Mathematical modelling of dynamics and control in metabolic networks. I. On Michaelis-Menten kinetics

As a starting point for modeling of metabolic networks this paper considers the simple Michaelis-Menten reaction mechanism. After the elimination of diffusional effects a mathematically intractable mass action kinetic model is obtained. The properties of this model are explored via scaling and linearization. The scaling is carried out such that kinetic properties, concentration parameters and external influences are clearly separated. We then try to obtain reasonable estimates for values of the dimensionless groups and examine the dynamic properties of the model over this part of the parameter space. Linear analysis is found to give excellent insight into reaction dynamics and it also gives a forum for understanding and justifying the two commonly used quasi-stationary and quasi-equilibrium analyses. The first finding is that there are two separate time scales inherent in the model existing over most of the parameter space, and in particular over the regions of importance here. Full modal analysis gives a new interpretation of quasi-stationary analysis, and its extension via singular perturbation theory, and a rationalization of the quasi-equilibrium approximation. The new interpretation of the quasi-steady state assumption is that the applicability is intimately related to dynamic interactions between the concentration variables rather than the traditional notion that a quasi-stationary state is reached, after a short transient period, where the rates of formation and decomposition of the enzyme intermediate are approximately equal. The modal analysis reveals that the generally used criterion for the applicability of quasi-stationary analysis that total enzyme concentration must be much less than total substrate concentration, et much less than St, is incomplete and that the criterion et much less than Km much less than St (Km is the well known Michaelis constant) is the appropriate one. The first inequality (et much less than Km) guarantees agreement over the longer time scale leading to quasi-stationary behavior or the applicability of the zeroth order outer singular perturbation solution but the second half of the criterion (Km much less than St) justifies zeroth order inner singular perturbation solution where the substrate concentration is assumed to be invariant. Furthermore linear analysis shows that when a fast mode representing the binding of substrate to the enzyme is fast it can be relaxed leading to the quasi-equilibrium assumption. The influence of the dimensionless groups is ascertained by integrating the equations numerically, and the predictions made by the linear analysis are found to be accurate.(ABSTRACT TRUNCATED AT 400 WORDS)     

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.     

A kinetic analysis of enzyme inactivation as applied to the covalent modification of Na+ + K+ -ATPase and Ca2+ -ATPase

A kinetic analysis of enzyme inactivation due to the covalent binding of chemically modified ligands is presented. Reaction schemes similar to the Michaelis-Menten scheme have been studied as well as schemes with two states of the enzyme or two binding sites. The resulting kinetic equations lead to time courses of inactivation which can be represented by two exponential functions at least in a quasi-steady state approximation. These curves are frequently encountered in inactivation experiments. Since rapid methods for model selection and parameter estimation are desirable, but not available, a technique for a preliminary analysis of the experimental data is presented. A mere glance at the time courses shows what reaction schemes are inapplicable. For each family of inactivation curves, the construction of a line of intersections is proposed. This line contains essential kinetic information and can further be utilized for a rough parameter estimation. The technique is illustrated for three sets of experimental data where Na+ + K+ -ATPase and Ca2+ -ATPase have been inactivated by ATP-analogs.     

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.     

Introducing total substrates simplifies theoretical analysis at non-negligible enzyme concentrations: pseudo first-order kinetics and the loss of zero-order ultrasensitivity

The Briggs–Haldane standard quasi-steady state approximation and the resulting rate expressions for enzyme driven biochemical reactions provide crucial theoretical insight compared to the full set of equations describing the reactions, mainly because it reduces the number of variables and equations. When the enzyme is in excess of the substrate, a significant amount of substrate can be bound in intermediate complexes, so-called substrate sequestration. The standard quasi-steady state approximation is known to fail under such conditions, a main reason being that it neglects these intermediate complexes. Introducing total substrates, i.e., the sums of substrates and intermediate complexes, provides a similar reduction of the number of variables to consider but without neglecting the contribution from intermediate complexes. The present theoretical study illustrates the usefulness of such simplifications for the understanding of biochemical reaction schemes. We show how introducing the total substrates allows a simple analytical treatment of the relevance of significant enzyme concentrations for pseudo first-order kinetics and reconciles two proposed criteria for the validity of the pseudo first-order approximation. In addition, we show how the loss of zero-order ultrasensitivity in covalent modification cycles can be analyzed, in particular that approaches such as metabolic control analysis are immediately applicable to scenarios described by the total substrates with enzyme concentrations higher than or comparable to the substrate concentrations. A simple criterion which excludes the possibility of zero-order ultrasensitivity is presented.     

