Due to increased concentration of metabolic enzymes

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Prevents interference with other cell metabolic activity

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Stochastic Reaction-Diffusion models

Because of the relatively small number of molecules we are expecting to have in our cells (≈10^5 enzymes per cell and 10^3 per microcompartment), we developed stochastic reaction-diffusion models to predict the distribution of formaldehyde within the system. These stochastic models build in an element of randomness that reflects the nature of diffusion for systems with few elements in a way that deterministic relationships such as Fick’s law do not.

We approached this problem in a number of different ways. Initially, we built a system in which molecules would move a scaled random distance selected from a normal distribution at every time interval dt. This was adapted from the Smoluchowski equations which state:

X(t+ ∆t)=X(t)+ √2D∆t ε
Y(t+ ∆t)=Y(t)+ √2D∆t ε
Z(t+ ∆t)=Z(t)+ √2D∆t ε

X(t),Y(t),Z(t) = particle co-ordinates at time t

D = diffusion constant

ε = normally distributed random variable

∆t = small time interval

Plotted above are the trajectories of three molecules in 3-D diffusion according to the Smoluchowski equations.

Modelling the effect of introducing a microcompartment on reaction rate

Our main approach to reducing the accumulation of toxic intermediates in the DCM degradation pathway is the expression of Pdu microcompartments in our bacteria that will contain high concentrations of both the DcmA enzyme and FdhA enzyme. In doing so, this increases the likelihood of a formaldehyde molecule produced by the action of DcmA encountering an FdhA enzyme before it leaves the microcompartment thus reducing the accumulation of formaldehyde.

To model the increased likelihood of reaction, we started with the Smoluchowski diffusion model above but discretized the system such that molecules could only occupy a fixed set of co-ordinates. This was done in order to define a reaction as occurring whenever two molecules occupied the same co-ordinate at the same time. While this does not necessarily ensure a reaction in the real system, we are assuming that collision rate is proportional to reaction rate and can therefore be used as an analogy.

Plotted here are two systems- one with complete freedom of molecular movement and one in which spatial constraints have been placed such that molecules which encounter the boundaries are reflected back into the system to represent the presence of the microcompartment:



















In the models above, collisions are indicated through red marks. As suspected, the likelihood of a collision in a spatially constrained environment are far higher than in one where the molecules have complete freedom of movement. This is particularly true not only because of the region of movement allowed by the microcompartment but also the initial distance between the enzyme and substrate at the point of substrate formation. Note also that the rate of diffusion of the enzyme, in green, has been made substantially lower than that of the formaldehyde- the far lighter and therefore more diffusive compound.

3D collision model

The 1-D collision model described above was quite easily expanded into a 3-D collision model that is more representative of the actual system. This results in simulations which appear far more realistic.

Illustrated above are the molecular trajectories of two different species- enzyme and substrate. The enzymes (red) are constrained to move within the micro-compartment and are reflected at the micro-compartment boundaries while the substrates (in blue) can freely diffuse anywhere in the system. While these simulations are very similar to the 1-D simulations previously described, they are far more computationally laborious.

Although the original plan was to run the simulation for several hundred enzymes and substrates and study the total number of collisions achieved, collisions are used as an analogy for reaction, it was quickly apparent that using this simulation to provide analytic results on the scales we wanted required far too much computation.

Given more time, access to more powerful computers and economization of our Matlab scripts, it would be possible to run this simulation for greater numbers of molecules than have been done thus far. This would then yield more data on the proportional increase in rate that results from introducing the micro-compartment into the system.

What we suspect is harmful to cell in degradation pathway ie. Formaldehyde and why it is bad

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Colocosation of FdhA --> faster reaction rate

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Driving the equilibrium forward by removing product

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Predicted effect of microcompartments on formaldehyde concentration

Having deduced that microcompartments will increase the rate at which intermediate compounds are degraded, the next step was to create a simulation that would predict how microcompartments would therefore affect the concentration of formaldehyde molecules in the system. To do so, we created a 1-D simulation in which we started with a fixed number of molecules while constraining degradation and production of a species to within pre-defined spatial limits- representing the fact that both these phenomenon can only occur in the microcompartment in our actual system.

In our system, the relative likelihood of degradation is far greater than that of production- this is done in order to ensure that accumulation of toxic compounds does not occur. The increased rate of degradation is the result of several factors:

1. Greater relative expression of FdhA than of DcmA- their expression ratios have been defined as approximately 2.5:1

2. K_catin FdhA is substantially higher than in DcmA

3. The effect of the microcompartment will increase the relative likelihood of degradation of formaldehyde while leaving the rate of DCM degradation unchanged.

What our models suggest is that the microcompartments will not constrain the concentration of intermediates to be high only within the system. This is because the rate of diffusion of formaldehyde through the microcompartment barrier is unaffected because its molecular size is much smaller than the pore size of the microcompartment. The significance of introducing the microcompartment is in fact to further increase the relative probability of degradation. This results in, as we expected, a net decrease in the total number of formaldehyde molecules, even when a high initial concentration is introduced, coupled with a decrease in concentration gradient throughout the system.

Displayed below is one realization of a stochastic simulation (grey) alongside the deterministic response (red) of intermediate concentration at two points in time- one at 30 a.u. and the other at 200 a.u..

Oxford iGEM 2014