MAE 545: Lecture 17 (4/10) Random walks Random walks
Brownian motion Swimming of E. coli J. Phys. A: Math. Theor. 42 (2009) 434013 L Mirny et al
(A) (B)
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Figure 1. (A) Schematic representation of the protein–DNA search problem. The protein (yellow) must find its target site (red) on a long DNA molecule confined within the cell nucleoid (in bacteria) or cell nucleus (in eukaryotes). Compare with figure 9(A) which shows confined DNA. (B)The Polymer 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)
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Figure 2. (A)Themechanismoffacilitateddiffusion.Thesearchprocessconsistsofalternating 2 rounds of 3D and 1D diffusion, 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) Brownian motion 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 molecules [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 probability 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
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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 binomial distribution ✓ ◆
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 probability distribution 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 Variance averaged over all = x x = N 1 = 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/(2 2) 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 2 t 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 2 t | 2 t 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) Z 1 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 physics) 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