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The aim of this model was to predict a theoretical maximum number of enzyme molecules that can be packed into a single microcompartment. To get a first estimate, without taking into consideration whether this volume of protein would interrupt the biological processes in the cell, we approached this problem volumetrically.

Due to the complexity of the enzyme movements and their interactions, I simplified their structures by approximating them as ellipsoids with axes lengths calculated through modelling the monomers and predicting the structures of the FdhA tetramer and DcmA hexamer respectively.





Oxford iGEM 2014

The ellipsoid packing problem

Once treated as ellipsoids, the problem was then reduced to the classical ‘sand packing’ problem. Because the dimensions of these proteins was substantially smaller than the icosahedron (by approximately a factor of 20 in every dimension), I assumed that the geometry of the container i.e. the microcompartment, was not significant.

Another assumption made in these calculations was that the enzymes could be treated as homogenous. They are of very similar dimensions, varying by no more than 20-30% on any axis, and also have very similar sphericities- the key variable in determining the packing efficiency of the molecules. Sphericity is defined as:

φ = sphericity

V_p = volume of ellipsoid (nm^3)

A_p = surface area of ellipsoid (nm^2)

For ellipsoids, a surface area approximation was used:

a = length of axis 1 (nm)

b = length of axis 2 (nm)

c = length of axis 3 (nm)

After calculating the sphericities of the enzymes, the porosity of the system could then be determined through empirical data from literature. Because the DcmA and FdhA sphericities were very similar (0.953 and 0.981 respectively), we considered the system to be composed of a homogenous spheroid species of porosity 0.973 i.e. the weighted average of the two species.

The image in the top right is a visualization of the ellipsoids packing in the microcompartment

The graph on the right shows the empirical ellipsoid packing efficiencies as defined by sphericity

Results

Because enzyme-enzyme interactions are very difficult to predict, and it is difficult to assign an analogous friction factor to their movements, I took the average of the predicted porosities across a range of different friction factors and alternative methods giving a predicted porosity of 37.3%.

Combining this information with the relative expression rates of the two enzymes, we predict that:





We should note that these numbers are a first approximation and do not take into consideration the strain placed on the cell involved in expressing this amount of protein. This model is based entirely on volumetric and geometric laws rather than biological ones and should be considered a first assumption and theoretical maximum given optimum conditions.