Four-state models and regulation of contraction of smooth muscle. I. Physical considerations, stability, and solutions.

Cyclic four-state models are frequently used in biology to represent a variety of molecular behaviors. A common experimental strategy to test such models is to follow the behavior of the real system after some of the rate constants are changed in a stepwise manner. We analyze the mathematical behavior of a simple example of such a model applicable to the regulation of contraction of smooth muscle, but our results apply in general to any linear, cyclic four-state model. We discuss detailed balance and requirements for linearity. We find that the only way to have sustained oscillations is for the rate constants of the model to be themselves oscillatory. We state conditions for decaying oscillations and find that in models that do not follow strictly first-order kinetics and do not satisfy detailed balance, these conditions can hold. We show analytically that the response of any state to step changes in the rate constants is the sum of three weighted exponentials plus a constant term, the steady-state value. We provide explicit expressions for the time dependence of all state variables. We discuss a simple way to use these results to obtain numerical solutions in cases where the rate constants change in an arbitrary way.     

Theory of the kinetic analysis of patch-clamp data.

This paper describes a theory of the kinetic analysis of patch-clamp data. We assume that channel gating is a Markov process that can be described by a model consisting of n kinetic states and n(n - 1) rate constants at each voltage, and that patch-clamp data describe the occupancy of x different conductance levels over time. In general, all the kinetic information in a set of patch-clamp data is found in either two-dimensional dwell time histograms describing the frequency of observation of sequential dwell times of durations tau 1 and tau 2 (Fredkin, D. R., M. Montal, and J. A. Rice, 1985, Proceedings of the Berkeley Conference in Honor of Jerzy Neyman and Jack Kiefer, vol. 1, 269-289) or in three-point joint probability functions describing the probability that a channel is in a given conductance at time t, and at time t + tau 1, and at time t + tau 1 + tau 2. For the special case of channels with a single open state plus multiple closed states, one-dimensional analyses provide all of the kinetic information. Stationary patch-clamp data have information that can be used to determine H rate constants, where H = n(n - 1) - G and G is the number of intraconductance rate constants. Thus, to calculate H rate constants, G rate constants must be fixed. In general there are multiple sets of G rate constants that can be fixed to allow the calculation of H rate constants although not every set of G rate constants will work. Arbitrary assignment of the G intraconductance rate constants equal to zero always provides a solution and the calculation of H rate constants. Nonstationary patch-clamp data have information for the determination of H rate constants at a reference voltage plus n(n - 1) rate constants at all test voltages. Thus, nonstationary data have extra information about the voltage dependencies of rate constants that can be used to rule out kinetic models that cannot be disqualified on the basis of stationary data.     

Calculation of kinetic curves for the helix-coil transition of polypeptides

Methods for calculating the rate of cooperative transitions on a linear lattice, for which the helix–coil transition of polypeptides is an example, are reported. The problem is to determine the kinetic characteristics of the transition given the rate constants for a set of elementary steps: in this case, the transformations of individual segments between the helix and coil states. The most straightforward method is to store the state of a long lattice (in which helix and coil segments are represented by 1′s and 0′s) in a computer and to use random-number techniques to generate its behavior as a function of time. This is, in principle, a solution to the problem, but it requires very large amounts of computer time. We have devised a matrix iteration procedure which allows much faster computation while reproducing the results of the random-number method accurately. In this procedure the computer operates repeatedly with a transition probability matrix on a vector which represents the time-dependent state of a finite group of units. The choice of a finite group neglects kinetic correlations between the state changes of units inside and outside the group, but comparison with the random-number method indicates that these correlations are not important. Thus it is possible to generate the kinetic behavior of the model under essentially any conditions, for either relaxation or large perturbations. Examination of these calculated curves suggests a simple and quite generally applicable solution to the inverse problem—that of evaluating the rate constants from kinetic curves. The initial slope is well defined in almost every case, and since an analytic equation can be written relating this to the rate constants, these can be obtained directly from the initial rate. This latter is therefore the most convenient single measure of transition rate.     

Ionic channels with conformational substates.

