MAE 545: Lecture 17 (4/10) Random walks Random walks

Brownian Swimming of E. coli J. Phys. A: Math. Theor. 42 (2009) 434013 L Mirny et al

(A) (B)

..ATTATGCATGACGAT..

Figure 1. (A) Schematic representation of the protein–DNA search problem. The protein (yellow) must find its target site (red) on a long DNA confined within the cell nucleoid (in ) or cell nucleus (in eukaryotes). Compare with figure 9(A) which shows confined DNA. (B)The random coils targetProtein site must be recognized search with 1 base-pair for (0 .34a nm) precision, as displacement by 1 bp results in a different sequence and consequently a different site. binding site on DNA

(A)(1d 3d B)

n

Figure 2. (A)Themechanismoffacilitateddiffusion.Thesearchprocessconsistsofalternating 2 rounds of 3D and 1D , each with average duration τ3D and τ1D,respectively.(B)The antenna effect [9]. During 1D diffusion (sliding) along DNA, a protein visits on average n¯ sites. This allows the protein to associate some distance n¯ away from the target site and reach it by sliding, effectively increasing the reaction cross-section∼ from 1bp to n¯. The antenna effect is responsible for the speed-up by facilitated diffusion. ∼

1.3. History of the problem: theory To resolve this discrepancy, one possible mechanism of facilitated diffusion that includes both 3D diffusion and effectively 1D diffusion of protein along DNA (the 1D/3D mechanism)was suggested. This mechanism was first proposed and dismissed by Riggs et al [1]butwassoon revived and rigorously studied by Richter and Eigen [3], then further expanded and corrected by Berg and Blomberg [4]andfinallydevelopedbyBerget al [5]. The basic idea of the 1D/3D mechanism is that while searching for its target site, the protein repeatedly binds and unbinds DNA and, while bound non-specifically, slides along the DNA, undergoing one-dimensional (1D) or a random walk. Upon dissociation from the DNA, the protein diffuses three dimensionally in solution and binds to the DNA in a different place for the next round of one-dimensional searching (figure 2(A)). During 1D sliding the protein is kept on DNA by the binding energy to non-specific DNA. This energy has been measured for several DNA-binding proteins and has a range of 10–15 kBT (at physiological salt concentration), was shown to be driven primarily by screened electrostatic interactions between charged DNA and protein [6], and

3 Brownian motion of small particles 1827 Robert Brown: observed irregular motion of small pollen grains suspended in water

10µm ⇡

https://www.youtube.com/watch?v=R5t-oA796to

1905-06 Albert Einstein, Marian Smoluchowski: microscopic description of Brownian motion and relation to diffusion equation

3 Random walk on a 1D lattice 1 q q x 5` 4` 3` 2` ` 0 ` 2` 3` 4` 5` At each step particle jumps to the right with q and to the left with probability 1-q. sample trajectories for q=1/2 sample trajectories for q=2/3 (unbiased random walk) (biased random walk) 100 100

80 80

60 60 N N 40 40

20 20

0 0 -20 -10 0 10 20 -50 -25 0 25 50 x/ℓ 4 x/ℓ Random walk on a 1D lattice 1 q q x 5` 4` 3` 2` ` 0 ` 2` 3` 4` 5` At each step particle jumps to the right with probability q and to the left with probability 1-q.

What is the probability p(x,N) that we find particle at position x after N jumps?

Probability that particle makes k N k N k jumps to the right and N-k jumps to p(k, N)= q (1 q) k the left obeys the ✓ ◆

x = k` (N k)` =(2k N)` Relation between k and 1 x particle position x: k = N + 2 ` ⇣ ⌘ 5 Random walk on a 1D lattice 1 q q x 5` 4` 3` 2` ` 0 ` 2` 3` 4` 5` unbiased random walk biased random walk

N k N k p(k, N)= q (1 q) k ✓1 ◆ x k = N + 2 ` Note: exact⇣ discrete⌘ distribution has been made continuous by replacing discrete peaks with boxes whose area corresponds to the same probability.

after several steps the spreads out and becomes approximately Gaussian

6 Gaussian approximation for p(x,N) 1 q q x 5` 4` 3` 2` ` 0 ` 2` 3` 4` 5` Position x after N jumps can be expressed N as the sum of individual jumps x ` , ` . x = xi i 2 { } i=1 Mean value averaged over N X x = x = N x = N (q` (1 q)`) all possible random walks h i h ii h 1i i=1 X x = N` (2q 1) h i 2 2 2 2 2 2 averaged over all = x x = N1 = N x1 x1 possible random walks h i h i 2 = N⌦ ↵q`2 +(1 q)`2 x ⇣2⌦ ↵ ⌘ h 1i ⇣ ⌘ 2 =4N`2q(1 q) According to the central limit 1 (x x )2/(22) theorem p(x,N) approaches p(x, N) e h i ⇡ p2⇡2 Gaussian distribution for large N: 7 Number of distinct sites visited by unbiased random walks

Total number of sites inside explored region after N steps

1D N pN tot / In 1D and 2D every site gets visited after 2D N N a long time tot / In 3D some sites are 3D N NpN never visited even 2r pN tot / / after a very long time! Shizuo Kakutani: “A drunk man will find his way home, but a drunk bird may get lost forever.”

