Transcript Lecture 21 (4/15/13) "Diffusion"

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HW #9 assigned last Friday, due Wed
4/24.
Today:
Diffusion
Why x2 = #Dt (from Equipartition Function)
When directed motion (v ≈ constant, x = vt)
is better/worse than diffusion (v not constant)
depends on how far you have to move.
short distances,
diffusion of small molecules very good.
Biological examples
Bacterial vs. Eukaryotic Cells
Oxygen transport: how close cells need to be to
Oxygen in blood in Lungs
Stopping time of Bacteria.
Diffusion
For “small” things, diffusion is a great way to get around.
For somewhat larger things, need directed motors.
Inertia does not matter for bacteria or
anything that is small / microscopic levels.
Reminder
Translation & Equipartition Theorem
Equipartition Theorem
What is velocity of water molecule at room temperature?
tcollision = ??
Diffusion: x2 = # Dt
Diffusion as a Random Walk
1-D case (first)
Particle at x = 0 at t = 0
1.Assume equally likely to step to right
as step to left.
2. Takes steps of length L every t seconds
i.e. moving with velocity between collisions ±v
(L = ±vt)
R steps/sec; total of N steps
[For now take v, t as constants : they actually depend on
size of particle, nature of fluid, temp…]
In reality, there is a distribution of step sizes,
but this model works amazingly well.
Thermal Motion: Move L
How far do particles move due to thermal motion
Derivation of <x2> = 6Dt
We cannot predict motion of individual molecules, but
can make statistical (probabilistic) arguments about
average/mean properties, as well as distribution
(standard deviation) of these properties.
Position after N steps = x N
Position after N+1 steps = x N+1
x N = x N-1  L
<xN> = 0
0 by symmetry– equally likely to
step left as right
<xN2> = <(xN-1  L)2>
<xN2> = <x2N-1>  2L <xN-1> +L2
<xN2> = <x2N-1> +L2
# depends on dimension
1-D: # = 2
2-D: # = 4
3-D: # = 6
What values for D?
Diffusion Coefficient &
Brownian Noise
Einstein– one of 3 papers each which
should have received a Nobel Prize
Stokes-Einstein Equation
D = kbT/6phr
h = viscosity (1 centipoise for water)
True where low “Reynolds number”;
where flow is laminar, where viscous forces are dominant,
and is characterized by smooth, constant fluid motion;
In contrast turbulent flow occurs at high Reynolds numbers;
Where flow is dominated by inertial forces,
which tend to produce chaotic eddies,
vortices and other flow instabilities
http://en.wikipedia.org/wiki/Reynolds_number
D = 250 um2/sec for small molecule in water
Neurons: Signals transmitted via
synapses.
Your brain:
100 billion neurons,
100 trillion
synapses
Information
flow
Pre-synaptic
Bouton
Axon
Synapse
(30-100 nm)
Post-synaptic
Spine
Axon
Valtschanoff &
Weinberg, 2003
Dendrite
How long to cross a synapse?
D = 250 mm2/sec
Nerve synapse: 0.1 mm
<x2> = 2Dt
0.01 mm2 = (2)(250 mm2/sec)t
t = 20 msec (fast!)
Diffusion is fast enough
to go across narrow synapse
Size of eukaryotes limited by
size (diffusion time of O2). As
size gets bigger, everything
happens more slowly.
Large cell: frog oocytes–
basically everything happens
slowly.
Every cell needs to be within
50-100 mm of blood supply!
Oocyte:1-2 mm!
Efficiency of Diffusion
Diffusion moves things short distances very fast!
What’s wrong? Special Relativity doesn’t allow this!
How Bacteria move
Inertia doesn’t matter for microscopic world
Life at low Reynold’s number
Why study?
1.Simple Example of F= ma
2.Doesn’t need much biology
3.Results are broadly applicable to microscopic level.
If turn off “propeller,” how far Bacteria coast?
Once force is over, no forward motion!
Class evaluation
1. What was the most interesting thing you
learned in class today?
2. What are you confused about?
3. Related to today’s subject, what would you like
to know more about?
4. Any helpful comments.
Answer, and turn in at the end of class.