated motors with geometric features sat- isfying the inequality Rµ/l ¿ 1, then the flow around the system of two spheres can be obtained, by the linearity of the Stokes equations, as the superposition of the flow around the individual spheres [31]. By noticing that the flow produced by an expanding sphere increases with the time derivative of its volume and decreases with the square of the distance from its center [33], we find, under the previous assumption, that xc.m. and xr obey the following system of coupled Langevin equations FIGURE 1. Basic algorithm for a Brownian motor based on mor- µ ¶ dx k 1 1 1 ³ ´ phological changes. The sphere with radius Rmax acts as anchor, c.m. ˙ ˙ = − − (xr − l) + 2 V1 − V2 while the one with radius Rmin moves as a consequence of the elas- dt 2 β2 β1 8πxr tic interaction represented by the spring. Thermal forces produce µ ¶ 1 1 1 the changes in the elongation of the system and coordinated shape + A1 + A2 , (9) adaptations produce the center of mass displacement in the direc- 2 β1 β2 tion indicated by the broader arrows.

Rev. Mex. Fis. 63 (2017) 314-327 STOCHASTIC DYNAMICS OF A BROWNIAN MOTOR BASED ON MORPHOLOGICAL CHANGES 317 µ ¶ dxr 1 1 Brownian motion even when it does not have a strong geo- = −k + (xr − l) dt β1 β2 metrical asymmetry. ³ ´ In addition, when configuration changes occur at finite 1 ˙ ˙ 1 1 + 2 V1 + V2 + A2 − A1. (10) times, R1 and R2 will spend time far from their limiting val- 4πxr β2 β1 ues Rmin and Rmax. The departure of R1 and R2 from Rmin Here, V˙1 and V˙2 represent the rates of change of the vol- and Rmax, will depend on τR, as well as on the characteristic umes of spheres 1 and 2, respectively. The terms containing time for which the conditions in Eq. (8) subsist, τ¯. Notice these quantities in Eqs. (9) and (10) represent the influence that, physically, τ¯ is determined by the strength of the elastic that the expanding and contracting spheres have on each other interaction and the viscosity of the fluid, while τR is a pa- through the fluid that surrounds them. When those terms are rameter that depends on the motor’s structure and that, con- neglected, Eqs. (9) and (10) reduce to Eqs. (1) and (2). More- sequently, can be theorized. In the present paper, we will an- over, if morphological changes were such that the total vol- alyze the case of motors that adapt their morphology slowly, ume of the motor is conserved, then V˙1 = −V˙2, and Eq. (9) in such a way that τ¯ ¿ τR. Thus, the time-scale for mor- would take a form resembling Eq. (1.1) in Ref. 31, but sup- phological adaptations can be considered as a fast scale, and plemented with the contribution of the stochastic forces on Eqs. (9) and (10) can be simplified by averaging over these the spheres. Another important attribute of Eqs. (9) and (10) fast processes. In this situation and assuming that the system is that they are non-linear, in contrast with Eqs. (1) and (2), starts in a condition such that xr > l, we have as a consequence of the hydrodynamic interaction. Further- ˙ more, notice that in Eqs. (9) and (10) βµ and Vµ, depend on Z∞ time implicitly through their dependence on Rµ (t). R2 (t) − R1 (t) ' dτ P (τ)[R2 (τ) − R1 (τ)] It is worth stressing that during the operation of efficient Brownian motors, detailed balance symmetry, which governs 0 the dynamics around equilibrium, must be broken by forcing ' (−1)p ∆R (1 − 2ζτ¯) , (12) these motors to operate away from thermal equilibrium [7]. In the case of our motor based on morphological changes, and departure from equilibrium is promoted by the adaptations in Z∞ h i shape themselves. Put differently, if such adaptations were V˙2 (t) − V˙1 (t) ' dτ P (τ) V˙2 (τ) − V˙1 (τ) not performed, the coefficients β and β in Eqs. (9) and (10) 1 2 0 would take constant values, while V˙1 and V˙2 would vanish. ' (−1)p 8πζ∆RR2 , (13) In that case, xr and xc.m. would relax from their initial val- min ues towards thermodynamic equilibrium where all dynamics where P (τ) is the probability distribution function (PDF) for would be governed by detailed balance. There, as it can be in- observing that the conditions in Eq. (8) last a time interval τ, ferred from Eqs. (4) and (5), no net average displacement of to be specified at the end of this section. In order to obtain x would occur. In contrast, morphological changes take c.m. the previous results, we have used the explicit form of R the motor away from equilibrium persistently thus braking 1 and R , given by Eq. (8), neglected quadratic and higher or- detailed balance. They also provide the combination of asym- 2 der contributions in ∆R and ζτ¯, and assumed that R and R metry and non-equilibrium forces, necessary for the produc- 1 2 remain close to their limiting values R and R . tion of constructive transport. min max Another simplifying assumption will consist in consider- The appearance of the stochastic forces in Eqs. (9) and ing that the thermal fluctuations in x are small as compared (10) deserves a special comment, since when R and R r 1 2 with the total elongation of the system. In other words, if are time-dependent, the system is not in equilibrium and the x˜ (t) ≡ x (t) − l, represents the spontaneous change in the usual fluctuation-dissipation theorem, Eq. (3), is not suitable. r r length of the motor, then we will suppose that x˜ /l ¿ 1. We will assume, that R and R exhibit significant variations r 1 2 By expanding Eqs. (9) and (10) up to the first order in the over a time-scale that is large compared with the time-scale previously defined small quantities, it can be verified that in of thermal noise. Accordingly, it could be expected that the the interval t ≤ t ≤ t , they reduce to fluctuation-dissipation relation will be valid instantaneously, p p+1 and we propose to write it in the form dxc.m. p ¿ À = (−1) η˜1∆R (1 − 2ζτ¯)x ˜r 0 dt Aµ (t ) Aν (t) 2kBT 0 = δ (t − t) δµν . (11) µ ¶ β (t0) β (t) β (t) 1 A1 A2 µ ν µ − (−1)pε2ζ∆R + + , (14) 2 β β In this paper we will restrict ourselves to analyze the dy- 1 2 namics resulting from Eqs. (9) and (10) in the limiting case and of small morphological changes, determined by the condition dx˜r A2 A1 = −α1x˜r + − , (15) ∆R/Rmin ≡ ∆ ¿ 1. Apart from being very convenient from dt β2 β1 the mathematical point of view, this case could be interesting respectively, where we have introduced the definitions 2 since it will exhibit that our motor is able to perform directed η˜1=k/12πηRmin, ε=Rmin/l, and α1 = k (2 − ∆) /6πηRmin.

