random walks, friction & diffusion (part ii) · 2012. 7. 4. · biological physics nelson updated...
TRANSCRIPT
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Chapter 4 Lecture
Biological PhysicsNelson
Updated 1st Edition
Slide 1-1
Random Walks, Friction & Diffusion (part II)
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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
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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
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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).
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Slide 1-5©1961. Used by permission of Dover Publications.
Summary: Random Walks
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Slide 1-6
Outline
• Brownian motion
• Random walks
• Diffusion
• Friction
• Three important equations, leading to the
Fluctuation-Dissipation relation
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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
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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.
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Slide 1-9
Polymer Diffusion
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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*)
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Slide 1-11
Random Walks on Wall Street*
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Slide 1-12
4.4 – 4.6 Equations Summary
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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
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Slide 1-14
4.4 Diffusion cartoon
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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?
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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”
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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
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Slide 1-18
4.5 Functions and Derivatives
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Slide 1-19
And Snakes Under the Rug
Try to use Wolfram α to make some plots
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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)
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Slide 1-21
4.6.1 Membrane Diffusion
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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
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Slide 1-23
4.6.3 Nernst relation
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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
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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
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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
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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
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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”