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 CellsOxygen transport: how close cells need to be to

Oxygen in blood in LungsStopping time of Bacteria.

Diffusion

Inertia does not matter for bacteria or anything that is small / microscopic levels.

For “small” things, diffusion is a great way to get around.

For somewhat larger things, need directed motors.

Equipartition Theorem

Translation & Equipartition TheoremReminder

What is velocity of water molecule at room temperature?

collision = ??

Diffusion: x2 = # DtDiffusion 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 secondsi.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 LHow 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.

<xN> = 0 0 by symmetry– equally likely to step left as right

<xN2> = <(xN-1 L)2>

<xN2> = <x2

N-1> 2L <xN-1> +L2

<xN2> = <x2

N-1> +L2

Position after N steps = x N

Position after N+1 steps = x N+1

x N = x N-1 L

<…> means average if we look at many molecules, or for 1 molecule many times, after each time, for N steps

<x12> = <(x0

2> + L2

<x22> = <x1

2> + L2 = <x02> + 2L2

<x32> = <x2

2> + L2 = <x02> + 3L2

<xN2> = <xo

2> + NL2

xN2> = <xN

2> - <xo2> = NL2

<xN2> = NL2

The average distance <x> = √N L

If N =( steps/time)(time) = Rt (= t/where= 1/R)

xN2> = RtL2

So distance (average deviation) grows like the √ of time

Let’s define D = RL2/2 (= L2/2) : (2 is convention)

D = diffusion constant = distance2/time = cm2/sec

xN2> = 2Dt

In m-dimensions: rN2> = 2mDt (H.W.)

Continuing…

If in 1 sec, it has gone q distance, then in 2 sec it’s gone?

As a function of time, width increases as the √t

What values for D?Diffusion Coefficient & Brownian Noise

Einstein–one of three 1905 papers, each which should have received a Nobel Prize

Stokes-Einstein EquationD = kbT/6r = kbT/f

= viscosity (1 centipoise for water)r = radius of beadf = frictional coefficient

True where “Reynolds number” is low; Flow is sufficiently slow that don’t have eddies, vortexes…is characterized by smooth, constant fluid motion…flow is laminar, where viscous forces are dominant.

The Reynolds number is defined as the ratio of inertial forces to viscous forces and consequently quantifies the relative importance of these two types of forces for given flow conditions.

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

D = 250 um2/sec for small molecule in water

Your brain: 100 billion neurons, 100 trillion

synapsesInformation

flow Pre-synaptic Bouton

Post-synaptic Spine

Valtschanoff & Weinberg, 2003

Synapse (30-100 nm)

Neurons: Signals transmitted via synapses.

Dendrite

Axon

Axon

D = 250 m2/secNerve synapse: 0.1 m <x2> = 2Dt

0.01 m2 = (2)(250 m2/sec)tt = 20 sec (fast!)

How long to cross a synapse?

Diffusion is fast enough to go across narrow synapse

D: diffusion constant, <x2> = 6 Dt

If molecule gets bigger, x and D

or

D small molecule , e.g. O2 = 1000 um2/sec: (D=1/f)

Dsucrose = 300 um2/sec (D=1/f)

How long does it take for O2 to go from the edge of a cell to the middle? How big cell? 20 um

<x2> = 6Dt

t = <x2>/6D = 16 msec

Ultimately limits the maximum metabolic rate.

Bacterial cell ~ 1-3um in size, eukaryotic 10-50 um

Metabolism of bacteria much higher than eukaryotes.

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 m of blood supply!

Oocyte:1-2 mm!

Efficiency of DiffusionDiffusion moves things short distances very fast!

What’s wrong? Special Relativity doesn’t allow this!

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.