This paper reports an experimental study, of the interference of GTP hydrolysis in the mechanism of microtubule assembly, following the model and theory previously published [Hill, T. L. & Carlier, M.-F. (1983) Proc. Natl. Acad. Sci. USA 80,[7234][7235][7236][7237][7238]. Results from dilution experiments show that microtubules depolymerize faster below the critical concentration than expected with a reversible polymerization model. The experimental plot of flux versus tubulin concentration exhibits a slope discontinuity at the critical copcentration, in agreement with the theory. Theoretical points calculated by the Monte Carlo method can be fitted qualitatively to the data. A consequence of this peculiar dynamic behavior of microtubules is that the ratio of tubulin dissociation and association rate constants measured, respectively, below and above the critical concentration does not yield the true value of the critical concentration. It is emphasized that the presence of GTP at microtubule ends is necessary to maintain the stability of the polymer.It is striking that two essential polymer components of the cytoskeleton of eukaryotic cells, actin filaments and microtubules (MTs), share many structural and physicochemical properties. In particular, the NTP bound to the protomer (ATP to actin, GTP to tubulin) undergoes hydrolysis during polymerization (1-3). NTP hydrolysis is also associated with maintenance of the polymers. Indeed, in the presence of low concentrations of GTP, MTs eventually depolymerize (4) and GDP is unable to promote polymerization (2). Therefore, actin filaments and MTs must be considered as steadystate polymers. However, GTP hydrolysis associated with MT assembly does not appear necessary for polymerization because hydrolysis is a monomolecular kinetic process uncoupled from polymerization (5), a result also more recently reported on actin (6, 7). Correlatively, MTs can be obtained and stabilized in the presence of nonhydrolyzable analogs of GTP (8) and thus be maintained in an equilibrium state.Although GTP hydrolysis is involved in MT assembly and steady state, until now Oosawa's model, which applies to equilibrium polymers-i.e., those undergoing linear reversible polymerization-satisfactorily accounted for the kinetic and thermodynamic data for MT assembly (4, 9).In an effort to explore the consequences of nucleotide hydrolysis in the polymerization process, Wegner first developed a model (10) pointing to the possibility of "head-to-tail polymerization" driven by ATP hydrolysis in actin polymerization. The same phenomenon was observed experimentally on MTs and called "treadmilling" by Margolis and Wilson (11).Possible implications of treadmilling in the regulation of MTs in vivo have been suggested (12)(13)(14)(15). On the basis of Wegner's model, the thermokinetic theory of MT and actin polymerization has been developed (16). In this model, GTP hydrolysis is tightly coupled to the polymerization process. As a consequence, only GDP is bound to the polymer and the critical concen...

A kinetic model for insulin secretion in pancreatic beta-cells is adapted from a model for fast exocytosis in chromaffin cells. The fusion of primed granules with the plasma membrane is assumed to occur only in the "microdomain" near voltage-sensitive L-type Ca(2+)-channels, where [Ca(2+)] can reach micromolar levels. In contrast, resupply and priming of granules are assumed to depend on the cytosolic [Ca(2+)]. Adding a two-compartment model to handle the temporal distribution of Ca(2+) between the microdomain and the cytosol, we obtain a unified model that can generate both the fast granule fusion and the slow insulin secretion found experimentally in response to a step of membrane potential. The model can simulate the potentiation induced in islets by preincubation with glucose and the reduction in second-phase insulin secretion induced by blocking R-type Ca(2+)-channels (Ca(V)2.3). The model indicates that increased second-phase insulin secretion induced by the amplifying signal is controlled by the "resupply" step of the exocytosis cascade. In contrast, enhancement of priming is a good candidate for amplification of first-phase secretion by glucose, cyclic adenosine 3':5'-cyclic monophosphate, and protein kinase C. Finally, insulin secretion is enhanced when the amplifying signal oscillates in phase with the triggering Ca(2+)-signal.

