Some self-consistent two-state sliding filament models of muscle contraction.

The general formalism required to treat two-state sliding filament models of muscle contraction, including free energy considerations, is first reviewed and amplified. This formalism is then used to examine, and modify as needed, three models studied previously by Podolsky and Nolan, in which cross-bridge attachment-detachment and ATP turnover are not tightly coupled. No attempt is made here to establish an optimal, self-consistent model of this type because our interest is primarily in methadology rather than in fitting experimental results. But it appears from this preliminary study that such a model, with satisfactory mechanical and thermodynamic properties, could be found. An extremely simple but unrealistic two-state model is also studied which is of interest because it demonstrates the fact that it is possible, in principle at least, for sliding filament models to work with very high thermodynamic efficiencies (50-100 percent). An appendix is included that is concerned with the form of the dependence of certain first-order rate constants on the concentrations of ATP, ADP, and P.     

Impaired associative learning in schizophrenia: behavioral and computational studies

Associative learning is a central building block of human cognition and in large part depends on mechanisms of synaptic plasticity, memory capacity and fronto–hippocampal interactions. A disorder like schizophrenia is thought to be characterized by altered plasticity, and impaired frontal and hippocampal function. Understanding the expression of this dysfunction through appropriate experimental studies, and understanding the processes that may give rise to impaired behavior through biologically plausible computational models will help clarify the nature of these deficits. We present a preliminary computational model designed to capture learning dynamics in healthy control and schizophrenia subjects. Experimental data was collected on a spatial-object paired-associate learning task. The task evinces classic patterns of negatively accelerated learning in both healthy control subjects and patients, with patients demonstrating lower rates of learning than controls. Our rudimentary computational model of the task was based on biologically plausible assumptions, including the separation of dorsal/spatial and ventral/object visual streams, implementation of rules of learning, the explicit parameterization of learning rates (a plausible surrogate for synaptic plasticity), and learning capacity (a plausible surrogate for memory capacity). Reductions in learning dynamics in schizophrenia were well-modeled by reductions in learning rate and learning capacity. The synergy between experimental research and a detailed computational model of performance provides a framework within which to infer plausible biological bases of impaired learning dynamics in schizophrenia.     

Modeling protein density of states: additive hydrophobic effects are insufficient for calorimetric two-state cooperativity

A well-established experimental criterion for two-state thermodynamic cooperativity in protein folding is that the van't Hoff enthalpy DeltaH(vH) around the transition midpoint is equal, or very nearly so, to the calorimetric enthalpy DeltaH(cal) of the entire transition. This condition is satisfied by many small proteins. We use simple lattice models to provide a statistical mechanical framework to elucidate how this calorimetric two-state picture may be reconciled with the hierarchical multistate scenario emerging from recent hydrogen exchange experiments. We investigate the feasibility of using inverse Laplace transforms to recover the underlying density of states (i.e., enthalpy distribution) from calorimetric data. We find that the constraint imposed by DeltaH(vH)/DeltaH(cal) approximately 1 on densities of states of proteins is often more stringent than other "two-state" criteria proposed in recent theoretical studies. In conjunction with reasonable assumptions, the calorimetric two-state condition implies a narrow distribution of denatured-state enthalpies relative to the overall enthalpy difference between the native and the denatured conformations. This requirement does not always correlate with simple definitions of "sharpness" of a transition and has important ramifications for theoretical modeling. We find that protein models that assume capillarity cooperativity can exhibit overall calorimetric two-state-like behaviors. However, common heteropolymer models based on additive hydrophobic-like interactions, including highly specific two-dimensional Gō models, fail to produce proteinlike DeltaH(vH)/DeltaH(cal) approximately 1. A simple model is constructed to illustrate a proposed scenario in which physically plausible local and nonlocal cooperative terms, which mimic helical cooperativity and environment-dependent hydrogen bonding strength, can lead to thermodynamic behaviors closer to experiment. Our results suggest that proteinlike thermodynamic cooperativity may require a cooperative interplay between local and nonlocal interactions. The prospect of using calorimetric data to constrain Z-scores of knowledge-based potentials is discussed.     

