Network-level allosteric effects are elucidated by detailing how ligand-binding events modulate utilization of catalytic potentials

Allosteric regulation has traditionally been described by mathematically-complex allosteric rate laws in the form of ratios of polynomials derived from the application of simplifying kinetic assumptions. Alternatively, an approach that explicitly describes all known ligand-binding events requires no simplifying assumptions while allowing for the computation of enzymatic states. Here, we employ such a modeling approach to examine the “catalytic potential” of an enzyme—an enzyme’s capacity to catalyze a biochemical reaction. The catalytic potential is the fundamental result of multiple ligand-binding events that represents a “tug of war” among the various regulators and substrates within the network. This formalism allows for the assessment of interacting allosteric enzymes and development of a network-level understanding of regulation. We first define the catalytic potential and use it to characterize the response of three key kinases (hexokinase, phosphofructokinase, and pyruvate kinase) in human red blood cell glycolysis to perturbations in ATP utilization. Next, we examine the sensitivity of the catalytic potential by using existing personalized models, finding that the catalytic potential allows for the identification of subtle but important differences in how individuals respond to such perturbations. Finally, we explore how the catalytic potential can help to elucidate how enzymes work in tandem to maintain a homeostatic state. Taken together, this work provides an interpretation and visualization of the dynamic interactions and network-level effects of interacting allosteric enzymes.

HEX mechanism. Hexokinase (HEX) was modeled as a monomeric enzyme with a single active site. Inhibition was carried out by ADP, 2,3-DPG, and G6P (6)(7)(8). We have here used the notation HEX * to denote the relaxed (i.e., active) forms of HEX and the notation HEX † to denote the tense (i.e., inactive) forms of HEX. The reaction mechanism mimicked that used by Du et al. (2) and is detailed in Eqs.
The active fraction for HEX (f HEX A ) is given by the summation of all active forms of the enzyme (HEX * and bound species) over the total enzyme, E T , given by: where λ = HEX * +HEX * -ATP+HEX * -GLC+HEX * -ATP-GLC.
PYK mechanism. Pyruvate kinase (PYK) was modeled as a tetrameric enzyme with allosteric activation by FDP and allosteric inhibition by ATP. This reaction mechanism allows these regulators to bind to four sites distal to the active sites. Because the reaction catalyzed by PYK is very close to irreversible (9), we have here modeled the last catalytic reaction to be irreversible.
We have here used the notation PYK * i to denote the relaxed (i.e., active) form i of PYK and the notation PYK † i to denote the tense (i.e., inactive) form i of PYK; i indicates the number of allosteric activators (FDP) or inhibitors (ATP) bound to the enzyme. The Monod-Wyman-Changeux (MWC) reaction framework (5) was adopted for PYK, wherein the allosteric activator and inhibitor can only bind to the relaxed and tense state, respectively. The full reaction schema is detailed in Eqs. (35)-(68): The active fraction for PYK (f PYK A ) is given by the summation of all active forms of the enzyme (PYK * i and bound species) over the total enzyme, E T , given by: Hemoglobin and the Rapoport-Luebering Shunt. The Rapoport-Luebering (RL) Shunt accounts for the presence of hemoglobin (HB), whose binding to oxygen is regulated by 2,3-diphosphoglycerate (2,3-DPG). We have modeled HB using a cooperative mechanism (i.e., the affinity for oxygen increases with more bound oxygen) with allosteric inhibition by 2,3-DPG (3). We have here used the notation HB * i to denote the relaxed (i.e., active) form i of HB and the notation HB † to denote the tense (i.e., inactive) form; i indicates the number of oxygen species bound to HB. The full reaction schema is detailed in Eqs. (70)-(77): Here, equation 77 represents an exchange reaction with extracellular oxygen. The active fraction for HB (f Hb A ) is given by the summation of all active forms of the enzyme over the total enzyme, E T , given by: where λ = HB * 0 + HB * 1 + HB * 2 + HB * 3 + HB *

Disturbance rejection capabilities of models with regulation.
The inclusion of feedback and other regulatory mechanisms are designed to improve the disturbance rejection capabilities of a system (10). For biological systems, regulatory mechanisms are expected to enable organisms to maintain a robust homeostatic state in the face of environmental perturbations (11,12). We therefore constructed nine different models representing every combination of enzyme modules (e.g., base model plus PFK, base model plus PFK and HEX). We then (1) investigated the capacity for each of these models to help maintain the homeostatic state and (2) examined how understanding the catalytic potentials help elucidate this ability. We simulated a 50% increase in ATP utilization for 100 hours and calculated the total ATP flux in the network (i.e., total flux through ATPproducing reactions minus total flux through ATP-consuming reactions) for each of the models constructed (Fig A). All systems were able to maintain a stable homeostatic state following the perturbation (Fig A). We calculated the sum of squared error (SSE) for each model in order to quantify the disturbance rejection capabilities of each model (Fig A). As expected, the models with little or no regulation performed the worst, while increased regulation generally lowered the SSE. The base glycolytic model with the PYK module performed the worst, while the model containing the PFK and HEX modules with hemoglobin performed the best. The final steady-state values for the energy charge differed with the inclusion of hemoglobin in the model, although the magnitude of these differences was small (Fig A).
The disturbance rejection capabilities of the models improved with the incorporation of additional regulatory mechanisms (Fig A). We simulated physiologically-relevant perturbations, observing that systems with regulation are improved over those with less regulation (i.e., fewer modules) as shown by quantifying the total deviation of the model output from the setpoint (i.e., the SSE). This increase in robustness with the addition of regulatory reactions corroborate findings from previous studies (13), demonstrating the utility and effectiveness of enzyme modules in capturing the subtle regulatory actions of enzymatic entities. It is notable that models with hemoglobin and either HEX or PFK performed well despite not accounting for all regulatory mechanisms. In particular, the model with PFK, HEX, and HB outperformed the model with all three enzyme modules together; these results indicate that while increased regulation generally improve the system's disturbance rejection capabilities, there is more complicated interplay among the enzyme modules when more than one is present. The most striking result is that the presence of the HB enzyme module (and the RL Shunt) drastically improved the disturbance rejection capabilities of the models. We observe that models that containing the PYK module exhibit lower SSE than models without PYK, likely due to the fact that PYK represents one of the last steps in the model and therefore has a smaller impact on the rest of the system.