Chapter 8 – Theoretical Biophysics 359
ΔW is the rotational energy that is required to work against the load to bring about rotation.
The associated free energy released by hydrolysis of an ATP molecule to ADP and the inor
ganic phosphate Pi can also be estimated as
(8.82)
∆
∆
G
G
k T
=
+
[
]
[
]
0
B
i
ln
ADP
P
ATP
where G0 is the standard free energy for a hydrolysis of a single mole of ATP molecule at pH
7, equivalent to around −50 pN·nm per molecule (the minus sign indicates a spontaneously
favorable reaction, and again square brackets indicate concentrations).
8.4.3 DIFFUSION-LIMITED REGIMES
In either the case of very slow diffusional processes compared to reaction kinetics or instances
where there is no chemical reaction element but only diffusion, then diffusion-limited ana
lysis can be applied. A useful parameter is the time scale of diffusion τ. For a particle of mass
m experience a viscous drag γ this time scale is given by m/γ. If the time t following obser
vation of the position of this tracked particle is significantly smaller than τ, then the particle
will exhibit ballistic motion, such that its mean square displacement is proportional to ~t2.
This simply implies that an average particle would not yet have had sufficient time to collide
with a molecule from the surrounding solvent. In the case of t being much greater than τ,
collisions with the solvent are abundant, and the motion in a heterogeneous environment is
one of regular Brownian diffusion such that the mean square displacement of the particle is
proportional to ~t.
Regular diffusive motion can be analyzed by solving the diffusion equation, which is a
subset of the reaction–diffusion equation but with no reaction component, and so the exact
analytical solution will depend upon the geometry of the model of the biological system
and the boundary conditions applied. The universal mathematical method used to solve this,
however, is separation of variables into separate spatial and temporal components. The gen
eral diffusion equation is derived from the two Fick’s equations. Fick’s first equation simply
describes the net flux of diffusing material, which in its 1D form indicates that the flux Jx of
material diffusing parallel to the x-axis at rate D and concentration C is given by
(8.83)
J
D C
x
x = −
∂
∂
Fick’s second equation simply indicates conservation of matter for the diffusing material:
(8.84)
∂
∂
= −∂
∂
C
t
J
x
x
Combining Equations 8.81 and 8.82 gives the diffusion equation, which can then also be
written in terms of a probability density function P, which is proportional to C, generalized
to all coordinate systems as
(8.85)
∂
∂=
∇
P
t
D
P
2
Thus, the 1D form for purely radial diffusion at a distance r from the origin is
(8.86)
∂(
)
∂
=
∂
∂
∂(
)
∂
P r t
t
D r
r r
P r t
r
,
,
1
2
2