Mathematical modeling, molecular

Publish or perish is a proverb that academicians take seriously. If you do not publish your work for the rest of the community to evaluate, then you have no business in academia (and if you don t already have tenure, you will be banished). But the saying can be applied to theories as well. If a theory claims to be able to explain some phenomenon but does not generate even an attempt at an explanation, then it should be banished. Despite comparing sequences and mathematical modeling, molecular evolution has never addressed the question of how complex structures came to be. In effect, the theory of Darwinian molecular evolution has not published, and so it should perish. 

Keywords polymerization kinetics, polymerization reactors, mathematical modelling, molecular weight distribution (MWD), chemical composition distribution (CCD), Ziegler-Natta catalysts, metallocenes, microstructure, isotacticity distribution, mass transfer resistances, heat transfer resistances, effects of multiple site types. 

A mathematical model, preferably based on molecular consi-... 

The classical microscopic description of molecular processes leads to a mathematical model in terms of Hamiltonian differential equations. In principle, the discretization of such systems permits a simulation of the dynamics. However, as will be worked out below in Section 2, both forward and backward numerical analysis restrict such simulations to only short time spans and to comparatively small discretization steps. Fortunately, most questions of chemical relevance just require the computation of averages of physical observables, of stable conformations or of conformational changes. The computation of averages is usually performed on a statistical physics basis. In the subsequent Section 3 we advocate a new computational approach on the basis of the mathematical theory of dynamical systems we directly solve a... 

A series of monographs and correlation tables exist for the interpretation of vibrational spectra [52-55]. However, the relationship of frequency characteristics and structural features is rather complicated and the number of known correlations between IR spectra and structures is very large. In many cases, it is almost impossible to analyze a molecular structure without the aid of computational techniques. Existing approaches are mainly based on the interpretation of vibrational spectra by mathematical models, rule sets, and decision trees or fuzzy logic approaches. 

Empirical energy functions can fulfill the demands required by computational studies of biochemical and biophysical systems. The mathematical equations in empirical energy functions include relatively simple terms to describe the physical interactions that dictate the structure and dynamic properties of biological molecules. In addition, empirical force fields use atomistic models, in which atoms are the smallest particles in the system rather than the electrons and nuclei used in quantum mechanics. These two simplifications allow for the computational speed required to perform the required number of energy calculations on biomolecules in their environments to be attained, and, more important, via the use of properly optimized parameters in the mathematical models the required chemical accuracy can be achieved. The use of empirical energy functions was initially applied to small organic molecules, where it was referred to as molecular mechanics [4], and more recently to biological systems [2,3]. 

One of the major uses of molecular simulation is to provide useful theoretical interpretation of experimental data. Before the advent of simulation this had to be done by directly comparing experiment with analytical (mathematical) models. The analytical approach has the advantage of simplicity, in that the models are derived from first principles with only a few, if any, adjustable parameters. However, the chemical complexity of biological systems often precludes the direct application of meaningful analytical models or leads to the situation where more than one model can be invoked to explain the same experimental data. 

Over the years there have been many attempts to simulate the behaviour of viscoelastic materials. This has been aimed at (i) facilitating analysis of the behaviour of plastic products, (ii) assisting with extrapolation and interpolation of experimental data and (iii) reducing the need for extensive, time-consuming creep tests. The most successful of the mathematical models have been based on spring and dashpot elements to represent, respectively, the elastic and viscous responses of plastic materials. Although there are no discrete molecular structures which behave like the individual elements of the models, nevertheless... 

With the experimental observation of constitutive activity for GPCRs by Costa and Herz [2], a modification was needed. Subsequently, Samama and colleagues [3] presented the extended ternary complex model to fill the void. This chapter discusses relevant mathematical models and generally offers a linkage between empirical measures of activity and molecular mechanisms. 