Recent studies of protein dynamics suggest that ionic channels can assume many conformational substates. Long-lived substates have been directly observed in single-channel current records. In many cases, however, the lifetimes of conformational states will be far below the theoretical limit of time resolution of single-channel experiments. The existence of such hidden substates may strongly influence the observable (time-averaged) properties of a channel, such as the concentration dependence of conductance. A channel exhibiting fast, voltage-dependent transitions between different conductance states may behave as an intrinsic rectifier. In the presence of more than one permeable ion species, coupling between ionic fluxes may occur, even when the channel has only a single ion-binding site. In special situations the rate of ion translocation becomes limited by the rate of conformational transitions, meaning that the channel approaches the kinetic behavior of a carrier. As a result of the strong coulombic interaction between an ion in a binding site and polar groups of the protein, rate constants of conformational transitions may depend on the occupancy of the binding site. Under this condition a nonequilibrium distribution of conformational states is created when ions are driven through the channel by an external force. This may lead to an apparent violation of microscopic reversibility, i.e., to a situation in which the frequency of transitions from state A to state B is no longer equal to the transition frequency from state B to state A.     

Time-dependent outward current in guinea pig ventricular myocytes. Gating kinetics of the delayed rectifier

Several conflicting models have been used to characterize the gating behavior of the cardiac delayed rectifier. In this study, whole-cell delayed rectifier currents were measured in voltage-clamped guinea pig ventricular myocytes, and a minimal model which reproduced the observed kinetic behavior was identified. First, whole-cell potassium currents between -10 and +70 mV were recorded using external solutions designed to eliminate Na and Ca currents and two components of time-dependent outward current were found. One component was a La3(+)-sensitive current which inactivated and resembled the transient outward current described in other cell types; single-channel observations confirmed the presence of a transient outward current in these guinea pig ventricular cells (gamma = 9.9 pS, [K]o = 4.5 mM). Analysis of envelopes of tail amplitudes demonstrated that this component was absent in solutions containing 30-100 microM La3+. The remaining time-dependent current, IK, activated with a sigmoidal time course that was well-characterized by three time constants. Nonlinear least-squares fits of a four-state Markovian chain model (closed - closed - closed - open) to IK activation were therefore compared to other models previously used to characterize IK gating: n2 and n4 Hodgkin-Huxley models and a Markovian chain model with only two closed states. In each case the four-state model was significantly better (P less than 0.05). The failure of the Hodgkin-Huxley models to adequately describe the macroscopic current indicates that identical and independent gating particles should not be assumed for this K channel. The voltage-dependent terms describing the rate constants for the four-state model were then derived using a global fitting approach for IK data obtained over a wide range of potentials (-80 to +70 mV). The fit was significantly improved by including a term representing the membrane dipole forces (P less than 0.01). The resulting rate constants predicted long single-channel openings (greater than 1 s) at voltages greater than 0 mV. In cell-attached patches, single delayed rectifier channels which had a mean chord conductance of 5.4 pS at +60 mV ([K]o = 4.5 mM) were recorded for brief periods. These channels exhibited behavior predicted by the four-state model: long openings and latency distributions with delayed peaks. These results suggest that the cardiac delayed rectifier undergoes at least two major transitions between closed states before opening upon depolarization.     

Four-state models and regulation of contraction of smooth muscle. II. Properties of the solutions and identification.

We analyze the behavior and the identification problem of cyclic four-state models. We find that for any state, or a weighted combination of two states, there can be at most one maximum, or one minimum, and two inflection points. We obtain necessary conditions for overshoot and undershoot and give examples. We describe procedures to estimate all the rate constants and discuss certain experimental aspects of the identification problem. Finally, we give an example of identification by obtaining the 10 model parameters from experimental data on skinned fibers from smooth muscle. These results, in conjunction with the results of the previous paper, can help in testing four-state models of regulation of contraction of smooth muscle and of a variety of other physiological and biochemical phenomena.     

Asymmetry and free energy transduction in biology

This paper is an extension of our earlier theoretical studies on the relationship between kinetic asymmetry and free-energy transductions in biological systems induced by external fluctuations. In the first part of the paper, the asymmetry conditions necessary for external-noise-induced free-energy transductions to occur are derived for a special cyclic, four-state model in which only one reaction step is perturbed by the fluctuations. The results can be used to explain the earlier findings that asymmetry in rate constants was not required in the uphill transport of ligands induced by externally fluctuating the ligand concentrations. In the second part of the paper, the coupling between two enzyme systems through direct enzyme-enzyme inter-actions is studied. The existence of kinetic asymmetry in both the driving and the driven enzyme systems is found necessary for coupling and free-energy transductions to occur.     

