Cannabis: X Diffusion – Ficks Laws

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In the first part of this course we assumed that the state of a cell at a particular point in
time could be described by the concentrations (or number of molecules in the stochastic
case) of key components. These concentrations could change over time but we implicitly
assumed that within the cell these concentrations do not vary as a function of position
inside the cell. In other words, we considered the cell as a well‐stirred biochemical
reactor. For small cells such as Escherichia coli this is often a valid assumption
particularly when we are interested in dynamical phenomena that occur at a much
slower time‐scale than the relatively fast random mixing by diffusion. However in larger
cells such as eukaryotic cells and even sometimes in bacteria we cannot assume that the
cell interior is well‐stirred. As a starting point for building up the mathematical tools
necessary to model spatial inhomogeneous system we use Fick’s laws:

Fick’s first law

Consider Fig. 24 illustrating particles moving along one dimension x. The particles are
randomly moving. Assume there are N(x) particles in the gray region on the left of area
A and N(x+Δx) particles in the gray region on the right of area A. How many particles will

cross the area A to the right? Since the probability to travel to right or the left is
identical 0.5N(x) particles will travel to the right. However 0.5N(x+Δx) will travel to the
left and cross area A. Therefore the net number of crossing to the right is:

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