Transient dynamics of genetic regulatory networks

We present an approximation scheme for deriving reaction rate equations of genetic regulatory networks. This scheme predicts the timescales of transient dynamics of such networks more accurately than does standard quasi-steady state analysis by introducing prefactors to the ODEs that govern the dynamics of the protein concentrations. These prefactors render the ODE systems slower than their quasi-steady state approximation counterparts. We introduce the method by examining a positive feedback gene regulatory network, and show how the transient dynamics of this network are more accurately modeled when the prefactor is included. Next, we examine the repressilator, a genetic oscillator, and show that the period, amplitude, and bifurcation diagram defining the onset of the oscillations are better estimated by the prefactor method. Finally, we examine the consequences of the method to the dynamics of reduced models of the phage lambda switch, and show that the switching times between the two states is slowed by the presence of the prefactor that arises from protein multimerization and DNA binding.     

The EPR signals from the S0 and S2 states of the Mn cluster in photosystem II relax differently

The oxygen evolving complex (OEC) of photosystem II (PSII) gives rise to manganese-derived electron paramagnetic resonance (EPR) signals in the S0 and S2 oxidation states. These signals exhibit different microwave power saturation behavior between 4 and 10 K. Below 8 K, the S0 state EPR signal is a faster relaxer than the S2 multiline signal, but above 8 K, the S0 signal is the slower relaxer of the two. The different temperature dependencies of the relaxation of the S0 and S2 ground-state Mn signals are due to differences in the spin-lattice relaxation process. The dominating spin-lattice relaxation mechanism is concluded to be a Raman mechanism in the S0 state, with a T(4.1) temperature dependence of the relaxation rate. It is proposed that the relaxation of the S2 state arises from a Raman mechanism as well, with a T(6.8) temperature dependence of the relaxation rate, although the data also fit an Orbach process. If both signals relax through a Raman mechanism, the different exponents are proposed to reflect structural differences in the proteins surrounding the Mn cluster between the S0 and S2 states. The saturation of SII(slow) from the Y(D)(ox) radical on the D2 protein was also studied, and found to vary between the S0 and the S2 states of the enzyme in a manner similar to the EPR signals from the OEC. Furthermore, we found that the S2 multiline signal in the second turnover of the enzyme is significantly more difficult to saturate than in the first turnover. This suggests differences in the OEC between the first and second cycles of the enzyme. The increased relaxation rate may be caused by the appearance of a relaxation enhancer, or it may be due to subtle structural changes as the OEC is brought into an active state.     

Dynamics of Epidemiological Models

We study the SIS and SIRI epidemic models discussing different approaches to compute the thresholds that determine the appearance of an epidemic disease. The stochastic SIS model is a well known mathematical model, studied in several contexts. Here, we present recursively derivations of the dynamic equations for all the moments and we derive the stationary states of the state variables using the moment closure method. We observe that the steady states give a good approximation of the quasi-stationary states of the SIS model. We present the relation between the SIS stochastic model and the contact process introducing creation and annihilation operators. For the spatial stochastic epidemic reinfection model SIRI, where susceptibles S can become infected I, then recover and remain only partial immune against reinfection R, we present the phase transition lines using the mean field and the pair approximation for the moments. We use a scaling argument that allow us to determine analytically an explicit formula for the phase transition lines in pair approximation.     

