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

Kinetic and thermodynamic models of biological systems are commonly used to connect microscopic features to system function in a bottom-up multiscale approach. 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 from 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 Hill. If left uncorrected, such models can produce arbitrary outputs that are physically inconsistent. Here, we develop a maximum likelihood approach (named multibind) based on the so-called potential graph to combine kinetic or thermodynamic measurements to yield state-resolved models that are thermodynamically consistent while being most consistent with the provided data and their uncertainties. We demonstrate the approach with two theoretical models, a generic two-proton binding site and a simplified model of a sodium/proton antiporter. We also describe an algorithm to use the multibind approach to solve the inverse problem of determining microscopic quantities from macroscopic measurements and, as an example, we predict the microscopic $$ values and protonation states of a small organic molecule from 1D NMR data. The multibind approach 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 package multibind, which implements the approach described here, is made publicly available under the MIT Open Source license.