Estimating the voltage-dependent free energy change of ion channels using the median voltage for activation

Voltage-gated ion channels are crucial for electrical activity and chemical signaling in a variety of cell types. Structure-activity studies involving electrophysiological characterization of mutants are widely used and allow us to quickly realize the energetic effects of a mutation by measuring macroscopic currents and fitting the observed voltage dependence of conductance to a Boltzmann equation. However, such an approach is somewhat limiting, principally because of the inherent assumption that the channel activation is a two-state process. In this analysis, we show that the area delineated by the gating charge displacement curve and its ordinate axis is related to the free energy of activation of a voltage-gated ion channel. We derive a parameter, the median voltage of charge transfer (V(m)), which is proportional to this area, and prove that the chemical component of free energy change of a system can be obtained from the knowledge of V(m) and the maximum number of charges transferred. Our method is not constrained by the number or connectivity of intermediate states and is applicable to instances in which the observed responses show a multiphasic behavior. We consider various models of ion channel gating with voltage-dependent steps, latent charge movement, inactivation, etc. and discuss the applicability of this approach in each case. Notably, our method estimates a net free energy change of approximately -14 kcal/mol associated with the full-scale activation of the Shaker potassium channel, in contrast to -2 to -3 kcal/mol estimated from a single Boltzmann fit. Our estimate of the net free energy change in the system is consistent with those derived from detailed kinetic models (Zagotta et al. 1994. J. Gen. Physiol. doi:10.1085/jgp.103.2.321). The median voltage method can reliably quantify the magnitude of free energy change associated with activation of a voltage-dependent system from macroscopic equilibrium measurements. This will be particularly useful in scanning mutagenesis experiments.     

Exploring voltage-dependent ion channels in silico by hysteretic conductance

Kinetic models of voltage-dependent ion channels are normally inferred from time records of macroscopic current relaxation or microscopic single channel data. A complementary explorative approach is outlined. Hysteretic conductance refers to conductance delays in response to voltage changes, delays at either macroscopic or microscopic levels of observation. It enables complementary assessments of model assumptions and gating schemes of voltage-dependent channels, e.g. independent versus cooperative gating, and multiple gating modes. Under the Hodgkin-Huxley condition of independent gating, and under ideal measurement conditions, hysteretic conductance makes it also possible to estimate voltage-dependent rate functions. The argument is mainly theoretical, based on experimental observations, and illustrated by simulations of Markov kinetic models.     

Phosphorylation of K+ channels in the squid giant axon. A mechanistic analysis

Protein phosphorylation is an important mechanism in the modulation of voltage-dependent ionic channels. In squid giant axons, the potassium delayed rectifier channel is modulated by an ATP-mediated phosphorylation mechanism, producing important changes in amplitude and kinetics of the outward current. The characteristics and biophysical basis for the phosphorylation effects have been extensively studied in this preparation using macroscopic, single-channel and gating current experiments. Phosphorylation produces a shift in the voltage dependence of all voltage-dependent parameters including open probability, slow inactivation, first latency, and gating charge transferred. The locus of the effect seems to be located in a fast 20 pS channel, with characteristics of delayed rectifier, but at least another channel is phosphorylated under our experimental conditions. These results are interpreted quantitatively with a mechanistic model that explains all the data. In this model the shift in voltage dependence is produced by electrostatic interactions between the transferred phosphate and the voltage sensor of the channel.     

