The Structure, Function, and
Evolution of Biological Systems
Instructor: Van Savage
Spring 2011 Quarter
5/5/2011
Global signature of diffusion
Random walk
x(t+1)=x(t)±1
->x2(t+1)=x2(t)+2x(t)+1(1/2 of time)
=x2(t)-2x(t)+1(1/2 of time)
<x2(t+1)>=(1/2)* <(x2(t)+2x(t)+1)>
+(1/2)*<(x2(t)-2x(t)+1)>
=<x2(t)>+1=<x2(t-1)>+2
Iterating this gives: <x2(t+1)>=Number of time steps~t
| x | x  t
2
Combined Effects
Person trying to walk north (directional) through
a busy intersection (nondirectional)
Net Flow=Directional Flow+Nondirectional Flow
Diffusion Equation
(Also known as Kolmogorov forward equation)
f
 (vf )  (Df )


2
t
x
x
2
Schienbein et al.
Migration of cells
Cells move randomly while also guided in direction of
inflammation, chemical signal, or electric field.
Movement observed in wound healing,
embryogenesis, and granulocytes. Can also modify
this for chemotaxis.
Langevin equation can describe this
White noise defined by the following properties
Langevin movement corresponds
to Kolmogorov forward equation/
Fokker-Planck for probability
Steady state solution
Match of theory and data
Solution to Langevin and correlation
function
Correlation
Relative Correlation
Match of theory and data for
correlation
Similar equations and analyses apply
for the distribution of angles
Langevin
Steady state probability disribution
Distribution for different electric fields
Expected value for cosine angle:
theory and data
Expected square of displacement
Predictions and Data
Time evolution of probability
distribution with no electric field
Time evolution of probability distribution
with electric field turned on
Some discrepancies between theory
and data for full case
Sources of discrepancy
1. Not at steady state during shift in electric field
2. Noise may not be white noise
3. Deterministic part may have frequency dependence
4. There is some weak memory to the “noise”