A systems-biology approach to molecular machines: Exploration of alternative transporter mechanisms

Kinetic Diagram Analysis: A Python Library for Calculating Steady-State Observables of Biochemical Systems Analytically

Kinetic diagrams are commonly used to represent biochemical systems in order to study phenomena such as free energy transduction and ion selectivity. While numerical methods are commonly used to analyze such kinetic networks, the diagram method by King, Altman and Hill makes it possible to construct exact algebraic expressions for steady-state observables in terms of the rate constants of the kinetic diagram. However, manually obtaining these expressions becomes infeasible for models of even modest complexity as the number of the required intermediate diagrams grows with the factorial of the number of states in the diagram. We developed Kinetic Diagram Analysis (KDA), a Python library that programmatically generates the relevant diagrams and expressions from a user-defined kinetic diagram. KDA outputs symbolic expressions for state probabilities and cycle fluxes at steady-state that can be symbolically manipulated and evaluated to quantify macroscopic system observables. We demonstrate the KDA approach for examples drawn from the biophysics of active secondary transmembrane transporters. For a generic 6-state antiporter model, we show how the introduction of a single leakage transition reduces transport efficiency by quantifying substrate turnover. We apply KDA to a real-world example, the 8-state free exchange model of the small multidrug resistance transporter EmrE of Hussey et al (J General Physiology 152 (2020), e201912437), where a change in transporter phenotype is achieved by biasing two different subsets of kinetic rates: alternating access and substrate unbinding rates. KDA is made available as open source software under the GNU General Public License version 3.

Multiscale Responsive Kinetic Modeling: Quantifying Biomolecular Reaction Flux under Varying Electrochemical Conditions

Attaining a complete thermodynamic and kinetic characterization for processes involving multiple interconnected rare-event transitions remains a central challenge in molecular biophysics. This challenge is amplified when the process must be understood under a range of reaction conditions. Herein, we present a condition-responsive kinetic modeling framework that can combine the strengths of bottom-up rate quantification from multiscale simulations with top-down solution refinement using experimental data. Although this framework can be applied to any process, we demonstrate its use for electrochemically driven transport through channels and transporters. Using the Cl - /H + antiporter ClC-ec1 as a model system, we show how robust and predictive kinetic solutions can be obtained when the solution space is grounded by thermodynamic constraints, seeded through multiscale rate quantification, and further refined with experimental data, such as electrophysiology assays. Turning to the Shaker K + channel, we demonstrate that robust solutions and biophysical insights can also be obtained with sufficient experimental data. This multi-pathway method proves capable of identifying single-pathway dominant mechanisms but also highlights that competing and off-pathway flux is still essential to replicate experimental findings and to describe concentration-dependent channel rectification.

From Average Transient Transporter Currents to Microscopic Mechanism─A Bayesian Analysis

Electrophysiology studies of secondary active transporters have revealed quantitative mechanistic insights over many decades of research. However, the emergence of new experimental and analytical approaches calls for investigation of the capabilities and limitations of the newer methods. We examine the ability of solid-supported membrane electrophysiology (SSME) to characterize discrete-state kinetic models with >10 rate constants. We use a Bayesian framework applied to synthetic data for three tasks: to quantify and check (i) the precision of parameter estimates under different assumptions, (ii) the ability of computation to guide the selection of experimental conditions, and (iii) the ability of our approach to distinguish among mechanisms based on SSME data. When the general mechanism, i.e., event order, is known in advance, we show that a subset of kinetic parameters can be "practically identified" within ∼1 order of magnitude, based on SSME current traces that visually appear to exhibit simple exponential behavior. This remains true even when accounting for systematic measurement bias and realistic uncertainties in experimental inputs (concentrations) are incorporated into the analysis. When experimental conditions are optimized or different experiments are combined, the number of practically identifiable parameters can be increased substantially. Some parameters remain intrinsically difficult to estimate through SSME data alone, suggesting that additional experiments are required to fully characterize parameters. We also demonstrate the ability to perform model selection and determine the order of events when that is not known in advance, comparing Bayesian and maximum-likelihood approaches. Finally, our studies elucidate good practices for the increasingly popular but subtly challenging Bayesian calculations for structural and systems biology.

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 p K a 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.

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 T. 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 p K a s 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.

WHY IT MATTERS

The 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 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.

Methodology for development of single cell dendritic spine (SCDS) synaptic tagging and capture model using Virtual Cell (VCell)

Single cell dendritic spine modelling methodology has been adopted to explain structural plasticity and respective change in the neuronal volume previously. However, the single cell dendrite methodology has not been employed previously to explain one of the important aspects of memory allocation i.e., Synaptic tagging and Capture (STC) hypothesis. It is difficult to relate the physical properties of STC pathways to structural changes and synaptic strength. We create a mathematical model based on earlier reported synaptic tagging networks. We built the model using Virtual Cell (VCell) software and used it to interpret experimental data and investigate the behavior and characteristics of known Synaptic tagging candidates.•We investigate processes associated with synaptic tagging candidates and compare them to the assumptions based on the STC hypothesis.•We assess the behavior of several reported synaptic tagging candidates against the requirements outlined in the synaptic tagging hypothesis.