Improved free energy parameters for RNA pseudoknotted secondary structure prediction

Accurate prediction of RNA pseudoknotted secondary structures from the base sequence is a challenging computational problem. Since prediction algorithms rely on thermodynamic energy models to identify low-energy structures, prediction accuracy relies in large part on the quality of free energy change parameters. In this work, we use our earlier constraint generation and Boltzmann likelihood parameter estimation methods to obtain new energy parameters for two energy models for secondary structures with pseudoknots, namely, the Dirks–Pierce (DP) and the Cao–Chen (CC) models. To train our parameters, and also to test their accuracy, we create a large data set of both pseudoknotted and pseudoknot-free secondary structures. In addition to structural data our training data set also includes thermodynamic data, for which experimentally determined free energy changes are available for sequences and their reference structures. When incorporated into the HotKnots prediction algorithm, our new parameters result in significantly improved secondary structure prediction on our test data set. Specifically, the prediction accuracy when using our new parameters improves from 68% to 79% for the DP model, and from 70% to 77% for the CC model.     

Solvation effects and driving forces for protein thermodynamic and kinetic cooperativity: how adequate is native-centric topological modeling?

What energetic and solvation effects underlie the remarkable two-state thermodynamics and folding/unfolding kinetics of small single-domain proteins? To address this question, we investigate the folding and unfolding of a hierarchy of continuum Langevin dynamics models of chymotrypsin inhibitor 2. We find that residue-based additive Gō-like contact energies, although native-centric, are by themselves insufficient for protein-like calorimetric two-state cooperativity. Further native biases by local conformational preferences are necessary for protein-like thermodynamics. Kinetically, however, even models with both contact and local native-centric energies do not produce simple two-state chevron plots. Thus a model protein's thermodynamic cooperativity is not sufficient for simple two-state kinetics. The models tested appear to have increasing internal friction with increasing native stability, leading to chevron rollovers that typify kinetics that are commonly referred to as non-two-state. The free energy profiles of these models are found to be sensitive to the choice of native contacts and the presumed spatial ranges of the contact interactions. Motivated by explicit-water considerations, we explore recent treatments of solvent granularity that incorporate desolvation free energy barriers into effective implicit-solvent intraprotein interactions. This additional feature reduces both folding and unfolding rates vis-à-vis that of the corresponding models without desolvation barriers, but the kinetics remain non-two-state. Taken together, our observations suggest that interaction mechanisms more intricate than simple Gō-like constructs and pairwise additive solvation-like contributions are needed to rationalize some of the most basic generic protein properties. Therefore, as experimental constraints on protein chain models, requiring a consistent account of protein-like thermodynamic and kinetic cooperativity can be more stringent and productive for some applications than simply requiring a model heteropolymer to fold to a target structure.     

Natural selection for kinetic stability is a likely origin of correlations between mutational effects on protein energetics and frequencies of amino acid occurrences in sequence alignments

It appears plausible that natural selection constrains, to some extent at least, the stability in many natural proteins. If, during protein evolution, stability fluctuates within a comparatively narrow range, then mutations are expected to be fixed with frequencies that reflect mutational effects on stability. Indeed, we recently reported a robust correlation between the effect of 27 conservative mutations on the thermodynamic stability (unfolding free energy) of Escherichia coli thioredoxin and the frequencies of residues occurrences in sequence alignments. We show here that this correlation likely implies a lower limit to thermodynamic stability of only a few kJ/mol below the unfolding free energy of the wild-type (WT) protein. We suggest, therefore, that the correlation does not reflect natural selection of thermodynamic stability by itself, but of some other factor which is linked to thermodynamic stability for the mutations under study. We propose that this other factor is the kinetic stability of thioredoxin in vivo, since( i) kinetic stability relates to irreversible denaturation, (ii) the rate of irreversible denaturation in a crowded cellular environment (or in a harsh extracellular environment) is probably determined by the rate of unfolding, and (iii) the half-life for unfolding changes in an exponential manner with activation free energy and, consequently, comparatively small free energy effects can have deleterious consequences for kinetic stability. This proposal is supported by the results of a kinetic study of the WT form and the 27 single-mutant variants of E. coli thioredoxin based on the global analyses of chevron plots and equilibrium unfolding profiles determined from double-jump unfolding assays. This kinetic study suggests, furthermore, one of the factors that may contribute to the high activation free energy for unfolding in thioredoxin (required for kinetic stability), namely the energetic optimization of native-state residue environments in regions, which become disrupted in the transition state for unfolding.     

