Mathematical theory of biological periodicities II. Effect of active transport

A previous study (Bull. Math. Biophysics,30, 735–749) is generalized to the case of active transport, which acts together in general with ordinary diffusion. The basic results obtained are the same except for an additional important conclusion. In principle it is possible to obtain sustained oscillations even when the secretions of the different glands do not affect the rates of formation or decay of each other at all, but affect the “molecular pumps,” which are responsible for the active transports in various parts of the system. Thus no biochemical interactions need necessarily take place between then-metabolites to make sustained oscillations possible in principle. This is an addition to a previous finding (Bull. Math. Biophysics,30, 751–760) that due to effects of the secreted hormones on target organs, non-linearity of biochemical interactions is not needed for production of sustained oscillations.     

Note on biological periodicities

The approximation method, recently developed by N. Rashevsky, is applied to a problem of periodic cell reactions, studied exactly in a previous publication. The fundamental features of that previous investigation are thus obtained in a rather simple way.     

A suggestion of a new approach to the theory of some biological periodicities

Previous studies of L. Danziger and G. Elmergreen (Bull. Math. Biophysics,16, 15–21, 1954;18, 1–13, 1956) of possible biochemical periodicities in organisms assumed non-linear biochemical interaction between different metabolites, because linear systems do not lead to undamped ocsillations. They treated homogeneous systems. Later N. Rashevsky generalized their results to a more realistic case where the non-homogeneity due to the histological structure is considered. (Some Medical Aspects of Mathematical Biology, Springfield, Illinois: Charles C. Thomas, Publisher, 1964;Bull. Math. Biophysics,29, 389–393, 1967.) As long as the histological structure remains constant, the existence of sustained periodicities requires the assumption of non-linearity of biochemical interactions. If, however, the secretions of an endocrine gland affect the histological structure of the target organ, notably as in the menstrual cycle, and if there is a feed-back, the equations become non-linear and may admit sustained periodic solutions even if the purely biochemical interactions are linear.     

Mathematical theory of the possible role of intercellular fluid and of vascularization on physiological periodicities

In continuation of previous work (Rashevsky,Some Medical Aspects of Mathematical Biology, Springfield, Ill.: Charles C. Thomas, 1964, Chap. 23 and Appendix 14), the study of the effects of the physical parameters of the cells of endocrine glands on the onset of sustained periodical oscillations in the interaction between the anterior pituitary and the thyroid hormones is generalized to include the possible effect of the intercellular fluid and of the degree of vascularization. Some conclusions of the previous study remain valid although some modifications must be made. A decreased relative volume of the intercellular fluid and an increased vascularization favor the conditions for sustained oscillations. The permeability of the cells and the permeability of the capillaries appear explicitly in the expressions which show the conditions for sustained periodicities.     

A note on biological periodicities

The approximation method for solving diffusion problems is generalized to enable computation not only of the fundamental frequency, but also of higher harmonics in certain types of periodic solutions.     

Mathematical biological of social behavior: II

A group of individuals is considered in which each individual has tendencies to exhibit one or another of two mutually exclusive behaviors. Neurobiophysically this may be described in terms of Landahl's reciprocally inhibited parallel reaction chains. The spontaneous excitations ε₁ and ε₂ at the central connections of each chain are a measure of the “natural” tendency of the individual toward one or the other of the two behaviors. According to equations derived by H. D. Landahl, the probability of one or the other behavior is determined by the difference ε₁ − ε₂. A population of individuals is considered in which ε₁ − ε₂ is distributed in some continuous way, and therefore in which the probability of a given behavior is distributed continuously between 0 and 1. The effect of other individuals exhibiting a given behavior is to increase the corresponding ε of the individual. Thus behavior of others affects the probability for a given behavior of each individual. It is shown that the equations describing the behavior of the population on the basis of this neurobiophysical picture reduce in the first approximation to the differential equations which were postulated by the author in his previous work on social behavior.     

