Validity of transfer-function representation of input-output relation in allosteric models

A transfer-function representation of reaction velocity is devised to describe analytically and approximately an input-output response of allosteric enzyme around a steady state. The transfer function is derived on assuming an exponential change in reaction velocity for the indicial response to substrate influx rate. The validity of the representation with variation in the kinetic parameters and flow rates is examined for the response of Koshland-Nemethy-Filmer (KNF) and Monod-Wyman-Changeux (MWC) dimeric models by comparing with the exact response obtained from the computer simulation, that is, by numerical integration of the rate equation. The representation has a wider valid region with a decrease in influx rate than with an increase. For the KNF model the representation is valid for negative cooperativity, but invalid for positive cooperativity. For the MWC model the validity decreases with stronger cooperativity. With the transfer functions valid for the Michaelis-Menten and allosteric reactions, we may derive the transfer-function representation for many metabolic pathways.     

Enzyme kinetics at high enzyme concentration

We re-visit previous analyses of the classical Michaelis-Menten substrate-enzyme reaction and, with the aid of the reverse quasi-steady-state assumption, we challenge the approximation d[C]/dt ≈ 0 for the basic enzyme reaction at high enzyme concentration. For the first time, an approximate solution for the concentrations of the reactants uniformly valid in time is reported. Numerical simulations are presented to verify this solution. We show that an analytical approximation can be found for the reactants for each initial condition using the appropriate quasi-steady-state assumption. An advantage of the present formalism is that it provides a new procedure for fitting experimental data to determine reaction constants. Finally, a new necessary criterion is found that ensures the validity of the reverse quasi-steady-state assumption. This is verified numerically.     

About and beyond the Henri-Michaelis-Menten rate equation for single-substrate enzyme kinetics

For more than a century the simple single-substrate enzyme kinetics model and related Henri-Michaelis-Menten (HMM) rate equation have been thoroughly explored in various directions. In the present paper we are concerned with a possible generalization of this rate equation recently proposed by F. Kargi (BBRC 382 (2009) 157-159), which is assumed to be valid both in the case that the total substrate or enzyme is in excess and the quasi-steady-state is achieved. We demonstrate that this generalization is grossly inadequate and propose another generalization based on application of the quasi-steady-state condition and conservation equations for both enzyme and substrate. The standard HMM equation is derived by (a) assuming the quasi-steady-state condition, (b) applying the conservation equation only for the enzyme, and (c) assuming that the substrate concentration at quasi-steady-state can be approximated by the total substrate concentration [S](0). In our formula the rate is already expressed through [S](0), and we only assume that when quasi-steady-state is achieved the amount of product formed is negligible compared to [S](0). Numerical simulations show that our formula is generally more accurate than the HMM formula and also can provide a good approximation when the enzyme is in excess, which is not the case for the HMM formula. We show that the HMM formula can be derived from our expression by further assuming that the total enzyme concentration is negligible compared to [S](0).     

Quasi-steady-state kinetics at enzyme and substrate concentrations in excess of the Michaelis-Menten constant

In vitro enzyme reactions are traditionally conducted under conditions of pronounced substrate excess since this guarantees that the bound enzyme is at quasi-steady-state (QSS) with respect to the free substrate, thereby justifying the Briggs-Haldane approximation (BHA). In contrast, intracellular reactions, amplification assays, allergen digestion assays and industrial applications span a range of enzyme-to-substrate ratios for which the BHA is invalid, including the extreme of enzyme excess. The quasi-equilibrium approximation (QEA) is valid for a subset of enzyme excess states. Previously, we showed that the total QSSA (tQSSA) overlaps and extends the validity of the BHA and the QEA, and that it is at least roughly valid for any total substrate and enzyme concentrations. The analysis of the tQSSA is hampered by square root nonlinearity. Previous simplifications of the tQSSA rate law are valid in a parameter domain that overlaps the validity domains of the BHA and the QEA and only slightly extends them. We now integrate the tQSSA rate equation in closed form, without resorting to further approximations. Moreover, we introduce a complimentary simplification of the tQSSA rate law that is valid in states of enzyme excess when the absolute difference between total enzyme and substrate concentrations greatly exceeds the Michaelis-Menten constant. This includes a wide range of enzyme and substrate concentrations where both the BHA and the QEA are invalid and allows us to define precisely the conditions for zero-order and first-order product formation. Remarkably, analytical approximations provided by the tQSSA closely match the expected stochastic kinetics for as few as 15 reactant molecules, suggesting that the conditions for the validity of the tQSSA and for its various simplifications are also of relevance at low molecule numbers.     

