Chapter 4 Lecture

Biological PhysicsNelson

Updated 1st Edition

Slide 1-1

Random Walks, Friction & Diffusion (part II)

Slide 1-2

Important Dates

• Extra class

– Wednesday May 6th (Self Study)

• Midterm report presentation

– Tuesday May 12th (5th Period)

– Presentation on Chapter 5 in book

• See next slide

• Final Report

– Topic of you choice based on research

papers related to biophysics

Slide 1-3

Announcement: Midterm Presentations

• Midterm presentation are Week 7/8

– May 12th, 5th period (1620-1800)

– Each group (3 students) will give a short 30

min. prezi from 3 subsections:-

5.1+5.3.x1; 5.2+5.3x2; and

5.3.1, 5.3.2, 5.3.3, 5.3.4, 5.3.5

(choose 3 –x1,x2)

– Each student ~10 min. (template on GDrive)

– Make mini-group-report (ShareLaTeX)

• Deadline May 26th

Slide 1-4

Biophysics quote

Humans are to a large degree sensitive to energy fluxes rather

than temperatures, which you can verify for yourself on a cold,

dark morning in the outhouse of a mountain cabin equipped with

wooden and metal toilet seats. Both seats are at the same

temperature, but your backside, which is not a very good

thermometer, is nevertheless very effective at telling you which is

which.

-Craig F. Bohren and Bruce A. Albrecht, Atmospheric

Thermodynamics (Oxford University Press, New York, 1998).

Slide 1-5©1961. Used by permission of Dover Publications.

Summary: Random Walks

Slide 1-6

Outline

• Brownian motion

• Random walks

• Diffusion

• Friction

• Three important equations, leading to the

Fluctuation-Dissipation relation

Slide 1-7

Homework

1. Read 4.1.3:- Understand statement: “Random

Walk is model independent!”

2. Read 4.2:- What Einstein did?

3. Make a diagram for 1D case of four steps

4. Extra:- Are two elevator shafts better when

stopping at odd and even floors only?

• Assume the cost of the elevator is only to

start and stop ~ 50 Yen per ride

Slide 1-8

4.3 Other Random Walks (Discussion)

If we synthesize polymers made from various numbers of the

same units, then the coil size increases proportionally as the

square root of the molar mass.

Slide 1-9

Polymer Diffusion

Slide 1-10

Figure 4.8 (Schematic; experimental data; photomicrograph.) Caption: See text.

©1999. Used by permission of the American Physical Society.

Polymer Random Walks (Problem 7.9*)

Slide 1-11

Random Walks on Wall Street*

Slide 1-12

4.4 – 4.6 Equations Summary

Slide 1-13

4.4 The diffusion equations: Fick’s 1st Law

• First let’s derive Fick’s first law: consider 4.10

and release a trillion random walkers and

compare P(x,0) with P(x,t) at time steps Δt

• Flow from L ー> R isand when bin size is shrunk we get

• No. density c(x) is just N(x) in a slot divided by

LYZ (vol. of slot) = N/(LYZ) implies

Slide 1-14

4.4 Diffusion cartoon

Slide 1-15

4.4 Fick’s Law (1st Law)

• From last time we know D = L2/Δt so we have

• Q:- What drives the flux?

Slide 1-16

4.4 Fick’s Law (1st Law)

• From last time we know D = L2/Δt so we have

• Q:- What drives the flux?

– Mere probability is “pushing” the particles (cf.

entropic forces)

• Fick’s (1st law) is not enough. We need his 2nd

law; otherwise known as the “Diffusion Equation”

Slide 1-17

4.4. Diffusion Equation

• Let’s look at how N(x) and hence c(x) vary in

time:

• Now dividing by LYZ gives the “continuity

equation”

• Now take derivative of

w.r.t. time and use continuity to show that

• Later our goal will be to solve this equation

Slide 1-18

4.5 Functions and Derivatives

Slide 1-19

And Snakes Under the Rug

Try to use Wolfram α to make some plots

Slide 1-20

4.6.1 Membrane Diffusion*

• Imagine a long thin membrane/tube of Length L,

with one end in ink C(0)=c0 and in water C(L)=0

• This leads to a quasi-steady state so we set

dc/dt =0 and hence d2c/dx2=0

• This means that c is constant and js=-DΔc/L

where Δc0=cL-c0 and subscript s means the flux

of solute not water

• Now define js=-PsΔc where Ps is the permeability

of the membrane. In simple cases Ps roughly

relates to the width of the pore and thickness of

the membrane (length of pore)

• Using dN/dt=-Ajs leads to (next slide)

Slide 1-21

4.6.1 Membrane Diffusion

Slide 1-22

4.6.2 Diffusion sets fundamental limit on

bacterial metabolism

• In class exercise:

– Example on pg. 138 of book

– Follow steps and present your derivation

• And also try to do Your Turn 4F

– a) Find I (mass per unit time) ...

– b) Estimating metabolic rate

Slide 1-23

4.6.3 Nernst relation

Slide 1-24

4.6.3 Nernst relation & scale of cell

membrane potentials

• Consider now a charged situation like many cell

membranes in biology (see Fig. 4.14)

• The electric field E = ΔV/l and hence the drift

velocity is

• Now consider a flux trough area A (Fig. 4.14)

and we argue that j = c vdrift (check units) which

implies that

• Now including dissipation in Fick’s law we find

and using the Einstein relation we find

Slide 1-25

The Nernst-Planck Formula

• FQ:- what electric field will cancel out non-

uniformity in a solution?

• Ans:- Set j=0 implies which has

solution

where ΔV = EΔx

• Using real values we estimate ΔV~58 mV. Not

far off voltages observed in real cell membranes

Slide 1-26

4.6.3 Comment (from Nelson)

• D has dropped out because we are considering

an equilibrium problem

• In reality in cell membranes are non-equilibrium

Slide 1-27

4.6.4 Electrical Resistivity from Nernst

• Show that electrical resistance in solution is due

to dissipation D of random walkers (amazing)

• In Fig. 4.14 now consider placing electrodes in

NaCl solution separation d

• Now the ions in the solution won’t pile up and we

will assume c(x) is uniform which from Nernst-

Planck means that E=ΔV/d= kBT/(Dqc) j (check)

and since j is no. of ions per unit time we have

current I = qAj and hence

• Ohm’s law ΔV=IR with electrical conductivity

κ=d/(RA) where

Slide 1-28

Homework: Section 4.6.5

• Read Section 4.6.5 and do “Your Turn 4G”

– Also “Your Turn 4F” on bacterium

• Solution of diffusion equation is a Gaussian

profile (Gaussians again)

– In 1D the solution is

– In 3D follow “Your Turn 4G” or do 1D case.

• Homework question 4.7:- “Vascular Design”