Protein dynamics. A time-resolved fluorescence, energetic and molecular dynamics study of ribonuclease T1

Studies using time-resolved fluorescence depolarization were performed on the internal motion of Trp 59 of ribonuclease T1 (EC 3.1.27.3) in the free enzyme, 2'-GMP-enzyme complex and 3'-GMP-enzyme complex. The Trp 59 motion was also studied in the free enzyme using molecular dynamics simulations. Energetic analysis of activation barriers to the Trp 59 motion was performed using both the transition state theory and Kramers' theory. The activation parameters showed a dependence on solvent viscosity indicating the transition state approach in aqueous solution to be inadequate. When taking solvent viscosity contributions into account agreement between the transition state and Kramers' theories was obtained. The results indicate the three enzyme forms to have different conformations with the free enzyme and 3'-GMP-enzyme complex being similar. Comparison of the experimental and theoretical results showed a good agreement on the Trp 59 motion in the free enzyme. Trp 59 appears to vibrate rapidly, with a relaxation time of the order of 1 ps, within free space in the protein matrix and to have a slower motion, with a relaxation time of the order of 100 ps, which is related to breathing of the surrounding protein matrix. Molecular dynamics results indicate high mobility in regions of the enzyme involved in the interaction with the guanine base of the inhibitor or substrate while much lower mobility occurred in residues involved in the catalytic mechanism of ribonuclease T1.     

Dynamics of non-linear biochemical systems and the evolutionary significance of time hierarchy

Several non-linear reaction networks are analyzed in order to study the influence of time hierarchy on the dynamics of biochemical systems. The analysis is based on the assumption that the separation of the time constants within metabolic systems is a direct consequence of strong differences of enzyme concentrations. Therefore, as variable system parameters, only the enzyme concentrations are considered. By investigation of the stationary states of various three-component models with feedback-activation bifurcation diagrams within a two-dimensional parameter space, the enzyme simplex, are constructed. The diagrams contain the information about the number of stationary states, their stability properties as well as the type of motion expected at different parameter combinations. The results support the hypothesis that systems with separated time constants generally show a simple dynamic behaviour. Complex motions can be expected mainly for systems without time hierarchy, which are characterized by parameters located within the centre of the enzyme simplex. Quantitative measures for the time hierarchy and the complexity of the dynamics are derived. It is supposed that the separation of time constants is a main feature of the evolution of biochemical systems. The hypothesis is further supported by the consideration of the time hierarchy of the glycolytic pathway.     

Possible mechanism of ATP formation in energy-transforming biological membranes

An "elementary act" of ATP formation from ADP and Pi in energy-transducing organels (mitochondria, chloroplasts and chromatophores) can be realized without closed membrane vesicles, pieces of membranes and F0-component of H+ATPase. The "elementary act" is initiated by a rather fast deprotonation of several acid groups of the coupling factor F1 (or CF1), this process leads to structurally non-equilibrium state of the enzyme due to the appearance of "additional" negative charges in unchanged protein globula. The endergonic step of ATP synthesis, i. e. release of tightly-bound ATP into the aqueous medium, occurs during conformational relaxation of the non-equilibrium state of H+ATPase. Closed membrane vesicles are necessary for a cyclic return of the enzyme to the initial state with protonized functional groups, this provides multiple synthesis of ATP under the steady state and quasi-stationary conditions. The energetical aspects and details of possible schemes of ATP synthesis initiated by artificial electrochemical gradient of protons, as well as ATP formation during oxidative and photophosphorylation are discussed here.     

Quasi-stationary and ratio of expectations distributions: A comparative study

Many stochastic systems, including biological applications, use Markov chains in which there is a set of absorbing states. It is then needed to consider analogs of the stationary distribution of an irreducible chain. In this context, quasi-stationary distributions play a fundamental role to describe the long-term behavior of the system. The rationale for using quasi-stationary distribution is well established in the abundant existing literature. The aim of this study is to reformulate the ratio of means approach ( [Darroch and Seneta, 1965] and [Darroch and Seneta, 1967]) which provides a simple alternative. We have a two-fold objective. The first objective is viewing quasi-stationarity and ratio of expectations as two different approaches for understanding the dynamics of the system before absorption. At this point, we remark that the quasi-stationary distribution and a ratio of means distribution may give or not give similar information. In this way, we arrive to the second objective; namely, to investigate the possibility of using the ratio of expectations distribution as an approximation to the quasi-stationary distribution. This second objective is explored by comparing both distributions in some selected scenarios, which are mainly inspired in stochastic epidemic models. Previously, the rate of convergence to the quasi-stationary regime is taking into account in order to make meaningful the comparison.     

