Structure-Based Models of NO Diffusion in the Nervous System - Computational Neuroscience: A Comprehensive Approach - page 98

4.2 Methods
This section gives a detailed overview of the methods used to model the diffusion of
NO in the CNS.

4.2.1 Equations governing NO diffusion in the brain

The rate of change of concentration in a volume element of a membrane,
within the diffusional field, is proportional to the rate of change of con-
centration gradient at that point in the field. Fick's second law (Fick
1855)

The equations governing diffusive movement can be understood by considering
the motion of individual molecules. In a dilute solution, each molecule behaves
independently of the others as it rarely meets them, but is constantly undergoing col-
lisions with solvent molecules which move it in random directions. Thus its path
can be described as a random walk ∗ resulting in a net transfer of molecules from
high to low concentrations at a rate proportional to the concentration gradient. This
process is captured by what is commonly known as Fick's first law, that in isotropic
substances the rate of transfer of diffusing substance through unit area of a section
is equal to the product of the diffusion coefficient, D , and the concentration gradient
measured normal to the section [8]. While in some cases D depends on concentra-
tion, it can be taken to be constant for dilute solutions [8]. As this is the case for
diffusion of NO in the brain [45], we will only consider these situations. Represent-
ing the concentration at a point x and time t as C , the following equation for

diffusion in the brain (Fick's second law) can then be derived from Fick's first law:

( x , t )

D ∂

2 C 2 C 2 C

C ( x , t ) ( x , t )

+ ∂

( x , t )

+ ∂

( x , t )

= (4.1)

t x 2 y 2 z 2

or, more generally:

C ( x , t )

2 C

= D ( x , t ) (4.2)

t

While the above equations govern the diffusive element of NO's spread, they do
not take into account its destruction. NO does not have a specific inactivating mech-
anism, and is lost through reaction with oxygen species and metals, as well as heme
containing proteins [25, 44]. This means that the movement of other molecules and

∗ Diffusion processes are thus amenable to Monte Carlo methods, where a (in the case of diffusion) uni-
form probability distribution, representing the probability of a molecule moving in a given direction,
together with a random number generator are used to calculate the path of each molecule. However, the
relatively long running times to achieve a good approximation render this method inappropriate for our
needs [1].