Enzyme activities predicted by metabolite concentrations and solvent capacity in the cell

Experimental measurements or computational model predictions of the post-translational regulation of enzymes needed in a metabolic pathway is a difficult problem. Consequently, regulation is mostly known only for well-studied reactions of central metabolism in various model organisms. In this study, we use two approaches to predict enzyme regulation policies and investigate the hypothesis that regulation is driven by the need to maintain the solvent capacity in the cell. The first predictive method uses a statistical thermodynamics and metabolic control theory framework while the second method is performed using a hybrid optimization–reinforcement learning approach. Efficient regulation schemes were learned from experimental data that either agree with theoretical calculations or result in a higher cell fitness using maximum useful work as a metric. As previously hypothesized, regulation is herein shown to control the concentrations of both immediate and downstream product concentrations at physiological levels. Model predictions provide the following two novel general principles: (1) the regulation itself causes the reactions to be much further from equilibrium instead of the common assumption that highly non-equilibrium reactions are the targets for regulation; and (2) the minimal regulation needed to maintain metabolite levels at physiological concentrations maximizes the free energy dissipation rate instead of preserving a specific energy charge. The resulting energy dissipation rate is an emergent property of regulation which may be represented by a high value of the adenylate energy charge. In addition, the predictions demonstrate that the amount of regulation needed can be minimized if it is applied at the beginning or branch point of a pathway, in agreement with common notions. The approach is demonstrated for three pathways in the central metabolism of E. coli (gluconeogenesis, glycolysis-tricarboxylic acid (TCA) and pentose phosphate-TCA) that each require different regulation schemes. It is shown quantitatively that hexokinase, glucose 6-phosphate dehydrogenase and glyceraldehyde phosphate dehydrogenase, all branch points of pathways, play the largest roles in regulating central metabolism.

where B = A −1 . Note that the calculation of C n i, j assumes metabolite concentrations are linearly dependent on enzyme activities. This assumption can be used to isolate the change in activity, j, needed to make a change in the product concentration n i : In practice, when C n i, j >> 0.0 the assumption of a small change n i used in MCA is no longer valid and instead the current activity j is instead updated using α j,new = α j,current /5. As the cost function L (Methods Eqn. 20) approaches zero, then Eqn. 3 can be applied.

Analysis of gluconeogenesis pathway
The gluconeogenesis pathway is analyzed at low NAD/NADH ratio (0.02). The pathway has two known regulation sites fructose 1,6-bisphosphatase (FBP) and pyruvate carboxylase (PC). While both can be utilized to bring steady state metabolite concentrations into agreement with experimentally observed values, regulation of pyruvate carboxylase results in a larger energy dissipation rate (dE/dt). Regulation of alternative enzymes results in lower flux through the pathway and less energy available for use. Optimal predicted enzyme activities are in agreement for each method ( Figure S2A). Table S4 lists the complete reaction activity, flux and free energy for each respective prediction method. Figure S2. Gluconeogenesis cycle predictions with low NAD/NADH initial conditions. Predicted enzyme activities (A) and free energy (B) at terminal states are calculated using concentration control theory, shown as red 'plus's and green 'X's, respectively. Results are compared to those found using a RL approach (black square). Grey dots (C) represent the population of terminal states found while training the RL agent.

Analysis of glycolysis-TCA pathway
The glycolysis-TCA pathway is a subset of the larger glycolysis-PPP-TCA pathway discussed in the Results which includes the pentose phosphate pathway (PPP). Reducing the number of reactions limits the possible regulation schemes. When utilizing the same initial metabolite concentrations as the larger pathway when appropriate, i.e. high NAD/NADH (31.3), the regulation schemes for various methods show closer agreement. Both HEX1 and GAPD are regulated by every method as in the glycolysis-PPP-TCA pathway. The local MCA method, however, regulates PFK, PGK, and PDH, while the RL method additionally regulates PGI. Both methods regulate more reactions than the unrestriced MCA method and therefore result in a lower energy dissipation rate. Table S5 lists the complete reaction activity, flux and free energy for each respective prediction method. Figure S3. Glycolysis-TCA cycle predictions with high NAD/NADH initial conditions. Predicted enzyme activities (A) and free energy (B) at terminal states are calculated using concentration control theory, shown as red 'plus's and green 'X's, respectively. Results are compared to those found using a RL approach (black square). Grey dots (C) represent the population of terminal states found while training the RL agent.

Analysis of pathways with proxy data
When no metabolomics data is available, the methods presented here are still able to perform accurate predictive measurements in terms of enzyme regulation, steady-state metabolite concentrations and reaction flux. Instead of utilizing known metabolomics data measurements, we instead assume the target values of previously measured variable metabolites are fixed at 0.1 mM. Predictive learning is performed for the glycolysis-TCA cycle ( Figure S4) and glycolysis-PPP-TCA pathway under the same three initial conditions ( Figures S5, S6 and S7). In all initial conditions, for both pathways, the unrestricted MCA method maintains the same regulation of enzymes GAPD and HEX1. Variations occur only in the amount of regulation applied to the respective reactions. The other two methods show more variation. In the glycolysis-TCA cycle, the local MCA method regulates PYK in addition to the reactions previously regulated, while the RL method predicts additional regulation to PFK and PGM ( Figure S4).
The glycolysis-PPP-TCA pathway, on the other hand, shows more variation. Specifically, in the high NAD/NADH and low NADP/NADPH ratio initial condition, the local MCA regulates PYK in addition to the reactions previously regulated. The RL method predicts increased regulation to PGI and PGM but no longer regulates PDH ( Figure S5). Under the high NAD/NADH and high NADP/NADPH ratio initial conditions, both the local MCA and RL methods predict regulation schemes with additional reduction in activity of PYK. The RL additionally regulates G6PDH, while neither method regulates TKT1 or PYRt2m ( Figure S6). Only slight alterations are observed in enzyme activity when PFK has zero activity ( Figure S7). Figure S4. Glycolysis-TCA cycle predictions with high NAD/NADH initial conditions without experimental metabolomics data. Predicted enzyme activities (A) and free energy (B) at terminal states are calculated using concentration control theory, shown as red 'plus's and green 'X's, respectively. Results are compared to those found using a RL approach (black square). Grey dots (C) represent the population of terminal states found while training the RL agent.     Table S1. Reaction fluxes at predicted enzyme activities from MCA-local, MCA, and RL methods for the glycolysis-PPP-TCA pathway under different boundary conditions. Under unregulated conditions (main text, Figure 2A) and under High/Low conditions, flux through lower glycolysis and the TCA cycle is twice that of upper glycolysis. The reason for this is that for each molecule of glucose entering upper glycolysis, two molecules of glyceraldehyde 3-phosphate enter lower glycolysis (and consequently, the TCA cycle). Comparing across boundary conditions, the flux through HEX1, the entry point for carbon, is slightly higher under High/High conditions compared to the High/Low condition. This increased flux, however, does not flow into upper glycolysis but rather into the PPP (HEX1 is necessary for both upper glycolysis and PPP), a pathway that was not thermodynamically feasible under High/Low conditions. That is, flux at HEX1 appears to be 'up-regulated' in High/High conditions relative to High/Low conditions, but the reason for this is the more favorable overall thermodynamics of the High/High condition and cannot be attributed directly to regulation.   Table S5. Predicted enzyme activities, fluxes, and free energy for glycolysis-TCA pathway from MCA-local, MCA, and RL methods.