Nonequilibrium chemical potentials and free energies for enzyme-catalyzed reactions

Using the statistical theory of nonequilibrium thermodynamics we explore the nature of nonequilibrium corrections to chemical potentials in simple enzyme-catalyzed reactions. The statistical definition of the chemical potential, which pertains to systems that are at stable steady states, is applied to the Michaelis-Menten reaction scheme in a cellular-sized compartment that communicates with out-side reservoirs. Calculations based on the kinetic parameters for hexokinase and triose phosphate isomerase show that substantial corrections to the chemical potential of product (the order of 25 mV) are possible if the reaction is sufficiently far from equilibrium. The dependence of the corrections to the chemical potentials on the size of the cellular compartment are explored, and the relevance of the corrections for understanding the thermodynamics of metabolites is discussed.     

Linear relation between rate and thermodynamic force in enzyme-catalyzed reactions

Starting from enzyme kinetics, it is shown that generally a linear rather than a proportional relationship exists between rate and free energy changes in biochemical processes. In the derivation the boundary condition of constant substrate plus product is used, which is appropriate for many cellular systems. An example is the ADP plus ATP concentration in mitochondrial oxidative phosphorylation, as is illustrated experimentally.     

Determination of kinetic parameters of enzyme-catalyzed reactions with a minimum number of velocity measurements

Duggleby [Duggleby, R.G., 1979. Experimental designs for estimating kinetic parameters for enzyme-catalyzed reactions. J. Theor. Biol. 81, 672-684] discussed the “design of several replicate measurements of the velocity at as many experimental conditions as there are parameters to be estimated.” He discussed the application of this method to A→products, without and with competitive inhibition, and commented briefly on A+B→products. The availability of computer applications that can solve large sets of simultaneous equations makes it possible to use this method to calculate kinetic parameters for more complicated enzyme mechanisms. This article is concerned with rapid-equilibrium rate equations, but this method can also be used with steady-state rate equations. Computer programs are provided for the calculation of the three kinetic parameters for ordered A+B→products from three velocity measurements and for the calculation of the four kinetic parameters for random A+B→products from four velocity measurements. Computer programs are also provided for competitive inhibition, uncompetitive inhibition, and mixed inhibition of ordered A+B→products.     

Determination of relative rate constants for in vitro RNA processing reactions by internal competition

Studies of RNA recognition and catalysis typically involve measurement of rate constants for reactions of individual RNA sequence variants by fitting changes in substrate or product concentration to exponential or linear functions. A complementary approach is determination of relative rate constants by internal competition, which involves quantifying the time-dependent changes in substrate or product ratios in reactions containing multiple substrates. Here, we review approaches for determining relative rate constants by analysis of both substrate and product ratios and illustrate their application using the in vitro processing of precursor transfer RNA (tRNA) by ribonuclease P as a model system. The presence of inactive substrate populations is a common complicating factor in analysis of reactions involving RNA substrates, and approaches for quantitative correction of observed rate constants for these effects are illustrated. These results, together with recent applications in the literature, indicate that internal competition offers an alternate method for analyzing RNA processing kinetics using standard molecular biology methods that directly quantifies substrate specificity and may be extended to a range of applications.     

Dual-color fluorescence cross-correlation spectroscopy for monitoring the kinetics of enzyme-catalyzed reactions

Dual-color fluorescence correlation spectroscopy is a biophysical technique that enables precise and sensitive analyzes of molecular interactions. It is unique in its ability to analyze reactions in real time at nanomolar substrate concentrations and below, especially when applied to the monitoring of enzyme-catalyzed reactions. Furthermore, it offers a wide range of accessible reactions, restricted only by the prerequisite that a chemical bond or a physical interaction between two spectrally distinguishable fluorophores is established or broken. Recently, the optical setup of dual-color fluorescence correlation spectroscopy has been extended toward two-photon excitation, resulting in several advantages compared with standard excitation, such as lower fluorescence background, an even larger spectrum of potential fluorescence dyes to be used, as well as a more stable and simplified optical setup. So far, the method has been successfully employed to analyze the kinetics of nucleic acid and peptide modifications catalyzed by nucleases, polymerases, and proteases.     

Generalized rate equation for single-substrate enzyme catalyzed reactions

The most widely used rate expression for single-substrate enzyme catalyzed reactions, namely the Michaelis-Menten kinetics is based upon the assumption that enzyme concentration is in excess of the substrate in the medium and the rate is mainly limited by the substrate concentration according to saturation kinetics. However, this is only a special case and the actual rate expression varies depending on the initial enzyme/substrate ratio (E₀/S₀). When the substrate concentration exceeds the enzyme concentration the limitation is due to low enzyme concentration and the rate increases with the enzyme concentration according to saturation kinetics. The maximum rate is obtained when the initial concentrations of the enzyme and the substrate are equal. A generalized rate equation was developed in this study and special cases were discussed for enzyme catalyzed reactions.     