Kinetics of a model of autocatalysis, coupling of a reaction in which the enzyme acts on one of its substrates.

A global kinetic analysis is presented of a model of an enzyme autocatalytic process, to which a reaction is coupled, in which the enzyme acts upon one of its substrates. The kinetic equations of both the transient phase and the steady state are derived for this mechanism. In addition, we determine the corresponding kinetic equations for several particular cases which are characterized by certain relations between the rate constants. Finally, a kinetic data analysis is proposed for one of these particular cases. It can easily be extended to any of the other cases.     

An evolutionary approach to enzyme kinetics: Optimization of ordered mechanisms

A theoretical investigation is presented which allows the calculation of rate constants and phenomenological parameters in states of maximal reaction rates for unbranched enzymic reactions. The analysis is based on the assumption that an increase in reaction rates was an important characteristic of the evolution of the kinetic properties of enzymes. The corresponding nonlinear optimization problem is solved taking into account the constraint that the rate constants of the elementary processes do not exceed certain upper limits. One-substrate-one-product reactions with two, three and four steps are treated in detail. Generalizations concern ordered uni-uni-reactions involving an arbitrary number of elementary steps. It could be shown that depending on the substrate and product concentrations different types of solutions can be found which are classified according to the number of rate constants assuming in the optimal state submaximal values. A general rule is derived concerning the number of possible solutions of the given optimization problem. For high values of the equilibrium constant one solution always applies to a very large range of the concentrations of the reactants. This solution is characterized by maximal values of the rate constants of all forward reactions and by non-maximal values of the rate constants of all backward reactions. Optimal kinetic parameters of ordered enzymic mechanisms with two substrates and one product (bi-uni-mechanisms) are calculated for the first time. Depending on the substrate and product concentrations a complete set of solutions is found. In all cases studied the model predicts a matching of the concentrations of the reactants and the corresponding Michaelis constants, which is in good accordance with the experimental data. It is discussed how the model can be applied to the calculation of the optimal kinetic design of real enzymes.     

Analysis of fractal ion channel gating kinetics: kinetic rates, energy levels, and activation energies

Ion channels in the cell membranes of the corneal endothelium, hippocampal neurons, and fibroblasts, and gramicidin channels in lipid bilayers have open and closed times that can be fit, in whole or part, by power law distributions. That is, the gating is self-similar when viewed at different time scales. Hence, kinetic processes at slow and fast time scales are not independent but rather are interrelated. To study how such a relationship can arise we analyze a closed in equilibrium open channel with the fractal dimension for leaving the closed state DCO approximately 2 and the fractal dimension for leaving the open state DOC approximately 1. This special case can be analyzed because it can be represented by equivalent Markov processes. We show that it is equivalent to Markov chains with forward and backward kinetic rate constants approximately equal at each stage, and forming an approximate geometric progression along the different stages. These kinetic rates determine the energy levels and activation energy barriers separating those levels. We find that there are many conformational states (substates) separated by high activation energy barriers. This is similar to the energy structure found for globular proteins such as myoglobin. However, the novel feature reported here is that the activation energy barriers are not independent but are interrelated and form an arithmetic progression. Because of this relationship the fast processes across the low activation energy barriers are linked to slow processes across the high activation energy barriers.     

Cross-bridge scheme and force per cross-bridge state in skinned rabbit psoas muscle fibers.

