Hyperbolic equilibrium

Center manifold theory has become an indispensable tool for the study of ODEs. An equilibrium is called hyperbolic if the linearization of the vector field at that equilibrium does not possess spectrum on the imaginary axis. The local dynamics of ODEs near a hyperbolic equilibrium is determined by that linearization. In particular there exist stable and un-... 

That is, the solution started near to z stays near z for all time. The Hartman-Grobman theorem clearly implies that the stability of a given hyperbolic equilibrium point z of a nonlinear system can be inferred from the stability of the origin for the linearization of the system around z. ... 

Vertebrate myoglobins, as we have pointed out, have a much higher oxygen afOnity than the corresponding hemoglobins and exhibit a hyperbolic equilibrium curve. 

The description of phenomena in a continuous medium such as a gas or a fluid often leads to partial differential equations. In particular, phenomena of wave propagation are described by a class of partial differential equations called hyperbolic, and these are essentially different in their properties from other classes such as those that describe equilibrium ( elhptic ) or diffusion and heat transfer ( para-bohc ). Prototypes are ... 

These results show that if the relationship between the concentration of an agonist and the proportion of receptors that it occupies is measured directly (e.g., using a radioligand binding method), the outcome should be a simple hyperbolic curve. Although the curve is describable by the Hill-Langmuir equation, the dissociation equilibrium constant for the binding will be not KA but Ke, which is determined by both E and KA. 

A plot of the initial reaction rate, v, as a function of the substrate concentration [S], shows a hyperbolic relationship (Figure 4). As the [S] becomes very large and the enzyme is saturated with the substrate, the reaction rate will not increase indefinitely but, for a fixed amount of [E], it reaches a plateau at a limiting value named the maximal velocity (vmax). This behavior can be explained using the equilibrium model of Michaelis-Menten (1913) or the steady-state model of Briggs and Haldane (1926). The first one is based on the assumption that the rate of breakdown of the ES complex to yield the product is much slower that the dissociation of ES. This means that k2 tj. 

The observed hyperbolic dependences suggest a mechanism that involves a pre-equilibrium binding of two pyridine carboxylates to the Fem of lm, followed by the intramolecular proton transfer from the coordinated acid (Scheme 3). This option has been supported by measurements of the binding constants for py and related ligands. The data in Fig. 8 agree with the mechanism in Scheme 3. The reactive intermediate for robust lm is the diaxially coordinated species, one of the two ligands being picolinic acid (L). 

An enzyme is said to obey Michaelis-Menten kinetics, if a plot of the initial reaction rate (in which the substrate concentration is in great excess over the total enzyme concentration) versus substrate concentration(s) produces a hyperbolic curve. There should be no cooperativity apparent in the rate-saturation process, and the initial rate behavior should comply with the Michaelis-Menten equation, v = Emax[A]/(7 a + [A]), where v is the initial velocity, [A] is the initial substrate concentration, Umax is the maximum velocity, and is the dissociation constant for the substrate. A, binding to the free enzyme. The original formulation of the Michaelis-Menten treatment assumed a rapid pre-equilibrium of E and S with the central complex EX. However, the steady-state or Briggs-Haldane derivation yields an equation that is iso-... 

In glycerol monooleate/decane bilayers we find the steady-state conductance at zero current to be proportional to the first power of the ion concentration and to the second power of the ionophore concentration, as illustrated in Fig. 1. (The current-voltage characteristic is hyperbolic for all ionic species indicating that this molecule is in the equilibrium domain for the interfacial reactions, with the rate-limiting step being the ion translocation across the membrane interior.) The conductance selectivity sequence is seen to be Na>K>Rb>Cs, Li. 

Langmuir equation for a uniform surface, but by the Zel dovich and Rogin-skii equation or by the Bangham equation. Adsorption equilibrium is described not by the hyperbolic Langmuir isotherm, but by the Freundlich isotherm or the logarithmic isotherm (40). 

Figure Cl. 1.2 shows a typical time course resulting from a continuous assay of product formation in an enzyme-catalyzed reaction. The hyperbolic nature of the curve illustrates that the reaction rate decreases as the reaction nears completion. The reaction rate, at any given time, is the slope of the line tangent to the curve at the point corresponding to the time of interest. Reaction rates decrease as reactions progress for several reasons, including substrate depletion, reactant concentrations approaching equilibrium values (i.e., the reverse reaction becomes relevant), product inhibition, enzyme inactivation, and/or a change in reaction conditions (e.g., pH as the reaction proceeds). With respect to each of these reasons, their effects will be at a minimum in the initial phase of the reaction—i.e., under conditions corresponding to initial velocity measurements. Hence, the interpretation of initial velocity data is relatively simple and thus widely used in enzyme-related assays.

The hyperbolic saturation curve that is commonly seen with enzymatic reactions led Leonor Michaelis and Maude Men-ten in 1913 to develop a general treatment for kinetic analysis of these reactions. Following earlier work by Victor Henri, Michaelis and Menten assumed that an enzyme-substrate complex (ES) is in equilibrium with free enzyme... 

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