An evaluation of Poisson-Boltzmann electrostatic free energy calculations through comparison with experimental mutagenesis data

For systems involving highly and oppositely charged proteins, electrostatic forces dominate association and contribute to biomolecular complex stability. Using experimental or theoretical alanine-scanning mutagenesis, it is possible to elucidate the contribution of individual ionizable amino acids to protein association. We evaluated our electrostatic free energy calculations by comparing calculated and experimental data for alanine mutants of five protein complexes. We calculated Poisson-Boltzmann electrostatic free energies based on a thermodynamic cycle, which incorporates association in a reference (Coulombic) and solvated (solution) state, as well as solvation effects. We observe that Coulombic and solvation free energy values correlate with experimental data in highly and oppositely charged systems, but not in systems comprised of similarly charged proteins. We also observe that correlation between solution and experimental free energies is dependent on dielectric coefficient selection for the protein interior. Free energy correlations improve as protein dielectric coefficient increases, suggesting that the protein interior experiences moderate dielectric screening, despite being shielded from solvent. We propose that higher dielectric coefficients may be necessary to more accurately predict protein-protein association. Additionally, our data suggest that Coulombic potential calculations alone may be sufficient to predict relative binding of protein mutants.     

Optimal combination of theory and experiment for the characterization of the protein folding landscape of S6: how far can a minimalist model go?

The detailed characterization of the overall free energy landscape associated with the folding process of a protein is the ultimate goal in protein folding studies. Modern experimental techniques provide accurate thermodynamic and kinetic measurements on restricted regions of a protein landscape. Although simplified protein models can access larger regions of the landscape, they are oftentimes built on assumptions and approximations that affect the accuracy of the results. We present a new methodology that allows to combine the complementary strengths of theory and experiment for a more complete characterization of a protein folding landscape. We prove that this new procedure allows a simplified protein model to reproduce remarkably well (correlation coefficient > 0.9) all experimental data available on free energies differences upon single mutations for S6 ribosomal protein and two circular permutants. Our results confirm and quantify the hypothesis, recently formulated on the basis of experimental data, that the folding landscape of protein S6 is strongly affected by an atypical distribution of contact energies.     

Three-state combinatorial switch models as applied to the binding of oxygen by human hemoglobin

We have generated a series of all 6561 unique, discrete three-state combinatorial switch models to describe the partitioning of the cooperative oxygen-binding free change among the 10 variously ligated forms of human hemoglobin tetramers. These models were inspired by the experimental observation of Smith and Ackers that the cooperative free energy of the intersubunit contact regions of the 10 possible ligated forms of human hemoglobin tetramers can be represented by a particular distribution of three distinct energy levels [Smith, F. R., & Ackers, G. K. (1985) Proc. Natl. Acad. Sci. U.S.A. 82, 5347-5351]. A statistical thermodynamic formulation accounting for both dimer-tetramer equilibria and ligand binding properties of hemoglobin solutions as a function of oxygen and protein concentrations was utilized to exhaustively test these thermodynamic models. In this series of models each of the 10 ligated forms of the hemoglobin tetramer can exist in one, and only one, of three possible energy levels; i.e., each ligated form was assumed to be associated with a discrete energy state. This series of models includes all possible ways that the 10 ligation states of hemoglobin can be distributed into three distinct cooperative energy levels. The mathematical models, as presented here, do not permit equilibria between energy states to exist for any of the 10 unique ligated forms of hemoglobin tetramers. These models were analyzed by nonlinear least-squares estimation of the free energy parameters characteristic of this statistical thermodynamic development.(ABSTRACT TRUNCATED AT 250 WORDS)     

Stability of proteins in the presence of carbohydrates; experiments and modeling using scaled particle theory

