Current-voltage curve of electrogenic Cl− pump predicts voltage-dependent Cl− efflux inAcetabularia

Summary The current-voltage relationship of carrier-mediated, passive and active ion transport systems with one charge-carrying pathway can exactly be described by a simple reaction kinetic model. This model consists of two carrier states (one inside, one outside) and two pairs (forwards and backwards) of rate constants: a voltage-dependent one, describing the transport of charge and a voltage-insensitive one, summarizing all the other (voltage-independent) reactions. For the electrogenic Cl^– pump inAcetabularia these four rate constants have been determined from electrical measurements of the current-voltage relationship of the pump (Gradmann, Hansen & Slayman, 1981;in: Electrogenic Ion Pumps, Academic Press, New York). The unidirectional Cl^– efflux through the pump can also be calculated by the availiable reaction kinetic parameters.³⁶Cl^– efflux experiments on singleAcetabularia cells with simultaneous electrical stimulation (action potentials) and recording, demonstrate the unidirectional Cl^– efflux to depend on the membrane potential. After subtraction of an efflux portion which bypasses the pump, agreement is found between the measured flux-voltage relationship and the theoretical one as obtained from the reaction kinetic model and its parameters from the electrical data.     

Global parameter optimization for cardiac potassium channel gating models.

Quantitative ion channel model evaluation requires the estimation of voltage dependent rate constants. We have tested whether a unique set of rate constants can be reliably extracted from nonstationary macroscopic voltage clamp potassium current data. For many models, the rate constants derived independently at different membrane potentials are not unique. Therefore, our approach has been to use the exponential voltage dependence predicted from reaction rate theory (Stevens, C. F. 1978. Biophys. J. 22:295-306; Eyring, H., S. H. Lin, and S. M. Lin. 1980. Basic Chemical Kinetics. Wiley and Sons, New York) to couple the rate constants derived at different membrane potentials. This constrained the solution set of rate constants to only those that also obeyed this additional set of equations, which was sufficient to obtain a unique solution. We have tested this approach with data obtained from macroscopic delayed rectifier potassium channel currents in voltage-clamped guinea pig ventricular myocyte membranes. This potassium channel has relatively simple kinetics without an inactivation process and provided a convenient system to determine a globally optimized set of voltage-dependent rate constants for a Markov kinetic model. The ability of the fitting algorithm to extract rate constants from the macroscopic current data was tested using "data" synthesized from known rate constants. The simulated data sets were analyzed with the global fitting procedure and the fitted rate constants were compared with the rate constants used to generate the data. Monte Carlo methods were used to examine the accuracy of the estimated kinetic parameters. This global fitting approach provided a useful and convenient method for reliably extracting Markov rate constants from macroscopic voltage clamp data over a broad range of membrane potentials. The limitations of the method and the dependence on initial guesses are described.     

Kinetics of the creatine kinase reaction in neonatal rabbit heart: an empirical analysis of the rate equation

Here we define the kinetics of the creatine kinase (CK) reaction in an intact mammalian heart containing the full range of CK isoenzymes. Previously derived kinetic constants [Schimerlik, M. I., & Cleland, W. W. (1973) J. Biol. Chem. 248, 8418-8423] were refit for the reaction occurring at 37 degrees C. Steady-state metabolite concentrations from 31P NMR and standard biochemical techniques were determined. 31P magnetization transfer data were obtained to determine unidirectional creatine kinase fluxes in hearts with differing total creatine contents and differing mitochondrial CK activities during KCl arrest and isovolumic work for both the forward reaction (MgATP synthesis) and reverse reaction (phosphocreatine synthesis). The NMR kinetic data and substrate concentration data were used in conjunction with a kinetic model based on MM-CK in solution to determine the applicability of the solution-based kinetic models to the CK kinetics of the intact heart. Our results indicated that no single set of rate equation constants could describe both the KCl-arrested and working hearts. We used our experimental data to constrain the solution-derived kinetic model and derived a second set of rate equation constants, which describe the isovolumic work state. Analysis of our results indicates that the CK reaction is rate limited in the direction of ATP synthesis, the size of the guanidino substrate pool drives the measured CK flux in the intact heart, and during isovolumic work the CK reaction operates under saturating conditions; that is, the substrate concentrations are at least 2-fold greater than the Km or Kim for each substrate.(ABSTRACT TRUNCATED AT 250 WORDS)     

