Reaction free energy surfaces in myosin-actin-ATP systems.

If we select for consideration any reaction M1 in equilibrium M2 in the myosin-ATPase cycle, the question arises as to the relations between the rate constants for (1) M1 equilibrium M2, (2) AM1 in equilibrium AM2 (A = actin), (3) A + M1 in equilibrium AM1, and (4) A + M2 equilibrium AM2, with actin and myosin either (a) in solution or (b) in the myofilament structure. It is shown here, by means of examples, that a single so-called potential of mean force, W, and structural free energy, Am, suffice to determine the reaction free energy surfaces for all of these transitions (W for the solution case, W + Am for the structured case). In fact, Am is the same for all reactions in the myosin-ATPase cycle. Of course, though indispensable as the starting point and adequate for qualitative understanding, the reaction free energy surface does not provide (without additional theory) the actual values of the rate constants or of the corresponding basic free energy changes in the myosin states involved. These rate constants and free energies are discussed, in a preliminary way, in two other papers.     

The frequency of cyclic processes in biological multistate systems

Summary Cyclic processes in stochastic models of macromolecular biological systems are considered. The diagram solution of the model equations (master equation) gives rise to special functions of the rate constants, called the circuit (or one-way cycle) fluxes. As Hill has shown, these functions are the fundamental theoretical components of the operational fluxes, i.e., of the rates of reaction, of transport, of energy conversion, etc. Evidence recently has been found by Monte Carlo simulations that the circuit fluxes can be interpreted as the frequencies of circuit completions. Making use of the theory of graphs, we prove that this physical interpretation of the circuit fluxes is generally valid.     

Stability of Radiation-Induced Organic Free Radicals: Decay by Heat

The rate of free radical decay was measured at various temperatures using electron paramagnetic resonance spectroscopy. Rate constants determined from first-order decay kinetics were used to determine the activation energy for the process of free radical decay. The similarity between the temperature dependence of free radical decay by heat and that of electrical conductivity has led us to consider the possibility that the two processes may be related. Mechanisms by which a population of electron-hole conducting states may lead to free radical decay are outlined and experimental data relating to these mechanisms are discussed.     

Stochastic Bistability and Bifurcation in a Mesoscopic Signaling System with Autocatalytic Kinase

Bistability is a nonlinear phenomenon widely observed in nature including in biochemical reaction networks. Deterministic chemical kinetics studied in the past has shown that bistability occurs in systems with strong (cubic) nonlinearity. For certain mesoscopic, weakly nonlinear (quadratic) biochemical reaction systems in a small volume, however, stochasticity can induce bistability and bifurcation that have no macroscopic counterpart. We report the simplest yet known reactions involving driven phosphorylation-dephosphorylation cycle kinetics with autocatalytic kinase. We show that the noise-induced phenomenon is correlated with free energy dissipation and thus conforms with the open-chemical system theory. A previous reported noise-induced bistability in futile cycles is found to have originated from the kinase synchronization in a bistable system with slow transitions, as reported here.     

Gating kinetics of batrachotoxin-modified Na+ channels in the squid giant axon. Voltage and temperature effects.

The gating kinetics of batrachotoxin-modified Na+ channels were studied in outside-out patches of axolemma from the squid giant axon by means of the cut-open axon technique. Single channel kinetics were characterized at different membrane voltages and temperatures. The probability of channel opening (Po) as a function of voltage was well described by a Boltzmann distribution with an equivalent number of gating particles of 3.58. The voltage at which the channel was open 50% of the time was a function of [Na+] and temperature. A decrease in the internal [Na+] induced a shift to the right of the Po vs. V curve, suggesting the presence of an integral negative fixed charge near the activation gate. An increase in temperature decreased Po, indicating a stabilization of the closed configuration of the channel and also a decrease in entropy upon channel opening. Probability density analysis of dwell times in the closed and open states of the channel at 0 degrees C revealed the presence of three closed and three open states. The slowest open kinetic component constituted only a small fraction of the total number of transitions and became negligible at voltages greater than -65 mV. Adjacent interval analysis showed that there is no correlation in the duration of successive open and closed events. Consistent with this analysis, maximum likelihood estimation of the rate constants for nine different single-channel models produced a preferred model (model 1) having a linear sequence of closed states and two open states emerging from the last closed state. The effect of temperature on the rate constants of model 1 was studied. An increase in temperature increased all rate constants; the shift in Po would be the result of an increase in the closing rates predominant over the change in the opening rates. The temperature study also provided the basis for building an energy diagram for the transitions between channel states.     

