1. Introduction to Biophysics Lecture 8 Brownian motion Diffusion

2. Diffusion is the dominant form of material transport on sub-micrometer scales.

3. Calculating probabilities
2 steps walk, probability of coming back to starting point P0 = 2/22=0.5
4 steps walk P0=6/24=0.375
How to do calculation for large number (10000 steps)?
M0 – number of different outcomes that land us at the starting point. Should have equal number of heads and tails (5000). Describe them as list containing 5000 different integers (n1, …, n5000); n1 = any number between 1 and 10000, n2 any of the 9999 remaining choices, and so on. We have a total of 10000x9999x…x5001 lists = 10000!/5000! But any two lists differing by exchange (or permutation) of the ni’s are not really different, so we must divide our answer by total number of possible permutations, which is 5000x4999x…x1.
M0 = 10000!/(5000!x5000!)
M0/M= 10000!/(5000!x5000!x210000)  (about 1%)
Stirling’s formula: ln M!  MlnM – M +1/2(ln(2M))

4. Probability to go far right/left or come back are low.

5. How far you are likely to go on a random walk?
<x32> = 3L2
The mean displacement <x> = 0 but <x2> is not 0.

6. How to find <(xN)2> =? (Home work) Nelson page 115
<(xN)2> = NL2 = 2Dt - one dimensional diffusion law, D[m2/s]
N=t/t and D=L2/2t

7. <(rN)2> = 6Dt - diffusion in three dimensions, D[m2/s]

8. Friction is quantitatively relates to diffusion – the same random collisions responsible for spread of molecules also give rise to friction.
Collision ones per t
In between kicks particle is free of random influence
x(t)= 0,x+F*t/m , F-external force,
x=0,xt +1/2(F/m)*(t)2
Each collision obliterates all memory of the previous step. Thus, after each step,
0,x is randomly pointing left or right, so the its average value < 0,x> =0
<x>=1/2(F/m)*(t)2
The particle, although buffered about by random collisions, nevertheless acquires a net drift velocity
drift = F/ , where =2m/t is a viscous friction coefficient
Both D and  are experimentally measurable
For round object:  = 6R where  is viscosity of the fluid (kg/m*sec), R –radius of the particle (Stokes formula)

9. L-effective step size
(L/t)2 = (0,x)2
From ideal gas law
<(0,x)2 > = kBT/m
Now we have all we need to demonstrate Einstein relation
D= kBT
This equation is universal.

10. The conformation of polymers
Length of the link

12. Compact (N1/3) versus extended (N1/2) polymers (macromolecules)

13. Homework: Problem 7.9 (Nelson pages 125,126 and 292)
Figure 4.8 (Schematic; experimental data; photomicrograph.) Caption: See text.
©1999. Used by permission of the American Physical Society.

14. 1. Distribution is uniform in Y and Z but not in X.
2. In t particle can move a L step. x
Thus, half hops to the left, half to the right.
N(x)- total number of particles in the slot centered at x.
½(N(x-L)-N(x)) - the net number of particles crossing the bin boundary a from left to right
take bins to be very narrow: N(x-L)-N(x)  -L(dN/dx)
the number density of particles: c(x) = N(x)/Vslot = N(x)/LYZ
How many particles cross per unit area per unit time (flux)?
j = ½*(1/(YZt))*L(-d(LYZc(x)/dx) = - ½ *(1/t)*L2*(dc/dx)

15. Fick’s law
Question: we would like to be able to predict evolution of c(r,t) if we know c(r,0)
Fick’s law is not enough…
dN(x)/dt = (YZ*j(x-L/2) – YZ*j(x+L/2))
Make bins small, get to derivative shape
Continuity equation
Diffusion equation
is deterministic

16. Reading Nelson Chapter 4
Check T2:
Home work:
show that diffusion law is model independent (Nelson page 117).
problem 7.9 (Nelson pages 125,126 and 292)