Rev. Mex. Fis. 63 (2017) 314-327 318 F. AMBIA´ AND H. HIJAR´

First, we notice that within the range of applicability of and the performed approximations, x˜ follows the usual stochas- r 1 ∆ ¡ ¢ tic process for a harmonically bound Brownian particle in Ψ (t; α ) = + (−1)p (1 − 2ζτ¯) 1 − e−α1t . (20) 2 1 2 4 an equilibrium environment in the overdamped regime [34]. 0 0 Therefore, the PDF for observing x˜r (t ) at time t , given that From Eqs. (18)-(20) we find the PDF for observing 0 0 0 it was found to be x˜r (t) at time t is xc.m. (t ) at time t , given the initial conditions xc.m. (t ) and xr (t), which is 0 1 W (˜xr (t ) |x˜r (t)) = p 2 0 2πσr (t − t) 0 1 W (xc.m. (t ) |xc.m. (t); xr (t)) = √  h i  2 0 2 2πσ  0 −α1(t −t)   x˜r (t ) − l − e x˜r (t)  ( ) 0 2 × exp − 2 0 , (16) [xc.m. (t ) − xc.m. (t) − h |xr − l|]  2σr (t − t)  × exp − , (21) 2σ2 ³ ´ 0 2 0 −2α1(t −t) 0 with σr (t − t) = kBT 1 − e /k. In the asymp- where h is a function of τ¯ and t − t, defined as h = 0 totic limit, this distribution reduces to the stationary distribu- h(¯τ; t0 − t) = (1 − 2ζτ¯)(1 − e−α(t −t))∆/4. tion In order to give an approximated description of the r ( ) 2 stochastic process followed by xc.m. over a long time inter- k k (xr − l) W (xr) = exp − . (17) val, [0, t], we subdivide it into N smaller regular intervals of 2πkBT 2kBT size τ¯ = t/N, and identify the intermediate times as tq = qτ¯,

Notice that the fluctuations of x˜r according to this distri- with q = 0, 1,...,N. We assume that we perform N obser- bution are responsible for promoting the advance of the motor vations of the state of the system at times t0, t1, . . . , tN−1, at when they are coupled with morphological changes though which |xr − l| takes values sampled from its equilibrium dis- Eq. (14). For the sake of quantifying this effect, we will con- tribution, Eq. (17). Then, we calculate the conditional prob- sider the evolution of xc.m. over a long period of time. ability for observing the motor at position xc.m. (tq+1) before It should be remarked, that under such circumstances, the beginning of the next interval, given the initial center of the hydrodynamic forces do not produce a net displacement mass position at time tq, of xc.m., because they are symmetric and act along oppo- Z site directions for the two possible configurations of the W (xc.m. (tq+1) |xc.m. (tq)) = dxr motor (elongated or contracted). More precisely, hydrody- namic forces always point in the direction of the contracting × W (xc.m. (tq+1) |xc.m. (tq); xr (tq)) W (xr (tq)) . (22) sphere [31]. Now in this simplyfied version of Eq. (14), over a long evolution time, equal number of symmetric contrac- By using Eqs. (17), and (18)-(22) we find that in the limit tions and expansions contribute with hydrodynamic forces ∆ ¿ 1, this probability can be approximated by a Gaussian and the average of these forces will vanish. This will not that explicitly reads as be the case in the extended model for a molecular Brownian 1 motor to be considered in Sec. 4. By the moment, in favor W (xc.m. (tq+1) |xc.m. (tq)) = p 2 of simplifying the subsequent analysis, the term in Eq. (14) 2πσc.m. (¯τ) containing the factor ε2 will be neglected. In this case, the ( ) [x (t ) − x (t ) − χ (¯τ)]2 formal solution of Eq. (14) is, up to first order in the small c.m. q+1 c.m. q × exp − 2 , (23) quantity ∆, 2σc.m. (¯τ) µ ¶ 0 p ∆R where µ ¶ xc.m. (t ) = xc.m. (t) + (−1) (1 − 2ζτ¯) 1 + k T ∆ 2Rmin 2 B σc.m. (¯τ) = 1 − τ,¯ (24) ³ ´ 6πηRmin 2 ∆ 0 × 1 − e−α1(t −t) |x (t) − l| 4 r and r 2 kBT Zt0 " χ (¯τ) = h (¯τ, τ¯) . (25) 0 A1 (ξ) π k + dξ Ψ1 (t − ξ; α1) β1 (ξ) Finally, we assume that process described by xc.m. is of t # the Markov type and using the Chapman-Kolmogorov equa- tion, 0 A2 (ξ) + Ψ2 (t − ξ; α1) , (18) β2 (ξ) Z NY−1 W (x (t) |x (0)) = dx dx ··· dx where c.m. c.m. N−1 N−2 1 q=0 1 ∆ Ψ (t; α ) = − (−1)p(1 − 2ζτ¯) (1 − e−α1t), (19) 1 1 2 4 × W (xc.m. (tq+1) |xc.m. (tq)) , (26)