The dissociation of 1-N6-ethenoadenosine diphosphate from regulated actomyosin subfragment 1

The effective rate of dissociation of 1-N6-ethenoadenosine diphosphate (epsilon ADP) from the regulated actin X subfragment 1 X epsilon ADP complex of rabbit skeletal muscle is approximately 10-15 times smaller in the absence of calcium ion compared to the presence of calcium ion. The decrease in fluorescence emission with dissociation of the bound epsilon ADP fitted two exponential terms. The evidence is consistent with a kinetic scheme in which two first-order transitions precede the dissociation step: (Formula: see text) where D is epsilon ADP, A is regulated actin, M is subfragment 1, the asterisks refer to the degree of fluorescence enhancement, and AM(D) is a collision complex in equilibrium with free epsilon ADP. Both rate constants k-2 and k-1 were reduced approximately 15-fold in the absence of calcium ion. The rate constants for the dissociation of epsilon ATP, epsilon ADP X Pi, formed in the enzyme cycle, and epsilon ADP are all reduced in the absence of calcium ion; consequently, the primary effect in calcium regulation of the actin-subfragment 1 ATPase is on the rate constant of a transition (or transitions) between actomyosin-nucleoside phosphate complexes.     

The sliding filament model of muscle contraction. II. The energetic and dynamical predictions of a quantum mechanical transducer model

A theoretical model of a molecular energy transducing unit designed for the production of mechanical work is constructed and its consequences examined and compared with the experimentally determined myothermal and dynamic properties of vertebrate striated muscle. The model rests on a number of independent assumptions which include: the almost instantaneous generation of mechanical force by the occurrence of a radiationless transition between vibronic states of the transducer (crossbridge) at a point of potential energy surface crossing; transmission of this force to the load via the active sites on the thin filament by means of non-bonding repulsive forces, no energy being required for detachment; “detachment” consists of a second radiationless transition at a lower energy point than the first force generating transition, the energy difference appearing largely as work. The method of force generation completely avoids problems such as the “force-rate dilemma” which occur repeatedly in any discussion where state populations are near-Boltzmann and also leads without further arbitrary assumptions to such concepts as “attached but non force-producing states” and strongly position dependent “attachment” and “detachment” rate constants since these can only be appreciable near potential energy surface crossings. The kinetics and energetics of a transducer of this type operating cyclically and converting ATP → ADP + P_i are considered and shown to lead to length-tension and energetic behaviour very similar to that exhibited by vertebrate striated muscle, both for contraction and stretching. The existence of a limiting tension for stretching is predicted by the model as is the decrease of the rate of enthalpy release rate below the isometric value. At the limiting tension the rate of enthalpy release by the transducers is virtually zero, as observed. However, the stretching only inhibits the ATP hydrolysis, the cyclic synthesis from ADP and work being impossible with this model. The response to rapid length step changes automatically contains the asymmetry observed experimentally (with respect to lengthening and shortening) and arbitrary assumptions over and above those giving adequate explanation of the steady-state properties are not required. The asymmetry arises mainly as a consequence of the non-bonded pushing action of the crossbridges. This same assumption predicts the occurrence of an asymmetric thermoelastic ratio for active muscle with respect to stretching and contraction. The quantitative aspects of the model are satisfactory as it simultaneously reconciles the numerical magnitudes of macroscopic quantities such as isometric tension, maximum contraction velocity, limiting tension sustainable on stretching, isometric heat rate and resting heat rate with molecular parameters such as the filament and crossbridge periodicities, molecular vibrational relaxation rates, recurrence times for the radiationless transitions occurring, etc. This is achieved without any parameter optimization and only a very much smaller number of unknown parameters than the number of observed results accounted for. Many of the entities occurring in the model cycle (vibronic states of crossbridges, ATP, etc.) appear to be in one-to-one correspondence with many of the kinetic entities postulated to account for the biochemical kinetic results obtained for the actomyosin ATPase system in vitro. Finally, the rigor state has to be viewed in a different way from the conventional one; on the basis that the present model states which are part of the contraction cycle but sparsely populated during the latter (and hence are of chemical kinetic but not dynamical importance) are heavily populated during the rigor state. The mechanical properties of the rigor state would then be determined by these molecular states which would be very short-lived during the contraction cycle. If this is correct the rigor state could yield much more information about inaccessible parts of the contraction cycle than is presently supposed. The model leads one to expect a rather different response to quick length step changes in the rigor state from that of the active state, in contrast to current interpretations in terms of a large number of attached crossbridges, unable to detach due to the absence of ATP.     

