Coquaternions, Metric Invariants of Biologic Systems and Malignant Transformations

Abstract

Different hypotheses of carcinogenesis have been proposed based on local genetic factors and physiologic mechanisms. It is assumed that changes in the metric invariants of a biologic system (BS) determine the general mechanisms of cancer development. Numerous pieces of data demonstrate the existence of three invariant feedback patterns of BS: negative feedback (NFB), positive feedback (PFB) and reciprocal links (RL). These base patterns represent basis elements of a Lie algebra sl(2,R) and an imaginary part of coquaternion. Considering coquaternion as a model of a functional core of a BS, in this work a new geometric approach has been introduced. Based on this approach, conditions of the system are identified with the points of three families of hypersurfaces in R42: hyperboloids of one sheet, hyperboloids of two sheets and double cones. The obtained results also demonstrated the correspondence of an indefinite metric of coquaternion quadratic form with negative and positive entropy contributions of the base elements to the energy level of the system. From that, it can be further concluded that the anabolic states of the system will correspond to the points of a hyperboloid of one sheet, whereas catabolic conditions correspond to the points of a hyperboloid of two sheets. Equilibrium states will lie in a double cone. Physiologically anabolic and catabolic states dominate intermittently oscillating around the equilibrium. Deterioration of base elements increases positive entropy and causes domination of catabolic states, which is the main metabolic determinant of cancer. Based on these observations and the geometric representation of a BS’s behavior, it was shown that conditions related to cancer metabolic malfunction will have a tendency to remain inside the double cone.