PHYSICO‐MATHEMATICAL METHODS IN BIOLOGICAL SCIENCES

Summary1. The fundamental method of exact physico‐mathematical sciences, that of abstraction and of a systematic study of abstract, idealised cases, is outlined and the timeliness of its application to biology indicated.2. This method is applicable to the study of general biology. The most general property of all cells being metabolism, the mathematical study of metabolising systems in general is indicated.First it is shown that regardless of the special character of metabolic reactions in and around any metabolising system, the concentrations of the various substances involved in metabolism is not uniform, the non‐uniformities being determined by the rates of reactions, size and shape of the system, etc.Making use of the general physical laws connecting the concentration of a dissolved substance with the osmotic pressure, the conclusion is reached that in and around any metabolising system the distribution of osmotic pressure is not uniform.Applying next a fundamental theorem of mechanics, we find that therefore any metabolising system is the seat of mechanical forces, the distribution of which is determined by the rate and type of reactions and other factors. A further mathematical study of the effects of non‐uniformities of concentration shows also that the osmotic pressure is not the only factor that produces mechanical forces. Inter‐molecular attractions and repulsions also result, in cases of non‐uniform concentration, in mechanical forces acting on each element of volume of the system.A closer consideration of the system of forces thus produced by metabolism shows that, for substances produced in the cell and diffusing outwards, these forces are generally also directed outwards. One of their effects is the tendency to expand the system, and to contribute to its growth. For substances diffusing into the cell and consumed there, the forces have in general the opposite direction and inhibit the growth of the system.A detailed mathematical investigation of the other effects of those forces shows that, for the case of produced substances, those forces cause a spontaneous division of the system when the latter exceeds a critical size. Calculations of this size gives values identical with the average size of actual cells. The forces due to consumed substances inhibit spontaneous division.In a system which, like ah actual cell, produces and consumes a great number of substances, the effect will depend on which type of forces prevail. Calculations show that forces produced by reactions connected with cell respiration considerably exceed the forces due to all other reactions. In the first approximation therefore only respiratory reactions may be considered. Mathematical analysis shows that when oxydation of sugar is complete and no appreciable amount of lactic acid is formed, the forces inhibiting growth and division prevail. When glycolysis is strong, the forces which produce division and accelerate growth prevail. This is pointed out to be in agreement with O. Warburg's findings that abnormally growing and dividing tumor cells have an abnormally high glycolytic coefficient.A further study of other possible effects of the forces produced by metabolism shows that they also will in general affect the permeability of the cell. Since the forces exist only as long as the cell metabolises, the death of the cell must result in sudden change of permeability, as is actually the case. The study of still more complex cases shows that the cell may possess two configurations of equilibrium. One is characterised by relatively low permeability and low glycolysis, hence by low rate of growth and multiplication. The other is characterised by a higher permeability and high glycolysis, hence rapid growth and multiplication. The cell, in such a case, can be brought irreversibly from the first configuration into the second, by temporary asphyxiation. This again is in agreement with Warburg's experiments on production of tumor‐like growth by asphyxiation.3. The non‐uniformities of concentrations, and therefore the forces, are present not only within the cell, but also outside it. This results in forces of repulsion and attraction between cells. Cells in which the dividing forces prevail usually repel each other. Cells in which the inhibiting forces prevail, attract. The mathematical theory of the configurations assumed by cellular aggregates under the influence of such forces indicates a way to a physico‐mathematical theory of organic forms of metazoans.4. A physico‐mathematical theory of nerve excitation and nerve conduction accounts for a number of empirical data. The generalisation of the ionic theory of excitation to the case of two types of ions, one exciting, the other inhibiting, gives a natural explanation for the excitation at the anode on opening a constant current, for the non‐excitability by slowly rising currents, and for various electrotonic phenomena. A formula is derived for the velocity of nervous conduction, which is verified by experimental data.5. The generalisation of the above results and their application to the central nervous system opens the way to a physico‐mathematical theory of brain activity.