The temperature-induced conformational transition of immobilized chymotrypsinogen

A method has been suggested for obtaining kinetic thermodynamic data on conformational transitions of insoluble proteins by fluorescence measurements. The method was used for treatment of the temperature-induced conformational transition of chymotrypsinogen bonded to the hydroxyethyl methacrylate Spheron matrix. The bonding to Spheron causes destabilization of the native conformation of chymotrypsinogen. Two types of transition of immobilized chymotrypsinogen have been found which are controlled by first-order kinetics with different rate constants. The entropy and enthalpy changes were smaller than for free chymotrypsinogen in solution. The data obtained are interpreted as an effect of the physical interaction of the protein in the activated and denatured states with the polymeric matrix.     

Reactions of 1-N6-ethenoadenosine nucleotides with myosin subfragment 1 and acto-subfragment 1 of skeletal and smooth muscle

The fluorescent nucleotides epsilon ADP and epsilon ATP were used to study the binding and hydrolysis mechanisms of subfragment 1 (S-1) and acto-subfragment 1 from striated and smooth muscle. The quenching of the enhanced fluorescence emission of bound nucleotide by acrylamide analyzed either by the Stern-Volmer method or by fluorescence lifetime measurements showed the presence of two bound nucleotide states for 1-N6-ethenoadenosine triphosphate (epsilon ATP), 1-N6-ethenoadenosine diphosphate (epsilon ADP), and epsilon ADP-vanadate complexes with S-1. The equilibrium constant relating the two bound nucleotide states was close to unity. Transient kinetic studies showed two first-order transitions with rate constants of approximately 500 and 100 s-1 for both epsilon ATP and epsilon ADP and striated muscle S-1 and 300 and 30 s-1, respectively, for smooth muscle S-1. The hydrolysis of [gamma-32P] epsilon ATP yielded a transient phase of small amplitude (less than 0.2 mol/site) with a rate constant of 5-10 s-1. Consequently, the hydrolysis of the substrate is a step in the mechanism which is distinct from the two conformational changes induced by the binding of epsilon ATP. An essentially symmetric reaction mechanism is proposed in which two structural changes accompany substrate binding and the reversal of these steps occurs in product release. epsilon ATP dissociates acto-S-1 as effectively as ATP. For smooth muscle acto-S-1, dissociation proceeds in two steps, each accompanied by enhancement of fluorescence emission. A symmetric reaction scheme is proposed for the acto-S-1 epsilon ATPase cycle. The very similar kinetic properties of the reactions of epsilon ATP and ATP with S-1 and acto-S-1 suggest that two ATP intermediate states also occur in the ATPase reaction mechanism.     

