Structures of steady states of an enzyme reaction in a CSTR

Structures at steady states have been investigated for an enzyme reaction in a continuous stirred tank reactor that has a random bi-uni reaction mechanism, using the imperfect bifurcation theory via singularities. The analysis has shown that two types of singular points exist. One of these points has the two types of transition states characterized by the hysteresis and double-limit points. It is obtained when the derivative of the steady-state equation with respect to the bifurcation parameter does not vanish. When the derivative vanishes, the other type of singular point is obtained. This point has the two transition states of hysteresis and bifurcation points.     

Computational multiple steady states for enzymatic esterification of ethanol and oleic acid in an isothermal CSTR

The capacity of complex biochemical reaction networks (consisting of 11 coupled non-linear ordinary differential equations) to show multiple steady states, was investigated. The system involved esterification of ethanol and oleic acid by lipase in an isothermal continuous stirred tank reactor (CSTR). The Deficiency One Algorithm and the Subnetwork Analysis were applied to determine the steady state multiplicity. A set of rate constants and two corresponding steady states are computed. The phenomena of bistability, hysteresis and bifurcation are discussed. Moreover, the capacity of steady state multiplicity is extended to the family of the studied reaction networks.     

On the number of steady states in a multiple futile cycle

The multisite phosphorylation-dephosphorylation cycle is a motif repeatedly used in cell signaling. This motif itself can generate a variety of dynamic behaviors like bistability and ultrasensitivity without direct positive feedbacks. In this paper, we study the number of positive steady states of a general multisite phosphorylation–dephosphorylation cycle, and how the number of positive steady states varies by changing the biological parameters. We show analytically that (1) for some parameter ranges, there are at least n + 1 (if n is even) or n (if n is odd) steady states; (2) there never are more than 2n − 1 steady states (in particular, this implies that for n = 2, including single levels of MAPK cascades, there are at most three steady states); (3) for parameters near the standard Michaelis–Menten quasi-steady state conditions, there are at most n + 1 steady states; and (4) for parameters far from the standard Michaelis–Menten quasi-steady state conditions, there is at most one steady state.      

Graphic rules in steady and non-steady state enzyme kinetics

Graphic methods, when applied to enzyme kinetics, can provide a visually intuitive relation between calculations and reaction graphs. This will not only greatly raise the efficiency of calculations but also significantly help the analysis of enzyme kinetic mechanisms. In this paper, four graphic rules are presented. Rules 1-3 are established for steady state enzyme-catalyzed reaction systems and Rule 4 is for non-steady state ones. In comparison with conventional graphic methods which can only be applied to steady state systems, the present rules have the following merits. 1) Complicated and tedious calculations can be greatly simplified; for example, in calculating the concentrations of enzyme species for the bi-bi random mechanism, the calculation work can be reduced 8-fold compared with the King-Altman's method. 2) A great deal of wasted labor can be avoided; for example, in calculating the rate of product formation for the same mechanism, the operation of finding and removing the 96 reciprocally canceled terms is no longer needed because they automatically disappear during the derivation. 3) Final results can be easily and safely checked by a formula provided in each of the graphic rules. 4) Non-steady state systems can also be treated by the present graphic method; for example, applying Rule 4, one can directly write out the solution for a non-steady state enzyme-catalyzed system, without the need to follow more difficult and complicated operations to solve differential equations. The mathematical proofs of Rules 1-4 are given in Appendices A-D (in the Miniprint), respectively.     

Kinetics of substrate reaction during irreversible modification of enzyme activity for enzymes involving two substrates

Kinetic equations for the substrate reaction during simultaneous irreversible inhibition of enzyme activity for enzymes involving two substrates have been derived. It has been shown that the method proposed previously (Tsou, Acta Biochim, Biophys. Sinica 5, 398-417, 1965) for the determination of the apparent inhibition rate constants in the cases of single substrate enzymes can also be used in the present situation. Moreover, the criteria proposed to distinguish between different substrate competition types and to detect the formation of a reversible enzyme-inhibitor complex prior to the irreversible inhibition step also apply. Methods for the estimation of the microscopic rate constants have been proposed and it has been shown that irreversible inhibition kinetics can be used to distinguish between different mechanisms for substrate binding sequences.     

Oscillations and multiple steady states in a cyclic gene model with repression

In this paper we study the cyclic gene model with repression considered by H. T. Banks and J. M. Mahaffy. Roughly, the model describes a biochemical feedback loop consisting of an integer number G of single gene reaction sequences in series. The model leads to a system of functional differential equations. We show that there is a qualitative difference in the dynamics between even and odd G if the feedback repression is sufficiently large. For even G, multiple stable steady states can coexist while for odd G, periodic orbits exist.This research was supported in part by the Air Force Office of Scientific Research under Contract #AFOSR-84-0376 and by the US Army Research Office under Contract #DAAG29-84-K-0082     

Impossibility of multiple steady states in a family of active membrane transport models

To study the dynamical behavior of active membrane transport models, Vieira and Bisch proposed a complex chemical network (model 3) with two cycles. One cycle involves monomers as pump units while the other cycle uses dimers. In their work, the stoichiometric network analysis was used to study the stability of steady states and the bifurcation analysis was done through numerical methods. They concluded that the possibility of multiple steady states in the model 3 could not be discarded. Here, a zero eigenvalue analysis is applied to prove the impossibility of multiple positive steady states in the model 3. (A positive steady state is one for which all species have positive concentrations.) Moreover, the result is generalized to its family networks. Received: 6 April 1998 / Revised version: 16 October 1998 / Accepted: 28 October 1998     

A kinetic analysis of enzyme systems involving four substrates

A treatment of kinetic data for enzyme mechanisms involving four substrates is described. The initial-rate equations and product-inhibition patterns for such mechanisms are presented. The treatment is extended to include analysis of enzyme mechanisms involving three substrates in which two molecules of one substrate are used.     

