Functional Dimensions of Biomolecules

New structures of biomolecules have been constructed: carbohydrates, proteins, nucleic acids. It is shown that glucose molecules and ribose molecules have dimensions of 15 and 12, respectively. The enantiomorphic forms of biomolecules in space of higher dimension make it possible to explain the experimentally observed facts of branching of chains of biomolecules in one of the enantiomorphic forms and the absence of chain branching in another enantiomorphic form. The enantiomorphic forms of the tartaric acid molecule in a space of higher dimension reveal the cause of the reversal in different directions of the polarization plane of light in two opposite forms.

Dimension of Molecules of Compounds of Biogenic Elements

Chemical compounds of biogenic elements are considered (i.e., chemical elements present in living organisms and ensuring the successful functioning of their various organs and systems). Biogenic elements are divided into s-, p-, and d-elements, in which respectively are completed with s-, p-, and d-electronic orbitals. In each of these groups, the structure of compounds of biogenic elements is investigated, and the dimension of the corresponding molecules is determined. It is proved that s- and d-biogenic elements exhibit increased chemical activity (higher than the standard valence) due to participation in the formation of a chemical bond of electrons of the preceding level. This leads to the creation of complex molecules of higher dimension. The chemical compounds of biogenic p-elements, which are the building blocks for the formation of biomolecules (elements of life), will be specifically investigated in subsequent chapters.

Spatial Dissipative Structures in Excitable Media (Plankton, Soil Bacteria. . . .)

A general, the simplest model of a spatial dissipative structure arising in an excitable medium is constructed, containing at least two components interacting with each other with their own mobility. One of these components (active) uses the other component as food. It is shown that such a model leads to a stationary stable spatial distribution of the components in the form of Liesegang bands. As specific examples of the formation of spatial dissipative structures, structures arising in plankton consisting of phytoplankton and zooplankton and in the soil containing the bacterial population and the nutrient substrate are considered. Bifurcation diagrams are constructed in the parameter space, characteristic for each of the considered excitable media, which determine the conditions for the formation of dissipative structures in these media. The existence in the plankton of a strange attractor of a previously unknown shape in four-dimensional phase space has been discovered.

Plankton Models and Its Attractors in a Local Approximation

The previously accepted models of plankton consisting of two interacting populations—phytoplankton and zooplankton—are considered in a local approximation. The analysis of models is carried out with the help of a qualitative study of systems of differential equations as a whole (i.e., in the entire phase space of systems, not limited to a neighborhood of equilibrium positions). Analytical conditions for the occurrence of a Hopf bifurcation are obtained for each model using the Lyapunov stability theory. A comparison of various models is given, and their shortcomings associated with the incompleteness of research are indicated. It has been established that in some cases the loss of stability of the equilibrium position does not lead to the formation of a limit cycle (Hopf bifurcation) but to the formation of a limit continuum with a chaotic behavior of the trajectories in a large part of the phase space. It is shown that the parameters significantly influencing the dynamics of the development of plankton are the natural mortality of populations as an environmental characteristic of the environment.

The Geometry of the Structure of Nucleic Acids With Regard to the Higher Dimension of the Components

Using three-dimensional visualization of nucleic acid molecules, obtained in the previous chapter, an analysis of the geometry of nucleic acid molecules in the space of higher dimension is carried out. It is shown that phosphoric acid residues and five-carbon sugar molecules in a double-stranded nucleic acid form polytopes of higher dimension with anti-parallel edges. These polytopes are of type n-cross-polytope (n = 5 for phosphoric acid residues, n = 13 for sugar molecules). It was found that these n-cross-polytopes located in right- and left-twisted spirals are enantiomorphic. It has been found that in cross-polytopes constructed of two sugar molecules there are 12 coordinate planes, each of which may contain a bond of nitrogenous bases (one of the 12 known ones). The formation of codons (triplets) corresponds to the separation in space of the highest dimension of nucleic acids of three-dimensional regions. This also occurs in the ribosomes upon contact with transport and adapter RNA during protein synthesis.

Equilibrium in the Population of the Plants

On the basis Mendel's experiments, a mathematical model is constructed that describes the results of these experiments in a wide range of parameters. This model is compared with the Hardy-Weinberg logistic model based only on probabilistic ideas about the presence of dominant and recessive alleles in the chromosomes of living organisms. There is shown that in the mathematical model of Mendel's experiments, based on real patterns of plant development, there are equilibrium positions between the dominant and recessive forms. It consists in the fact that with an increase in the number of generations all dominant and recessive phenotypes of organisms, with any number of sings, quickly equalize and then synchronously (in the absence of death of organisms) increase together, seeking asymptotically to a stable isolated equilibrium position of the type of a multidimensional node. This newly discovered behavior of the dominant and recessive forms in the vicinity of the equilibrium position (true) differs significantly from the logistic equilibrium position in the Hardy-Weinberg principle, built without taking into account the real patterns in the plant population.

Three-Dimensional Models of a Five-Carbon Sugar Molecule and Nucleic Acids

Three-dimensional images of five-carbon sugar molecules and single-stranded nucleic acid molecules (DNA and RNA) were obtained. The geometrical cause of the formation of different form by molecules nucleic acids (right and left spirals with different number of D-ribose and ribose molecules in the period, including closed chains) has been determined. Substituting the known effective values of the lengths of chemical bonds (carbon-carbon, oxygen-oxygen, phosphorus-oxygen) into the structure of polytopes, the values of the characteristic geometric parameters of molecules nucleic acids were calculated: their effective diameter and period. It turned out that the calculated values of these parameters are in good agreement with their values, determined earlier experimentally. It is shown that the set of single-stranded nucleic acids (both DNA and RNA) is broken into two sets of chiral forms. Each form in one set contains a chiral form in another set. Moreover, in each set there are possible rotation of the spirals both in the right and in the left direction.

Waves in Biological Populations

Self-regulating nonlinear waves in various biological populations are considered as moving attractors in excitable media. Mathematically, waves in populations are solutions of nonstationary parabolic systems of differential diffusion equations with source terms, and the velocity of the wave is an eigenvalue of the problem, and its profile is an eigenvalue function of the problem. There is no general exact method for solving such a problem. An approximate method for its solution is proposed (the semi-infinite reaction zone method), which essentially reduces to solving an algebraic system of equations. The method is used to calculate the waves in various biological populations. It is shown that there are two types of waves: a wave of conquest and a solitary wave. In all cases considered, formulas for calculating the velocity of the wave and its profile were obtained. One of the important examples considered is the analysis of solitary waves in populations of the herd locust.