Kinetics and quantum yield of photoconversion of protochlorophyll(ide) to chlorophyll(ide) a

The kinetics of photoconversion of protochlorophyll(ide) to chlorophyll(ide) a were investigated in dark-grown barley leaves and in a preparation of protochlorophyll holochrome subunits. In the subunits the conversion obeyed first-order kinetics. This indicates that the excitation of protochlorophyll(ide), energy loss through deexcitation, and the reduction of excited protochlorophyll(ide) are all reactions that follow first-order kinetics with respect to protochlorophyll(ide) in protochlorophyll holochrome subunits.In contrast, photoconversion in leaves obeyed neither first- nor second-order kinetics. This prompted the postulation of an additional route within macromolecular units of protochlorophyll holochrome, whereby energy is lost from excited protochlorophyll(ide) by a reaction that is not first order. Such a process might be energy transfer from excited protochlorophyll(ide) to newly-formed chlorophyll(ide) a.A dynamic model describing photoconversion in macromolecular units was derived. The model is consistent with the observed progress of photoconversion in barley leaves and in protochlorophyll holochrome subunits from barley.Determinations of the quantum yield of photoconversion in protochlorophyll holochrome subunits gave values of 0.4–0.5 molecules · quantum^?1. Estimates of the initial quantum yield of the photoconversion process in leaves fell into the same range. The dynamic model allows predictions on the progressively decreasing quantum yield as the photoconversion proceeds in macromolecular units.     

Millisecond-scale biochemical response to change in strain

Muscle fiber contraction involves the cyclical interaction of myosin cross-bridges with actin filaments, linked to hydrolysis of ATP that provides the required energy. We show here the relationship between cross-bridge states, force generation, and Pi release during ramp stretches of active mammalian skeletal muscle fibers at 20°C. The results show that force and Pi release respond quickly to the application of stretch: force rises rapidly, whereas the rate of Pi release decreases abruptly and remains low for the duration of the stretch. These measurements show that biochemical change on the millisecond timescale accompanies the mechanical and structural responses in active muscle fibers. A cross-bridge model is used to simulate the effect of stretch on the distribution of actomyosin cross-bridges, force, and Pi release, with explicit inclusion of ATP, ADP, and Pi in the biochemical states and length-dependence of transitions. In the simulation, stretch causes rapid detachment and reattachment of cross-bridges without release of Pi or ATP hydrolysis.     

First-order kinetics of muscle oxygen consumption, and an equivalent proportionality between QO2 and phosphorylcreatine level. Implications for the control of respiration

In frog sartorius muscle, after a tetanus at 20 degrees C, during which an impulse-like increase occurs in the rate of ATP hydrolysis, the rate of O2 consumption (QO2) reaches a peak relatively quickly and then declines monoexponentially, with a time constant not dependent on the tetanus duration (tau = 2.6 min in Rana pipiens and 2.1 min in Rana temporaria). To a good approximation, these kinetics are those of a first-order impulse response, and the scheme of reactions that couple O2 consumption to extramitochondrial ATP hydrolysis thus behaves as a first-order system. It is first deduced and then demonstrated directly that while QO2(t) is monoexponential, it changes in parallel with the levels of creatine and phosphorylcreatine, with proportionality constants +/- 1/tau p, where p is the P/O2 ratio in vivo. From this, it is further deduced that the mitochondrial creatine kinase (CK) reaction is pseudo-first order in vivo. The relationship between [creatine] and QO2 predicted by published models of the control of respiration is markedly different from that actually observed. As shown here, the first-order kinetics of QO2 are consistent with the hypothesis that respiration is rate-limited by the mitochondrial CK reaction; this has as a corollary the "creatine shuttle" hypothesis.     