Recently, it has been demonstrated that free energy from an alternating electric field can drive the active transport of Rb+ by way of the Na+,K+-ATPase. In the present work, it is shown why many transmembrane enzymes can be expected to absorb free energy from an oscillating electric field and transduce that to chemical or transport work. In the theoretical analysis it turned out to be sufficient that (i) the catalytic process be accompanied by either net or cyclic charge translocation across the membrane and (i) the stability of the enzyme states involved be asymmetric. Calculations based on a four-state model reveal that free-energy transduction occurs with sinusoidal, square-wave, and positive-only oscillating electric fields and for cases that exhibit either linear or exponential field-dependent rate constants. The results suggest that in addition to oscillating electric field-driven transport, the proposed mechanism can also be used to explain, in part, the "missing" free energy term in the cases in which ATP synthesis has been observed with insufficient transmembrane proton electrochemical potential difference.In oxidative phosphorylation and in ion transport, cases are found in which the output free energy seems to exceed the input free energy. The most prominent such case in oxidative phosphorylation is that of alkalophilic bacteria (1). Here the transmembrane proton electrochemical potential difference seems too low to account for the observed ATP synthesis. An early explanation of this type of phenomenon was that there existed an additional input term in the free-energy balance sheet for ATP synthesis due to a direct coupling of the conformational energy of the electron transport chain to the FOF1-ATPase (2). Alternatively, it was proposed that the protons involved in energy coupling are confined to small domains near the membrane, and thus the equation used for the free-energy balance based on bulk parameters (i.e., spatial averages) would be inappropriate (for review, see ref.3; see also ref. 4).Recent results obtained in a different experimental context exhibit a similar deficit in the free-energy balance (for review, see ref. 5). Serpersu and Tsong (6, 7) reported that when an alternating electric field (=1 kHz) was applied to an erythrocyte suspension, the Na',K+-ATPase catalyzed the active transport of Rb+ without detectable hydrolysis of ATP even though the time average of the electric field was zero.The suggested solution was that the Na ,K+-ATPase had directly extracted free energy from the oscillations in the field and transduced this to the uphill transport of Rb+ (5). A crucial role for an oscillating electric field has also been proposed for ATP synthesis driven by a pulsed dc field (5). Since, especially locally, electric fields across biological membranes may well have a large oscillating component, the experimental results of Serpersu and Tsong (6, 7) may also have a more general implication for cases in which input free energy seems to be insufficient to explain output work.We ...

It was previously shown that a one-dimensional Ising model could successfully simulate the equilibrium binding of myosin S1 to regulated actin filaments (T. L. Hill, E. Eisenberg and L. Greene, Proc. Natl. Acad. Sci. U.S.A. 77:3186-3190, 1980). However, the time course of myosin S1 binding to regulated actin was thought to be incompatible with this model, and a three-state model was subsequently developed (D. F. McKillop and M. A. Geeves, Biophys. J. 65:693-701, 1993). A quantitative analysis of the predicted time course of myosin S1 binding to regulated actin, however, was never done for either model. Here we present the procedure for the theoretical evaluation of the time course of myosin S1 binding for both models and then show that 1) the Hill model can predict the "lag" in the binding of myosin S1 to regulated actin that is observed in the absence of Ca++ when S1 is in excess of actin, and 2) both models generate very similar families of binding curves when [S1]/[actin] is varied. This result shows that, just based on the equilibrium and pre-steady-state kinetic binding data alone, it is not possible to differentiate between the two models. Thus, the model of Hill et al. cannot be ruled out on the basis of existing pre-steady-state and equilibrium binding data. Physical mechanisms underlying the generation of the lag in the Hill model are discussed.