The rate and association constants (kinetic constants) which comprise a seven state cross-bridge scheme were deduced by sinusoidal analysis in chemically skinned rabbit psoas muscle fibers at 20 degrees C, 200 mM ionic strength, and during maximal Ca2+ activation (pCa 4.54-4.82). The kinetic constants were then used to calculate the steady state probability of cross-bridges in each state as the function of MgATP, MgADP, and phosphate (Pi) concentrations. This calculation showed that 72% of available cross-bridges were (strongly) attached during our control activation (5 mM MgATP, 8 mM Pi), which agreed approximately with the stiffness ratio (active:rigor, 69 +/- 3%); active stiffness was measured during the control activation, and rigor stiffness after an induction of the rigor state. By assuming that isometric tension is a linear combination of probabilities of cross-bridges in each state, and by measuring tension as the function of MgATP, MgADP, and Pi concentrations, we deduced the force associated with each cross-bridge state. Data from the osmotic compression of muscle fibers by dextran T500 were used to deduce the force associated with one of the cross-bridge states. Our results show that force is highest in the AM*ADP.Pi state (A = actin, M = myosin). Since the state which leads into the AM*ADP.Pi state is the weakly attached AM.ADP.Pi state, we confirm that the force development occurs on Pi isomerization (AM.ADP.Pi --> AM*ADP.Pi). Our results also show that a minimal force change occurs with the release of Pi or MgADP, and that force declines gradually with ADP isomerization (AM*ADP -->AM.ADP), ATP isomerization (AM+ATP-->AM*ATP), and with cross-bridge detachment. Force of the AM state agreed well with force measured after induction of the rigor state, indicating that the AM state is a close approximation of the rigor state. The stiffness results obtained as functions of MgATP, MgADP, and Pi concentrations were generally consistent with the cross-bridge scheme.     

Modeling Facilitative Sugar Transporters: Transitions Between Single and Double Ligand Occupancy of Multiconformational Channel Models Explain Anomalous Kinetics

The four-state simple carrier model (SCM) is employed to describe ligand translocation by diverse passive membrane transporters. However, its application to systems like facilitative sugar transporters (GLUTs) is controversial: unidirectional fluxes under zero-trans and equilibrium-exchange experimental conditions fit a SCM, but flux data from infinite-cis and infinite-trans experiments appear not to fit the same SCM. More complex kinetic models have been proposed to explain this ``anomalous' behavior of GLUTs, but none of them accounts for all the experimental findings. We propose an alternative model in which GLUTs are channels subject to conformational transitions, and further assume that the results from zero-trans and equilibrium-exchange experiments as well as trans-effects corresponds to a single-occupancy channel regime, whereas the results from the infinite-cis and infinite-trans experiments correspond to a regime including higher channel occupancies. We test the plausibility of this hypothesis by studying a kinetic model of a two-site channel with two conformational states. In each state, the channel can bind the ligand from only one of the compartments. Under single-occupancy, for conditions corresponding to zero-trans and equilibrium-exchange experiments, the model behaves as a SCM capable of exhibiting trans-stimulations. For a regime including higher degrees of occupancy and infinite-cis and infinite-trans conditions, the same channel model can exhibit a behavior qualitatively similar to a SCM, albeit with kinetic parameters different from those for the single-occupancy regime. Numerical results obtained with our model are consistent with available experimental data on facilitative glucose transport across erythrocyte membranes. Hence, if GLUTs are multiconformational channels, their particular kinetic properties can result from transitions between single and double channel occupancies. Received: 12 April 1995/Revised: 28 August 1995     

A comparative tool for the validity of rate kinetics in ion channels by Onsager reciprocity theorem

The precise form of the rate constant functions of ion channels is very crucial for reproducing the electrophysiological behavior. Therefore, how well they account for experimental data plays an important role in the behavior of the model. In this study, we derive kinetic coefficients of activation and inactivation gates in ion channels by Onsager reciprocity theorem for an ensemble of gating particles, and propose that the obtained kinetic coefficients can be used as a comparative tool for the empirical validity of fitted rate constant functions to experimental data. We also illustrate its applicability based on the activation and inactivation kinetics of T-type calcium channel in thalamic relay neurons. We show that the shape of the steady-state curve by itself seems to be a poor indicator of the functional form of the rate functions, but the time constant curves reflect considerable variation depending on the particular form of the rate functions, and that the kinetic coefficients related to the time constants provide a powerful tool to determine the empirical validity of the fitted rate constants.     

Kinetic model of regulation of muscle protein activity.

