Functional Dimensions of Biomolecules

Functional Dimensions of Biomolecules

DOI: 10.4018/978-1-5225-9651-6.ch006
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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.
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Higher Dimension Of Polyatomic Molecules From Elements Of Live As A Result Of The Interaction Of The Electron Orbitals Of Atoms In A Molecules

The most common in biomolecules is a carbon atom, the main role of which to be binding in the center of biomolecules. Consider, for example, methane molecule CH4. The carbon atom in this molecule binds around four hydrogen atoms. Geometrically, this molecule is a tetrahedron, whose vertices are located of the hydrogen atoms, and in the center is carbon atom.

The carbon atom in the center of the methane molecule has the valence electrons978-1-5225-9651-6.ch006.m01. Valence electron orbitals of carbon atoms and hydrogen atoms 1s overlap and form four hybrid orbitals978-1-5225-9651-6.ch006.m02, directed from the carbon atom to the hydrogen atoms (Gray, 1965). Let the distance from hydrogen atoms to carbon atoms is taken as unity, for the origin of coordinates to take the carbon atom, and the directions hybrid orbitals send on four coordinates x, y, z, t. Then the coordinates of the hydrogen atoms equal to (0, 0, 0, 1), (0, 0, 1, 0), (1, 0, 0, 0), (0, 1, 0, 0), and the carbon atom coordinates equal to (0, 0, 0, 0). So one have the integer coordinates of vertex in the four -dimensional space (Figure 1).

Figure 1.

The methane molecule (CH4)


This is consistent with the evidence of four - dimensional convex hull of the methane molecule on the Euler – Poincare equation (1) in Chapter 5. It is easy to see that the body in Figure 1, seen in the four - dimensional space, convex, because its edges belong to the body and enter into his boundary complex (Grunboum, 1967). Polytope in Figure 1 is a 4 - simplex, since each vertex of the polytope associated edges with all the other vertices of this polytope (Zhizhin, 2014).

If in the methane molecule a hydrogen atom replaced by a hydroxyl group - OH, then can to get the simplest alcohol - methanol. If the hydroxyl group considered as the vertex of the polytope, then the dimension of this molecule will also be equal to 4. If each atom of the molecule of methanol is considered the vertex of the polytope, then connecting each vertex to all other vertices edges, it turns out that it is equal to the dimension of the polytope to 5 and one have 5 - simplex. However, here must remember that the accession of the hydroxyl group does not change the hybridization of the carbon atom, as the binding site as the place of one hydrogen atom took one oxygen atom of the hydroxyl group. Therefore, as a separate vertex in methanol molecule should take hydroxyl group entirely. Then the dimension of the methanol molecules is equal to 4 (Zhizhin, 2017).

In the biomolecules can find a lot of examples of molecules or ions in the form of a tetrahedron with the center (978-1-5225-9651-6.ch006.m03, 978-1-5225-9651-6.ch006.m04, etc.). All of them have dimension 4. If the binding site appears d - element it is formed around the coordination sphere ligands with more than 4 of the amount due to of d - orbitals of the element. One can show that in this case the dimension of the molecule is equal to the number of hybrid electron orbitals directed from the center to the ligands. The ligand may act as no individual atoms or ions, and some functional groups, which may be regarded as corresponding vertices of the polytope. This is consistent with the need to describe more convenient biomolecules, molecular structures consisting of different complexity. Therefore, the dimension of the group of atoms in biomolecules, one call a functional dimension. In addition, when such descriptions of specific dimensions one will not be considered distances between atoms in molecules. Therefore, a certain dimension of the molecules so called topological dimension.

Key Terms in this Chapter

Branching of the Chain of the D-Glucose Molecule: The formation of branches in a chain of carbon atoms in the molecule of a – D -glucose. Such branches in a chain of the carbon atoms of the molecule ß – D -glucose are impossible. This follows from the representation of glucose molecules in the form of a polytope of higher dimension.

Spiral Peptide Chain: The formation of a spiral chain of protein molecules, as a consequence of the higher dimension of protein molecules.

First Coordination Sphere of Fe-Porphyrin: A set of nitrogen atoms bound by a covalent bond to an iron atom. The dimension of the coordination sphere upon addition of the oxygen atom increases from 5 to 6.

Hybridization of Electronic Orbitals: This is the interaction of the electronic orbitals of atoms entering the molecule, leading under certain conditions to the formation of higher dimensionality of molecules.

Functional (Topological) Dimension of a Molecule: The dimension of a convex polytope, as a model of a molecule, at the vertices of which not only individual atoms but also functional groups of the molecule can be located.

Enantiomorphism (Chirality) of Biomolecules: The possibility of changing the mutual arrangement of hydrogen atoms and hydroxyl groups in biomolecules (polytopes of higher dimension), leading to a change in their properties.

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