A general framework for thermodynamically consistent parameterization and efficient sampling of enzymatic reactions

Abstract

Kinetic models provide the means to understand and predict the dynamic behaviour of enzymes upon different perturbations. Despite their obvious advantages, classical parameterizations require large amounts of data to fit their parameters. Particularly, enzymes displaying complex reaction and regulatory (allosteric) mechanisms require a great number of parameters and are therefore often represented by approximate formulae, thereby facilitating the fitting but ignoring many real kinetic behaviours. Here, we show that full exploration of the plausible kinetic space for any enzyme can be achieved using sampling strategies provided a thermodynamically feasible parameterization is used. To this end, we developed a General Reaction Assembly and Sampling Platform (GRASP) capable of consistently parameterizing and sampling accurate kinetic models using minimal reference data. The former integrates the generalized MWC model and the elementary reaction formalism. By formulating the appropriate thermodynamic constraints, our framework enables parameterization of any oligomeric enzyme kinetics without sacrificing complexity or using simplifying assumptions. This thermodynamically safe parameterization relies on the definition of a reference state upon which feasible parameter sets can be efficiently sampled. Uniform sampling of the kinetics space enabled dissecting enzyme catalysis and revealing the impact of thermodynamics on reaction kinetics. Our analysis distinguished three reaction elasticity regions for common biochemical reactions: a steep linear region (0> ΔGr >-2 kJ/mol), a transition region (-2> ΔGr >-20 kJ/mol) and a constant elasticity region (ΔGr <-20 kJ/mol). We also applied this framework to model more complex kinetic behaviours such as the monomeric cooperativity of the mammalian glucokinase and the ultrasensitive response of the phosphoenolpyruvate carboxylase of Escherichia coli. In both cases, our approach described appropriately not only the kinetic behaviour of these enzymes, but it also provided insights about the particular features underpinning the observed kinetics. Overall, this framework will enable systematic parameterization and sampling of enzymatic reactions.

Fig 1. General framework for thermodynamically consistent parameterization and efficient sampling of metabolic reactions.

(A) General Reaction Assembly and Sampling Platform (GRASP) workflow. The steps indicated by * are only required for parameterizing and sampling allosteric reactions. (B) Example of pattern constraints present in a random-order Uni-Bi mechanism with the formation of a ternary complex (EPQ). The intermediate EPQ splits to the EP and EQ enzyme intermediates. This behaviour can be modelled by uniformly sampling the solution space for the steady-state elementary net fluxes $v_{\text{elem}}^{\text{net}}$, which captures the stoichiometric properties of the reaction pattern. (C) Illustration of energetic constrains in the previous mechanism. There are 2 possible cycles converting A into P+Q, namely: E → EA → EPQ → EP → E and E → EA → EPQ → EQ → E. According to the principle of microscopic reversibility both pathways must be energetically equivalent as they execute the same reaction. Thermodynamic constraints for both paths are illustrated in the free energy graph. (D) Thermodynamic constraints on the equilibrium allosteric constant (L) within the generalized MWC model. The value of L depends on the ligand affinity of the active and tense states. Notably, in the absence of ligand the allosteric constant L favours the tense state, whereas with increasing concentrations of ligand the active state becomes more favoured (lower conformational energy).

Fig 2. Energetic landscape of the contributions of the catalytic and binding terms for ordered reaction mechanisms with different molecularities.

Molecularities analysed include (A) Uni-Uni, (B) Bi-Bi and (C) Ter-Ter reaction mechanisms. In each case, the following Gibbs free energy differences were considered: -1, -5, -10, -20, -30, -50 and -80 kJ/mol and a reference flux of 1 mM/min was assumed. For this analysis, 10⁴ parameter sets were sampled. The blue and red lines denote the median of the distributions for the catalytic and binding terms of the reaction energetics, respectively, while the shaded areas represent the 95% confidence regions for the respective contributions. The black line represents the sum of the two contributions which yields ΔG _r/RT. As it can be observed, both contributions decrease linearly with decreasing ΔG _r/RT. In addition, the higher the molecularity the higher the relative importance of the binding term. The numbers showed above each line denote their slope and quantify the extent of their energetic contributions. Notably, the sum of all contributions must be one.

Fig 3. Revealing the impact of thermodynamics on enzyme kinetics.