The effects of sucrose and fructose on the free energy of unfolding, DeltaG(N-->D), and on the change in hydrodynamic radius, R(H), upon unfolding were measured for RNase A and alpha-lactalbumin. Recently we analyzed the results for RNase A and showed that the effects of the carbohydrates on the protein's thermal stability can be accurately accounted for by scaled particle theory (SPT), and are thus largely entropic in nature. In this paper we extend this analysis to alpha-lactalbumin and demonstrate the generality of this finding. We also investigate the relationship between SPT and the thermodynamic formalism of preferential interactions. The preferential binding parameters calculated using SPT are in excellent agreement with experimentally measured values available in the literature. This agreement is expected to hold as long as enthalpic interactions between the cosolute and the protein are not important, as appears to be the case here. Finally we use the experimental data and SPT to calculate the change in the number of sugar molecules excluded from the protein surface during unfolding from knowledge of the preferential binding parameter for the native and denatured state of the protein.     

Thermodynamically feasible kinetic models of reaction networks

The dynamics of biological reaction networks are strongly constrained by thermodynamics. An holistic understanding of their behavior and regulation requires mathematical models that observe these constraints. However, kinetic models may easily violate the constraints imposed by the principle of detailed balance, if no special care is taken. Detailed balance demands that in thermodynamic equilibrium all fluxes vanish. We introduce a thermodynamic-kinetic modeling (TKM) formalism that adapts the concepts of potentials and forces from irreversible thermodynamics to kinetic modeling. In the proposed formalism, the thermokinetic potential of a compound is proportional to its concentration. The proportionality factor is a compound-specific parameter called capacity. The thermokinetic force of a reaction is a function of the potentials. Every reaction has a resistance that is the ratio of thermokinetic force and reaction rate. For mass-action type kinetics, the resistances are constant. Since it relies on the thermodynamic concept of potentials and forces, the TKM formalism structurally observes detailed balance for all values of capacities and resistances. Thus, it provides an easy way to formulate physically feasible, kinetic models of biological reaction networks. The TKM formalism is useful for modeling large biological networks that are subject to many detailed balance relations.     

Extending the Conditions of Application of an Inversion of the Hodgkin–Huxley Gating Model

We present an inversion of the Hodgkin–Huxley formalism to estimate initial conditions and model parameters, including functions of voltage, from the solutions of the underlying ordinary differential equation (ODE) subjected to multiple voltage step stimulations. As such, the procedure constitutes a means to estimate the parameters including functions of voltage of an Hodgkin–Huxley formalism from experimental data. The basic idea was developed in a previous communication (SIAM J. Appl. Math. 64:1264–1274, 2009). The inversion in question applies to currents exhibiting activation and inactivation, but the version, as published previously, cannot estimate the unknowns for channels that rapidly inactivate just after a brief opening. In such cases, the amplitude of the current, in a given voltage range, is too small to be detectable by the instrumentation using previously applied experimental protocols. This is, for example, the case for the sodium channels in a number of excitable tissue for potential in the vicinity of the cell resting potential. The current communication extends the inversion procedure in a manner to overcome this limitation. Furthermore, within the inversion framework, we can determine whether the data at the basis of the estimation sufficiently constrains the estimation problem, i.e., whether it is complete. We exploit this element of our method to document a set of stimulation protocols that constitute a complete data set for the purpose of inverting the Hodgkin–Huxley formalism.     

Evidence for nonlinear capacitance in biomembrane channel system.

The electrophysiological properties of voltage-dependent anion channels from mitochondrial membrane have been studied in a bilayer membrane system. It was observed that the probability of opening of the membrane channel depends on externally applied voltage and the plot is a bell-shaped curve symmetric around probability axis. A scheme of conformational energy levels under varying externally applied voltage was formulated. Assuming that the probability follows Boltzmann distribution, we arrive at an expression of change in energy containing a separate term identical to the energy of a capacitor. This fact indicates the possibility of existence of an added capacitance due to the channel protein. Further it was shown that the aforesaid channel capacitor could be a function of voltage leading to nonlinearity. We have offered a general method of calculating nonlinear capacitance from the experimental data on opening probability of a membrane channel. In case of voltage-dependent anion channel the voltage dependence of the capacitor has a power 0.786. The results have been interpreted in view of the structural organization of the channel protein in the membrane. Our hypothesis is that the phenomenon of capacitor behaviour is a general one for membrane channels.     

