Lecture 1: Random walk

Transcription

1 Lecture : Random walk Paul C Bressloff (Spring 209). D random walk q p r- r r+ Figure 2: A random walk on a D lattice. Consider a particle that hops at discrete times between neighboring sites on a one dimensional (D) lattice with unit spacing, see Fig. 2. At each step, the random walker moves a unit distance to the right with probability p or to the left with probability q = p. Let P N (r) denote the probability that the particle is at site r at the Nth time step. The discrete-time master equation is P N (r) = pp N (r ) + qp N (r + ), integer r, N. (.) If q = p = /2 then the random walk is symmetric or unbiased, whereas for p > q (p < q) it is biased to the right (left). It can be shown that P N (r) = ( ) ( ) p (N+r)/2 q (N r)/2 (.2) N + r N r!! 2 2 when N +r is an even integer and zero otherwise. Also = N(N )(N 2)... etc. (factorials). Setting p = q = /2 and evaluating log P N (r) for large N using Stirling s formula, ln N ln N N, one finds that (after exponentiating) P N (r) 2πN e r2 /2N. (.3) Characteristic function. In order to derive the formula for P N (r), we introduce the characteristic function (discrete Fourier transform) for fixed N G N (k) = r= e ikr P N (r), k [, π]. (.4) The characteristic function generates moments of the random displacement variable r according to ( i d ) m G N (k) dk = r m P N (r) = r m, (.5) k=0 r= where r m is the mth order moment of r. Multiplying both sides of the master equation (.) by e ikr and summing over r gives G N (k) = (pe ik + qe ik )G N (k)

2 Random walks Paul C Bressloff (Spring 209) Iterating this equation and setting u(k) = pe ik + qe ik, we have G N (k) = u(k)g N (k) = u(k) 2 G N 2 (k) = u(k) N G 0 (k). Finally, assuming that the particle starts at the origin, P 0 (r) = δ r,0, so that G 0 (r) =, it follows that G N (k) = u(k) N. Now take the inverse Fourier transform and apply the Binomial theorem to expand (pe kr + qe ikr ) N : P N (r) = e ikr u(k) N dk 2π = 2π = = = N m=0 N m=0 e ikr [ N m=0 (N m)!m! pm q N m ] (N m)!m! pm q N m e ik(n 2m) dk (N m)!m! pm q N m δ N 2m+r,0 [ π ] e ik(n 2m+r) dk 2π ( ) ( ) p (N+r)/2 q (N r)/2. (.6) N + r N r!! 2 2 when N +r is an even integer and zero otherwise. Setting N 2m+r = n, we have used the results ikn dk e 2π = 2iπn (einπ e inπ ) = 0 for n 0, dk 2π =, and the definition of a Kroenecker delta: δ n,0 = if n = 0 and is zero otherwise. The distribution (.6) is known as the Binomial distribution. In the unbiased case p = q = /2, it gives the probability of a total of r heads in tossing a fair coin N times and is known as the Bernouilli distribution..2 Transient and recurrent random walks Another useful quantity when analyzing random walks is the generating function (discrete Laplace transform or one-sided z-transform) Γ(r, z) = z N P N (r). (.7) N=0 It is often simpler to evaluate the generating function in Fourier space, Γ(k, z) e ikr Γ(r, z) = z N G N (k). r= N=0 Page 2

3 Random walks Paul C Bressloff (Spring 209) assuming that we can reverse the order of summations. Since G N (k) = u(k) N, we can sum the resulting geometric series to obtain the result Γ(k, z) = zu(k). The generating function is thus given by the inverse Fourier transform Γ(r, z) = dk zu(k) 2π. e ikr It can be shown that for r = 0 and p = q = /2 (unbiased random walk), Γ(0, z) = ( z 2 ) /2. One immediate consequence of this result is that an unbiased D random walk is recurrent, which means that the walker is certain to return to the origin; a random walk is said to be transient if the probability of returning to the origin is less than one. Recurrence follows from the observation that Γ(0, ) = N=0 P N(0) is the mean number of times that the walker visits the origin, and lim Γ(0, z) = z for the D random walk. Interestingly, although the D random walk is recurrent, the mean time to return to the origin for the first time is infinite. This result can also be established using transform methods and generating functions. An unbiased random walk in 2D is also recurrent but in 3D it is transient..3 Continuum limit Having analyzed the discrete random walk, it is now possible to take an appropriate continuum limit to obtain a diffusion equation in continuous space and time. First, introduce infinitesimal step lengths δx and δt for space and time and set P N (r) = p(x, t)δx with x = rδx, t = Nδt. Substituting into the master equation (.) for p = q = /2 gives the following equation for the probability density p(x, t): p(x, t) = 2 p(x δx, t δt) + p(x + δx, t δt) [ 2 p(x, t) p ] t δt + 2 p 2 x 2 δx2, where p has been Taylor expanded to first order in δt and to second order in δx. Dividing through by δt and taking the continuum limit δx, δt 0 such that the quantity D is finite, where D = yields the diffusion equation with diffusivity D δx 2 lim δx,δt 0 2δt, p(x, t) t = D 2 p(x, t) x 2. (.8) Page 3

