Diffusion
Why is diffusion important?
Fundamental physical property of biological signals
Sets a fundamental limit to signal speed
•Single channel currents
•Second messenger signalling
•Chemical signalling across the synaptic cleft
•Action potential generation
•Action potential propagation
The basic framework:
The random walk– Brownian motion
What you should take home:
When is diffusion sufficient as a signalling mechanism?
-- how far does something diffuse in a given time?
-- how fast can diffusion move stuff around?
How do we go from single molecule dynamics to a macroscopic description?
Diffusion and random walks
1 mm
Nature (2002) 417: 649-653
How fast does the particle move?
How far does the particle move in time t ?
How fast does the particle move?
Its motion, or kinetic energy, is determined by thermal energy
• Thermal energy (1D) is kT/2:
k is Boltzmann’s constant
T is absolute temperature (in Kelvin)
at room temperature, kT ≈ 4 x 10-14 g/cm2/sec2
• Kinetic energy (1D) is mv2/2:
m is mass (e.g. Na+ ion, m = 10-22 g)
v is velocity
Thus, the mean square velocity is
vrms 2 = <v2> ≈ kT/m
With numbers similar to above, the root-mean-square velocity is
vrms ≈ 2 x 104 cm/sec
… very fast!! Is this consistent with what you saw in the movie?
How far does the particle move?
Model diffusion as a random walk (Einstein).
Model assumptions:
-- in t seconds, the particle moves
= vrms t.
-- it may collide every t seconds
and has equal probability of being
bumped in either direction.
Collisions do not change speed
(perfectly elastic)
Additionally,
-- particles are assumed to be
independent
Random walk simulations
How far does the particle move?
Mean position at time t is <x(t)> = 0
RMS position at time t is √<x2(t)> = (2Dt)1/2
Diffusion constant D = 2/2t
For a typical ion, D ≈ 10-5 cm2/sec or 1mm2/msec
Diffusion simulation
Diffusion is effective for short distances and timescales on the order of hundreds of
milliseconds
A “soma”, 500 msec random walk
An axon-like structure, 500 msec random walk
Particle can diffuse across a cell (10-4 cm) in ~ 0.5 msec
But to travel 1 cm takes 5 x 104 sec, or 14 hours!
What other factors might matter for the effectiveness of diffusion?
Dimensionality
We have thought mostly about the 1D case, but we could have either 1, 2 or 3D situations.
Same rules of motion apply in each direction.
In 2D:
<r2> = <x2> + <y2> = 4Dt
In 3D:
<r2> = 6Dt
Note that the distance covered increases linearly,
but the volume to be explored increases exponentially
Macroscopic equations for diffusion
The random walk model is for single particles.
What happens when we have many of them?
Fick’s first law:
J(x) = -D dC(x)/dx
Fick’s second law:
∂C(x,t)/ ∂t = D ∂2C(x,t)/ ∂2x
Flux:
J(x)
number of particles
per unit area per unit time
Diffusion from a point source
Inject a bolus of dye at r = 0 at t = 0:
C(r,t) = A exp(-r2/4Dt)
Comparison of simulations with solution of Fick’s equation
How often will diffusion lead to encounters?
Allow a spherical object to begin absorbing particles at time t = 0
Concentration C0
For a large container,
C(r) = C0 (1 – a/r)
where
r = distance from center of sphere
a = sphere radius
Rate of arrival at sphere surface-- flux from Fick’s law:
J = -D dC/dx = -D a C0 /r2
Rate of arrival is flux x surface area:
R = -J(r = a) * 4 p a2 = 4 p aD C0
Absorption by a disk: single channel currents
Rate of diffusional encounters: R = 4DaC0
Radius a
With C0 = 100mM = 10 -4 moles/cm 3
a = 10 Angstroms = 10 -7 cm
R = 4 x 10 -16 moles/sec
= 2.4 x 10 8 ions/sec
So pure diffusion can give a current of
I = 38pA for Na+