On the reality of transition state dissociation constants

Transition state dissociation constants are currently considered, utilizing stopped flow equipment. The underlying theory is briefly reviewed, relating the ideas to steady state kinetics of enzyme systems. The ideas are further analyzed under the consideration of chemical relaxation. Test conditions are described which would allow an investigation of the concepts of transition state dissociation constants by chemical relaxation techniques. A discussion concerning the way in which the concepts of transition state dissociation constants relate to other theories which assume short-lived, but real, dissociation constants is included. The theory is rigorously analyzed (in a second part), revealing the nature of the assumption of a transition state dissociation constant: While they may be written in a formal manner, they are not based on reality—on kinetic grounds direct interconversions between transition states are practically impossible. This applies also to transition state dissociation constants involving protons.     

Precise Estimates of Persistence Time for SIS Infections in Heterogeneous Populations

For a susceptible–infectious–susceptible infection model in a heterogeneous population, we derive simple and precise estimates of mean persistence time, from a quasi-stationary endemic state to extinction of infection. Heterogeneity may be in either individuals’ levels of infectiousness or of susceptibility, as well as in individuals’ infectious period distributions. Infectious periods are allowed to follow arbitrary non-negative distributions. We also obtain a new and accurate approximation to the quasi-stationary distribution of the process, as well as demonstrating the use of our estimates to investigate the effects of different forms of heterogeneity. Our model may alternatively be interpreted as describing an infection spreading through a heterogeneous directed network, under the annealed network approximation.     

Voltage-clamp experiments on oxidized cholesterol membranes modified with excitability-inducing material and comparison with a model

The conductance of oxidized cholesterol membranes modified with excitability-inducing material was observed in membranes containing either single conductance channels or 100–1000 channels. Membranes containing single channels have several conductance states for each voltage polarity, and the current through membranes containing many channels decays with at least two, and probably three, time constants following a step change in voltage (voltage-clamp). The time constants differ by about an order of magnitude. The multi-state behavior seems more pronounced in membranes made from highly oxidized cholesterol. Although a given conductance state could occur at either positive or negative voltages, each state was much more frequent at one polarity or the other. The most frequently observed single-channel conductance states in 0.1 M NaCl are about 0.3, 0.1, 0.03, 0.0 nΩ^-1 for negative voltages and 0.25, 0.05, 0.03, and 0.0 nΩ^-1 for positive voltages. The current following a voltage clamp decays to a quasi-steady state within 1 min for positive voltages and 1–15 min for negative voltages. When the holding voltage is −20 mV, the decay constants and quasi-steady state conductances as functions of clamping voltage are reasonably well described by either a three-state model of the conductance or a two-state model applied independently at negative and positive voltages. However, for high voltages, the quasi-steady state does not appear to approach a state in which all the channels are in a low conductance state.     

On the quasi-stationary distribution of the Ross malaria model.

Approximations are derived for the quasi-stationary distribution of the fully stochastic version of the classical Ross malaria model. The approximations are developed in two stages. In the first stage, the Ross process is approximated with a bivariate Markov chain without an absorbing state. The second stage of the approximation uses ideas from perturbation theory to derive explicit expressions that serve as approximations of the joint stationary distribution of the approximating process. Numerical comparisons are made between the approximations and the quasi-stationary distribution.     

Immobilized catalase: deactivation and reactor stability

The quasi-steady behavior of a continuous flow reactor in which hydrogen peroxide is decomposed by immobilized catalase is investigated. Under certain conditions, reactors involving such substrate-inhibited, self-poisoning reactions are susceptible to suddne failure and the reactor moves catastrophically from high- to low-conversion quasi-steady states. This exchange-of-steady-states phenomenon is ex-amined in the light of experimental evidence for the enzyme catalase from bovine liver. (c) 1993 John Wiley & Sons, Inc.     

