Modelling compartmentalization towards elucidation and engineering of spatial organization in biochemical pathways

Abstract

Compartmentalization is a fundamental ingredient, central to the functioning of biological systems at multiple levels. At the cellular level, compartmentalization is a key aspect of the functioning of biochemical pathways and an important element used in evolution. It is also being exploited in multiple contexts in synthetic biology. Accurate understanding of the role of compartments and designing compartmentalized systems needs reliable modelling/systems frameworks. We examine a series of building blocks of signalling and metabolic pathways with compartmental organization. We systematically analyze when compartmental ODE models can be used in these contexts, by comparing these models with detailed reaction-transport models, and establishing a correspondence between the two. We build on this to examine additional complexities associated with these pathways, and also examine sample problems in the engineering of these pathways. Our results indicate under which conditions compartmental models can and cannot be used, why this is the case, and what augmentations are needed to make them reliable and predictive. We also uncover other hidden consequences of employing compartmental models in these contexts. Or results contribute a number of insights relevant to the modelling, elucidation, and engineering of biochemical pathways with compartmentalization, at the core of systems and synthetic biology.

Figure 1

Compartmentalized single modification cycle. (a) Schematic for the compartmentalized pathway. The PDE description yields steady state concentration profiles of X and X*. The compartmental ODE gives steady state compartmental concentrations of X and X*. We examine the error between the PDE compartmental averages and the ODE steady state. (b) Percentage error for each species, between the steady state compartmental averages of the PDE model, and the steady state of a compartmental model using the same total amount of substrate as the PDE model. We consider a system with two compartments, each occupying 5 percent of the whole domain. (c) In the mass action regime, if the total amount of substrate in the compartments is exactly accounted for, transport parameters can be computed, which result in an exact match. (d) In the non-mass action regime, an exact fit cannot be obtained, even if the total amount of substrate in the compartments is accounted for. Errors are shown for both the basic compartmental model calibrated with the exact total amount of substrate in the compartments under basal conditions - green circles, and the compartmental model with modified conservation to account for amount of substrate in the intermediate space – blue triangles. For the basal kinetics, both cases match the PDE closely. When the kinetic parameter is changed (phosphatase concentration in compartment 2 increases), the modified model (blue triangles) continues to give a good fit, while the basic compartmental model with calibration under basal conditions (green circles) produces large errors.

Figure 2

The transport parameters that produce an exact match with the PDE depend on kinetic parameters (non-thin compartment regime). (a) In the mass action regime, transport parameters can be computed, which can guarantee an exact match, provided that sequestration in the intermediate space is exactly accounted for. However, this exact match is not robust to changes in kinetics, even if sequestration is correctly accounted for. On the other hand, if we account for the kinetic dependence of the transport parameters, the compartmental model continues to produce an exact fit. (b) The percentage error in X ₁ (average concentration of X in compartment 1) is zero at the basal value of kinetic parameter k ₁, and increases when k ₁ is perturbed. (c) Away from the thin compartment regime, D/LL ₁ is a poor choice of transport parameter (triangles). In the non- mass action regime, an exact fit is impossible, but transport parameters may be computed by minimizing the error (sum-squared error). Transport parameters obtained by minimizing the error produce a good match (green circles - maximum error 0.6 percent). However, for these transport parameters, the ability to match the PDE is lost if the kinetics is perturbed.

Figure 3

Kinetic dependence of the transport parameters in the mass action regime. (a) The transport parameters that produce an exact match between the compartmental model and the PDE, depend on kinetics (see text). For a given change in kinetics, the corresponding variation of transport parameters depends on compartment sizes relative to the intermediate space- green line: $\frac{L}{L_1}=1$; blue line: $\frac{L}{L_1}=0.1$. (b) Using D/LL ₁ as the transport parameter produces a good approximation in the thin compartment regime (the maximum error here is about 1 percent). (c) Using D/LL ₁ as the transport parameter, away from the thin compartment regime (small L/L ₁), can result in a significant mismatch between the compartmental model and the PDE model at steady state. (d) The effect of varying $L$ and consequently L/L ₁, keeping D/L constant. Away from the thin compartment regime (small L/L ₁), the transport parameter that produces an exact match differs significantly from D/LL ₁. It approaches this value as L/L ₁ becomes large (i.e. as the compartment sizes become smaller relative to the intermediate space).