Kinetic mechanism of adenine phosphoribosyltransferase from Leishmania donovani

Adenine phosphoribosyltransferase (APRT, EC 2.4.2.7) catalyzes the reversible phosphoribosylation of adenine from alpha-D-5-phosphoribosyl-1-pyrophosphate (PRPP) to form AMP and PP(i). Three-dimensional structures of the dimeric APRT enzyme from Leishmania donovani (LdAPRT) bear many similarities to other members of the type 1 phosphoribosyltransferase family but do not reveal the structural basis for catalysis (Phillips, C. L., Ullman, B., Brennan, R. G., and Hill, C. P. (1999) EMBO J. 18, 3533-3545). To address this issue, a steady state and transient kinetic analysis of the enzyme was performed in order to determine the catalytic mechanism. Initial velocity and product inhibition studies indicated that LdAPRT follows an ordered sequential mechanism in which PRPP is the first substrate to bind and AMP is the last product to leave. This mechanistic model was substantiated by equilibrium isotope exchange and fluorescence binding studies, which provided dissociation constants for the LdAPRT-PRPP and LdAPRT-AMP binary complexes. Pre-steady-state kinetic analysis of the forward reaction revealed a burst in product formation indicating that phosphoribosyl transfer proceeds rapidly relative to some rate-limiting product release event. Transient fluorescence competition experiments enabled measurement of rates of binary complex dissociation that implicated AMP release as rate-limiting for the forward reaction. Kinetics of product ternary complex formation were evaluated using the fluorophore formycin AMP and established rate constants for pyrophosphate binding to the LdAPRT-formycin AMP complex. Taken together, these data enabled the complete formulation of an ordered bi-bi kinetic mechanism for LdAPRT in which all of the rate constants were either measured or calculated.     

Kinetic study of the enzymatic cycling reaction conducted with 3alpha-hydroxysteroid dehydrogenase in the presence of excessive thio-NAD(+) and NADH

We have established a simple kinetic model applicable to the enzyme cycling reaction for the determination of 3alpha-hydroxysteroids. This reaction was conducted under the reversible catalytic function of a single 3alpha-hydroxysteroid dehydrogenase (3alpha-HSD) with nucleotide cofactors, thio-NAD(+) (one of the NAD(+) analogues) for the oxidation of 3alpha-hydroxysteroids and NADH for the reduction of 3-oxosteroids. This model was constructed based on the reaction mechanism of 3alpha-HSD, following an ordered bi-bi mechanism with cofactor binding first, under the assumption that the respective enzyme-cofactor complexes were distributed according to the initial ratio of thio-NAD(+) to NADH by the rapid equilibrium of both enzyme-cofactor complexes. The cycling rate in the new kinetic model could be expressed with the dissociation constants of enzyme-cofactor complexes and the initial concentrations of cofactors and enzyme. The cycling rate was verified by a comparison with the experimental data using 3alpha-HSD from Pseudomonas sp. B-0831. The results showed that the experimental data corresponded well with the results obtained from the kinetic model.     

Concurrent binding to multiple ligands: kinetic rates of CD16b for membrane-bound IgG1 and IgG2

CD16b (FcgammaRIIIb) is the most common receptor for the Fc domain of IgG on leukocytes. The binding of Fc receptors to immunoglobulin triggers a wide array of immune responses. In published assays measuring the reaction of CD16b with isotypes of soluble IgG, the affinity for IgG1 was low and that for IgG2 was undetectable. Here we report the first measurement of kinetic rates of CD16b binding to membrane-bound IgG isotypes-a physically distinct and physiologically more relevant presentation-using a recently developed micropipette method. In contrast to the soluble data, we found clearly measurable IgG2 binding, with a forward kinetic rate six-fold lower than that of IgG1 but with an equilibrium affinity only threefold lower. This suggests a nonnegligible role for IgG2 in Fc-mediated immune responses, particularly in longer duration contacts. The binding constants were measured from two sets of experiments. Single-isotype experiments were analyzed by an existing model (, Biophys. J. 75:1553-1572). The resulting kinetic rates were used as input to an extended model (, Biophys. J. 79:1850-1857.) to predict the results of mixed-isotype experiments. This design enabled rigorous validation of the concurrent binding model through a test of its predictive ability.     

