Michaelis-Menten dynamics

The following is the principal theorem for competition between two adequate competitors under Michaelis-Menten dynamics. Proofs (with varying degrees of mathematical rigor) may be found in [AH HHW P SLl]. Extensions will be discussed in the next chapter. 

This question was studied by Hsu [HsulJ in the chemostat with Michaelis-Menten dynamics, and his work is presented here. The equations take the form (ignoring the yield constants)... 

The biological conclusion is, of course, that differing removal rates do not alter competitive exclusion in the chemostat. One anticipates that a similar conclusion is true if the Michaelis-Menten dynamics is replaced by the general monotone term f,(S) used in Section 3. However, the Liapunov calculations depend on this form and the general question is still unresolved. 

As noted in the introduction, the model is that of a standard chemostat with two competitors, but with the added feature that an inhibitor is also input from an external source. The nutrient (and inhibitor) uptake and conversion (in the case of nutrient) are assumed to follow Michaelis-Menten dynamics. The results are probably valid for general monotone dynamics, although this has not been established. 

Many questions remain open. Key among these are the questions of uniqueness of the interior steady state in the case of Michaelis-Menten dynamics and of sufficient conditions for uniqueness in the general case. The model does not contain any transport terms, although their inclusion would be important to model a moving stream. Periodic coefficients, as discussed in Chapter 7, are certainly relevant to this model and have not been considered. In addition, the case of non-equal diffusion is not considered at all by these methods. Hence further modeling and mathematics are still needed. 

As in the model of Section 2, the problem can be studied on its omega limit set with three rest points Eq,Ei,E2. A local stability analysis and, for some special cases, the asymptotic behavior of solutions were given in [E]. However, the populations cannot invade each other simultaneously El and E2 cannot be simultaneously unstable), so the persistence theory does not hold [E]. Moreover, for Michaelis-Menten dynamics, when one of the boundary rest points is locally stable and the other unstable, the locally stable one is globally stable [HWE]. In particular, the oscillation observed in the case of system (3.2) does not occur with (3.4). Indeed, the delayed system seems to behave much like the simple chemostat. 

The term f(P)Z in (3.71), when f P) is given by (3.74), leads formally to the Michaelis-Menten dynamics (3.39), if Et is identified with the predator density and P with the substrate. This analogy has been elaborated in the literature. For example Real (1977) describes predator-prey dynamics with the Michaelis-Menten scheme (3.27), with S the prey, C the intermediate state of the prey when it is eaten, E is the predator searching for food and P is the new predator biomass produced during the consumption process, so that Et = E + P is the total amount of predator. This leads to a justification... 

Similar to Eq. (67), the first reaction (incorporating the enzyme phosphofructo-kinase) exhibits a Hill-type inhibition by its substrate ATP [126]. The overall ATP utilization v3 (ATP) is modeled by a saturable Michaelis Menten function. The system is specified by five kinetic parameters (with Gx lumped into Vm ), the Hill coefficient n, and the total concentration, 4 / = [ATP] + [ADP]. Note that the model is not intended to capture biological realism, rather it serves as a paradigmatic example to identify dynamic behavior in metabolic pathways. 

Most problems associated with approximate kinetics are avoided when Michaelis Menten-type rate equations are utilized. Though this choice sacrifices the possibility of analytical treatment, reversible Michaelis Menten-type equations are straightforwardly consistent with fundamental thermodynamic constraints, have intuitively interpretable parameters, are computationally no more demanding than logarithmic functions, and are well known to give an excellent account of biochemical kinetics. Consequently, Michaelis Menten-type kinetics are an obvious choice to translate large-scale metabolic networks into (approximate) dynamic models. It should also be emphasized that simplified Michaelis Menten kinetics are common in biochemical practice almost all rate equations discussed in Section III.C are simplified instances of more complicated rate functions. 

The quantity of any given solute being presented to the reabsorptive mechanisms is determined by the product of the GFR and the solute concentration in plasma. One of the features of any carrier-mediated process is its limited capacity. Binding of a substance to its transport protein follows the same principles as substrate binding to an enzyme or hormone binding to its receptor so we may appropriately liken the dynamics to Michaelis-Menten kinetics. 

ASME CFSTR CFD CFM DIERS exp IR HA/AN HAZOP MM MMM American Society of Mechanical Engineers Continuous flow stirred tank reactor Computational fluid dynamics Computational fluid mixing Design Institute for Emergency Relief Systems exponential Infrared (spectroscopy) Hazard analysis Hazard and operability studies Michaelis-Menten Maximum-mixedness model... 

The dynamics were run for several concentrations of substrate and variations in the Pc values. Initial velocities of the reaction were recorded. The Michaelis-Menten model was observed and characteristic Lineweaver-Burk plots were found from the model. Systematic variation of the lipophilicity of substrates and products showed that a lower affinity between a substrate and water leads to more of the S —> P reaction at a common point along the reaction progress curve. This influence is greater than that of the affinity between the substrate and the enzyme. The study created a model in which the more lipophilic substrates are more reactive. The water-substrate affinity appears... 