The kinetics of enzyme systems involving activation of zymogens

A general model of zymogen activation is proposed and explicit kinetic equations for the time courses of the various species and products involved are given. These equations are valid for the whole course of the reaction and therefore for both the transient phase and the steady state. This model is sufficiently general to include mechanisms possessing one or more steps of zymogen activation besides possible steps of inhibition (reversible or irreversible) or inactivation.     

Concentration multiplicity in a draft tube fluidized-bed bioreactor involving two limiting substrates

Concentration multiplicity in a two-phase or three-phase draft tube fluidized-bed bioreactor containing biofloc particles is studied. The kinetics of biological reactions considered involve two limiting substrates. The necessary and sufficient conditions for concentration multiplicity in both the biofilm and bioreactor are examined in terms of effectiveness factor, inlet and bulk concentration of substrates, and liquid flow rate. Hysteresis behavior in both the biofilm and bioreactor and multiplicity of concentration profiles in the biofilm are also discussed.     

Transient-phase kinetics of enzyme inactivation induced by suicide substrates

This paper deals with the kinetic study of reaction mechanisms with enzyme inactivation induced by a suicide substrate in the presence or absence of an auxiliary substrate and in conditions of excess of substrates in relation to the enzyme concentration and vice versa. A transient-phase approach has been developed that enables explicit equations with one or two significant exponentials to be obtained, thereby showing the dependence of product concentration on time. The validity of these equations has been checked, and a comparison made with those previously obtained by other authors. We propose an experimental design to determine the corresponding parameters and kinetic constants. The simplicity of our method allows a systematic application to more complex mechanisms.     

Steady state kinetics of consecutive enzyme catalysed reactions involving single substrates: procedures for the interpretation of coupled assays

Sets of differential rate equations are written describing a linear sequence of reactions occurring in solution each catalysed by a control enzyme or one of the Michaelis-Menten type. It is shown that the solutions of these equations may be formulated as a set of Maclaurin polynomials, expressing the concentration of each reactant and of final product as a function of time. From arrays of such polynomials, general expressions are induced for the first non-zero term of the series. These are used to formulate a procedure (illustrated with an example simulated by numerical integration) by which results of coupled enzymic assays may be analysed in terms of maximal velocities and apparent Michaelis constants: correlation is made with other established methods for conducting coupled assays. The present procedure assumes a steady state of enzyme-substrate complexes but not of intermediate reactants.     

On the validity of the steady state assumption of enzyme kinetics

By estimating relevant time scales, a simple new condition can be found that ensures the validity of the steady state assumption for a standard enzyme-substrate reaction. The generality of the approach is demonstrated by applying it to the determination of validity criteria for the steady state assumption applied to an enzyme-substrate-inhibitor system.     

Reversible inhibition of penicillinase by quinacillin: evaluation of mechanisms involving two conformational states of the enzyme.

Staphylococcal penicillinase (EC 3.5.2.6) is shown to undergo partial, fully reversible inactivation of benzylpenicillinase activity on incubation with the substrate quinacillin, the hydrolysis of which follows a corresponding biphasic time-course. The kinetics fit a scheme involving slow isomerization of the enzyme between conformational states that differ in K_m and V_max for quinacillin. The possibility that inactivation is related to formation of a previously observed covalent enzyme-quinacillin conjugate is ruled out because the kinetics of its formation differ from those of inactivation. This implies that the conjugate arises from a state of the enzyme substrate complex present during the normal catalytic cycle. The multiplicity of binding sites found suggests that a reactive catalytic intermediate substitutes several amino-acid side chains during denaturation of the enzyme-quinacillin mixture, thus providing an explanation of earlier results.     

Experimental investigations of multiple steady states in aerobic continuous cultivations of Saccharomyces cerevisiae

The steady-state behavior of a glucose-limited, aerobic, continuous cultivation of Saccharomyces cerevisiae CEN.PK113-7D was investigated around the critical dilution rate. Oxido-reductive steady states were obtained at dilution rates up to 0.09 h(-1) lower than the critical dilution rate by operating the bioreactor as a productostat, where the dilution rate was controlled on the basis of an ethanol measurement. Thus, the experimental investigations revealed that multiple steady states exist in a region of dilution rates below the critical dilution rate. The existence of multiple steady states was attributed to two distinct physiological effects occurring when growth changed from oxidative to oxido-reductive: (i) a decrease in the efficiency of ATP production and utilization (at ethanol concentrations below 3 g/L) and (ii) repression of the oxidative metabolism (at higher ethanol concentrations). The first effect was best observed at low ethanol concentrations, where multiple steady states were observed even when no repression of the oxidative metabolism was evident, i.e., the oxidative capacity was constant. However, at higher ethanol concentrations repression of the oxidative metabolism was observed (the oxidative capacity decreased), and this resulted in a broader range of dilution rates where multiple steady states could be found.