360 8.4 Reaction, Diffusion, and Flow
Similarly, the 1D form in Cartesian coordinates parallel to the x-axis is
(8.87)
∂(
)
∂
=
∂
(
)
∂
P x t
t
D
P x t
x
,
,
2
2
This latter form has the most general utility in terms of modeling many real biophysical
systems. Separation of variables to this equation indicates solutions of the form
(8.88)
P x t
A x B t
D
B t
B t
A
x
A x
,
(
) =
( ) ( )
∴
( )
( )
=
( )
( )
′
′′
1
The left-hand side and right-hand side of this equation depend only on t and x, respectively,
which can only be true if each is a constant. An example of this process is 1D passive diffusion
of a molecule along a filament of finite length. If the filament is modeled as a line of length L
and sensible boundary conditions are imposed such that P is zero at the ends of the rod (e.g.,
where x = 0 or L), this leads to a general solution of a Fourier series:
(8.89)
P x t
A
nx
L
D
n
L
t
n
n
,
(
) =
=
∞
∑
1
2
sin
exp
π
π
Another common 1D diffusion process is that of a narrow pulse of dissolved molecules at
zero x and t followed by diffusive spreading of this pulse. If we model the pulse as a Dirac
delta function and assume the “container” of the molecules is relatively large such that typical
molecules will not encounter the container boundaries over the time scale of observation,
then the solution to the diffusion equation becomes a 1D Gaussian function, such that
(8.90)
P x t
Dt
x
Dt
,
(
) =
−
1
4
4
2
π
exp
Comparing this to a standard Gaussian function of ~exp(−x2/2σ2), P is a Gaussian function
in x of σ width equal to √(2Dt). This is identical to the rms displacement of a particle with
diffusion coefficient D after a time interval of t (see Equation 2.12). Several other geometries
and boundary conditions can also be applied that are relevant to biological processes, and for
these, the reader is guided to Berg (1983).
Fick’s equations can also be easily modified to incorporate diffusion with an additional
scaler drift component of speed vd, yielding a modified version of the diffusion equation, the
advection–diffusion equation, as in 1D Cartesian coordinates as
(8.91)
∂(
)
∂
=
∂
(
)
∂
−
∂(
)
∂
P x t
t
D
P x t
x
v
P x t
x
d
,
,
,
2
2
It can be seen that the advection–diffusion equation is a special case of the Fokker–Planck
equation for which the external force F is independent of x, for example, if a biomolecule
experiences a constant drift speed, then this will be manifested as a constant drag force. The
resulting relation of mean square displacement with time interval for advection–diffusion is
parabolic as opposed to linear as is the case for regular Brownian diffusion in the absence of
drift, which is similar to the case for ballistic motion but over larger time scales.
A valuable, simple result can be obtained for the case of a particle exhibiting diffusion-
to-capture. We can model this as, for example, a particle exhibiting regular Brownian