4 part ii

Chapter 4 Lecture

Biological PhysicsNelson

Updated 1st Edition

Slide 1-1

Random Walks, Friction & Diffusion (part II)

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

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

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).

5©1961. Used by permission of Dover Publications.

Summary: Random Walks

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Outline

• Brownian motion

• Random walks

• Diffusion

• Friction

• Three important equations, leading to the

Fluctuation-Dissipation relation

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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

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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.

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Polymer Diffusion

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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*)

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Random Walks on Wall Street*

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4.4 – 4.6 Equations Summary

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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 is

and 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

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4.4 Diffusion cartoon

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4.4 Fick’s Law (1st Law)

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

• Q:- What drives the flux?

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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”

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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

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4.5 Functions and Derivatives

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And Snakes Under the Rug

Try to use Wolfram α to make some plots

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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)

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4.6.1 Membrane Diffusion

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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 <= I/m =

3Dc0/(ρR2) = ...

– c) use the actual metabolic activity of a

bacterium is 0.001 mole kg-1s-1 to estimate

size of bacterium (ans: - 24 μm)

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4.6.3 Nernst relation

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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

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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

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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

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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

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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”

4 part ii

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