Logistic kinetics

A front corresponds to a traveling wave solution, which maintains its shape, travels with a constant velocity v, p x, t) = p(x - v t), and joins two steady states of the system. The latter are uniform stationary states, p(x, t) = p, where Ffp) = 0. For the logistic kinetics, the steady states are = 0 and jo2 = 1- While the logistic kinetics has only two steady states, three or more stationary states can exist for a broad class of systems in nonlinear chemistry and population dynamics with Alice effect, but a front can only connect two of them. To determine the propagation direction of the front, we need to evaluate the stability of the stationary states, see Sect. 1.2. The steady state jo is stable if P (fp) < 0 and unstable if F (jo) > 0. Let the initial particle density p x,0) be such that on a certain finite interval, p x,0) is different from 0 and 1, and to the left of this interval p(x,0) = 1, while to the right p x, 0) = 0. In this case, the initial condition is said to have compact support. Kolmogorov et al. [232] showed for Fisher s equation that due to the combined effects of diffusion and reaction, the region of density close to 1 expands to the... 

To be specific, we consider logistic kinetics and Nagumo kinetics [274], F(p) = rp(l —p) p—b), as examples for cases A and B, respectively. For the logistic case, a linear stability analysis of the stationary states (0,0) and (1,0) provides their eigen-... 

To test whether the logistic kinetic equation (82), which is a natural generalization of the Michaelis-Menten equation, may provide a workable analytical solution in an elementary form, we first integrate it tmder the form... 

This time-dependent solution (89) substitutes an elementary logarithinic dependency for the W-Lambert function. It is nevertheless remarkable that the solutirm of a generalized logistic kinetic version of the Michaelis-Menten instantaneous equation provides an analytically exact solution. It clearly reduces to the above Eq. (74) in the first order expansion of the chemical concentration time evolution with respect to the 50-effect concentration (EC50) observed. 

Firstly, we showed that the quantum fluctuation and tunneling may be simulated when among the Michaelis-Menten also the logistic kinetics and its solution is considered, a feature confirmed by applying the Beer-Eambert law of absorption spectroscopy. Such picture is in accordance with the observed enhanced rate of the vibrationally states of ES by means of quantum tunneling when considered within the Brownian mechanism (1.168) and Figure 1.13. 

However, by following the considerations of the Section 1.3 one may further generalizes Eq. (3.213) such that the range of reaction rates is expanded and provides the logistic kinetic equation (1.37), reloaded here as the logistic expression... 

If an approximate Km value for the enzyme-substrate combination of interest is known, a full-scale kinetic assay may be done immediately. However, often an approximate value is not known and it is necessary first to do a range finding or suck and see preliminary assay. For such an assay, a concentrated substrate solution is prepared and tenfold serial dilutions of the substrate are made so that a range of substrate concentrations is available within which the experimenter is confident the Km value lies. Initial velocities are determined at each substrate concentration, and data may he plotted either hyperholically (as V versus [S]) or with [S] values expressed as logio values. In the latter case, a sigmoidal curve is fitted to data with a three parameter logistic equation (O Eq. 4) ... 

A number of reports also describe the prediction of mechanism-based inhibition (MBI) [17,18]. In this type of model, MBI is determined in part by spectral shift and inactivation kinetics. Jones et al. applied computational pharmacophores, recursive partitioning and logistic regression in attempts to predict metabolic intermediate complex (MIC) formation from structural inputs [17]. The development of models that accurately predict MIC formation will provide another tool to help reduce the overall risk of DDI [19]. 

Recently, Peleg el al.129 have made a case for applying non-Arrhenius and non-Williams-Landel-Ferry kinetics to the Maillard reaction, since its reactivity is only noticeable above a certain temperature. They show that a relatively simple, empirical log-logistic relationship,... 

In other cases the kinetics of tumors growth is described by equation of auto-catalysis (in biology it is logistic Verhulst equation) ... 

Most of the theoretical details of the material covered in this chapter can be found in Coveil et al. (4), Jacquez and Simon (5), and Jacquez (6). Of particular importance to this chapter is the material covered in Coveil et al. (4) in which the relationships between the calculation of kinetic parameters from statistical moments and the same parameters calculated from the rate constants of a linear, constant-coefficient compartmental model are derived. Jacquez and Simon (5) discuss in detail the mathematical properties of systems that depend upon local mass balance this forms the basis for understanding compartmental models and the simplifications that result from certain assumptions about a system under study. Berman (7) gives examples using metabolic turnover data, while the examples provided in Gibaldi and Perrier (8) and Rowland and Tozer (9) are more familiar to clinical phar maco logists. 

Yano Y, Oguma T, Nagata H, Sasaki S. Application of a logistic growth model to pharmacodynamic analysis of in vitro bactericidal kinetics. J Pharm Sci 1998 87 1177-83. 

Additionally, the technique is capable of delivering in vivo enzyme kinetics data as dose response curves, which can be fitted to the four-parameter logistic equation (Eq. (4)) ... 

Note that the reduction of the 18 steps of the FKN mechanisms to the 5 kinetic equations of the Oregonator does not allow the correct stoechiometry to be obtained. We note the analogy of the first step of the process B (A+X -> 2X+Z) with the "chemical" interpretation of Verhulst s logistic equation (A + X 2X+Z) described in section 2. 

Enzyme production kinetics in SSF have the potential to be quite complex, with complex patterns of induction and repression resulting from the multisubstrate environment. As a result, no mechanistic model of enzyme production in SSF has yet been proposed. Ramesh et al. [120] modeled the production of a-amylase and neutral protease by Bacillus licheniformis in an SSF system. They showed that production profiles of the two enzymes could be described by the logistic equation. However, although they claimed to derive the logistic equation from first principles, the derivation was based on a questionable initial assumption about the form of the equation describing product formation kinetics They did not justify why the rate of enzyme production should be independent of biomass concentration but directly proportional to the multiple of the enzyme concentration and the substrate concentration. As a result their equation must be considered as simply empirical. 

For packed beds without internal heat transfer a modified Damkohler number (DaM) can be proposed assiuning logistic growth kinetics without maintenance metabohsm [142] ... 

Figure 1. Growth Kinetics of A. nidulcms strains cultivated on liquid vinasse medium. Approximately 9 x 10 spores (3 replications) were grow on lOOmL of vinasse medium at 37 X) with agitation of ISO rpm. Samples were taken every 12 hours for 60 hours. The results were evaluated by logistic regression analysis.

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