A systems-level model reveals that 1,2-Propanediol utilization microcompartments enhance pathway flux through intermediate sequestration

The spatial organization of metabolism is common to all domains of life. Enteric and other bacteria use subcellular organelles known as bacterial microcompartments to spatially organize the metabolism of pathogenicity-relevant carbon sources, such as 1,2-propanediol. The organelles are thought to sequester a private cofactor pool, minimize the effects of toxic intermediates, and enhance flux through the encapsulated metabolic pathways. We develop a mathematical model of the function of the 1,2-propanediol utilization microcompartment of Salmonella enterica and use it to analyze the function of the microcompartment organelles in detail. Our model makes accurate estimates of doubling times based on an optimized compartment shell permeability determined by maximizing metabolic flux in the model. The compartments function primarily to decouple cytosolic intermediate concentrations from the concentrations in the microcompartment, allowing significant enhancement in pathway flux by the generation of large concentration gradients across the microcompartment shell. We find that selective permeability of the microcompartment shell is not absolutely necessary, but is often beneficial in establishing this intermediate-trapping function. Our findings also implicate active transport of the 1,2-propanediol substrate under conditions of low external substrate concentration, and we present a mathematical bound, in terms of external 1,2-propanediol substrate concentration and diffusive rates, on when active transport of the substrate is advantageous. By allowing us to predict experimentally inaccessible aspects of microcompartment function, such as intra-microcompartment metabolite concentrations, our model presents avenues for future research and underscores the importance of carefully considering changes in external metabolite concentrations and other conditions during batch cultures. Our results also suggest that the encapsulation of heterologous pathways in bacterial microcompartments might yield significant benefits for pathway flux, as well as for toxicity mitigation.

We can then use the solution in the cytosol to generate the following boundary condition at the MCP membrane: First, consider the mass balance on A M CP : And similarly for P M CP : We now assume that the concentrations in the MCP are constant since ξ >> 1. First we solve for P M CP , as this does not depend on A M CP due to the irreversibility of PduCDE. We simplify the solution by defining the following important timescales, assuming that k c = k a c = k p c and k m = k a m = k p m (Table 2): Letting p = P M CP K M CDE , λ = 1 + jc km , ρ = Rc R b , and p * = Pout K M CDE , the solution for p is therefore as follows: Furthermore, if PduCDE is saturated, We can estimate the magnitudes of these various timescales based on the baseline model parameters (Table 2) and thence analyze the magnitude of the various terms in Γ CDE .
Therefore, for the baseline model parameter values in Table 1, Suggesting that in the vicinity of the baseline parameter values, the solution for P in the MCP is governed by the relative timescales of the transport of 1,2-PD in and out of the MCP and the reaction of 1,2-PD to propionaldehyde by PduCDE, as well as by the external 1,2-PD concentration. Now we can find A M CP similarly, given the solution for If PduCDE is saturated and A out is negligible, then And if both PduCDE and PduPQ are saturated and A out is negligible, then We can analyze the relative magnitudes of the timescales in Γ P Q as above, assuming the baseline parameter values in Table 1, and we find that Suggesting that in the vicinity of the baseline parameter values, the solution for A in the MCP is governed by the relative timescales of the transport of propionaldehyde in and out of the MCP and the reaction of propionaldehyde by PduP/Q, as well as by the relative rates of PduCDE and PduP/Q. Again, the solutions in the cytosol follow directly from these MCP solutions.

Governing equations for computation
As described in the Models section, the equations describing the concentrations of P and A in the MCP are as follows: And the concentrations in the cytosol are described by the following: The following boundary conditions hold at the cell and MCP membranes, respectively:

Non-dimensional equations
We then recast the system in terms of the following non-dimensional variables: Applying the non-dimensionalization and letting κ = K CDE K P Q , we obtain the following governing equation for A in the MCP: Similarly for P in the MCP, We can then nondimensionalize the boundary conditions as follows: Similarly for P, PLOS 4/6 These nondimensional equations can then be solved numerically by a finite-difference approach to find the steady-state concentrations in the MCP, and the solutions in the cytosol follow directly. We solve the spherical finite-difference equations using the ODE15s solver in MATLAB.

Analytical solution
For ease of computation, we cast the analytical solution (assuming constant concentrations in the MCP) differently than in the Models section.
First, consider the mass balance on A M CP : And similarly for P M CP : First we solve for P M CP , as this does not depend on A M CP due to the irreversibility of PduCDE: Let E = Y − Z + 1.
Now we can find A M CP similarly, given the solution for P M CP .