Catastrophes in molecular structures

A function/of the essential variables, is said to be of finite co-dimension if a small perturbation of/gives rise to only a finite number of topological types. It is in this case that a map / of co-rank k can be embedded in a family of deformations (/, x R), parametrized by q variables which form the control space a R. By definition, q, the dimension of the control space, is the co-dimension of the singularity represented by /. The family of functions (/, / X R R) thus defined, describes the universal unfolding of /, in which the function/itself is the member associated with the origin of control space. In our applications the control space is a subset of the nuclear configuration space. 

The number of elementary catastrophe types depends upon q. It has been shown that only for values of g 5 is the number of catastrophe types finite. Thom has classified these types by their co-rank k and co-dimension q for values of g 4. The concept of an unfolding and the accompanying definitions are illustrated first in terms of the simplest of all catastrophe types, the so-called fold catastrophe for which both the co-rank and co-dimension equal one. 

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The immediate consequence of the theorem is that a structural instability can be established through only one of two possible mechanisms which correspond to the bifurcation and conflict catastrophes. A change in molecular structure—the making and breaking of chemical bonds—can only be caused by the formation of a degenerate critical point in the electronic charge distribution or by the attainment of an unstable intersection of the submanifolds of bond and ring critical points. 

From the early advances in the quantum-chemical description of molecular electron densities [1-9] to modem approaches to the fundamental connections between experimental electron density analysis, such as crystallography [10-13] and density functional theories of electron densities [14-43], patterns of electron densities based on the theory of catastrophes and related methods [44-52], and to advances in combining theoretical and experimental conditions on electron densities [53-68], local approximations have played an important role. Considering either the formal charges in atomic regions or the representation of local electron densities in the structure refinement process, some degree of approximate transferability of at least some of the local structural features has been assumed. 

We here summarize the results of Bader et a/.118 which are concerned with the definition of molecular structure and with the extension of this concept, together with the associated concept of a bond, to the dynamic case. A precise description and physical interpretation of the making and breaking of chemical bonds is presented by these workers in a quantitative analysis of the evolution of molecular structure. The topological analysis of the dynamic system, as pointed out by Collard and Hall,119 falls naturally into the realm of an existing and elegant mathematical theory, the catastrophe theory of Thom.120... 

As discussed above and illustrated in Fig. 3.4, the partitioning of nuclear configuration space obtained as a result of the definition of molecular structure leads to the concept of a structure diagram. The space R is partitioned into a finite number of structural regions with their boundaries, as defined by the catastrophe set, denoting the configurations of unstable structures. This information constitutes a system s structure diagram, a... 

In general, the differential description is useful for processes where there is a wide separation of scales between the smallest macroscopic scales of interest and the microscopic scales associated with the internal structure of the fluid. If the micro-scales were always of molecular magnitude then questions of scale separation would seldom arise. But, in many of the models employed for engineering purposes, the characteristic scales of the internal structure being described are themselves macroscopic in nature. In such situations the desired separation between the calculated and modeled scales is much less clear cut, and one must be careful not to attribute quantitative significance to any predicted solution features with scales comparable to the internal micro-scale. When a continuum description is pushed to far, i.e. applied on scales too small, one can only hope that such inaccuracies are not catastrophic in nature. 

A chemical reaction represents a molecular catastrophe, in which the electronic structure, as well as the nuclear framework of the system changes qualitatively. Most often a chemical reaction corresponds to the breaking of an old and creation of a new bond. 

For selected correlation models, in either structure-driven or molecular mechanistic waves, one employs them to compute the associated predicted activities for test molecules and to provide the statistics regarding the observed activity. If the obtained relative statistical power is close to those characteristics for the trial set of molecules, then these models may be validated for the specific eco-, bio-, or pharmacological problem. Moreover, further insight will be provided by the analysis of the catastrophe shape of the models involved and discussed accordingly. 

A system that contains heavy protoribosomes can avoid error catastrophes because high-molecular-weight structures absorb thermal noise, and are immune to a wide range of perturbations. This conclusion is based on a general engineering principle that Burks (1970) expressed in this way There exists a direct correlation between the size of an automaton - as measured roughly by number of components - and the accuracy of its function. In the case of protein synthesis, this means that, in order to be precise, ribosomes must be immune to thermal noise and must therefore be heavy. 

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