Upload
others
View
9
Download
0
Embed Size (px)
Citation preview
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”