Simplified velocity equations for characterizing the partial inhibition or nonessential activation of bireactant enzymes

The steady state velocity equation for a bireactant enzyme in the presence of a partial inhibitor or nonessential activator, M, contains squared substrate concentration and higher-ordered M concentration terms. The equation is too complex to be useful in kinetic analyses. Simplification by the method of Cha (J. Biol. Chem. 243, 820 825 (1968)) eliminates squared substrate concentration terms, but retains higher-ordered terms in [M]. It is shown that if strict equilibrium is assumed between free E, M, and EM and for all but one other M-binding reaction, a velocity equation is obtained for an ordered bireactant enzyme that is first degree in all ligands in the absence of products. The equation is an approximation (because it was derived assuming only one M-binding reaction in the steady state), but it contains five inhibition (or activation) constants associated with M, all of which can be obtained by diagnostic replots and/or curve-fitting procedures. The equation also provides a framework for obtaining limiting constants (V'max, K'ia, K'mA, K'mB) that characterize the enzyme at saturating M. The same approach is applicable to an enzyme that catalyzes a steady state ping pong reaction.     

The total quasi-steady-state approximation is valid for reversible enzyme kinetics

The Briggs-Haldane approximation of the irreversible Michaelis-Menten scheme of enzyme kinetics is cited in virtually every biochemistry textbook and is widely considered the classic example of a quasi-steady-state approximation. Though of similar importance, the reversible Michaelis-Menten scheme is not as well characterized. This is a serious limitation since even enzymatic reactions that go to completion may be reversible. The current work derives a total quasi-steady-state approximation (tQSSA) for the reversible Michaelis-Menten and delineates its validity domain. The tQSSA allows the derivation of uniformly valid approximations for the limit of low enzyme concentrations, ET相似文献   

A transfer-function representation for regulatory responses of a controlled metabolic pathway

A transfer-function representation for the response of a controlled metabolic pathway to the changes in influx and efflux rates of metabolites is formulated to describe analytically and approximately the regulatory behavior of the pathway around a steady state. The pathway model analysed is an open and homogeneous system which consists of two consecutive enzymatic reactions catalyzed by an allosteric enzyme of Monod-Wyman-Changeux (MWC) dimeric model and a Michaelis-Menten-type enzyme, respectively, and undergoes the feedback inhibition by the end product. The rate equation for the system (a system of ordinary differential equations) is linearized about a steady state, so that the responses of the reaction rates to the changes in influx rate of the substrate and efflux rate of the end product are expressed in a form of transfer function. The formulation leads to the transfer function for the response of production rate of the end product to the change in its efflux rate to clarify the regulatory response of feedback mechanism in controlled metabolic pathways. The relationship among the chemical species in the system at steady stete also supports a reasonable assumption that the regulatory mechanisms in metabolic pathways are to control the production of end product against the change in its demand from the cellular environments.     