Slow conformational changes of a Neurospora glutamate dehydrogenase studied by protein fluorescence

1. The NADP-dependent glutamate dehydrogenase of Neurospora crassa undergoes slow reversible structural transitions, with half-times in the order of a few minutes, between active and inactive states. The inactive state of the enzyme, which predominates at pH values below 7.0, has an intrinsic tryptophan fluorescence 25% lower than that of the active state, which predominates at pH values above 7.6. The inactive state can be activated either by an increase in pH or by addition of activators such as succinate. 2. The kinetics of the slow transitions that follow activating and inactivating rapid changes in conditions have been monitored by measurements of protein fluorescence. The results show that the slow reversible conformational change detected by the change in fluorescence is the rate-limiting process for enzyme activation and inactivation. 3. In both directions this conformational change follows apparent first-order kinetics and the rate constant is independent of protein concentration. These kinetics and published measurements of molecular weight are indicative of an isomerization process. 4. In both directions the changes show a large energy of activation and a large positive entropy of activation, consistent with a considerable disturbance of conformation in the transition state. 5. Comparisons of the fluorescence emission spectra of the active and inactive states indicate that the difference in fluorescence is produced by quenching, possibly intramolecular, in the inactive conformation. Iodide ions cause similar quenching. 6. In some mutationally altered forms of the enzyme comparable but modified conformational changes can be followed by protein fluorescence.     

Frequency response model of skeletal muscle and its association with contractile properties of skeletal muscle

The aims of the present study were to develop a mathematical model of the skeletal muscle based on the frequency transfer function, referred to as frequency response model, and to presume the relationship between the model elements and skeletal muscle contractile properties. Twitch force in elbow flexion was elicited by applying a single electrical stimulation to the motor point of biceps brachii muscles, and then analyzed visually by the Bode gain and phase diagram of the force signal. The frequency response model was represented by a frequency transfer function consisting of five basic control elements (proportional element, dead time element, and three first-order lag elements). The model element constants were estimated by best-fitting to the Bode gain and phase diagram of the twitch force signal. The proportional constant and the dead time in the frequency response model correlated significantly with the peak torque and the latency in the actual twitch force, respectively. In addition, the time constants of the three first-order lag elements in the model correlated strongly with the contraction time and the half relaxation time in the actual twitch force. The results suggested a possibility that the individual elements in the frequency response model would reflect the biochemical and biomechanical properties in the excitation–contraction coupling process of skeletal muscle.     

The relationship between kinetics of substrate-limited transitions and steady-state growth in continuous cultures ofAquaspirillum autotrophicum limited by pyruvate

Heterotrophic growth at steady state and during transient states caused by the sudden change of the concentration of the limiting factor in the feed medium was investigated experimentally for continuous cultures ofAquaspirillum autotrophicum limited by pyruvate. A model for describing the growth at steady state was selected from three unstructured models after statistical tests of the data. This model postulates that the growth yield increases linearly with the growth rate. Growth during transitions where the substrate remained limiting at all times was fitted with first-order kinetics. Theoretical predictions of these kinetics were derived from the unstructured models used to describe steady state. The predicted rate coefficients of the transients were compared to the experimental coefficients. It appeared that the model which best described steady-state growth also provided the best predictions for growth during the transient state. It is a widespread opinion that unstructured models are adequate to describe growth under steady-state conditions but not to predict transitions in continuous culture. However, for the particular case studied here, no higher degree of complexity was required to describe transitions, provided the growth of the culture was always limited by the substrate.     

Duration time of a one-dimensional random walk as a function of the energies of the intermediate states: application for dissociation and relaxation processes in DNA hybrids.

