Team:CLS CLSG UK/Cocaine diffusion

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Cocaine diffusion

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Cocaine diffusion

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• Constants Model

Cocaine diffusion

Constants

J - the diffusion flux

D - the diffusion flux

C_outside - the concentration of cocaine outside the cell

C_inside - the concentration of cocaine inside the cell

x - the membrane width

P - permetivity coefficient

A - surface area of the cell

K - partition constant

n - number of moles of cocaine

Cocaine diffusion model

Our initial model aimed to calculate the rate of diffusion across the E.coli membrane. It is applicable for passive diffusion of any nonpolar molecule across a cell with a phospholipid bilayer. Cocaine is non polar and moderately hydrophobic, a quantity measured by its partition constant, K, the equilibrium constant for its partition between oil and water. Because the viscosity inside the phospholipid bilayer is far greater than water, movement across the layer is the rate determining step in the process.

Diffusion is normally modelled using Fick’s Law but this is a complicated partial differential equation. In one dimension the equation is:

J here is the diffusion flux (i.e. the amount of a substance passing through a certain cross-sectional area in a certain amount of time, which is essentially 1/A∙dn/dt). Equally for steady state diffusion, across a membrane the flux is equal to -P(C_inside-C_outside) which means equation 1.1 can be rewritten as:

Equation 1.2 becomes:

Both sides of the equation can then de divided by the volume of the E. coli to give the differential dC_Inside/dt on the left hand side.

If we assume that the concentration outside the cell is constant, since the cocaine being constantly replenished, the only variable becomes C_inside. Rearranging gives:

Where B is the constant of integration. Initially the time is 0 and the concentration inside the cell is 0. With these values equation 1.6 gives:

We couldn’t find a definite value for K but we knew from our research that it was greater than 1. With 1 being the minimum value, diffusion still occurs rapidly and assuming it is more on the scale of hundreds, as most seem to agree, then diffusion is almost instant. Putting in the relevant constants gives this graph 1 which shows that all the diffusion occurs extremely rapidly. We used this to draw the approximation that diffusion across the cell membrane is instant and so every time cocaine is broken down in the cell it is instantly replaced.

Graph 1

With this approximation, the concentration of cocaine inside the cell is always constant at a value equal to that outside the cell, 3.297 nM. This is because every time cocaine binds to the substrate, under our assumption, more cocaine will instantly replace it and if it unbinds, cocaine will diffuse out of the cell instantly.