(A) Schematic representation of the bimolecular mechanisms considered for the analysis. (B) Substrate elasticities for three mechanisms considered: ordered (green), ping-pong (blue) and random-order (red) at different Gibbs free energy differences of reaction ranging from -1 to -80 kJ/mol. Elasticities were calculated every -1 kJ/mol interval by sampling 10⁴ instances each time. In this panel, each line represents the median of the respective elasticity distribution, while the shaded areas denote the 95% confidence regions. Depending on the chosen thermodynamic reference state, the elasticities can be almost constant for regions far from equilibrium (ΔG _r <-20 kJ/mol) or highly variable close to equilibrium (dashed blue line). The dashed green line represents the limit for the lineal regime of substrate elasticity variation, while the dashed red line denotes the zero limit. (C) Product elasticities for the same representative bimolecular mechanisms. The lines and shaded areas represent the same as for the substrate elasticities. Product inhibition is on average negligible for favourable thermodynamic conditions, i.e. ΔG _r < -30 kJ/mol (right side of the dashed blue line).

Fig 4. Schematic representation of the mnemonic model for the mammalian glucokinase.

This model proposes a slow conformational transition from a low-affinity state (E ^*) to a high-affinity state (E) that can be enhanced with increasing glucose concentration yielding the observed cooperative behaviour.

Fig 5. Sampling monomeric cooperativity of mammalian glucokinase.

(A) Probability distribution of the Hill coefficient for the glucokinase. Approximately 93% out of 10⁴ sampled kinetics displays cooperative behaviour. (B) Probability distribution for the ratio of the forward and reverse rate constants from the low- to the high-affinity enzyme state. The great majority (98.1%) of the models exhibiting positive cooperativity agrees with a slow transition from the low- to the high-affinity enzyme states. (C) Comparison of the kinetic space described by the model developed by Storer and Cornish-Bowden [48] (blue) and the sampled kinetics using the mnemonic model (red). Each line represents the real kinetic behaviour described by the two models, while the shaded areas denote one-standard-deviation confidence regions for each approach.

Fig 6. Complex allosteric regulatory interactions control the kinetic behaviour of phosphoenolpyruvate carboxylase (PEPC) in E. coli.

(A) PEPC regulates anaplerotic metabolism in E. coli through several allosteric interactions. This enzyme carries out the conversion of phosphoenolpyruvate and bicarbonate into oxaloacetate and inorganic phosphate. All reactants are highlighted in black. The red and green dashed lines denote the inhibition and activation exerted by different effectors, respectively. (B) Ordered Bi-Bi mechanism for PEPC catalysis. Phosphoenolpyruvate, bicarbonate, oxaloacetate and orthophosphate are denoted by the abbreviations pep, hco ₃ ⁻, oaa and p _i, respectively. (C) Mechanism of synergistic activation of PEPC transitions proposed by Smith et al. [57]. This mechanism considers two separate binding sites for the two types of activators (allosteric sites) and another for the substrates (catalytic site), each of them being capable of independently interacting with different enzymatic complexes. Notably, this model assumes the existence of the relaxed enzyme (active) only in the presence of activators, one of which may be the substrate pep.

Fig 7. Modelling the regulatory co-activation behaviour of PEPC upon FBP and Acetyl-CoA binding.

(A) Comparison of the kinetic behaviour described by the empirical model of Lee et al. [59] (blue line) and the best sampled kinetics using our framework (red line) with and without Acetyl-CoA. The substrates concentrations used in each case were: 2 mM PEP, 10 mM bicarbonate, 15 mM FBP and 0.63 mM acetyl-CoA, which correspond to the physiological concentrations found in E. coli in glucose-limited cultures [58]. The red shaded area denotes a 99% confidence region for 10⁴ instances sampled. (B) Comparison of the kinetic behaviours including additional information during the sampling. The legends and concentrations used are the same as in (A). (C) Hill coefficient curves for the most accurate PEPC model sampled in the presence and absence of FBP and acetyl-CoA at physiological concentrations. The coloured circles denote experimentally determined Hill coefficients under conditions resembling physiological concentrations of PEP, FBP and acetyl-CoA [55]. (D) Regulatory kinetic behaviour described by the PEPC model with 15 mM of FBP (black circles), i.e. physiological condition, and 0.45 mM of FBP (black squares), i.e. carbon-starvation condition, for different acetyl-CoA concentrations. The dashed line represents the physiological concentration of 0.63 mM Acetyl-CoA found in E. coli under normal physiological conditions.