Kinetic parameters of macromolecular systems are important for their function in vitro and in vivo. These parameters describe how fast the system dissociates (the characteristic dissociation time), and how fast the system reaches equilibrium (characteristic relaxation time). For many macromolecular systems, the transitions within the systems are described as a random walk through a number of states with various free energies. The rate of transition between two given states within the system is characterized by the average time which passes between starting the movement from one state, and reaching the other state. This time is referred to as the mean first-passage time between two given states. The characteristic dissociation and relaxation times of the system depend on the first-passages times between the states within the system. Here, for a one-dimensional random walk we derived an equation, which connects the mean first-passage time between two states with the free energies of the states within the system. We also derived the general equation, which is not restricted to one-dimensional systems, connecting the relaxation time of the system with the first-passage times between states. The application of these equations to DNA branch migration, DNA structural transitions and other processes is discussed.     

Reduced kinetic models of facilitative transport

In spite of the highly complex structural dynamics of globular proteins, the processes mediated by them can usually be described in terms of relatively simple kinetic diagrams. How do complex proteins, characterized by undergoing transitions among a possibly very large number of intermediate states, exhibit functional properties that can be interpreted in terms of kinetic diagrams consisting of only a small number of states? One possible way of explaining this apparent contradiction is that, under some conditions, a reduction of the actual complete kinetic diagram that describes all of the macromolecular states and transitions takes place. In this work, we contribute with a formal basis to this interpretation, by generalizing the procedure of diagram reduction to the case of multicyclic kinetic diagrams. As an example, we apply the procedure to a complex kinetic model of facilitative transport. We develop Monte Carlo simulations to obtain the kinetic parameters of the complex model and we compare them with the ones analytically obtained from the reduced model. We confirm that, under some conditions, the kinetic behavior of the complex transporter is indistinguishable from the one of a four-state simple carrier model, derived from the former by diagram reduction. Besides introducing some novel methodological aspects, this work further contributes to the idea that, under many physiological and experimental conditions, a reduction occurs of the complete kinetic diagram that describes the dynamics of a globular protein.     

Dynamics of muscle glycogenolysis modeled with pH time course computation and pH-dependent reaction equilibria and enzyme kinetics

Cellular metabolites are moieties defined by their specific binding constants to H+, Mg2+, and K+ or anions without ligands. As a consequence, every biochemical reaction in the cytoplasm has an associated proton stoichiometry that is generally noninteger- and pH-dependent. Therefore, with metabolic flux, pH is altered in a medium with finite buffer capacity. Apparent equilibrium constants and maximum enzyme velocities, which are functions of pH, are also altered. We augmented an earlier mathematical model of skeletal muscle glycogenolysis with pH-dependent enzyme kinetics and reaction equilibria to compute the time course of pH changes. Analysis shows that kinetics and final equilibrium states of the closed system are highly constrained by the pH-dependent parameters. This kinetic model of glycogenolysis, coupled to creatine kinase and adenylate kinase, simulated published experiments made with a cell-free enzyme mixture to reconstitute the network and to synthesize PCr and lactate in vitro. Using the enzyme kinetic and thermodynamic data in the literature, the simulations required minimal adjustments of parameters to describe the data. These results show that incorporation of appropriate physical chemistry of the reactions with accurate kinetic modeling gives a reasonable simulation of experimental data and is necessary for a physically correct representation of the metabolic network. The approach is general for modeling metabolic networks beyond the specific pathway and conditions presented here.     

Predicting Secondary Structural Folding Kinetics for Nucleic Acids

We report a new computational approach to the prediction of RNA secondary structure folding kinetics. In this approach, each elementary kinetic step is represented as the transformation between two secondary structures that differ by a helix. Based on the free energy landscape analysis, we identify three types of dominant pathways and the rate constants for the kinetic steps: 1), formation; 2), disruption of a helix stem; and 3), helix formation with concomitant partial melting of a competing (incompatible) helix. The third pathway, termed the tunneling pathway, is the low-barrier dominant pathway for the conversion between two incompatible helices. Comparisons with experimental data indicate that this new method is quite reliable in predicting the kinetics for RNA secondary structural folding and structural rearrangements. The approach presented here may provide a robust first step for further systematic development of a predictive theory for the folding kinetics for large RNAs.