Brownian Motion as a Free Energy Minimizing Process#
Consider a suspension of a large number of ideal beads:
Goal:#
Construct a phenomenological model for the dynamics of the particle density
Based on simple and natural assumptions, we will be led to the generalized diffusion equation. One nice (and strange) feature of these dynamics is that they continuously lower the free energy.
Conservation of Particle Number#
The conservation of particle number requires that the dynamical equation of motion is described by the continuity equation:
where
The probability current has two contributions:#
Deterministic motion (drift due to potential
):where we introduced the mean velocity
of an overdamped particle in a potential . The friction coefficient for a spherical particle of diameter in a fluid of viscosity is given by:Random motion (diffusion):
Total current:#
Combining both contributions yields
which can be rewritten as
where
Note that
Generalized Diffusion Equation#
Substituting this into the continuity equation gives
Equilibrium ( ):#
At equilibrium,
Solving for
Since this must reproduce the Boltzmann distribution:
we require the Stokes-Einstein relation:
This is a key example of a fluctuation-dissipation theorem. So, indeed,
Example values:#
For a spherical particle with radius
: .Proteins (
nm): .Bacteria:
.
(For physical intuition, it’s helpful to anticipate
Generalization: Multiple degrees of freedom#
Knowing
As before, we demand a continuity equation,
where
with mobility matrix
This is known as the Smoluchowski equation.
Free Energy Minimization#
Define the free energy functional:
The free energy is monotonically decreasing under Brownian motion:
Using integration by parts and boundary conditions at infinity, we obtain:
which is negative unless
Note: The dynamics are time-irreversible due to coarse-graining and underlying microscopic chaos (see next lecture on ergodicity).
Diffusion Equation – Basic Solution:#
For
Assume
Fourier transform of (2):
Solution:
Inverse Fourier transform:
with
Note: This is the Green’s function solution. For arbitrary initial conditions, convolve with this Green’s function.
Since particles are non-interacting:
where
Practical Example: Nerve Cell Transport#
Is diffusion fast enough to transport vesicles?
Given
Example: Bacteria#
For
which is much shorter than the
Application: Diffusion-limited Reaction Rates#
Ligand capture rate:#
How many ligands are captured per sec?
Use diffusion equation to model the ligands:
$
Steady state:
Perfect absorber limit:
The particle current is
Thus, the total flux of captured ligands is given by the expression
(Check dimensions!)
Universal Bound on Reaction Rates#
Key question: How fast is
In the extreme case, where each A-B collision leads to a C particle we have
Since
where
for spherical particles. Note that this is a nice universal bound, dependent only on temperature and viscosity. For water at room temperature, one estimates
Notes:
Most enzymes are 100-1000 times slower
they require more than just one collision to form a product.… could be modeled by assuming a finite absorption rate.
Relation to Michaelis-Menten kinetics: Rate of product formation =
.
Comparing with the above equation, we see that the enzyme efficience is bounded by
For
Thus,
with:
Estimate:
Most enzymes operate 100–1000x slower.
Enzyme efficiency is thus bounded by
.
Relation to Michaelis-Menten:
where