STADEERS: a software package for the statistical design of experiments pertaining to the estimation of parameters in rate expressions that describe enzyme-catalyzed processes

This paper describes a computational algorithm (STADEERS-STAtisticalDesign of Exeriments in Enzyme ReactorS) for the statisticaldesign of biochemical engineering experiments. The type of experimentthat qualifies for this package involves a batch reaction catalyzedby a soluble enzyme where the activity of the enzyme decayswith time. Assuming that both the catalytic action and the deactivationof the enzyme obey known rate expressions, the present codeis helpful in the process of obtaining estimates of the kineticparameters by providing as output the times at which samplesshould be withdrawn from the reacting mixture. Starting D-optimaldesign is used as a basis for the statistical approach. ThisBASIC code is a powerful tool when fitting a rate expressionto data because it increases the effectiveness of experimentationby helping the biochemical kineticist obtain data points withthe largest possible informa tional content.     

Steric hindrance effects on surface reactions: applications to BIAcore

Because surface-volume reactions occur in many biological and industrial processes, understanding the rate of such reactions is important. The BIAcore surface plasmon resonance (SPR) biosensor for measuring rate constants has such a geometry. Though several models of the BIAcore have been presented, few take into account that large ligand molecules can block multiple receptor sites, thus skewing the sensogram data. In this paper some general mathematical principles are stated for handling this phenomenon, and a surface-reaction model is presented explicitly. An integro-partial differential equation results, which can be simplified greatly using perturbation techniques, yielding linear and nonlinear integrodifferential equations. Explicit and asymptotic solutions are constructed for cases motivated by experimental design. The general analysis can provide insight into surface-volume reactions occurring in various contexts. In particular, the steric hindrance effect can be quantified with a single dimensionless parameter. This work was supported in part by NIGMS Grant 1R01GM067244-01.     

Determination of the optimum operating time for batch isothermal performance of enzyme-catalyzed multisubstrate reactions

This communication consists of a mathematical analysis encompassing the maximization of the average rate of monomer production in a batch reactor performing an enzymatic reaction in a system consisting of a multiplicity of polymeric substrates which compete with one another for the active site of a soluble enzyme, under the assumption that the form of the rate expression is consistent with the Michaelis-Menten mechanism. The general form for the functional dependence of the various substrate concentrations on time is obtained in dimensionless form using matrix terminology; the optimum batch time is found for a simpler situation and the effect of various process and system variables thereon is discussed. The reasoning developed here emphasizes, in a quantitative fashion, the fact that the commonly used lumped substrate approaches lead to nonconservative decisions in industrial practice, and hence should be avoided when searching for trustworthy estimates of optimum operation.List of Symbols O 1/s row vector of zeros - a 1/s row vector of rate constants k _i(i = 2,...,N) - A 1/s matrix of rate constants k _i and k_–i (i=2,...,N) - b 1/s row vector of rate constant k ₂ and zeros - C mol/m³ molar concentration of S - C mol/m³ vector of molar concentrations of C _i (i=0, 1, 2, ..., N) - C ₀ mol/m³ column vector of initial molar concentrations of C _i(i=0, 1, 2,.., N) - C _–01 mol/m³ column vector of initial molar concentrations of C _i(i=2,..., N) - C _{E, tot} mol/m³ total molar concentration of enzyme molecules - C _i mol/m³ molar concentration of S _i (i=0,1,2,...,N) - C _{i, o} mol/m³ initial molar concentration of S _i(i=0, 1, 2, ..., N) - E enzyme molecule - I identity matrix - K 1/s matrix of lumped rate constants - k _i 1/s pseudo-first order lumped rate constant associated with the formation of S _i -1 (i=1, 2, ...,N) - k _{cat, i} 1/s first order rate constant associated with the formation of S _i-1 (i=1, 2, ..., N) - K _m mol/m³ Michaelis-Menten constant - L number of distinct eigenvalues - M _i multiplicity of the i-th eigenvalue - N maximum number of monomer residues in a single polymeric molecule - r ₁ mol/m³ s rate of formation of S ₀ - r _i mol/m³ s rate of release of S _i -1 - r _opt maximum average dimensionless rate of production of monomer S₀ - S lumped, pseudo substrate - S₁ inert moiety - S _i substrate containing i monomer residues, each labile to detachment as - S₀ by enzymatic action (i=1,2,...,N) - t s time elapsed since startup of batch reaction - t _lag s time interval required for cleaning, loading, and unloading the batch reactor - t _opt s time interval leading to the maximum average rate of monomer production - v _ij s^1-j eigenvectors associated with eigenvalue imi (i=1, 2, ..., L; j =1, 2, ..., M_i) Greek Symbols _ij mol/m³ arbitrary constant associated with eigenvalue _i (i=1, 2, ..., L; j=1, 2, ..., M _i ) - 1/s generic eigenvalue - _i 1/s i-th eigenvalue     