The widely accepted steric model of calcium regulation of actin-myosin interactions in vertebrate muscles has to be completed to fit the kinetic data. It should be supposed that: (1) the thin filaments consist of functionally independent units, containing seven actin sites regulated by one troponin-tropomyosin complex; (2) actin sites become available for myosin heads only due to fluctuations of tropomyosin position; (3) binding of calcium to troponin results either in the shift of the tropomyosin equilibrium position or in the weakening of its interactions with actin strand so that the probability of effective fluctuations increases; (4) link formation between myosin head and some of the available actin site fixates the tropomyosin in such a position that the other six actin sites of the same functional unit become available for myosin too.The model gives linear kinetic scheme for the transitions of a functional unit between nine states (a “turned off” state, and eight “turned on” ones with different occupancy by myosin heads). The dependences of the apparent rate constants of actomyosin formation and dissociation upon the myosin head and substrate concentrations are obtained from the Lymn-Taylor scheme. The frequency of the actomyosin complexes dissociation is assumed to give the ATPase rate.The model fits the kinetic data on the ATP hydrolysis by myosin subfragment-1 with regulated or unregulated actin as a cofactor under various conditions. It shows a sharp dependence of activation upon the apparent affinity of the actin and myosin sites. Therefore, the model appears to be applicable to myosin controlled systems.     

On the mechanistic origin of damped oscillations in biochemical reaction systems

A generalized reaction scheme for the kinetic interaction of two reactants in a metabolic pathway has been examined in order to establish what minimal mechanistic patterns are required to support a damped oscillatory transient-state kinetic behaviour of such a two-component system when operating near a steady state. All potentially oscillating sub-systems inherent in this scheme are listed and briefly characterized. The list includes several mechanistic patterns that may be frequently encountered in biological system (e.g. involving feedback inhibition, feed-forward activation, substrate inhibition or product activation), but also draw attention to some hitherto unforeseen mechanisms by which the kinetic interaction of two metabolites may trigger damped oscillations. The results can be used to identify possible sources of oscillations in metabolic pathways without detailed knowledge about the explicit rate equations that apply.     

Reaction kinetic parameters for ion transport from steady-state current-voltage curves.

This study demonstrates possible ways to estimate the rate constants of reaction kinetic models for ion transport from steady-state current-voltage data as measured at various substrate concentrations. This issue is treated theoretically by algebraic reduction and extension of a reaction kinetic four-state model for uniport. Furthermore, an example for application is given; current-voltage data from an open K+ selective channel (Schroeder, J.I., R. Hedrich, and J.M. Fernandez, 1984, Nature (Lond.), 312:361-362) supplemented by some new data have been evaluated. The analysis yields absolute numerical estimates of the 14 rate constants of a six-state model, which is discussed in a wider context.     

Presteady-state kinetics and carrier-mediated transport: A theoretical analysis

Summary Kinetic studies of cotransport mechanisms have so far been limited to the conventional steady-state approach which does not allow in general to resolve either isomerization or ratelimiting steps and to determine the values of the individual rate constants for the elementary reactions involved along a given transport pathway. Such questions can only be answered using presteady-state or relaxation experiments which, for technical reasons, have not yet been introduced into the field of cotransport kinetics. However, since two recent reports seem compatible with the observation of such transient kinetics, it would appear that theoretical studies are needed to evaluate the validity of such claims and to critically evaluate the expectations from a presteady-state approach. We thus report such a study which was performed on a simple four-state mechanism of carrier-mediated transport. The time-dependent equation for zero-trans substrate uptake was thus derived and then extended to models withp intermediary steps. It is concluded that (p-1) exponential terms will describe the approach to the steady state but that such equations have low analytical value since the parameters of the flux equation cannot be expressed in terms of the individual rate constants of the elementary reactions for models withp>5. We thus propose realistic simplifications based on the time-scale separation hypothesis which allows replacement of the rate constants of the rapid steps by their equilibrium constants, thereby reducing the complexity of the kinetic system. Assuming that only one relaxation can be observed, this treatment generates approximate models for which analytical expressions can easily be derived and simulated through computer modeling. When performed on the four-state mechanism of carrier-mediated transport, the simulations demonstrate the validity of the approximate solutions derived according to this hypothesis. Moreover, our approach clearly shows that presteady-state kinetics, should they become applicable to (co)transport kinetics, could be invaluable in determining more precise transport mechanisms.     