Phosphorylation reaction of vertebrate smooth muscle myosin: an enzyme kinetic analysis

Phosphorylation of vertebrate smooth muscle myosin or its isolated 20 000-dalton light chains by myosin light-chain kinase (MLCK) was found to follow first-order kinetics not only at low ([M] much less than Km) but also at high ([M] greater than or equal to Km) substrate concentration. This observation can most simply be explained by a product inhibition for which the Michaelis constants (Km) of the enzyme for the substrate (dephosphorylated myosin) and for the product (phosphorylated myosin) are approximately the same. For such a case, integration of the kinetic velocity equation gives an exponential formula similar to that of a true first-order reaction, the only difference being that its rate constant (k) depends additionally on the initial substrate concentration ([M]0). The standard kinetic constants (k, Km, Vmax) have been calculated by using this pseudo-first-order relationship. Independent evidence for the validity of the derived kinetic relationship was obtained from binding studies with myosin and MLCK. These showed that MLCK binds to phosphorylated and dephosphorylated myosin with approximately equal affinity (Ks = 30 X 10(-9) M). The possible applicability of the same kinetic relationship to other enzyme systems is discussed.     

Time-dependent closed form solutions for fully competitive enzyme reactions

An analytic formalism developed earlier to describe the time evolution of the basic enzyme reaction is extended to fully competitive systems. Time-dependent closed form solutions are derived for the three nominal cases of competition: even, slow and fast inhibitors, allowing for the first time the complete characterization of the reactions. In agreement with previous work, the time-independent Michaelis-Menten approach is shown to be inaccurate when a fast inhibitor is present. The validity of the quasi-steady-state approximation on which the present framework is based is also revised.     

Simplified Velocity Equations for Characterizing the Partial Inhibition or Nonessential Activation of Bireactant Enzymes

The steady state velocity equation for a bireactant enzyme in the presence of a partial inhibitor or nonessential activator, M, contains squared substrate concentration and higher-ordered M concentration terms. The equation is too complex to be useful in kinetic analyses. Simplification by the method of Cha (J. Biol. Chem. 243, 820–825 (1968)) eliminates squared substrate concentration terms, but retains higher-ordered terms in [M]. It is shown that if strict equilibrium is assumed between free E, M, and EM and for all but one other M-binding reaction, a velocity equation is obtained for an ordered bireactant enzyme that is first degree in all ligands in the absence of products. The equation is an approximation (because it was derived assuming only one M-binding reaction in the steady state), but it contains five inhibition (or activation) constants associated with M, all of which can be obtained by diagnostic replots and/or curve-fitting procedures. The equation also provides a framework for obtaining limiting constants (V¹_max, K¹_ia, K¹_mA,K¹_mB) that characterize the enzyme at saturating M. The same approach is applicable to an enzyme that catalyzes a steady state ping pong reaction.     

The Total Quasi-Steady-State Approximation for Fully Competitive Enzyme Reactions

The validity of the Michaelis–Menten–Briggs–Haldane approximation for single enzyme reactions has recently been improved by the formalism of the total quasi-steady-state approximation. This approach is here extended to fully competitive systems, and a criterion for its validity is provided. We show that it extends the Michaelis–Menten–Briggs–Haldane approximation for such systems for a wide range of parameters very convincingly, and investigate special cases. It is demonstrated that our method is at least roughly valid in the case of identical affinities. The results presented should be useful for numerical simulations of many in vivo reactions.     

Two theoretical problems concerning the irreversible modification kinetics of enzyme activity

Some years ago, a systematic study on the kinetics of irreversible modification of enzyme activity was presented by Tsou (1988, Adv. Enzymol. Relat. Areas molec. Biol. 61, 381-436). This paper presents a discussion of the theoretical aspects of Tsou's method for the determination of the rate constants for the irreversible modification reaction. Range of validity for the kinetic equations of substrate reaction in the presence of the modifier is discussed quantitatively. The expressions of A, B, v* and [P]infinity for any enzyme catalyzed reaction mechanism are derived.     

Use and abuse of the quasi-steady-state approximation

The transient kinetic behaviour of an open single enzyme, single substrate reaction is examined. The reaction follows the Van Slyke-Cullen mechanism, a spacial case of the Michaelis-Menten reaction. The analysis is performed both with and without applying the quasi-steady-state approximation. The analysis of the full system shows conditions for biochemical pathway coupling, which yield sustained oscillatory behaviour in the enzyme reaction. The reduced model does not demonstrate this behaviour. The results have important implications in the analysis of open biochemical reactions and the modelling of metabolic systems.     