A simple model is devised to show that an enzymatic Brownian particle in a static electric field can undergo directional movement when coupled with a nonequilibrium chemical reaction which the particle catalyzes, if at least one of the intermediate states of the catalytic cycle is charged. The direction of the movement depends not only on the asymmetry of the electric field, but also on the direction of the chemical reaction and the mechanism of the catalytic cycle. The Brownian particle can also move against an external load and thus do mechanical work. This study suggests that enzyme molecules could be separated based on their enzymatic activities. The formalism developed in this paper can be extended and applied to biological motors. [S0031-9007(96) PACS numbers: 87.15.-v, 05.40.+j Recently the directional movement of Brownian particles in a periodic potential has attracted considerable attention [1][2][3][4][5][6][7]. It is known that the long-time movement of a Brownian particle is not directionally biased in the presence of a periodic potential if the potential is static. But, if the potential is asymmetric within a period and is randomly or regularly switched on and off (so that the force acting on the particles fluctuates), then a net directional movement of the particle can be achieved [1][2][3][4][5]. As in other cases of external fluctuation-induced free energy transduction [8][9][10][11][12], energy from a fluctuating force field is thus found to be able to do mechanical work. Experiments using oscillating electric fields have not only confirmed the existence of directional movement but also shown that particles with different charges or electric properties could be separated [3,4]. In this Letter, we demonstrate with a simple model that a Brownian particle can execute directional movement in a static (nonfluctuating) periodic electric field when coupled with a nonequilibrium chemical reaction. In other words, the free energy of a nonequilibrium chemical reaction can be directly transduced by a Brownian particle to do mechanical work. The general principle of the model can be tested experimentally and should prove useful in biomolecular separation. Moreover, the formalism developed here can be generalized and used for analyzing the motility of single biological motors in vitro, where the effect of Brownian motion may be important.As shown in Fig. 1(a), the Brownian particle E is considered as an enzyme that catalyzes the breakdown of AB into A 1 and B 2 . For simplicity, we consider only the reduced scheme in Fig. 1(b). However, the general conclusions discussed below are expected to be applicable to Fig. 1(a) or other more complicated kinetic schemes. The chemical reaction ͑AB ! A 1 1 B 2 ͒ is assumed to be inhibited in the absence of E and the concentrations of AB, A 1 , and B 2 in solution are assumed to be time independent. We want to study the movement of this chemi- 2 which is catalyzed by the enzymatic Brownian particle (symbolized by a circle). (b) The reduced kinetic scheme after neglect...

Recently, it was shown that free energy can be transduced from a regularly oscillating electric field to do chemical or transport work when coupled through an enzyme with appropriate electrical characteristics. Here we report that randomly pulsed electric fields can also lead to work being done, giving rise to speculation as to whether appropriately designed enzymes can extract and convert free energy from the inherent fluctuations in their environment. The paradox is resolved by showing that equilibrium electrical noise resulting from the environment around an enzyme cannot be completely random but is correlated to the state that the enzyme is in. If the noise has the appropriate reciprocal interaction with the enzyme, its potential to serve as a free-energy source disappears. This is shown by Monte Carlo and other numerical calculations and is proven analytically by use of the diagram method. This method also is used to provide an explicit equation showing that, under a range of conditions, our model enzyme will be induced by uncorrelated ("autonomous") noise to undergo net cyclic flux. That work can be transduced from the "random" noise is demonstrated by using numerical methods.Recently, it was demonstrated (1, 2) that an oscillating electric field is competent to do chemical work when coupled through an enzyme with differences in macroscopic polarization and basic free energy between its conformational states. These results might well account for the observation that the Na+/K+-ATPase mediates active transport when subjected to an oscillating electric field (1, 3, 4).In vivo, fluctuating electric fields have been observed around cells (reviewed in ref. 5), and, certainly, large potential fluctuations must occur in the vicinity of ion channels upon opening or closing. This has been used (1, 6) as the basis for a model ofATP synthesis via the FO/Fl-ATPase, in which coupling factor Fo serves as a field-modulating channel, the opening and closing of which is linked to the binding and release of ligands in F1. Also, it was suggested that the electric field around HW-transporting ATPase in free energytransducing membranes might oscillate because of turnover of neighboring electron-transfer chains, and that these oscillations might contain the free energy often missing in free energy balances (2).In preliminary calculations, we found that totally random noise, when applied to the system described previously (1, 2), led to work being done. This result suggests that many (indeed most) forms offield fluctuations can do work. Yet we must realize that, in keeping with the laws of thermodynamics, internal noise arising in a system at equilibrium certainly cannot do work.Although the effects of field fluctuations have been studied before for both noncyclic (7) and cyclic (8) systems, our results demonstrate that energy can be transduced from a randomly fluctuating field, allowing an enzyme to do work.In a subsequent paper (9), the general asymmetry requirements for a four-state enzyme to work will be studied i...