Substrate depletion analysis as an approach to the pre-steady-state anticooperative kinetics of aminoacyl adenylate formation by tryptophanyl-tRNA synthetase from beef pancreas

The formation of tryptophanyl adenylate catalyzed by tryptophanyl-tRNA synthetase from beef pancreas has been studied by stopped-flow analysis under conditions where the concentration of one of the substrates was largely decreasing during the time course of the reaction. Under such conditions a nonlinear regression analysis of the formation of the adenylate (adenylate vs. time curve) at several initial tryptophan and enzyme concentrations gave an accurate determination of both binding constants of this substrate. The use of the jackknife procedure according to Cornish - Bowden & Wong [ Cornish - Bowden , A., & Wong , J.J. (1978) Biochem. J. 175, 969-976] gave the limit of confidence of these constants. This approach confirmed that tryptophanyl-tRNA synthetase presents a kinetic anticooperativity toward tryptophan in the activation reaction that closely parallels the anticooperativity found for tryptophan binding at equilibrium. Both sites are simultaneously forming the adenylate. The dissociation constants obtained under the present pre-steady-state conditions for tryptophan are KT1 = 1.6 +/- 0.5 microM and KT2 = 18.5 +/- 3.0 microM at pH 8.0, 25 degrees C. The rate constant kf of adenylate formation is identical for both active sites (kf = 42 +/- 5 s-1). The substrate depletion method presently used, linked to the jackknife procedure, proves to be particularly suitable for the determination of the kinetic constants and for the discrimination between different possible kinetic models of dimeric enzyme with high substrate affinity. In such a case this method is more reliable than the conventional method using substrate concentrations in high excess over that of the enzyme.     

Kinetic study of the reduction mechanism for Desulfovibrio gigas cytochrome c3.

The kinetic aspects of the reduction process in cytochrome c3 from Desulfovibrio gigas have been investigated over a wide range of pH values ranging between pH 5.8 and pH 9.8. The data have been analyzed in the framework of an I2H4 interaction network coupled to a proton-linked equilibrium between two tertiary structures (Cornish-Bowden, A. & Koshland, D.E. Jr (1970) J. Biol. Chem. 245, 6241-6250). The kinetic rate constants for the reduction of the four hemes for the two tertiary conformations have been characterized in the framework of the thermodynamic network obtained from the equilibrium analysis (Coletta, M., Catarino, T., LeGall, J.J. & Xavier, A.V. (1991) Eur. J. Biochem. 202, 1101-1106). The intrinsic reduction rate constants determined by reaction with sodium dithionite for two hemes (namely heme 4 and heme 1) are significantly faster than those for the other two heme residues. In view of the equilibrium redox properties, heme 4 (with the fastest reduction rate) may then work as the kinetic electron-capturing site for the electrons from sodium dithionite. The transfer to hemes 2 and 3 then occurs by virtue of their free-energy levels at equilibrium. At our experimental conditions, there is also transfer of electrons to hemes 2 and 3 from heme 1, which is reduced at a slower rate than heme 4, thus contributing to the biphasic kinetics observed for the overall process. The kinetic parameters obtained are discussed in terms of the mechanism proposed for the coupling between the electron and proton transfer, as induced by the heme/heme cooperativity network.     

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.     

Electrogenic Cl- pump in Acetabularia

Measurements of this transmembrane potential difference (V) under various conditions have demonstrated the operation of an electrogenic Cl- pump in the outer plasma membrane (plasmalemma) of the unicellular marine alga Acetabularia. In preparations of partly purified membranes (containing plasmalemma), there is Cl- stimulated, N,N'-dicyclohexylcarbodiimide-insensitive, vanadate-sensitive ATPase activity with a pH optimum around pH 6.5. These properties are consistent with the assumption that the electrogenic Cl- pump is an ATPase. In order to investigate electrical details of the "Mitchellian" type of charge-translocating enzyme, steady-state current-voltage curves of the electrogenic pump (Ip(V)) were measured in vivo under dark and light conditions and analysed by two-state reaction kinetic model. This model with the resulting parameters predicts V-sensitive, undirectional Cl- effluxes through the pump. The predictions of this model agree with the experimental results. Green light causes a fast decrease of V, which is explained as a disturbance of the pump cycle. Relaxation studies on this effect and reaction kinetic analysis of Ip(V) under different external Cl- concentrations are used to develop a consistent three-state model of the pump that includes the order of and absolute rate constants of individual reactions, states of charge, stoichiometry, voltage-sensitivity and density of the pump molecules in the membrane.     

An extended kinetic analysis of valinomycin-induced Rb-transport through monoglyceride membranes.

The time course of the current following a voltage jump, which is applied to monoglyceride bilayers in the presence of valinomycin, shows two relaxation times. This is basically in agreement with a simple carrier model which has been described in full detail formerly. Relaxation times and amplitudes allow a calculation of the rate constants of the transport model. The presented data supplement an analysis which was hitherto based only on the slower relaxation process and on information derived from the nonlinearity of current-voltage characteristics. The additional resolution of the faster relaxation time allowed an approximate determination of the voltage dependence of the translocation rate constant for carrier-ion-complex and provided evidence for a small voltage dependence of the interfacial reaction. The dependence of the relaxation parameters on the ion concentration in the aqueous phase was interpreted assuming a saturation of the ion concentration at the reaction plane at high bulk concentrations.     