Michaelis-Menten kinetics at high enzyme concentrations

The total quasi-steady state approximation (tQSSA) for the irreversible Michaelis-Menten scheme is derived in a consistent manner. It is found that self-consistency of the initial transient guarantees the uniform validity of the tQSSA, but does not guarantee the validity of the linearization in the original derivation of Borghans et al. (1996, Bull. Math. Biol., 58, 43–63). Moreover, the present rederivation yielded the noteworthy result that the tQSSA is at least roughly valid for any substrate and enzyme concentrations. This reinforces and extends the original assertion that the parameter domain for which the tQSSA is valid overlaps the domain of validity of the standard quasi-steady state approximation and includes the limit of high enzyme concentrations. The criteria for the uniform validity of the original (linearized) tQSSA are corrected, and are used to derive approximate solutions that are uniformly valid in time. These approximations overlap and extend the domains of validity of the standard and reverse quasi-steady state approximations.     

Analysis of the quasi-steady-state approximation for an enzymatic one-substrate reaction

The validity of the quasi-steady state approximation for the calculation of the rate function of an isolated enzyme reaction is analysed by a detailed consideration of the time dependent process. For the characterization of the deviations of the real motion from the quasi-stationary state three kinds of error functions are used, the relaxation deficit, the relative relaxation time and the relaxation error. An improved approximation procedure is developed to calculate the transient states of the system. The maximum distance of the original motion from the quasi-stationary states is estimated by a general method. By consideration of different enzyme and substrate concentrations as well as different kinetic constants those parameter regions have been determined, where the errors of the quasi-steady state approximation do not exceed tolerated values. It is suggested how the methods can be applied to metabolic pathways.     

Reaction diffusion model of the enzymatic erosion of insoluble fibrillar matrices

Predicting the time course of in vivo biodegradation is a key issue in the design of an increasing number of biomedical applications such as sutures, tissue analogs and drug-delivery devices. The design of such biodegradable devices is hampered by the absence of quantitative models for the enzymatic erosion of solid protein matrices. In this work, we derive and simulate a reaction diffusion model for the enzymatic erosion of fibrillar gels that successfully reproduces the main qualitative features of this process. A key aspect of the proposed model is the incorporation of steric hindrance into the standard Michaelis-Menten scheme for enzyme kinetics. In the limit of instantaneous diffusion, the model equations are analogous to the standard equations for enzymatic degradation in solution. Invoking this analogy, the total quasi-steady-state approximation is used to derive approximate analytical solutions that are valid for a wide range of in vitro conditions. Using these analytical approximations, an experimental-theoretical method is derived to unambiguously estimate all the kinetic model parameters. Moreover, the analytical approximations correctly describe the characteristic hyperbolic dependence of the erosion rate on enzyme concentration and the zero-order erosion of thin fibers. For definiteness, the analysis of published experimental results of enzymatic degradation of fibrillar collagen is demonstrated, and the role of diffusion in these experiments is elucidated.     

A Kinetic Analysis of Coupled (or Auxiliary) Enzyme Reactions

As a case study, we consider a coupled (or auxiliary) enzyme assay of two reactions obeying the Michaelis–Menten mechanism. The coupled reaction consists of a single-substrate, single-enzyme non-observable reaction followed by another single-substrate, single-enzyme observable reaction (indicator reaction). In this assay, the product of the non-observable reaction is the substrate of the indicator reaction. A mathematical analysis of the reaction kinetics is performed, and it is found that after an initial fast transient, the coupled reaction is described by a pair of interacting Michaelis–Menten equations. Moreover, we show that when the indicator reaction is fast, the quasi-steady-state dynamics are governed by three fast variables and one slow variable. Timescales that approximate the respective lengths of the indicator and non-observable reactions, as well as conditions for the validity of the Michaelis–Menten equations, are derived. The theory can be extended to deal with more complex sequences of enzyme-catalyzed reactions.     