Rev. Mex. Fis. 63 (2017) 314-327 STOCHASTIC DYNAMICS OF A BROWNIAN MOTOR BASED ON MORPHOLOGICAL CHANGES 319 we find the conditional PDF to observe the motor at a posi- This result is valid only for τ > τmin, where τmin is a tion xc.m. (t) at the end of the process, given that it was at quantity introduced to guarantee that P (τ) is correctly nor- xc.m. (0) at the beginning. Such PDF is malized. Physically, it could be expected that such a mini- mum time for the validity of Eq. (31) must exist, since the 1 W (xc.m. (t) |xc.m. (0)) = p analysis that conducted to this result is valid only in a time- 2 2πσc.m. (t) scale that is larger than the one corresponding to the rapidly ( ) fluctuating force of the solvent [36]. [x (t) − x (0) − Nχ (¯τ)]2 c.m. c.m. It is well known that for the power law distribution func- × exp − 2 , (27) 2σc.m. (t) tion given by Eq. (31), it is not possible to find an analytical expression for its mean value [37]. Therefore, we will esti- It will be discussed in the subsequent Sec. 3 that, in gen- −1 mate τ¯ following a numerical procedure based on the results eral, the relaxation time α1 takes values that are much larger of BD simulations. To be specific, we will show in the subse- than τ¯ and, consequently, the approximation α1τ¯ ¿ 1 is quent section 3. that the PDF for τ can be very well adjusted valid, from which we can write Nχ (¯τ) = vc.m.t, that defines by Eq. (31), and use the time series of values of xr to deter- the average velocity of the motor mine both, τmin and τ¯. √ 1 − 2ζτ¯ ∆R k k T v = √ B . (28) c.m. 3/2 2 6 2π Rmin η 3. Brownian Dynamics Simulations In order to finish the analysis of the present model, With the purpose of analyzing the validity of the model de- we will discuss how the characteristic time τ¯, appearing in scribed in Sec. 2, we implemented a BD simulation scheme Eqs. (27) and (28), can be estimated. It can be seen from that allowed us to solve Eqs. (9) and (10) numerically. Eq. (8), that τ¯ can be interpreted as the average of the time Our implementation consisted of a temporal discretization elapsed between two successive observations of the value method in which we fixed Rmin = 1, m = 1, and kBT = 1, xr = l. Under the approximations leading to Eq. (17), τ¯ as the units of length, mass, and , respectively. No- is also the mean time needed by a harmonically constrained tice that under such conditions, unitsp of time, ut, are not in- Brownian particle for consecutively crossing over the poten- dependent but given by ut = Rmin m/kBT . The numeri- tial’s minimum. According to general theory for first exit cal integration of Eqs. (9) and (10) was carried out using a −3 times in harmonic potentials [35], the PDF for observing two time-step with size ∆t = 10 ut. In the numerical scheme, consecutive values xr = l, occurring in a time interval τ is stochastic forces were sampled from a Gaussian distribution given by and actualized every two simulation steps. We performed nu- ∂S (τ) merical experiments varying R from R = 1.0 R , P (τ) = − , (29) max max min ∂τ to 1.25 Rmin, with the purpose of showing that the coupled where S (τ) is the survival probability, i.e., the probability Brownian system depicted in Fig. 1 acquires directed mo- for observing the confined particle at xr = l, at time t = 0, tion even for small configuration changes. On the other hand, and at the same position at a subsequent time t = τ, un- the values of the restitution coefficient and the viscosity were 2 √ 2 der the condition that it has not passed over this point at any fixed at k = 5 kBT/Rmin, and η = 5 mKBT /Rmin, respec- intermediate time. If we restrict ourselves to consider short tively. Notice that this selection of parameters gives charac- −1 times τ, i.e., times satisfying the condition α1τ ¿ 1, the teristic relaxation times α1 ∼ 10ut À ∆t, which guarantee function S (τ) can be obtained from the transition probabil- that the usual separation of time-scales of Brownian motion 0 ity W (˜xr (t ) |x˜r (t)), as it is given by Eq. (16). Specifically, is properly approximated. In addition, the characteristic time we have for radii adaptations was chosen to be τR = 2ut À ∆t. Let us show first, that directed Brownian motion indeed Z∞ appears in motors simulated under such conditions. This is 0 0 0 S (τ) = dxrW (xr (τ) = l |xr (τ1)) W (xr(τ1)|xr (0) = l) done in Fig. 2, in which we present six trajectories for Brow- 0 nian motors with morphological adaptations with different h ³ ´i−1/2 magnitude, reported in terms of the dimensionless parameter 2 −2α1(τ−τ1) 2 = 2π σr (τ − τ1) +e σr (τ1) , (30) ∆. The noisy curves in Fig. 2 represent the numerical re- sults. They prove that morphological changes in fact induce where 0 ≤ τ1 ≤ τ. By expanding Eq. (30) in terms of the directed motion. small quantity α1τ, and retaining only the leading contribu- On the other hand, the straight lines in Fig. 2 were drawn 1/2 tion, we find that S (τ) = (k/4πkBT α1τ) , and conse- using the average velocity, vc.m., calculated from Eq. (28). It quently, P (τ) turns out to be is worth noticing that in order to obtain an estimate of vc.m., it is necessary to provide the average time for configurational 1 P (τ) = τ 1/2τ −3/2, (31) changes, τ¯. As it was explained at the end of Sec. 2, τ¯ was 2 min calculated using the numerical results. More precisely, dur- which is a special case of the so-called Pareto distribution. ing the simulations that yielded the results depicted in Fig. 2,

