1. Random walk models in biology

2. 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

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

4. 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)

5. 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

6. 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

7. 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

8. 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

9. 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

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

11. 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.

12. Simple Isotropic Random Walk

13. 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

14. 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

15. 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

16. 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.

17. 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.

18. 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

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

20. 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

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

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

23. 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

24. 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

25. 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

26. 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)

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

28. 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:

29. Crws And The Telegraph Equation
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

30. Reference Random walk models in biology Random walk in biology
Random walk in biology
Diffusion
Brownian motion