1D Nvis 8N/⇡ Number of distinct visited ⇡ p 2D Nvis ⇡N/ ln(8N) sites after N steps ⇡ 3D N 0.66N 8 vis ⇡ Master equation 1 q q x 5` 4` 3` 2` ` 0 ` 2` 3` 4` 5` Master equation provides recursive relation for the evolution of probability distribution, where ⇧ ( x, y ) describes probability for a jump from y to x. p(x, N + 1) = ⇧(x, y)p(y, N) y X For our example the master equation reads: p(x, N + 1) = qp(x `,N)+(1 q) p(x + `,N) Initial condition: p(x, 0) = (x)

Probability distribution p(x,N) can be easily obtained numerically by iteratively advancing the master equation.

9 Master equation and Fokker-Planck equation 1 q q x 5` 4` 3` 2` ` 0 ` 2` 3` 4` 5` Assume that jumps occur in regular small time intervals: t Master equation: p(x, t + t)=qp(x `,t)+(1 q) p(x + `,t) In the limit of small jumps and small time intervals, we can Taylor expand the master equation to derive an approximate drift-diffusion equation: @p @p 1 @2p @p 1 @2p p + t = q p ` + `2 +(1 q) p + ` + `2 @t @x 2 @x2 @x 2 @x2 ✓ ◆ ✓ ◆ Fokker-Planck equation: ` drift velocity v =(2q 1) @p @p @2p t = v + D 2 diffusion `2 @t @x @x D = coefficient 2t 10 Fokker-Planck equation

x 5` 4` 3` 2` ` 0 ` 2` 3` 4` 5` In general the probability distribution ⇧ of jump ⇧(s x) lengths s can depend on the particle position x | Generalized master equation: p(x, t + t)= ⇧(s x s)p(x s, t) | s Again Taylor expand theX master equation above to derive the Fokker-Planck equation: @p(x, t) @ @2 = v(x)p(x, t) + D(x)p(x, t) @t @x @x2   drift velocity diffusion coefficient (external fluid flow, external potential) (e.g. position dependent temperature) s s(x) s2 s2(x) v(x)= ⇧(s x)=h i D(x)= ⇧(s x)= t | t 2t | 2t s 11 s ⌦ ↵ X X Lévy flights Lévy flight trajectory Probability of C ~s ↵, ~s >s jump lengths in ⇧(~s )= 0 | 0|, |~s |

Normalization dD~s ⇧(~s )=1 condition ↵ >D Z Moments of A s2, ↵ >D+2 ~s =0 ~s 2 = D 0 distribution h i , ↵

D. W. Sims et al.

Nature 451, 1098-1102 (2008) 12 Probability current Fokker-Planck equation @p(x, t) @ @2 = v(x)p(x, t) + D(x)p(x, t) @t @x @x2   Conservation law of probability (no particles created/removed) @p(x, t) @J(x, t) = @t @x Probability current: @ J(x, t)=v(x)p(x, t) D(x)p(x, t) @x  Note that for the steady state distribution, where @p⇤(x, t)/@t 0 ⌘ the steady state current is constant and independent of x @ J ⇤ v(x)p⇤(x) D(x)p⇤(x) = const ⌘ @x  Equilibrium probability distribution: If we don’t create/remove 1 x v(y) particles at boundaries then J*=0 p⇤(x) exp dy 13 / D(x) D(y) Z1 Spherical particle suspended U(x) in fluid in external potential Newton’s law: @2x @U(x) m = v(x) + F @t2 @x r fluid external random drag potential Brownian force force @2x For simplicity assume overdamped regime: 0 @t2 ⇡ 1 @U(x) x Drift velocity v(x) = averaged over time h i @x R particle radius Equilibrium probability distribution ⌘ fluid viscosity U(x)/D U(x)/k T p⇤(x)=Ce = Ce B =6⇡⌘R Stokes drag coefficient (see previous slide) (equilibrium ) kB Boltzmann constant Einstein - Stokes equation T temperature kBT kBT D diffusion constant D = = 14 6⇡⌘R Translational and rotational diffusion for particles suspended in liquid

Translational Rotational ✓ diffusion x diffusion

2 2 x =2D t ✓ =2DRt h i T 3 Stokes viscous drag: T =6⇡⌘R Stokes viscous ⌦drag:↵ R =8⇡⌘R k T k T Einstein - Stokes D = B Einstein - Stokes D = B relation T 6⇡⌘R relation R 8⇡⌘R3

Time to move one body length Time to rotate by 900 in water at room temperature in water at room temperature 3⇡⌘R3 4⇡⌘R3 x2 R2 t ✓2 1 t ⇠ ⇠ kBT ⇠ ⇠ kBT ⌦ ↵ R 1µm t 1s ⌦ ↵ 23 ⇠ ⇠ Boltzmann constant kB =1.38 10 J/K 3 ⇥ 1 1 water viscosity ⌘ 10 kg m s R 1mm t 100 years ⇡ ⇠ ⇠ 15 room temperature T = 300K Fick’s laws Adolf Fick 1855 N noninteracting Local concentration Brownian particles of particles c(x, t)=Np(x, t) Fick’s laws are equivalent to Fokker-Plank equation First Fick’s law @c Flux of particles J = vc D @x Second Fick’s law @c @J @ @ @c Diffusion of = = vc + D @t @x @x @x @x particles   Generalization to higher dimensions @c J~ = c~v D~ c = ~ J = ~ (c~v )+~ (D~ c) r @t r r · r · r

16 Further reading

17