Rev. Mex. Fis. 63 (2017) 314-327 320 F. AMBIA´ AND H. HIJAR´

FIGURE 3. Probability distribution function for the duration of FIGURE 2. Directed Brownian motion experienced by simulated morphological configurations, P (τ). Noisy black curves repre- motors with different asymmetry degrees represented by the pa- sent the results of numerical experiments for the indicated values rameter ∆. Straight lines correspond with the theoretical velocity of ∆, while the red straight lines are fits obtained from the power calculated from Eq. (28), while noise curves are the actual trajecto- law P (τ) ∝ τ −3/2. The numerical distributions are used to esti- ries of the motors. mate τ¯. we stored the values of the time intervals of elongated or con- tracted configuration, τ, from which we calculated numeri- TABLE I. Average time τ¯ obtained from numerical experiments of motors with different morphological asymmetry factors ∆. The es- cally the PDF P (τ). For the experiments conducted over the timation of the mean center of mass velocity, vc.m., obtained from time interval shown in Fig. 2, the available data for calculat- Eq. (28) is also presented. ing P (τ) ranged from 818, 342 for ∆ = 0.00, to 776, 089 for −4 the case ∆ = 0.25. We found that P (τ) actually follows the ∆τ ¯ (±0.002ut) vc.m. (10 ut/Rmin) power law behavior predicted in Eq. (31). This fact is illus- 0.00 0.306 0.0 trated in Fig. 3 in which we present the numerical evaluation 0.05 0.305 3.29 ± 0.01 of P (τ) obtained for the simulation cases ∆ = 0.00, 0.05 0.10 0.315 6.58 ± 0.02 and 0.25 (black noisy curves). 0.15 0.317 9.69 ± 0.03 The straight lines shown in Fig. 3 correspond to a non- 0.20 0.319 12.90 ± 0.04 linear fitting of the numerical data in which Eq. (31) was 0.25 0.322 16.04 ± 0.05 used by considering τmin as an adjustable parameter. The using the estimations of vc.m. presented in Table I. In addition, specific values of τmin were found to be independent on ∆, within the range of the numerical uncertainties, e.g., they we calculated the standard deviation σc.m. of the distributions were (1.85 ± 0.03) × 10−3, (1.84 ± 0.03) × 10−3, and observed for the ensembles as function of the simulation time −3 and compare the numerical results with the expected behav- (1.86±0.03)×10 ut, respectively, for the three cases illus- trated in Fig. 3. On the other hand, we summarize in Table I ior as given by Eq. (24). Our results are summarized in Fig. 4, the set of numerical values of τ¯ obtained from the numerical where a very good agreement between the numerical and ex- distributions P (τ), where it can be appreciated that, for the perimental results can be observed. selected range of simulation parameters, τ¯ increases with ∆ but exhibits small percental variations lower than 5%. The 4. Extended model for a molecular Brownian values of τ¯ shown in table I were used to calculate vc.m. from Eq. (28). The resulting values are shown in the third column motor of Table I. These were used to plot the straight lines appear- In Sec. 3, we considered a numerical method that allowed ing in Fig. 2. us to solve the system of stochastic Eqs. (9) and (10), and Finally, we will show that the stochastic process describ- to verify the predictions of the theoretical model introduced ing the spatial advance of an set of simulated motors ex- in Sec. 2. Here, we will extend our analysis to make it hibits the properties predicted in Sec. 2. With this purpose, able to describe the Brownian dynamics of a microscopic we conducted additional simulations for two ensembles of system which is more realistic from the physical point of 104 independent motors with characteristic size asymmetries view. Specifically, we will consider a model consisting of ∆ = 0.05, and ∆ = 0.25. For such ensembles, we calcu- two spherical colloidal particles with radii R1 (t) and R2 (t), lated, at different simulation times, the PDF for the center attached by a molecular chain with finite extension, lmax. We of mass position, W (xc.m. (t) | xc.m. (0) = 0), and compared will suppose that the elastic potential energy stored in the the results with those expected from Eqs. (24), (27) and (28), system when these particles are located at positions x1 and

Rev. Mex. Fis. 63 (2017) 314-327 STOCHASTIC DYNAMICS OF A BROWNIAN MOTOR BASED ON MORPHOLOGICAL CHANGES 321 x2 > x1, is given by the finite extension non-linear elastic We will assume that the colloidal particles do not in- formula teract at large distances but strongly repel each other when " # µ ¶2 they get in contact. Mathematically, such interaction can be 1 2 x2 − x1 Φe (x2 − x1) = − k l ln 1 + . (32) represented in terms of the Weeks-Chandler-Andersen poten- 2 max l max tial [38],

 · ¸ ³ ´12n ³ ´6n  σ σ 1 1/6n 4² − + , if x2 − x1 < 2 σ; Φ (x − x ) = x2−x1 x2−x1 4 (33) c 2 1  0, otherwise; where ² denotes the interaction strength, σ is the diameter of the interaction, and n is a positive integer that determines the An estimation of the center of mass velocity of the molec- hardness of the colloidal particles. Notice that when R1 (t) ular motors defined by Eqs. (8) and (32)-(33) can be obtained and R2 (t) change in time symmetrically, as it is the case es- by generalizing the theory presented in Sec. 2. The main dif- tablished by the exponential algorithm Eq. (8), σ can be con- ferences to be taken into account with respect to our previ- sidered to be a constant with σ = Rmax + Rmin. We consid- ous situation are that, in the presence of the potentials Φe ered that the radii R1 and R2 tend towards Rmin or Rmax expo- and Φc, the elongation of the system is distributed over an nentially as prescribed by Eq. (8), with a fixed characteristic asymmetric probability function around hxri, and that the time τR. Notice that in Eq. (8) the parameter l indicated the average time intervals that the motor spends in the contracted length at which the morphological changes of the harmonic (xr < hxri) or elongated (xr > hxri) configurations will be motor were activated. For the case of a motor that operates different. These facts have two major consequences in the under the internal potentials given by Eqs. (32) and (33), this dynamics of the motor. First, the morphological changes, parameter is absent. Therefore, we will consider that for this R2 (t) − R1 (t), have not the same average magnitude dur- case a particular elongation exists, l0, that signals the config- ing the time lapses for contraction and elongation. Second, uration of the motor at which structural adaptations are pro- the hydrodynamic forces occurring in such time intervals are moted. no longer balanced, as it happens in the symmetric case, and

We observed that the selection of l0 affects the efficiency they have a contribution to the center of mass velocity. of the motor, since when it is selected to be far from the av- In order to quantify these effects, we will introduce a se- erage elongation, hxri, a smaller number of morphological ries of practical simplifications that are analogous to those changes takes place and the velocity of the motor is reduced. leading to Eqs. (27) and (28). First, we will assume that the Accordingly, we propose to set l0 ' hxri. instantaneous morphological differences of the motor can be approximated by their average values. Using the same ap- proximations that yielded Eqs. (12) and (13), we find in the present case ¯ Z∞ ¯ ¯ [R2 (t) − R1 (t)]¯ ' dτP± (τ)[R2 (τ) − R1 (τ)] τ ± 0

' ±∆R (1 − 2ζτ¯∓) , (34) and ¯ Z∞ h i¯ h i ˙ ˙ ¯ ˙ ˙ V2 (t) − V1 (t) ¯ ' dτP± (τ) V2 (τ) − V1 (τ) τ ± 0 2 2 ' ±8πRminζ ∆Rτ¯∓, (35)

where P+ (τ) and P− (τ) represent, respectively, the PDF’s for observing the elongated and contracted configurations ex- tending over a period of time τ, Z FIGURE 4. Probability distribution function for the center of mass position of an ensemble of harmonic Brownian motors at different τ¯± = dτ τP± (τ) , times. Symbols correspond with the numerical results, while con- tinuous lines were obtained from Eqs. (24), (27), and (28). The are the corresponding mean times, and |τ± is used to indicate standard deviation of the distribution as function of the advance the evolution of a time-dependent function over the periods time is also shown. of contracted (τ−) or elongated (τ+) configuration.