Asymmetric Effect of Mechanical Stress on the Forward and Reverse Reaction Catalyzed by an Enzyme

The concept of modulating enzymatic activity by exerting a mechanical stress on the enzyme has been established in previous work. Mechanical perturbation is also a tool for probing conformational motion accompanying the enzymatic cycle. Here we report measurements of the forward and reverse kinetics of the enzyme Guanylate Kinase from yeast (Saccharomyces cerevisiae). The enzyme is held in a state of stress using the DNA spring method. The observation that mechanical stress has different effects on the forward and reverse reaction kinetics suggests that forward and reverse reactions follow different paths, on average, in the enzyme''s conformational space. Comparing the kinetics of the stressed and unstressed enzyme we also show that the maximum speed of the enzyme is comparable to the predictions of the relaxation model of enzyme action, where we use the independently determined dissipation coefficient for the enzyme''s conformational motion. The present experiments provide a mean to explore enzyme kinetics beyond the static energy landscape picture of transition state theory.     

Oxygen-17 and deuterium nuclear magnetic relaxation studies of lysozyme hydration in solution: field dispersion, concentration, pH/pD, and protein activity dependences

A comparison of 17O and 2H NMR relaxation rates of water in lysozyme solutions as a function of concentration, pH/pD, and magnetic field suggests that only 17O monitors directly the hydration of lysozyme in solution. NMR measurements are for the first time extended to 11.75 T. Lysozyme hydration data are analyzed in terms of an anisotropic, dual-motion model with fast exchange of water between the "bound" and "free" states. The analysis yields 180 mol "bound" water/mol lysozyme and two correlation times of 7.4 ns ("slow") and 29 ps ("fast") for the bound water population at 27 degrees C and pH 5.1, in the absence of salt, assuming anisotropic motions of water with an order parameter value for bound water of 0.12. Under these conditions, the value of the slow correlation time of bound water (7.4 ns) is consistent with the value of 8 ns obtained by frequency-domain fluorescence techniques for the correlation time associated with the lysozyme tumbling motion in solutions without salt. In the presence of 0.1 M NaCl the hydration number increases to 290 mol/mol lysozyme at pD 4.5 and 21 degrees C. The associated correlation times at 21 degrees C in the presence of 0.1 M NaCl are 4.7 ns and 15.5 ps, respectively. The value of the slow correlation time of 4.7 ns is consistent with the calculated value (4.9 ns) for the lysozyme monomer tumbling in solution. The systematic deviations of the relaxation rates, estimated with the single-exponential approximation, from the theoretical, multiexponential nuclear (I' + 1/2) spin relaxation are evaluated at various frequencies for 17O (I = 5/2) with the first-order, linear approximation (25). All NMR relaxation data for hydrated lysozymes are affected by protein activity and are sensitive both to the ionization of protein side chains and to the state of protein aggregation.     

Distribution of the Fittest Individuals and the Rate of Muller's Ratchet in a Model with Overlapping Generations

Muller''s ratchet is a paradigmatic model for the accumulation of deleterious mutations in a population of finite size. A click of the ratchet occurs when all individuals with the least number of deleterious mutations are lost irreversibly due to a stochastic fluctuation. In spite of the simplicity of the model, a quantitative understanding of the process remains an open challenge. In contrast to previous works, we here study a Moran model of the ratchet with overlapping generations. Employing an approximation which describes the fittest individuals as one class and the rest as a second class, we obtain closed analytical expressions of the ratchet rate in the rare clicking regime. As a click in this regime is caused by a rare, large fluctuation from a metastable state, we do not resort to a diffusion approximation but apply an approximation scheme which is especially well suited to describe extinction events from metastable states. This method also allows for a derivation of expressions for the quasi-stationary distribution of the fittest class. Additionally, we confirm numerically that the formulation with overlapping generations leads to the same results as the diffusion approximation and the corresponding Wright-Fisher model with non-overlapping generations.