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.     

Acyl group transfer by proteases forming acyl-enzyme intermediate: kinetic model analysis

A kinetic model of peptide synthesis via transfer of the acyl moiety from activated derivatives of amino acids (S) to nucleophiles (N) catalyzed by proteases forming an acyl-enzyme intermediate has been analyzed. The kinetic model takes into account the subsequent enzymatic hydrolysis of synthesized peptide (P), and so the kinetic curve for this compound shows a maximum (denoted as p(max)). Particular stress is placed on analyzing the effects of initial concentrations and of kinetic constants on the value of p(max).The analysis has demonstrated that at a given ratio of initial S and N concentrations, p(max) is affected only by (i) the ratio of the second-order rate constants for enzymatic hydrolysis of S and P(alpha) and (ii) the ratio of rate constants for an attack of the acyl-enzyme intermediate by the nucleophile and water (beta). These conclusions apply regardless of the existence of linear inhibition by the components of the reaction mixture. Thus, the kinetically controlled maximum yield of peptide (p(max)) can be calculated a priori from values of alpha and beta that can be estimated experimentally or from reference data. Simple analytical expressions were obtained, allowing a fairly accurate prediction of p(max) for a broad spectrum of S and N initial concentrations.     

Estimating single-channel kinetic parameters from idealized patch-clamp data containing missed events.

We present here a maximal likelihood algorithm for estimating single-channel kinetic parameters from idealized patch-clamp data. The algorithm takes into account missed events caused by limited time resolution of the recording system. Assuming a fixed dead time, we derive an explicit expression for the corrected transition rate matrix by generalizing the theory of Roux and Sauve (1985, Biophys. J. 48:149-158) to the case of multiple conductance levels. We use a variable metric optimizer with analytical derivatives for rapidly maximizing the likelihood. The algorithm is applicable to data containing substates and multiple identical or nonidentical channels. It allows multiple data sets obtained under different experimental conditions, e.g., concentration, voltage, and force, to be fit simultaneously. It also permits a variety of constraints on rate constants and provides standard errors for all estimates of model parameters. The algorithm has been tested extensively on a variety of kinetic models with both simulated and experimental data. It is very efficient and robust; rate constants for a multistate model can often be extracted in a processing time of approximately 1 min, largely independent of the starting values.     

Purine nucleoside phosphorylase. Kinetic mechanism of the enzyme from calf spleen.

Ribose 1-phosphate, phosphate, and acyclovir diphosphate quenched the fluorescence of purine nucleoside phosphorylase at pH 7.1 and 25 degrees C. The fluorescence of enzyme-bound guanine was similar to that of anionic guanine in ethanol. Guanine and ribose 1-phosphate bound to free enzyme, whereas inosine and guanosine were not bound to free enzyme in the absence of phosphate. Thus, synthesis proceeded by a random mechanism, and phosphorolysis proceeded by an ordered mechanism. Steady-state kinetic data for the phosphorolysis of 100 microM guanosine were fitted to a bifunctional kinetic model with catalytic rate constants of 22 and 1.3 s-1. The dissociation rate constants for guanine from the enzyme-guanine complex at high and low phosphate concentrations were similar to the catalytic rate constants. Fluorescence changes of the enzyme during phosphorolysis suggested that ribose 1-phosphate dissociated from the enzyme ribose 1-phosphate-guanine complex rapidly and that guanine dissociated from the enzyme-guanine complex slowly. The association and dissociation rate constants for acyclovir diphosphate, a potent inhibitor of the enzyme (Tuttle, J. V., and Krenitsky, T. A. (1984) J. Biol. Chem. 259, 4065-4069), were also dependent on phosphate concentration. The effects of phosphate are discussed in terms of a dual functional binding site for phosphate.     