Rev. Mex. Fis. 63 (2017) 314-327 322 F. AMBIA´ AND H. HIJAR´

As it was the case before, we will assume that the elon- gation of the system does not possess strong fluctuations. In this case, the contribution of the hydrodynamic forces to the dynamics of xc.m. can be approximated by

¯ h i¯ ¯ V˙ − V˙ ¯ ˙ ˙ ¯ 2 1 ¯ V2 (t) − V1 (t)¯ τ± − 2 ¯ ' − 2 8πxr (t) ¯ 8πhxr i± τ± 2 2 Rminζ ∆Rτ¯∓ ' ∓ 2 ≡ λ±. (36) hxr i±

FIGURE 5. Normalized correlations for the extension of Brownian Another restriction to be incorporated will consist in ana- motors simulated with different asymmetry degrees, ∆, (symbols). lyzing motors with strong repulsive interactions only, which The continuous curve represents a numerical fit obtained assuming could be realized using large values of n in Eq. (33), i.e. an exponential decay. n > 1 [39]. In this case, most of the time the colloidal spheres are separated and the dynamics is dominated by the nonlin- Now, let W+ (xr) and W− (xr) represent the partial sta- ear elastic potential. It could be expected that, in this limit, tionary PDF’s for finding xr at xr > hxri and xr < hxri, and x will evolve by following a similar relaxation behavior to 2 r hxr i± the mean square elongation of the motor calculated the one exhibited in the simple harmonic case. However, the over such distributions, i.e., Z effective characteristic relaxation time for elongated and con- 2 2 tracted configurations, α±, could be anticipated to be slightly hx i± = dxr x W± (xr) . r r different. Under such approximation and taking into account Eq. (36), an estimate for the formal solution of xc.m. is ³ ´ ∆ 0 x (t0) ' x (t) ± (1 − 2ζτ¯ ) 1 − e−α±(t −t) [x (t) − hx i ] + λ (t0 − t) c.m. c.m. ∓ 4 r r ± ± Zt0 · ¸ 0 A1 (ξ) 0 A2 (ξ) + dξ Ψ1 (t − ξ; αeff) + Ψ2 (t − ξ; αeff) , (37) β1 (ξ) β2 (ξ) t where we have introduced the effective relaxation parame- ter, αeff, under the assumption that α+ and α− are propor- tional to the corresponding inverse average times τ¯ and τ¯ , + − ∆ in such a way that α+ = αeff (¯τ+ +τ ¯−) /τ¯+, and α− = v = α [(1 − 2ζτ )x ˜ − (1 − 2ζτ )x ˜ ] c.m. eff 4 − r,+ + r,− αeff (¯τ+ +τ ¯−) /τ¯−. µ ¶ R2 ζ2∆Rτ¯ τ¯ 1 1 + min − + − . (39) If we assume, as it was done in Sec. 2.2, that we ob- τ¯ +τ ¯ hx2i hx2i serve the dynamics of the motor over regular time intervals − + r + r − at which xr is sampled from the stationary PDF’s W± (xr), In the previous expression, we have defined the averages that an expansion in terms of small quantities ∆ and α±τ± is x˜r,± as valid, and that xc.m. is a Markov process, we arrive at the fol- lowing expression for the PDF for the advance of the motor Z±∞ ∓1 operating over asymmetric morphological changes x˜r,± = (−1) dxrW± (xr)[xr − hxri] . (40)

hxri

1 In Eq. (39), the first term on the right hand side is the ve- W (xc.m. (t) |xc.m. (0)) = p 2 2πσc.m. (t) locity achieved by the motor due to the unbalanced viscous ( ) 2 drag and elastic effects, while the second term represents the [xc.m. (t) − xc.m. (0) − vc.m. t] contribution of the asymmetric hydrodynamic forces experi- × exp − 2 , (38) 2σc.m. (t) enced by the spheres over the time intervals τ¯+ and τ¯−. Equa- tion (39) reduces to Eq. (28) for symmetric morphological adaptations, i.e., in the case τ¯+ =τ ¯−, x˜r,+ = −x˜r,−, and 2 2 where the average center of mass velocity is given by hxr i+ = hxr i−.

Rev. Mex. Fis. 63 (2017) 314-327 STOCHASTIC DYNAMICS OF A BROWNIAN MOTOR BASED ON MORPHOLOGICAL CHANGES 323

TABLE II. Parameter l0 that indicates the length at which struc- tural adaptations are activated in simulated motors. Values were obtained from Eq. (41), and are expected to be close to hxri.

∆ l0 (Rmin) 0.00 2.3732 0.05 2.4176 0.10 2.4621 0.15 2.5067 0.20 2.5515 0.25 2.5964

We conducted a series of numerical experiments in order to assess the validity of Eq. (39). We used the same numer- ical integration method described in Sec. 3, where the sys- FIGURE 6. Average motion of Brownian motors based on morpho- tematic forces acting the Brownian particles were obtained logical changes and asymmetric molecular potentials. Noisy con- tinuous curves represent the average displacement of an ensemble from Fi = −∂ (Φe + Φc) /∂xi. Our new simulations were on 103 independent motors. Straight interrupted lines are theoreti- carried out with parameters√ ² = 2kBT , lmax = 10Rmin, 2 2 cal estimations obtained from Eq. (39). k = 1 kBT/Rmin, η = 5 mkBT /Rmin, and n = 4, which makes the interaction between the colloids steep [39]. We simulated motors with the same asymmetry factors used be- tained from a nonlinear fitting procedure. Our numerical fore, ∆ = 0.00, 0.05,..., 0.25. With the purpose of giving evidence revealed that α does not change systematically an approximated value for the parameter l , we considered eff 0 with the simulation parameter ∆, e.g., it took the values that the elongation of the motor is restricted to small values, 0.1384 u−1 for ∆ = 0.00, and 0.1386 u−1 for ∆ = 0.25. x ¿ l . Thus, we assumed Φ to be harmonic. In addition, t t r max e Thus, we decided to use the average value of this inverse Φ was assumed to represent a hard wall that restricts x to c r characteristic time, calculated over the six performed simu- be in the region x > σ. This allowed us to calculate l from r. 0 lations, which turned out to be α = (0.138 ± 0.008) u−1. the canonical distribution, using the approximated potential eff t , as