A kinetic model for lipoxygenases based on experimental data with the lipoxygenase of reticulocytes

A comprehensive kinetic model for lipoxygenase catalysis is proposed which includes the simultaneous occurrence of dioxygenase and hydroperoxidase activities and is based on the assumption of a single binding site for substrate fatty acid and product. The aerobic reaction of purified lipoxygenase from rabbit reticulocytes with 9,12(Z,Z)-octadecadienoic acid (linoleic acid) as substrate was studied. The rate constants and the dissociation constants of this enzyme were calculated for the model from progress curves; the model describes correctly the experimental data. The following kinetic features of the reticulocyte enzyme are assumed to apply generally to lipoxygenases. (a) The enzyme shows autoactivation by its product. (b) The rate-limiting step is the hydrogen abstraction. (c) Both substrate fatty acid and its product are competitive inhibitors of the lipoxygenase. (d) Lowering the oxygen concentration enhances the degree of substrate inhibition, whereas product inhibition is not influenced. (e) If substrate is in excess the oxygen concentration determines the share of dioxygenase and hydroperoxidase activities of the enzyme. As predicted from the model it was found that at low concentrations of oxygen the regio- and stereo-specificities of the dioxygenation are diminished. During the autoactivation phase the steady-state approximation does not hold.     

Stochastic simulation of enzyme-catalyzed reactions with disparate timescales

Many physiological characteristics of living cells are regulated by protein interaction networks. Because the total numbers of these protein species can be small, molecular noise can have significant effects on the dynamical properties of a regulatory network. Computing these stochastic effects is made difficult by the large timescale separations typical of protein interactions (e.g., complex formation may occur in fractions of a second, whereas catalytic conversions may take minutes). Exact stochastic simulation may be very inefficient under these circumstances, and methods for speeding up the simulation without sacrificing accuracy have been widely studied. We show that the “total quasi-steady-state approximation” for enzyme-catalyzed reactions provides a useful framework for efficient and accurate stochastic simulations. The method is applied to three examples: a simple enzyme-catalyzed reaction where enzyme and substrate have comparable abundances, a Goldbeter-Koshland switch, where a kinase and phosphatase regulate the phosphorylation state of a common substrate, and coupled Goldbeter-Koshland switches that exhibit bistability. Simulations based on the total quasi-steady-state approximation accurately capture the steady-state probability distributions of all components of these reaction networks. In many respects, the approximation also faithfully reproduces time-dependent aspects of the fluctuations. The method is accurate even under conditions of poor timescale separation.     

Comments on diffusive and electrostatic effects with immobilized enzymes

Shuler, Aris &; Tsuchiya (1972) have recently considered the combined effects of diffusive resistance and electrostatic field on the rate of reaction catalyzed by an enzyme immobilized on a non-porous surface. They employed a potential distribution for the electrical double layer which is asymptotically valid when surface potential is small.The complete Gouy-Chapman solution, which is valid for higher surface potential, is employed here. Numerical values of the effectiveness factor calculated with this potential distribution agree very closely with the results of Shuler et al. for most cases. It is shown that the effectiveness factor can (i) attain magnitudes much greater than unity in physically realizable systems, (ii) approach the solution for “infinite” surface potential at reasonable values of surface charge density, and (iii) pass through a maximum as bulk substrate concentration is varied. This behavior leads to the existence of an optimum surface concentration for enzyme immobilized on a highly charged non-porous support such that the most effective catalytic action on a charged substrate is ensured. Finally, it is established that significant electrical and/or diffusive effects result in non-linear Lineweaver-Burk plots of reciprocal observed reaction velocity against reciprocal bulk substrate concentration. These non-linear plots cannot be interpreted in the same way as linear plots obtained when enzyme is unbound.