Previous work has shown that Na,K-ATPase of human erythrocytes can extract free energy from sinusoidal electric fields to pump cations up their respective concentration gradients. Because regularly oscillating waveform is not a feature of the transmembrane electric potential of cells, questions have been raised whether these observed effects are biologically relevant. Here we show that a random-telegraph fluctuating electric field (RTF) consisting of alternating square electric pulses with random lifetimes can also stimulate the Rb(+)-pumping mode of the Na,K-ATPase. The net RTF-stimulated, ouabain-sensitive Rb+ pumping was monitored with 86Rb+. The tracer-measured, Rb+ influx exhibited frequency and amplitude dependencies that peaked at the mean frequency of 1.0 kHz and amplitude of 20 V/cm. At 4 degrees C, the maximal pumping activity under these optimal conditions was 28 Rb+/RBC-hr, which is approximately 50% higher than that obtained with the sinusoidal electric field. These findings indicate that Na,K-ATPase can recognize an electric signal, either regularly oscillatory or randomly fluctuating, for energy coupling, with high fidelity. The use of RTF for activation also allowed a quantitative theoretical analysis of kinetics of a membrane transport model of any complexity according to the theory of electroconformational coupling (ECC) by the diagram methods. A four-state ECC model was shown to produce the amplitude and the frequency windows of the Rb(+)-pumping if the free energy of interaction of the transporter with the membrane potential was to include a nonlinear quadratic term. Kinetic constants for the ECC model have been derived. These results indicate that the ECC is a plausible mechanism for the recognition and processing of electric signals by proteins of the cell membrane.

A simple theorem is derived relating the extent to which enzymes in a metabolic pathway control the steady-state concentration of metabolites to the kinetic properties of those enzymes. The theorem gives insight into the mechanism by which the concentration of a second messenger is controlled by the enzymes that form and degrade it, and provides an alternative to the 'cross-over theorem'.The theoretical principles developed by Kacser and Burns [I] and Heinrich and Rapoport [2] have greatly facilitated the quantitative determination of the extent to which certain enzymes control the flux through a metabolic pathway [3, 41 (the 'control strength' [2] of those enzymes). Through the 'connectivity theorem [I] for the control of fluxes', these principles allow the designation of the enzymic properties (i.e. elasticity coefficients [I, 51) that are responsible for a certain distribution of the control on such a flux between the enzymes in the pathway [I, 3,5 -81. In the present paper, a new theorem (the 'connectivity theorem for control of metabolite concentrations') is derived relating the extent to which metabolite concentrations are controlled by the different pathway enzymes to those enzymic properties (elasticity coefficients). THEORY The theorem .6).Remark : We shall assume throughout that both C and i: can bc defined. Cases with zero reaction rates or concentrations are formally excluded. For these cases alternative theorems are easily formulated.symbol for the elasticity coefficient of the reaction catalyzed by enzyme Ei with respect to metabolite M,, defined by Eqn (3) as (cf. [I, 2, 51):where ci is the (net) reaction rate through enzyme Ei. In the following the metabolite-concentration control coefficient C p and the elasticity coefficient cSkI will occassionally be abbre- ' [6, 10, 111, 'sensitization' [12], 'coefficient' [9]). For a review see [5]. The concentration control coefficient of a metabolite M, with respect to an enzyme Ei (identical to 'element of control matrix' [2]), i.e. Cg, is a measure of the relative change in concentration of metabolite M, occurring if the concentration of enzyme Ei is increased, whilst the concentrations of the other enzymes in the system are kept constant. It is not the immediate change in the metabolite concentration that is meant here, but the change in concentration from its initial value to the value it attains after the system has relaxed towards a new steady state. The concentrations of other metabolites may have changed as well. By dividing the relative change in metabolite concentration by the relative change in enzyme concentration, a measure is obtained for the importance of the enzyme Ei in the determination of the metabolite concentration [M,]. However, unless the relative change in enzyme concentration is small, the ratio depends on the magnitude of that relative change. To overcornc this complication, the changes are taken to be infinitesimally small in the definitions of control coefficients (and of elasticity coefficients).As an illustration of the...