Thermodynamically consistent determination of free energies and rates in kinetic cycle models

Kinetic and thermodynamic models of biological systems have been used to connect microscopic features to system function. The parameters of such models—free energy differences for equilibrium properties and in general rates for equilibrium and out-of-equilibrium observables—have to be measured by different experiments or calculated by multiple computer simulations. All such parameters necessarily come with uncertainties so that when they are naively combined in a full model of the process of interest, they will generally violate fundamental statistical mechanical equalities, namely detailed balance and an equality of forward/backward rate products in cycles due to T. Hill. If left uncorrected, such models can produce arbitrary outputs that are physically inconsistent. Here we develop a formalism (namedmultibind) to combine kinetic or thermodynamic measurements to yield state resolved models that are thermodynamically consistent. For equilibrium thermodynamic models, we transform the graph of given free energy differences between states to the potential graph of free energies of states using a maximum likelihood approach; the potential graph obeys path-independence of free energies and detailed balance by construction. Based on the thermodynamically consistent potential graph, we project the given rates of an equilibrium system onto a new set of rates that by construction obeys Hill’s rate cycle-product equality. This approach produces thermodynamically consistent models that are most consistent with the provided data and their uncertainties. We show examples for how observables can be calculated from these models to interpret the effects of microscopic features on system function for two toy systems: A two-proton binding site is modeled as a four-state model and we demonstrate how cooperative or anti-cooperative behavior emerges as a function of the microscopic pKas. For simplified model of an electroneutral sodium/proton antiporter we compute the turnover number (sodium ions transported across the membrane per second) as a function of sodium concentration, pH, and membrane potential to show the strong effect of the membrane potential on transport in this system. We also describe an algorithm to use themultibindapproach to solve the inverse problem of determining microscopic quantities from macroscopic measurements. As an example we predict the microscopic pKas and protonation states of a small organic molecule from 1D NMR data. Themultibindapproach is applicable to any thermodynamic or kinetic model that describes a system as transitions between well defined states with associated free energy differences or rates between these states. A Python packagemultibind, which implements the approach described here, is made publicly available under the MIT Open Source license.WHY IT MATTERSThe increase in computational efficiency and rapid advances in methodology for quantitative free energy and rate calculations has allowed for the construction of increasingly complex thermodynamic or kinetic “bottom-up” models of chemical and biological processes. These multi-scale models serve as a framework for analyzing aspects of cellular function in terms of microscopic, molecular properties and provide an opportunity to connect molecular mechanisms to cellular function. The underlying model parameters—free energy differences or rates—are constrained by thermodynamic identities over cycles of states but these identities are not necessarily obeyed during model construction, thus potentially leading to inconsistent models. We address these inconsistencies through the use of a maximum likelihood approach for free energies and an exact projection for rates to adjust the model parameters in such a way that they are maximally consistent with the input parameters and exactly fulfill the thermodynamic cycle constraints. This approach enables formulation of thermodynamically consistent multi-scale models from simulated or experimental measurements.