4 Random walks Paul C Bressloff (Spring 209) Given the initial probability distribution p(x, 0) = f(x), the solution of the diffusion equation is p(x, t) = 4πDt e (x y)2 /4Dt f(y)dy. (.9) In particular, if we take the particle starts at the origin then we obtain the so-called fundamental solution p(x, t) = e x2 /4Dt. (.0) 4πDt Here the solution to the diffusion equation describes the probability density function for the location of a single particle (rather than the concentration of many particles). In particular, Ω p(x, t)dx can be identified as the probability that the particle is in region Ω at time t. Fundamental solution of the diffusion equation. One method for solving the diffusion equation is to use Fourier transforms. Fourier transforming the diffusion equation (.8) with respect to x gives d p(k, t) = k 2 D p(k, t), p(k, t) = p(x, te ikx dx dt which is a differential equation in t with k treated as a parameter. Its solution is p(k, t) = c(k)e k2 Dt, with the coefficient c(k) determined by the initial data. That is, Fourier transforming the initial condition implies p(k, 0) = f(k) and, hence, p(k, t) = f(k)e k2 Dt. Since p(k, t) is the product of two Fourier transforms, its inverse Fourier transform is given by a convolution: p(x, t) = K(x y, t)f(y)dy, where K(x, t) is the inverse Fourier transform of e k2 Dt : K(x, t) = 2π e ikx e k2 Dt dk = 4πDt e x2 /4Dt..4 Mean-square displacement It immediately follows from properties of Gaussian distributions that the mean and variance, also known as the mean square displacement (MSD), are x = xp(x, t)dx = 0, (.) and x 2 = x 2 p(x, t)dx = 2Dt. (.2) Page 4

5 Annu. Rev. Biophys : Downloaded from by University of Utah - Marriot Library on 0/04/2. For personal use only. Random walks Paul C Bressloff (Spring 209) a b MSD Superdiffusion Normal Subdiffusion Time Normal Anomalous 0 Displacement c d Figure 3: Anomalous 0.6 diffusion. Two characteristic features of anomalous diffusion are shown: superlinear or sublinear variation of the mean-square displacement (MSD) with time; large tails in the probability density. MSD (μm 2 ) % 7% Similarly for normal diffusion in d spatial dimensions, one finds that x 2 = 2dDt. 9% Probability Probability Anomalous diffusion. 0 It turns out that the time-variation of the mean-square displacement (MSD) of a diffusing particle is an important quantity in experimental studies of cells. For example, Time (ms) a general signature of anomalous diffusion is the power law behavior MSD (μm 2 ) 0.5 x 2 = 2dDt α (.3) corresponding to either subdiffusion (α < ) or superdiffusion (α > ). In recent years a powerful Figure experimental method has been developed based on single-particle tracking (SPT), in which one Characteristics and simulations of anomalous diffusion. (a) MSD curves defining normal (Brownian) images diffusionthe and anomalous trajectory subdiffusion of a marker (downward attached curvature) and to asuperdiffusion diffusing(upward molecule. curvature). Various transport properties of the (b) particle Distributionare of displacements then derived for normal through and anomalous a statistical diffusion. Initial analysis particleof position the trajectory, is at the including a measurement origin. For normal diffusion, the distribution is Gaussian and gives rise to Brownian motion. The curve labeled ofanomalous the mean has long square tails and displacement. an infinite second moment, Visualization resulting inof nonlinear the diffusive MSD plots and behavior of single-membrane proteins anomalousindiffusion. living(c) cells MSD plots has for revealed simulations that of crowding theseinmolecules an aqueous phase. undergo Simulations a variety done of stochastic behaviors for 75-nm-radius spherical particles in a µm box for ms using the method of Dix et al. (0). including normal and anomalous diffusion, and confinement within subcellular compartments. Fig. (d) (Top) MSD distributions for simulation of 75-nm-radius particles for 0 ms at 9% volume exclusion. 4 The illustrates smooth curve one is fitted application using the expected of SPT, distribution namely, for normal studying diffusion. the (Bottom) role Difference of lateral membrane diffusion in delivering between fitted neurotransmitter and observed MSD distributions. receptors to synapses of a neuron. lipids and proteins in plasma membranes of cells can show anomalous diffusion (4, 24, 44) as well as normal diffusion (54). The variation in experimentally measured diffusion properties in the plasma membrane may relate, in part, to the timescales over which the measurements are made (24). The finding of anomalous diffusion of membrane proteins has led to an update to the Singer-Nicolson model (52), in which clusters of proteins or lipids, such as rafts, contribute to biological function. The presence of fixed barriers to diffusion can in effect produce volume exclusion Note. Although we have derived the diffusion equation from an unbiased random walk, it is more typically interpreted in terms of an evolution equation for a conserved quantity such as particle number rather than a probability density for a single random walker. In order to link these two interpretations, consider N non-interacting, identical diffusing particles and let u(x, t) = N p(x, t). Crowding Effects on Cellular Diffusion 25 For sufficiently large N, we can treat u(x, t)dx as the deterministic number of particles in the infinitesimal interval [x, x + dx] at time t, with u(x, t) evolving according to the diffusion equation written in the conservation form u t = J x, where J(x, t) is the Fickian flux of particles. u J(x, t) = D x, (.4) Integrating the diffusion equation (.4) over the Page 5

6 Random walks Paul C Bressloff (Spring 209) a b Figure 4: Membrane receptor diffusion in neurons measured by single particle tracking. (a) Superimposed image of the trajectory of 500 nm beads bound to glycine receptors (GlyRs) with the fluorescent image (green) of green fluorescent protein (GFP)- tagged gephyrin. Periods of free diffusion and confinement are indicated by blue and red lines, respectively. (b) Plots of the average mean squared displacement (MSD) function during periods of free diffusion (left panel) and confinement (right panel) for GlyRs. Note the difference in both shape and amplitude of the MSDs. The curved shape of the MSD is characteristic of movement in a confined space. [Adapted from Choquet and A. Triller (2003).] interval [x, x + dx] and reversing the order of integration and differentiation shows that d dt x+dx x u(y, t)dy = J(x, t) J(x + dx, t), which is an expression of particle conservation. That is, the rate of change of the number of particles in [x, x + dx] is equal to the net flux crossing the endpoints of the interval. Page 6