Pharmacokinetic studies are in general less variable than pharmacodynamic studies. This is so since simpler dynamics are associated with pharmacokinetic processes. According to van Rossum and de Bie [234], the phase space of a pharmacokinetic system is dominated by a point attractor since the drug leaves the body, i.e., the plasma drug concentration tends to zero. Even when the system is as simple as that, tools from dynamic systems theory are still useful. When a system has only one variable a plot referred to as a phase plane can be used to study its behavior. The phase plane is constructed by plotting the variable against its derivative. The most classical, quoted even in textbooks, phase plane is the c (f) vs. c (t) plot of the ubiquitous Michaelis-Menten kinetics. In the pharmaceutical literature the phase plane plot has been used by Dokoumetzidis and Macheras [235] for the discernment of absorption kinetics, Figure 6.21. The same type of plot has been used for the estimation of the elimination rate constant [236]. 

In the previous chapter it was shown that the simple chemostat produces competitive exclusion. It could be argued that the result was due to the two-dimensional nature of the limiting problem (and the applicability of the Poincare-Bendixson theorem) or that this was a result of the particular type of dynamics produced by the Michaelis-Menten hypothesis on the functional response. This last point was the focus of some controversy at one time, inducing the proposal of alternative responses. In this chapter it will be shown that neither additional populations nor the replacement of the Michaelis-Menten hypothesis by a monotone (or even nonmonotone) uptake function is sufficient to produce coexistence of the competitors in a chemostat. This illustrates the robustness of the results of Chapter 1. It will also be shown that the introduction of differing death rates (replacing the parameter D by D, in the equations) does not change the competitive exclusion result. 

In the previous chapter the gradostat was introduced as a model of competition along a nutrient gradient. The case of two competitors and two vessels with Michaelis-Menten uptake functions was explored in considerable detail. In this chapter the restriction to two vessels and to Michaelis-Menten uptake will be removed, and a much more general version of the gradostat will be introduced. The results in the previous chapter were obtained by a mixture of dynamical systems techniques and specific computations that established the uniqueness and stability of the coexistence rest point. When the number of vessels is increased and the restriction to Michaelis-Menten uptake functions is relaxed, these computations are inconclusive. It turns out that unstable positive rest points are possible and that non-uniqueness of the coexistence rest point cannot be excluded. The main result of this chapter is that coexistence of two microbial populations in a gradostat is possible in the sense that the concentration of each population in each vessel approaches a positive equilibrium value. The main difference with the previous chapter is that we cannot exclude the possibility of more than one coexistence rest point. 

We turn now to the question of competition. In the previous chapter, we established a classification of the dynamic behavior based on the set of rest points. Unfortunately, our computations - which established the stability of any interior rest point and thereby led to the conclusion that such an equilibrium is unique in the case of two vessels and Michaelis-Menten uptake functions - are extremely difficult for n vessels and general uptake functions [HSo], so the results in this case are not as simple as in that chapter. In the present context we attempt to classify the dynamics in terms of both the set 0 of rest points and the sign of the stability modulus of certain key matrices. The theory of monotone dynamics is then used to resolve global questions. The principal result is Theorem 4.4. There are three obvious candidates for equilibria ... 

Most ensemble enzymatic kinetics have been satisfactorily described by the classic Michaelis-Menten equation [31]. The observation of conformational dynamics occurring on multiple time scales raises an intriguing question why does the Michaelis-Menten equation work so well despite the broad distributions and dynamic fluctuations at the single-molecule level ... 

Thus, Kn, the Michaelis constant, is a dynamic or pseudo-equilibrium constant expressing the relationship between the actual steady-state concentrations, rather than the equilibrium.concentrations. If Aj, is very small compared to A-i, reduces to K. A steady-state treatment of the more realistic reaction sequence E+ S ES EP E + P yields the same final velocity equation although now Km is a more complex function, composed of the rate constants of all the steps. Thus, the physical significance of K cannot be stated with any certainty in the absence of other data concerning the relative magnitudes of the various rate constants. Nevertheless, represents a valuable constant that relates the velocity of an enzyme-catalyzed reaction to the substrate concentration. Inspection of the Henri-Michaelis-Menten equation shows that Km is equivalent to the substrate concentration that yields half-maximal velocity ... 

We assume that Ka, dependent mainly on polar bonding substituents, will be similar in magnitude for the carrier, whether or not it is associated with the hydrophobe. For the case of excess carrier in solution, ct c0 and elementary sorption dynamics (Langmuir, or Michaelis-Menten - depending on one s discipline) gives the fraction of total available adsorption sites T) as... 

Autoregulatory action helps to reduce nerve cell destruction resulting from brain tissue anoxia. Two possible mechanisms include flow controller dynamics in the form of pure delays and time constant lags and oxygen consumption control with Michaelis-Menten behavior. Response curves also suggest the possibility of facilitated or active transport of oxygen in tissue and resistance to the diffusion of oxygen from the tissue into the blood stream. 

Hydrolysis and fermentation models were developed using two hydrolysis datasets and two SSF datasets and by using modified Michaelis-Menten and Monod-type kinetics. Validation experiments made to represent typical kitchen waste correlated well with both models. The models were generated in Matlab Simulink and represent a simple method for implementing ODE system solvers and parameter estimation tools. These types of visual dynamic models may be useful for applying kinetic or linear-based metabolic engineering of bioconversion processes in the future. 

A minimum requirement for a true enzyme mimic is a binding interaction between two molecules preliminary to the catalytic reaction, indicated by Michaelis-Menten kinetics. Intramolecular systems can support very rapid reactions because we can use synthesis to bring groups together into close and unavoidable proximity. But an enzyme must select and bind its substrate non-covalently in a dynamic equilibrium. The chemistry of... 

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