Understanding Mechanisms of Secondary Active Transport by Analyzing the Effects of Mutations and Stoichiometry

Biological cells frequently exhibit a so-called secondary active transport by moving various species across their membranes. In this mode of transport, an energetically favorable transmembrane gradient of one type of molecule is used to drive another type of molecule in the energetically unfavorable direction against their gradient. Although it is well established that conformational transitions play a critical role in functioning of transporters, the molecular details of underlying mechanisms remain not well understood. Here, we utilize a recently developed theoretical method to understand better the microscopic picture of secondary active transport. Specifically, we evaluate how mutations in different parts of transporters affect their dynamic properties. In addition, we present a possible explanation on existence of different stoichiometries in the secondary active transport. Our theoretical analysis clarifies several important aspects of complex biological transport phenomena.

General principles of secondary active transporter function

Transport of ions and small molecules across the cell membrane against electrochemical gradients is catalyzed by integral membrane proteins that use a source of free energy to drive the energetically uphill flux of the transported substrate. Secondary active transporters couple the spontaneous influx of a "driving" ion such as Na+ or H+ to the flux of the substrate. The thermodynamics of such cyclical non-equilibrium systems are well understood, and recent work has focused on the molecular mechanism of secondary active transport. The fact that these transporters change their conformation between an inward-facing and outward-facing conformation in a cyclical fashion, called the alternating access model, is broadly recognized as the molecular framework in which to describe transporter function. However, only with the advent of high resolution crystal structures and detailed computer simulations, it has become possible to recognize common molecular-level principles between disparate transporter families. Inverted repeat symmetry in secondary active transporters has shed light onto how protein structures can encode a bi-stable two-state system. Based on structural data, three broad classes of alternating access transitions have been described as rocker-switch, rocking-bundle, and elevator mechanisms. More detailed analysis indicates that transporters can be understood as gated pores with at least two coupled gates. These gates are not just a convenient cartoon element to illustrate a putative mechanism but map to distinct parts of the transporter protein. Enumerating all distinct gate states naturally includes occluded states in the alternating access picture and also suggests what kind of protein conformations might be observable. By connecting the possible conformational states and ion/substrate bound states in a kinetic model, a unified picture emerges in which the symporter, antiporter, and uniporter functions are extremes in a continuum of functionality. As usual with biological systems, few principles and rules are absolute and exceptions are discussed as well as how biological complexity may be integrated in quantitative kinetic models that may provide a bridge from the structure to function.

Theoretical study of active secondary transport: Unexpected differences in molecular mechanisms for antiporters and symporters

Successful functioning of biological cells relies on efficient translocation of different materials across cellular membranes. An important part of this transportation system is membrane channels that are known as antiporters and symporters. They exploit the energy stored as a trans-membrane gradient of one type of molecules to transport the other types of molecules against their gradients. For symporters, the directions of both fluxes for driving and driven species coincide, while for antiporters, the fluxes move in opposite directions. There are surprising experimental observations that despite differing only by the direction of transport fluxes, the molecular mechanisms of translocation adopted by antiporters and symporters seem to be drastically different. We present chemical-kinetic models to quantitatively investigate this phenomenon. Our theoretical approach allows us to explain why antiporters mostly utilize a single-site transportation when only one molecule of any type might be associated with the channel. At the same time, the transport in symporters requires two molecules of different types to be simultaneously associated with the channel. In addition, we investigate the kinetic constraints and efficiency of symporters and compare them with the same properties of antiporters. Our theoretical analysis clarifies some important physical–chemical features of cellular trans-membrane transport.

Multiscale kinetic analysis of proteins

The stochasticity of molecular motion results in the existence of multiple kinetically relevant pathways in many biomolecular mechanisms. Because it is highly demanding to characterize them for complex systems, mechanisms are often described with a single-pathway perspective. However, kinetic network analysis and sub-ensemble experimental insight are increasingly demonstrating not only the existence of competing pathways but also the importance of kinetic selection in biology. This review focuses on advances in multiscale kinetic analysis of proteins, which connects molecular level information from simulations to macroscopic data to characterize mechanistic reaction networks and the reactive flux through them. We describe a range of methods used and highlight several examples where kinetic modeling has revealed functional importance of pathway heterogeneity.

Toward a Multipathway Perspective: pH-Dependent Kinetic Selection of Competing Pathways and the Role of the Internal Glutamate in Cl-/H+ Antiporters

Canonical descriptions of multistep biomolecular transformations generally follow a single-pathway viewpoint, with a series of transitions through intermediates converting reactants to products or repeating a conformational cycle. However, mounting evidence suggests that more complexity and pathway heterogeneity are mechanistically relevant due to the statistical distribution of multiple interconnected rate processes. Making sense of such pathway complexity remains a significant challenge. To better understand the role and relevance of pathway heterogeneity, we herein probe the chemical reaction network of a Cl-/H+ antiporter, ClC-ec1, and analyze reaction pathways using multiscale kinetic modeling (MKM). This approach allows us to describe the nature of the competing pathways and how they change as a function of pH. We reveal that although pH-dependent Cl-/H+ transport rates are largely regulated by the charge state of amino acid E148, the charge state of E203 determines relative contributions from coexisting pathways and can shift the flux pH-dependence. The selection of pathways via E203 explains how ionizable mutations (D/H/K/R) would impact the ClC-ec1 bioactivity from a kinetic perspective and lends further support to the indispensability of an internal glutamate in ClC antiporters. Our results demonstrate how quantifying the kinetic selection of competing pathways under varying conditions leads to a deeper understanding of the Cl-/H+ exchange mechanism and can suggest new approaches for mechanistic control.