∞ n o In our numerical experiments we verified that the molec- R kx2 dx x exp − r ular motors based on the use of molecular potentials exhibit r r 2kB T σ directed motion at constant average speeds. This is illus- l0 = ∞ n o R kx2 dx exp − r trated in Fig. 6 where we present the resulting average tra- r 2kB T σ jectories taken over ensembles of 103 molecular motors. In n o r 2 the same Fig. 6, the straight interrupted lines were drawn us- exp − kσ 2kBT 2kB T ing the velocities estimated from Eq. (39) and the parame- = nq o. (41) 2 πk 1 − erf kσ2 ters ∆xr,±, hxr i± and τ¯±, presented in Table III, as well as 2kB T −1 αeff = 0.138 ut . Our estimations of vc.m. are presented in Table IV where they are compared with the values resulting The corresponding values for l used in the adaptation algo- 0 from the simulation experiments, which were obtained from rithm, Eq. (8), were obtained from Eq. (41) and are explicitly a simple application of the least squares method to the noisy shown in Table II. curves in Fig. 6. We found a very good agreement between Firstly, we obtained the time series of x (t) for six in- r the calculations based on Eq. (39) and the observed average dividual motors simulated over a time interval with size speed of the simulated motors. It can be noticed from the 5 × 105 u . From these time series we calculated the parame- t data in Table IV that velocities derived from Eq. (39) coin- ters ∆x , hx2i , and τ¯ , appearing in Eq. (39). The result- r,± r ± ± cide with the experimental velocities, within the uncertainty ing values of these quantities are summarized in Table III. On ranges, for all the used parameters ∆, although deviations the other hand, α was obtained from the examination of the eff between the central values can be recognized, specially at decay of the autocorrelation functions of the variable x (t), r ∆ = 0.05 and ∆ = 0.25. Nevertheless, such deviations 6% h(xr (t) − hxri)(xr (0) − hxri)i are lower than , and could be considered small. In this g (t) = , case our results would allow us to propose Eq. (39) as a suffi- h(x (0) − hx i)2i r r ciently good expression for making estimations of the speed which can be well approximated by a decaying exponen- of molecular motors operating at asymmetric small morpho- tial function, whose inverse characteristic time, αeff, was ob- logical changes, ∆ ≤ 0.25.

Rev. Mex. Fis. 63 (2017) 314-327 324 F. AMBIA´ AND H. HIJAR´

TABLE III. Parameters used to estimate vc.m. from Eq. (39). These quantities were obtained from simulations of individual motors as it is −4 described through the text. Values are reported in simulation units. Statistical uncertainties were found to be δ∆xr,+ = ±3 × 10 Rmin, −4 2 −4 2 2 −4 2 δ∆xr,− = ±2 × 10 Rmin, δhxr i+ = ±2 × 10 Rmin, and δhxr i− = ±1 × 10 Rmin. 2 2 ∆ ∆xr,+ ∆xr,− hxr i+ hxr i− τ¯+ τ¯− 0.00 0.3035 0.1691 7.1774 4.8097 0.187 0.303 0.05 0.2983 0.1614 7.3542 5.0174 0.187 0.310 0.10 0.2929 0.1567 7.5584 5.2364 0.183 0.303 0.15 0.2990 0.1577 7.8940 5.4766 0.183 0.306 0.20 0.2895 0.1571 8.0919 5.7002 0.177 0.307 0.25 0.2827 0.1498 8.2682 5.9270 0.181 0.303

the spatial asymmetry of the ratchet, in such a way that for TABLE IV. Center of mass velocity of molecular motors with dif- arat < Lrat/2, it will rectify the Brownian motion to the pos- ferent morphological asymmetries. Theoretical estimations were itive direction, while for arat > Lrat/2, the mean stream in- carried out using Eq. (39) and the values of the parameters pre- duced by the ratchet will be negative. Finally, the parameter sented in Table III. trat is used to indicate the time interval in which the ratchet

∆ vc.m. (simulations, vc.m. (estimated, remains turned off. This classical operation of the ratchet −4 −4 mechanism is illustrated in Fig. 7. 10 Rmin/ut) 10 Rmin/ut) 0.00 0 0.00 A first group of simulations was conducted in molecular 0.05 5.200 ± 0.003 4.9 ± 0.3 motors that do not perform morphological changes, this with 0.10 10.410 ± 0.004 9.8 ± 0.7 the purpose of assessing the flow velocity induced by the 0.15 14.690 ± 0.003 15.0 ± 1.0 0.20 19.298 ± 0.004 19.9 ± 1.3 0.25 22.711 ± 0.003 24.0 ± 1.6

5. Competition against a ratchet potential

Finally, we carried out a comparison between the rectifying mechanism based on morphological changes and a classical ratchet potential, in order to show that the former could ex- hibit larger velocities, even for small values of ∆. We per- formed the comparison in such a way that our motor will try to overcome the backwards stream of particles created by the ratchet potential. This means that in order to overcome the ratchet’s stream, our Brownian motor needs to create a larger net displacement in the opposite direction. The ratchet potential acting on each of the constituent parts of the , V (xµ; t) = Φ (t) V (xµ), will be modeled as spatially asymmetric and intermittent. Specif- ically, it will be defined as in Ref. 46 through ( hrat a xµ, if , 0 < xµ < arat; V (xµ) = rat (42) FIGURE 7. The ratchet potential, V (xµ; t), used for competi- hrat (Lrat − xµ) , if arat < xµ < Lrat; Lrat−arat tion with morphological changes. It has two possible states, on and off, determined by the parameters trat and Trat in Eq. (43). In and ( the on-state, particles are driven to the minimum energy position, 0, if 0 < t < t ; Φ(t) = rat (43) while particle diffuse freely in the off-state. This mechanism to- 1, if trat < t < Trat. gether with the spatial asymmetry of V controlled by arat and Lrat in Eq. (42), induce the rectified motion, which for the presented In Eqs. (42) and (43), hrat represents the maximum en- case points to the left. Smooth curves above the x-axis illustrate ergy of the ratchet, while Lrat and Trat are, respectively, its schematically the PDF for finding the particles at given positions at spatial and temporal periods. The parameter arat controls different times.