Figure 4

Simple open system, and pathways with added complexity. (a) Schematic for a simple open system with production in the first compartment and degradation in both first and second compartments. (b) For a given change in kinetics, the corresponding variation of transport parameters, that gives an exact match, depends on compartment sizes relative to the intermediate space- red line: $\frac{L}{L_1}=1$; blue line: $\frac{L}{L_1}=0.1$. (c) Open system with X produced in compartment 1 and consumed in compartments 1, 2, and 3. (d) Using the analytical solution to the PDE model, we see that modulating the kinetics in compartment 3, can alter the transport parameter (that gives an exact match) associated with transport of X between compartments 1 and 2 (tr₁₂). (e) Schematic for the comparmentalized two-step cascade.

Figure 5

Multi-site modification. (a) Schematic for compartmentalized distributive two-site modification pathway (kinase in compartment 1 and phosphatase in compartment 2). (b) With multiple diffusing species in a pathway (two-site substrate modification: see text), their associated transport parameters, that produce an exact match, need not be in the same ratio as their diffusivities, even when the species belong to the same conserved pool. In (c) and (e), where steady state concentration profiles are shown, the dashed lines mark the compartment boundaries. (c) An example of steady state concentration profiles where [X*] is clearly non-monotonic, and the difference in edge concentrations and the difference in average concentrations between compartments, have opposite signs. Note that the maximum of [X*] is inside compartment 1 and not on the boundary. (d) Plot showing how the difference in average concentrations and the difference in edge concentrations vary with kinetics (rate constant k ₄). Notice that they have opposite signs over a certain range of k ₄. (e) The two-site modification pathway shows bistability with compartmentalized enzymes (The system considered is in the thin compartment regime: $\frac{L}{L_1}=18$). Spatial concentration profiles of the modified forms, for the two different stable steady states are shown. (f) Comparing a compartmental description that systematically accounts for sequestration in the intermediate space with another which is calibrated to account for total amount of species in the compartments in an ad hoc manner (the calibration is based on one of the two steady states at basal conditions: $E_1^{Total}$ = 0.6). We find that the model with ad hoc calibration introduces significant errors in capturing the other steady state. In fact, under basal conditions, the model with ad hoc calibration does not even exhibit a second steady state.

Figure 6

Sample design problem, using compartmental models to optimally target a metabolic enzyme to a compartment. (a) Schematic of the compartmentalized metabolic pathway, with upstream pathway in compartment 1 producing a metabolite X, and reactions consuming X in both compartments, including the desired conversion (by enzyme E) and competing pathways. In the text, we examine two scenarios, one with negative feedback inhibition (depicted) and one where this is absent. (b) The case of no feedback inhibition. The choice of transport parameter is crucial for the correct prediction of fluxes for the two extreme distributions (no feedback inhibition). An incorrect transport parameter might lead to a completely misleading impression as to which distribution is better. The fluxes are shown as computed using a range of transport parameters between D/LL ₁ (which is 0.25 here) and the transport parameter accounting for the basal reactions (which is 0.026). (c) The case with feedback inhibition. There is significant disparity between the flux of product obtained from the compartmental model and the PDE. The green dots represent the flux of the desired reaction, as given by the PDE. If a transport parameter D/LL ₁ is used in the compartmental model, the disparity is substantial (red dashed line), and further, even an improved estimate of the transport parameter (accounting for the basal reactions) still leaves a clear degree of disparity (blue dashed line).