Random walk models in biology

Outline Introduction Fundamentals Of Random Walks The Simple Isotropic Random Walk A Brw With Waiting Times Random Walks With A Barrier Crws And The Telegraph Equation Reference

Introduction Random Walk Brownian motion a particle repeatedly moves in all directions Brownian motion continuous irregular motion of individual pollen particles Robert Brown(1828 )

Introduction Brownian Motion Analysis Langevin Equation(1908) Einstein- Smoluchowski Equation (1905,1916) x2 = 2Dt (D is diffusion coefficient) Average Particles Actions (Probability) Langevin Equation(1908) Single Particle Action (F=ma)

Outline Introduction Fundamentals Of Random Walks The Simple Isotropic Random Walk A Brw With Waiting Times Random Walks With A Barrier Crws And The Telegraph Equation Reference

Fundamentals Of Random Walks Uncorrelated the direction of movement is completely independent of the previous directions Unbiased the direction moved at each step is completely random The Brown motion is uncorrelated & unbiased

Fundamentals Of Random Walks Fixed Step length moves a distance δ in a short time τ Variable Step length Finite variance (Brownian motion) Infinite variance (Lévy flight) Brownian motion Lévy flight

Outline Introduction Fundamentals Of Random Walks The Simple Isotropic Random Walk A Brw With Waiting Times Random Walks With A Barrier Crws And The Telegraph Equation Reference

Simple Isotropic Random Walk Consider a walker moving on an 1-D infinite uniform lattice One Dimensional Fixed Step length Uncorrelated Unbiased The walker starts at the origin (x=0) and then moves a distance δ either left or right in a short time τ

Simple Isotropic Random Walk Consider probability p(x , t) x is distance form x=o t is number of time step

Simple Isotropic Random Walk The probability a walker will be at a distance mδ to the right of the origin after nτ time steps This form is Binomial distribution, with mean displacement 0 and variance nδ2.

Simple Isotropic Random Walk

Simple Isotropic Random Walk For large n, this converges to a normal (or Gaussian) distribution after a sufficiently large amount of time t=nτ, the location x=mδ of the walker is normally distributed with mean 0 and variance δ2t/τ. (δ2/2τ = D) PDF for location of the walker after time t

Simple Isotropic Random Walk mean location E(Xt)=0 the absence of a preferred direction or bias mean square displacement (MSD) E(Xt2)=2Dt a system or process where the signal propagates as a wave in which MSD increases linearly with t2

Outline Introduction Fundamentals Of Random Walks The Simple Isotropic Random Walk A Brw With Waiting Times Random Walks With A Barrier Crws And The Telegraph Equation Reference

A Brw With Waiting Times 1-D Biased random walks preferred direction (or bias) and a possible waiting time between movement steps. at each time step τ, the walker moves a distance δ to the left or right with probabilities l and r, or stays in the same location (‘waits’) probabilities 1-l-r.

A Brw With Waiting Times the walker is at location x at time t+τ, then there are three possibilities for its location at time t. P(x, t+τ)= P(x, t)(1-l-r)+P(x-δ,t)r+ P(x+δ,t)l it was at x and did not move at all. it was at x - δ and then moved to the right. it was at x + δ and then moved to the left.

A Brw With Waiting Times P(x, t+τ )= P(x, t)(1-l-r)+P(x-δ,t)r+ P(x+δ,t)l expressed it as a Taylor series about (x, t) Fokker–Planck equation Special case D is constant

A Brw With Waiting Times E(Xt2)~ t2 is like wave σt 2=2Dt is a standard diffusive process

Outline Introduction Fundamentals Of Random Walks The Simple Isotropic Random Walk A Brw With Waiting Times Random Walks With A Barrier Crws And The Telegraph Equation Reference

Random Walks With A Barrier a walker reaching the barrier turn around and move away in the opposite direction or absorbed in barrier

Random Walks With A Barrier 1-D random walk process that satisfies the drift–diffusion

Random Walks With A Barrier At time t, either the walker has been absorbed or its location has PDF given by p(x, t) PDF of the absorbing time Ta

Random Walks With A Barrier the probability of absorption taking place in a finite time (Ta < ∞) the walker is certain to be absorbed within a finite time drift towards the barrier (u ≤0) (u >0) probability decreases exponentially as the rate of drift u, or the initial distance x0 from the barrier, increases. if the rate of diffusion D increases, the probability of absorption will increase

Outline Introduction Fundamentals Of Random Walks The Simple Isotropic Random Walk A Brw With Waiting Times Random Walks With A Barrier Crws And The Telegraph Equation Reference

Crws And The Telegraph Equation Correlated random walks (CRWs) involve a correlation between successive step orientations CRW is a velocity jump process population of individuals moving either left or right along an infinite line at a constant speed v total population density is p(x, t)=a(x, t)+b(x, t). (left + right-moving)

Crws And The Telegraph Equation CRW at each time step,turning events occur as a Poisson process with rate λ

Crws And The Telegraph Equation Expanding these as Taylor series and taking the limit δ, τ->0 such that δ/τ =v gives telegraph equation:

Crws And The Telegraph Equation http://www.math.ubc.ca/~feldman/apps/telegrph.pdf small t (i.e. t=1/ λ), E(Xt2)~O(v2t2) is a wave propagation process; for large t, E(Xt2)~ O(v2t/ λ), which is a diffusion process

Reference Random walk models in biology Random walk in biology http://privatewww.essex.ac.uk/~ecodling/Codling_et_al_2008.pdf Random walk in biology http://rieke-server.physiol.washington.edu/People/Fred/Classes/532/berg_randomwalk_ch1.pdf Diffusion http://www.che.ilstu.edu/standard/che38056/lecturenotes/380.56chapter13-S06.pdf Brownian motion http://sciweb.nybg.org/science2/pdfs/dws/Brownian.pdf