Rev. Mex. Fis. 63 (2017) 314-327 STOCHASTIC DYNAMICS OF A BROWNIAN MOTOR BASED ON MORPHOLOGICAL CHANGES 325

FIGURE 8. Drift velocity induced by the ratchet mechanism with FIGURE 9. Average trajectories of molecular motors under external parameters hrat = 10kB T , Lrat = 10Rmin, arat = 9Rmin, and ratchet forces and antagonistic morphological adaptations charac- Trat = 100ut, for different values of trat. terized by the asymmetry parameter ∆. ratchet mechanism. In these simulations the ratchet’s energy barrier was chosen to be hrat = 10kBT , which is highly un- likely to be overcome by particles in a thermal bath at temper- 6. Conclusions ature T . On the other hand, the spatial parameters were fixed We have analyzed the dynamics of a simple intuitive Brow- at the values L = 10R , and a = 9R . The time pe- rat min rat min nian motor that acquires directed motion from thermal un- riod was selected to be T = 100u , for which we performed rat t biased forces by coordinated morphological modifications. ten different experiments varying t from 0 to 100u in reg- rat t This motor needs to be elastic and to possess adaptable shape. ular intervals. The resulting stream velocities were obtained Thermal fluctuations of the surrounding medium promote from averages over ensembles of 104 inactivated motors. changes in the elongation of the motor and the information of They are shown in Fig. 8 from which it can be observed that the length of the system is used by a Maxwell’s demon con- the drift velocity is minimized for t ' 60u . Thus, in order rat t trol system to indicate that morphological adaptations must to perform a comparison against the motor based on morpho- be activated. In turn, the structural modifications of the mo- logical changes, we selected this latter value for all our subse- tor induce unbalanced drag forces that favor the displacement quent experiments. Under such conditions, we activated the of the motor’s center of mass in one direction. morphological changes in molecular motors with repulsive We have used the classical theory of Brownian motion and non-linear interactions, simulated with the same set of to give an expression for the mean velocity reached by a parameters described in Sec. 4. As it was the case before, we motor with simple harmonic restitution forces that performs varied the asymmetry ratio ∆ from 0.00 to 0.25 with incre- slow morphological adaptations. We expressed this velocity ments of 0.05. Figure 9 illustrates the mean trajectories ob- as function of the magnitude of its morphological changes. tained from averages of 103 motors. It can be noticed that for Our analysis was based on the assumption that these changes no morphological adaptations, ∆ = 0.00, the center of mass are small when compared with the length of the motor it- position moves at a constant speed to the left. However, v c.m. self. This assumption turned out to reduce enormously the increases as ∆ does. Morphological changes finally over- mathematical complexity of the problem and allowed us to come the ratchet mechanism for ∆ = 0.25. The specific arrive at closed expressions for the probability distribution velocities obtained in the experiments described above are functions describing the state of the motor. We verified the shown in Table V. applicability of our theoretical treatment using BD simula- tions. Afterwards, we conducted simulations of more real- TABLE V. Center of mass velocity of molecular motors with differ- istic molecular motors conceived as two colloidal particles ent morphological asymmetries in the presence of a ratchet poten- of controllable size attached by a finite extension chain. We tial. showed by means of BD simulations that they also exhibit di- −4 ∆ vc.m. (simulations, 10 Rmin/ut) rected Brownian motion. In the same manner, we were able 0.00 −5.074000 to give an expression for the velocity of the molecular motors that works very well for small morphological changes. Fi- 0.05 −2.699700 nally, we showed that morphological adaptations could pro- 0.10 −1.812000 duce larger velocities than those obtained from the classical 0.15 −0.471480 rectifying mechanism based on an external ratchet potential. 0.20 −0.027426 It is interesting to perform an estimation of the veloc- 0.25 +1.119100 ity that could be achieved by an actual motor possessing

Rev. Mex. Fis. 63 (2017) 314-327 326 F. AMBIA´ AND H. HIJAR´ the geometrical characteristics used in our model. For this tional changes that are induced in artificial molecular ma- purpose we could consider a molecular motor such that chines by diverse stimuli, e.g., , electrochemistry, heat, −1 −1 Rmin=0.1 µm, immersed in a fluid with η=0.01 gcm s , allosteric effects, or temperature [42]. Some specific molec- at T = 300 K. We can also consider a motor with ular structures with conformational changes have been pre- lmax = 10 Rmin, restitution coefficient selected ac- sented as potential solutions to proposed designs for theoreti- cording to the so-called extensibility parameter [41], cal nanoswimmers [42,43]. On the other hand, the activation b = klmax√2 /kBT = 50. By assuming that of these configurational changes could be controlled using a (1 − 2ζτ¯) ∆/6 2π3/2 ∼ 10−4, we obtain, from Eq. (28), Maxwell’s demon, similar to those that raise and lower an −2 −1 that vc.m. ∼ 10 µm s . energy barrier by using the information of the location of a We do not know whether natural or artificial microscopic macrocycle over a long rod-shaped [44]. Experi- systems exist that presents directed Brownian motor behav- mental implementations of such mechanism have been con- ior based on the mechanisms proposed in the present paper. ducted using photo and chemically activated barriers [45,46]. We stress that biological systems that propel themselves in In this paper, some issues that could be relevant for under- fluid environments by performing morphological changes re- standing the dynamics of the proposed Brownian motor have sembling those described in Fig. 1, such as certain protozoa not been considered. They include a detailed non-equilibrium and species of Euglena (see the discussion in Ref. 31 for fur- Thermodynamics analysis of its operation and efficiency, and ther details), can not be considered as physical realizations an analysis of the effects induced by the hydrodynamic mem- of our model, since in such systems contractions and expan- ory of its environment, similar to those calculated for single sions are not driven by thermal fluctuations but by an internal Brownian particles [47]. Such effects will be taken into ac- mechanism that can be theoretically treated as a piston [31]. count in subsequent publications. However, it is worth noticing that some microscopic systems exist that possess the characteristics needed by the compo- Acknowledgements nents of our theoretical motor. In particular, differences in viscous drag between different parts of a larger microstruc- H. H´ıjar thanks La Salle University Mexico for financial sup- ture could be obtained from conformational or configura- port under grant NEC-04/15.