Mathematical models are the link between what is observed experimentally and what is thought to occur at the molecular level. In physical sciences, such as chemistry, there is a direct correspondence between the experimental observation and the molecular world (i.e., a nuclear magnetic resonance spectrum directly reflects the interaction of hydrogen atoms on a molecule). In pharmacology the observations are much more indirect, leaving a much wider gap between the physical chemistry involved in drug-receptor interaction and what the cell does in response to those interactions (through the cellular veil ). Hence, models become uniquely important. 

Mathematical models of the reaction system were developed which enabled prediction of the molecular weight distribution (MWD). Direct and indirect methods were used, but only distributions obtained from moments are described here. Due to the stiffness of the model equations an improved numerical integrator was developed, in order to solve the equations in a reasonable time scale. 

Mathematical models of the reaction system have been developed, enabling prediction of the molecular weight... 

On the continuum level of gas flow, the Navier-Stokes equation forms the basic mathematical model, in which dependent variables are macroscopic properties such as the velocity, density, pressure, and temperature in spatial and time spaces instead of nf in the multi-dimensional phase space formed by the combination of physical space and velocity space in the microscopic model. As long as there are a sufficient number of gas molecules within the smallest significant volume of a flow, the macroscopic properties are equivalent to the average values of the appropriate molecular quantities at any location in a flow, and the Navier-Stokes equation is valid. However, when gradients of the macroscopic properties become so steep that their scale length is of the same order as the mean free path of gas molecules,, the Navier-Stokes model fails because conservation equations do not form a closed set in such situations. 

Zhao, Y. W., Chang, L., and Kim, S. H., "A Mathematical Model for Chemical-Mechanical Polishing Based on Formation and Removal of Weakly Bonded Molecular Species, Wear, Vol. 254,2003, pp. 332-339. 

In the 1960s it was discovered that a mathematical model could describe the relationship between simple calculated molecular properties for a series of... 

Previous reports on FMSZ catalysts have indicated that, in the absence of added H2, the isomerization activity exhibited a typical pattern when measured as a function of time on stream [8, 9], In all cases, the initial activity was very low, but as the reaction proceeded, the conversion slowly increased, reached a maximum, and then started to decrease. In a recent paper [7], we described the time evolution in terms of a simple mathematical model that includes induction and deactivation periods This model predicts the existence of two types of sites with different reactivity and stability. One type of site was responsible for most of the activity observed during the first few minutes on stream, but it rapidly deactivated. For the second type of site, both, the induction and deactivation processes, were significantly slower We proposed that the observed induction periods were due to the formation and accumulation of reaction intermediates that participate in the inter-molecular step described above. Here, we present new evidence to support this hypothesis for the particular case of Ni-promoted catalysts. 

This closure property is also inherent to a set of differential equations for arbitrary sequences Uk in macromolecules of linear copolymers as well as for analogous fragments in branched polymers. Hence, in principle, the kinetic method enables the determination of statistical characteristics of the chemical structure of noncyclic polymers, provided the Flory principle holds for all the chemical reactions involved in their synthesis. It is essential here that the Flory principle is meant not in its original version but in the extended one [2]. Hence under mathematical modeling the employment of the kinetic models of macro-molecular reactions where the violation of ideality is connected only with the short-range effects will not create new fundamental problems as compared with ideal models. 

A mathematical model has been developed which allows the calculation of the degradation of polymeric drug delivery systems. The model has been shown to accurately simulate both the drug release and molecular weight changes in such systems. The concentration of anhydride levels affect the erosion characteristics of... 

Alternative mechanisms have been recently proposed [78,79] based on a kinetic investigation of NO reduction by n-octane under isothermal (200°C) and steady-state conditions in the presence of H2. The authors built up a mathematical model based on supposed reaction pathways, which account for molecular adsorption of NO and CO and dissociative ones for H2 and 02. The elementary steps, which have been considered for modelling their results are reported in Table 10.3. Interesting kinetic information can be provided by the examination of this mechanism scheme in particular the fast bimolecular... 

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