Atypical kinetic behavior of chloroperoxidase-mediated oxidative halogenation of polycyclic aromatic hydrocarbons

We have identified an atypical kinetic behavior for the oxidative halogenation of several polycyclic aromatic hydrocarbons (PAHs) by chloroperoxidase (CPO) from Caldariomyces fumago. This behavior resembles the capacity of some members of the P450 family to simultaneously recognize several substrate molecules at their active sites. Indeed, fluorometric studies showed that PAHs exist in solution as monomers and π-π dimers that interact to different extents with CPO. The dissociation constants of dimerization were evaluated for every single PAH by spectrofluorometry. Furthermore, docking studies also suggest that CPO might recognize either one or two substrate molecules in its active site. The atypical sigmoidal kinetic behavior of CPO in the oxidative halogenation of PAHs is explained in terms of different kinetic models for non-heteroatomic PAHs (naphthalene, anthracene and pyrene). The results suggest that the actual substrate for CPO in this study was the π-π dimer for all evaluated PAHs.     

Intermediates and kinetics of membrane fusion.

Recently, it has become clear that the influenza virus fusion protein, hemagglutinin (HA), produces membrane destabilization and fusion by a multistep process, which involves the aggregation of the HAs to form a fusion site. While the details of this process are under debate, it is important to recognize that proposing any sequence of "microscopic" fusion intermediates encumbers general "macroscopic" kinetic consequences, i.e., with respect to membrane mixing rates. Using a kinetic scheme which incorporates the essential elements of several recently proposed models, some of these measurable properties have been elucidated. First, a rigorous mathematical relationship between fusion intermediates and the fusion event itself is defined. Second, it is shown that what is measured as the macroscopic "fusion rate constant" is a simple function of all of the rate constants governing the transitions between intermediates, whether or not one of the microscopic steps is rate limiting. Third, while this kinetic scheme predicts a delay (or lag) time for fusion, as has been observed, it will be very difficult to extract reliable microscopic information from these data. Furthermore, it is predicted that the delay time can depend upon HA surface density even when the HA aggregation step is very rapid compared with fusion, i.e., the delay time need not be due to HA aggregation. Fourth, the inactivation process observed for influenza virions at low pH can be described within this kinetic scheme simply, yet rigorously, via the loss of the fusion intermediates. Fifth, predicted Arrhenius plots of fusion rates can be linear for this multistep scheme, even though there is no single rate-determining step and even when a branched step is introduced, i.e., where one pathway predominates at low temperature and the other pathway predominates at high temperature. Furthermore, the apparent activation energies obtained from these plots bear little or no quantitative resemblance to the microscopic activation energies used to simulate the data. Overall, these results clearly show that the intermediates of protein mediated fusion can be studied only by using assays sensitive to the formation of each proposed intermediate.     

Theoretical effects of radioligand diffusional gradients and microscopic neuroreceptor distribution inin vivo kinetic studies

A simplified one-dimensional model system was used to test the possibility that physically realistic parameters would lead to the prediction of microscopic heterogeneity of radioligand distribution in the brain and that microscopic heterogeneity of radioligand and neuroreceptor distribution could influence the macroscopically observedin vivo kinetics. The model was represented mathematically by a partial differential equation which is similar to the heat diffusion equation, but with special boundary conditions. The equation was solved analytically under the condition of negligible receptor occupancy by inversion of the Laplace transform and in the more general case of arbitrary receptor occupancy by cubic spline approximation. In simulations with physically reasonable values for rate constants and parameters, we find that significant radioligand gradients can occur. Thus, the level of radioligand in the immediate vicinity of the receptor may be substantially different from the average level in a macroscopically measured region of interest. In order to analyze the simulated data, we derived a rigorous steady-state solution, including both a statement of necessary and sufficient conditions for the validity of the steady-state approximation as well as a demonstration of the proper technique for assessing the consistency of the derived parameter with the requirements of the approximation. The radioligand heterogeneity leads to significant errors in the parameters estimated in the steady-state kinetic analysis. In particular, the pseudo first-order rate constant for radioligand-neuroreceptor association, which is often used as a measure of the total amount of neuroreceptor, is underestimated. The first-order rate constant for radioligand-neuroreceptor dissociation is also underestimated. These effects can partially account for the experimentally-observed discrepancy betweenin vivo andin vitro estimates of these kinetic parameters.