A rate distortion approach to protein symmetry

A spontaneous symmetry breaking argument is applied to the problem of protein folding, via a rate distortion analysis of the relation between genome coding and the final condensation of the protein molten globule that is, in spirit, analogous to Tlusty’s (2007) exploration of the evolution of the genetic code. In the ‘energy’ picture, the average distortion between codon message and final protein structure, under constraints driven by evolutionary selection, serves as a temperature analog, so that low values limit the possible distribution of protein forms, producing the canonical folding funnel. A dual ‘developmental’ perspective sees the rate distortion function itself as the temperature analog, and permits incorporation of chaperons or toxic exposures as catalysts, driving the system to different possible outcomes or affecting the rate of convergence. The rate distortion function appears constrained by the availability of metabolic free energy, with implications for prebiotic evolution, and a nonequilibrium empirical Onsager treatment provides an adaptable statistical model that can be fitted to data, in the same manner as a regression equation. In sum, mechanistic models of protein folding fail to account for the observed spectrum of protein folding and aggregation disorders, suggesting that a biologically based cognitive paradigm describing folding will be needed for understanding the etiology, prevention, and treatment of these diseases. The developmental formalism introduced here may contribute substantially to such a paradigm.     

An on-line method for the collection and analysis of quasi-steady-state kinetic data for enzyme-catalyzed reactions.

A special mixing device for initiating enzyme-catalyzed reactions is used to rapidly achieve an unperturbed quasi-steady state. An on-line computer is employed to sample the initial conditions, the mixing time, and concentrations that change as a function of time during this quasi-steady state phase. A statistical method for estimating initial, quasi-steady state rates from the time course of the enzyme-catalyzed reaction is described. Practical considerations for using this parameter estimation system lead to the conclusion that for the enzyme-catalyzed reaction tested, the extent overall reaction should be above .2% for high initial substrate concentrations, and above 1% for initial substrate concentrations in the range of the Michaelis constant. Application of this method to a typical enzyme-catalyzed reaction suggests that objective estimates of initial rates from a given set of concentrations and corresponding times can be obtained with a standard error in the range of 2–3%, but that reproducibility is not better than about 10%. When this procedure was used to estimate initial rates for the glycerol dehydrogenase-catalyzed oxidation of glycerol by NAD, it was found that this enzyme did not behave according to the classical “Michaelis-Menten” mechanism of enzyme action.     

Determination of inhibition constants, I50 values and the type of inhibition for enzyme-catalyzed reactions

A procedure is proposed for determining whether an inhibitor of an enzyme-catalyzed reaction is competitive, noncompetitive, or uncompetitive with respect to the substrate. The method is based on fitting the equation for noncompetitive inhibition to data obtained by measuring the rate of the reaction over a range of substrate and inhibitor concentrations. The results of this fit may suggest that the inhibition may be either competitive or uncompetitive, whereupon the data are reanalyzed using the appropriate equation. Comparison of this second fit with the first using an F test permits a statistical decision to be made on the type of inhibition. The chosen fit yields values and standard errors for the Michaelis-Menten parameters (maximum velocity and Michaelis constant), as well as the inhibition constant(s). From these values it is then possible to predict the I50, and its standard error, at any chosen substrate concentration, thereby facilitating comparison with results obtained with similar inhibitors, for homologous enzymes, or in other laboratories.     

A potential role for isothermal calorimetry in studies of the effects of thermodynamic non-ideality in enzyme-catalyzed reactions

Attention is drawn to the feasibility of using isothermal calorimetry for the characterization of enzyme reactions under conditions bearing greater relevance to the crowded biological environment, where kinetic parameters are likely to differ significantly from those obtained by classical enzyme kinetic studies in dilute solution. An outline of the application of isothermal calorimetry to the determination of enzyme kinetic parameters is followed by considerations of the nature and consequences of crowding effects in enzyme catalysis. Some of those effects of thermodynamic non-ideality are then illustrated by means of experimental results from calorimetric studies of the effect of molecular crowding on the kinetics of catalysis by rabbit muscle pyruvate kinase. This review concludes with a discussion of the potential of isothermal calorimetry for the experimental determination of kinetic parameters for enzymes either in biological environments or at least in media that should provide reasonable approximations of the crowded conditions encountered in vivo.     

On the rate of molecular evolution

Summary There are at least two outstanding features that characterize the rate of evolution at the molecular level as compared with that at the phenotypic level. They are; (1) remarkable uniformity for each molecule, and (2) very high overall rate when extrapolated to the whole DNA content.The population dynamics for the rate of mutant substitution was developed, and it was shown that if mutant substitutions in the population are carried out mainly by natural selection, the rate of substitution is given byk = 4 N _e s ₁ v, whereN _e is the effective population number,s ₁ is the selective advantage of the mutants, andv is the mutation rate per gamete for such advantageous mutants (assuming that 4N _e s ₁ 1). On the other hand, if the substitutions are mainly carried out by random fixation of selectively neutral or nearly neutral mutants, we havek = v, wherev is the mutation rate per gamete for such mutants.Reasons were presented for the view that evolutionary change of amino acids in proteins has been mainly caused by random fixation of neutral mutants rather than by natural selection.It was concluded that if this view is correct, we should expect that genes of living fossils have undergone almost as many DNA base replacements as the corresponding genes of more rapidly evolving species.Contribution No. 789 from the National Institute of Genetics, Mishima, Shizuokaken 411 Japan. Aided in part by a grant-in-aid from the Ministry of Education, Japan.