Kinetic analysis of the nicking and hairpin formation steps in V(D)J recombination

The complete cleavage phase of V(D)J recombination includes four phases: binding of the active RAG complexes to the 12- or 23-signals, nicking of the signals, synapsis of the two signals, and hairpin formation at both signals concurrently. We have done time courses for the complete cleavage phase of the V(D)J recombination reaction and quantitated the amount of active RAG enzyme. We have also formulated a kinetic model for the binding, nicking, synapsis, and hairpin formation phases. We have utilized free solution enzymatic measurements for the binding and nicking phases as we do mathematical simulations of the kinetic model. This permits iteration of rate constants for the synapsis and hairpin formation phases until the model fits the observed overall cleavage time course. This process yields a rate constant for the hairpin formation that is 0.004 min(-1), which corresponds to an average catalytic cycle time of 250 min. This value is exceedingly close to a measured value of this constant that relied on wash-out of an inhibitory cofactor. The agreement indicates that this is likely to be the rate of the hairpin step over a wide range of range of conditions and irrespective of the DNA sequence of the V, D or J coding end located adjacent to the signal. These findings indicate that, under optimal in vitro conditions, the core RAG proteins carry out nicking at a rate which is nearly 150-fold faster than hairpin formation. The physiologic implications of this and other kinetic inferences of these time courses are discussed.     

Kinetics of the quinone binding reaction at the QB site of reaction centers from the purple bacteria Rhodobacter sphaeroides reconstituted in liposomes.

Transmembrane proton translocation in the photosynthetic membranes of the purple bacterium Rhodobacter sphaeroides is driven by light and performed by two transmembrane complexes; the photosynthetic reaction center and the ubiquinol-cytochrome c oxidoreductase complex, coupled by two mobile electron carriers; the cytochrome and the quinone. This paper focuses on the kinetics and thermodynamics of the interaction between the lipophylic electron carrier ubiquinone-10 and the photosynthetic enzyme reconstituted in liposomes. The collected data were simulated with an existing recognized kinetic scheme and the kinetic constants of the uptake (7.2 x 107 M(-1) x s(-1)) and release (40 s(-1)) processes of the ligand were inferred. The results obtained for the quinone release kinetic constant are comparable to the rate of the charge recombination reaction from the state D(+)QA(-). Values for the kinetic constants are discussed as part of the overall photocycle, suggesting that its bottleneck may not be the quinone uptake reaction in agreement with a previous report.     

Enzyme hydrolysis of plasma proteins in a CSTR ultrafiltration reactor: Performances and modeling

By investigating the effects of four operating variables-volume (V), Ultrafiltration flux (J), enzyme concentration (E), and substrate concentration (S)-on capacity (K) and conversion rate (epsilon) of a hollow fiber CSTR, the performances of the CSTR and the kinetic constants of the reaction were determined. A model which takes into account the course of fractional conversion (X) according to the modified space-time parameter, tau (integrated form of V, J, S, and E), was devised by employing the relationship to integrate the equation for the reaction rate of the CSTR and the expression of the modified space time. Correlation of this model and the experimentally obtained results demonstrates that the characteristics for an ultrafiltration membrane reactor for enzymatic hydrolysis by alcalase of plasma proteins are close to those of an ideal CSTR. Optimal scaling up, however, remains dependent on the compromise which may be obtained between capacity and the conversion rate.     

Transient kinetic experiments demonstrate the existence of a unique catalytic enzyme form in the peptide-stimulated ATPase mechanism of Escherichia coli Lon protease

Lon is an oligomeric serine protease whose proteolytic activity is mediated by ATP hydrolysis. Although each monomeric subunit has an identical sequence, Lon contains two types of ATPase sites that hydrolyze ATP at drastically different rates. The catalytic low-affinity sites display pre-steady-state burst kinetics and hydrolyze ATP prior to peptide cleavage. The high-affinity sites are able to hydrolyze ATP at a very slow rate. By utilizing the differing Kd's, the high-affinity site can be blocked with unlabeled nucleotide while the activity at the low-affinity site is monitored. Little kinetic data are available that describe microscopic events along the reaction pathway of Lon. In this study we utilize MANT-ATP, a fluorescent analogue of ATP, to monitor the rate constants for binding of ATP as well as the release of ADP from Escherichia coli Lon protease. All of the adenine nucleotides tested bound to Lon on the order of 10(5) M(-1) s(-1), and the previously proposed conformational change associated with nucleotide binding was also detected. On the basis of the data obtained in this study we propose that the rate of ADP release is slightly different for the two ATPase sites. As the model peptide substrate [S2; YRGITCSGRQK(Bz)] [Thomas-Wohlever, J., and Lee, I. (2002) Biochemistry 41, 9418-9425] or the protein substrate casein affects only the steady-state ATPase activity of the low-affinity sites, we propose that Lon adopts a different form after its first turnover as an ATP-dependent protease. Based on the obtained rate constants, a revised kinetic model is presented for ATPase activity in Lon protease in both the absence and presence of the model peptide substrate (S2).