1. S. Chandrasekhar, Rev. Mod. Phys. 15 (1943) 1. 15. D.S. Johnson, L. Bai, B.Y. Smith, S.S. Patel and M.D. Wang, 2. R.P. Feynman, R.B. Leighton and M. Sands, The Feynman Lec- 129 (2007) 1299. tures on (Addison Wesley, Reading, MA, 1966) Vol. 1 16. P.M. Hoffmann, Life’s Ratchet (Basic Books, 2012, New York). chapt. 46. 17. P. Reimann and P. Hanggi,¨ App. Phys. A 75 (2002) 169. 3. M.O. Magnasco, Phys. Rev. Lett. 71 (1993) 1477. 18. P. Reimann, Phys. Rep. 361 (2002) 57. 4. M. Von Smoluchowski, Phys. Z. XIII (1902) 1069. 19. G.G. Stokes, Trans. Camb. Philos. Soc. 8 (1847) 441. 5. P. Hanggi¨ and R. Bartussek, Brownian rectifiers: how to convert 20. H. Risken, The Fokker-Planck Equation (Springer, Berlin, Brownian motion into directed transport, in Nonlinear Physics 1984). of Complex Systems (Berlin: Springer 1996) p. 294. 6. P. Hanggi,¨ F. Marchesoni, and F. Nori, Ann. Phys. (Leipzig) 14 21. S.J. Ebbens and J.R. Howse, Soft Matter 6 (2010) 726. (2005) 51. 22. D. Patra, S. Sengupta, W. Duan, H. Zhang, R. Pavlick and A. 7. P. Hanggi¨ and F. Marchesoni Rev. Mod. Phys. 81 (2009) 387. Sen Nanoscale 5 (2013) 1273. 8. P. Nelson, Biological Physics. Energy, Information, Life W.H. 23. J. Rousselet, L. Salome, A. Ajdari and J. Prost, Nature 370 Freeman and Company, (New York 2014). (1994) 446. 9. K. Svoboda, C.F. Schmidt, B.J. Schnapp and S.M. Block, Na- 24. C. Kettner, P. Reimann, P. Hanggi¨ and F. Muller,¨ Phys. Rev. E ture 365 (1993) 721. 61 (2000) 312. 10. H. Lodish et al., Molecular Cell , 4th edition, (W. H. 25. S. Matthias and F. Muller,¨ Nature 424 (2003) 53. Freeman, New York, 2000). 26. S. Sundararajan, P.E. Lammert, A.W. Zudans, V.H. Crespi and 11. H. Noji, R. Yasuda, M. Yoshida and K. Kinosita Jr., Nature 386 A. Sen, Nano Lett. 8 (2008) 1271. (1997) 299. 27. L. Baraban et al., Nano 6 (2012) 3383. 12. D. Sabbert, S. Engelbrecht and W. Junge, Proc. Natl. Acad. Sci. 28. S. Martel, Magnetoactic Bacteria for Microrobotics, in Micro- U. S. A. 94 (1998) 4401. robotics. Biologically Inspired Microscale Robotic Systems, M. 13. P.D. Boyer, Annu. Rev. Biochem. 66 (1997) 717. Kim, A.A. Julius and E. Steager, editors, (Elsevier, Oxford, 14. S.S. Patel and I. Donmez, Journal of Biological 281 2012). (2006) 18265. 29. A. Najafi and R. Golestanian, Phys. Rev. E 69 (2004) 062901.

Rev. Mex. Fis. 63 (2017) 314-327 STOCHASTIC DYNAMICS OF A BROWNIAN MOTOR BASED ON MORPHOLOGICAL CHANGES 327

30. J.E. Avron, O. Kenneth and D.H. Oaknin, Phys. Rev. E 69 39. J.T. Padding and A.A. Louis, Phys. Rev. E 74 (2006) 041402. (2004) 232. 40. J. Prost and J.F. Chauwin, Phys. Rev. Lett. (1994). 31. J.E. Avron, O. Gat and O. Kenneth, Phys. Rev. Lett. 93 (2004) 41. A.P.G. van Heel, M.A. Hulsen and B.H. A.A. van den Brule, J. 186001. Non-Newtonian Fluid Mech. 75 (1998) 253. 32. F. Amb´ıa and H. H´ıjar, JPCS 738 (2016) 012037. 42. S. Erbas-Cakmak, D.A. Leigh, C.T. McTernan, and A.L. Nuss- 33. G.K. Batchelor, An introduction to fluid dynamics, (Cambridge baumer, Chem. Rev. 115 (2015) 10081. University Press, New York, 2000). 43. E.R. Kay, D.A. Leigh, and F. Zerbetto, Angew. Chem. Int. Ed. 34. H. H´ıjar, J. Chem. Phys. 139 (2013) 234903. 46(2007) 72. 35. D.S. Grebenkov, J. Phys. A 48 (2014) 013001. 44. R.D. Astumian, Annu. Rev. Biophys. 40 (2011) 289. 36. J.K.G. Dhont, An introduction to dynamics of Colloids, (Else- 45. M. Alvarez-Perez, S.M. Goldup, D.A. Leigh, and A.M.Z. vier, Amsterdam, 2003). Slawin, J. Am. Chem. Soc. 130 (2008) 1836. 37. M.E.J. Newman, Contemporary Physics 46 (2005) 323. 46. V. Serreli, C.F. Lee, E.R. Kay, and D. A. Leigh, Nature 445 38. D. Weeks, D. Chandler and H.C. Andersen, J. Chem. Phys. 54 (2007) 523. (1971) 5237. 47. H. H´ıjar, Phys. Rev. E 91 (2015) 022139.

Rev. Mex. Fis. 63 (2017) 314-327