Models of active transport of ions in biomembranes of various types of cells

Nonequilibrium statistical models of the active transport of ions in biomembranes have been constructed. Differences of chemical potentials of the ATP-ADP reaction and the electrochemical potential of ions were taken as the thermodynamic forces responsible for the flow of ions through the membrane. The active transport of ions was viewed as a cross phenomenon arising from the chemical reaction of the ATP hydrolysis. These models provide independent calculations of the resting potential at the biomembrane and concentrations of ions in a cell on the assumption the free energy of the ATP-ADP reaction is fully (without the dissipation loss) converted to the free energy of transported ions. They take into account the presence of nonpenetrating ions in a cell. It was shown that different concentrations of nonpenetrating ions have a considerable effect on the resting potential. The proposed models were compared with experimental data obtained for different types of cells including neurons, muscular cells, bacteria, plants, and mitochondria. Calculated values of the membrane potential and ion concentrations were in good qualitative agreement with experimental data.     

Alamethicin adsorption to a planar lipid bilayer.

The effect of alamethicin and its derivatives on the voltage-dependent capacitance of phosphatidylethanolamine (squalane) membranes was measured using two different methods: lock-in detection and voltage pulse. Alamethicin and its derivatives modulate the voltage-dependent capacitance at voltages lower than the voltage at which alamethicin-induced conductance is detected. The magnitude and sign of this alamethicin-induced capacitance change depends on the aqueous alamethicin concentration and the kind of alamethicin used. Our experimental data can be interpreted as a potential-dependent pseudocapacitance associated with adsorbed alamethicin. Pseudocapacitance is expressed as a function of alamethicin charge, its concentration in the bathing solution and the applied electric field. The theory describes the dependence of the capacitance on applied voltage and alamethicin concentration. When alamethicin is neutral the theory predicts no change of the voltage-dependent capacitance with either sign of applied voltage. Experimental data are consistent with the model in which alamethicin molecules interact with each other while being adsorbed to the membrane surface. The energy of this interaction depends on the alamethicin concentration.     

Simulation of the voltage dependence of the Na, K pump applied to cardiac cells

We use simulation to study the dependence of the Na, K pump on membrane potential. Two consecutive mechanisms for the Na, K pump, based on a reduced Post-Albers scheme, are examined: one with six steps called GV3 and one with seven steps called MGV3. In GV3, a single voltage-dependent step combines both Na+ translocation and Na+ release into the extracellular medium. In MGV3, these two processes are allocated to two separate consecutive steps, but only the Na+ translocation step is voltage-dependent. Using the optimization software SCoPfit, numerical values of rate coefficients, symmetry factor (beta), and pump site density were found by fitting the models to published experimental data so that both GV3 and MGV3 could quantitatively reproduce steady-state current-voltage relationships for both forward and backward running of the pump, as well as [Na+]in and [K+]out activation curves. Using the rate coefficient values found by SCoPfit, we simulated a voltage-clamp experiment with both models running under their Na(+)-Na+ exchange mode, and we computed the transient currents generated following voltage steps in both depolarizing and hyperpolarizing directions from a basic potential of -40 mV. The voltage dependence of the rate constant (1/tau) of decay of the transient currents could qualitatively be reproduced when beta = 0.884 for GV3, and 0.932 for MGV3. The quantitative discrepancy between published experimental data and the theoretical curve generated by GV3 at potentials more negative than -20 mV was considerably reduced by using model MGV3. This finding alone suggests that a more detailed mechanism containing a single voltage-dependent step may reproduce all major steady-state and transient characteristics of the Na, K pump without the need of a second voltage sensitive step. However, the quantitative discrepancy between published experimental data and the theoretical curve generated by MGV3 at potentials more negative than -60 mV may be fully removed if either beta itself is voltage-dependent, or if a second voltage-dependent step is included in the model.