Mathematical analysis of Paleodictyon: a graph theory approach

The graphoglyptid ichnogenus Paleodictyon has been alternatively interpreted as a foraging or farming trace; as a subsurface burrow for the habitation of one or more unknown organisms; the remains of a xenophyophore; and as the result of modular growth of an unknown organism. Graph theory and analysis of the geometry of the regular ichnospecies suggests that if the elements of Paleodictyon are interpreted as tunnels, then they are of extraordinary length relative to the size of any likely solitary tracemaker. In addition, because each vertex of the mesh is of degree three, any possible path through mesh requires revisiting in order to travel through the entire network; this makes the minimum path length even longer. These results suggest that it is unlikely that Paleodictyon is the result of subsurface burrowing.     

Mathematical theory of chemical synaptic transmission

Mathematical theory of chemical synaptic transmission is suggested in which the modes of operation of chemical synapses are given as consequencies of some fundamental theoretical principles presented in the form of systems of quantum and macroscopic postulates. These postulates establish transmitter transfer rules between 3 component parts — cytoplasmic, vesicular and external pools of neurotransmitter. The main features of the transfers are determined by special properties of the dividing membranes (synaptic and vesicle) which show high selectivity towards the direction of the transmitter quantum transfer. The formulation of a previously unknown effect of transmitter quantum transfer from the vesicular pool into the cytoplasmic one is introduced: it is postulated that each arriving presynaptic impulse not only releases a constant fraction of the current contents of the cytoplasmic pool into the synaptic cleft (external pool), but also realizes practically simultaneous transmitter transfer from the vesicular pool into the cytoplasmic one. Zone structure of the vesicular pool is postulated. In accordance with basic equations of the theory a nonlinear control system (dynamic synaptic modulator — DYSYM) of transmitter release from the terminal is constructed.Depending on the parameters relation two types of synapses are classified — those with rapid and slow demobilization. Analytical dependencies of the transmitter pools sizes on the stimulation frequency are introduced. By fitting the frequency dependencies to the empirical data model parameters are determined corresponding to a set of experimentally studied synaptic junctions. Different aspects of the chemical synapse behaviour under the influence of presynaptic stimulation are simulated.     

Mathematical theory of periodic relapsing catatonia

Two first-order, non-linear differential equations are presented to describe the interaction of the thyroid and pituitary glands and to explain some of the phenomena of periodic relapsing catatonia. Topographical study of these equations shows a mechanism in which either random fluctuations of hormone levels or system stability may occur. The variations in hormone concentrations are assumed to be caused by system malfunction, while stability is shown to result from the administration of exogenous thyroid which, if supplied at an optimum rate, suppresses the output of both glands. The system, under treatment, shows a zero level of thyrotropin and a thyroid concentration which is larger than the equilibrium level in the untreated system. The theory is consistent with what is known of the course and proper treament of periodic relapsing catatonia. Read before the American Chemical Society, Chicago, Ill., September 11, 1953.     

Mathematical theory of motivation interactions of two individuals: I

The behavior of two individuals is considered as consisting of an increase or a decrease of productive output. Motivation for increase is the derivative of a “satisfaction function”. This is an algebraic sum of the well-known Thurstone satisfaction curve and another essentially negative quantity, which is a product of a “reluctance parameter” and the “effort”. Each individual attempts to maximize his own total satisfaction. The resulting behavior is examined under a variety of conditions; namely, 1) equal sharing of produced without prescribed sharing of effort; 2) various contracts prescribing the sharing of effort; 3) situations in which one individual is more aware of the underlying motivations than the other. It is these latter situations which under the simplest assumptions of equal sharing, without prescribed sharing of effort, lead to parasitism, i.e. total cessation of effort on the part of one individual. This happens when one individual becomes aware of the other'sautomatic adjustment of his effort so as to bring about a total optimum output, which is a constant. Parasitism is prevented by various forms of contracts in which either the effort necessary for the total optimum output is shared according to a prescribed ratio or the effort of one individual is fixeda priori as a function of the effort of the other. In the latter case the respective efforts become a function of a single variable, and each of the satisfaction functions is maximized by a particular value or values of this variable. In general, these critical values do not coincide for the two satisfaction functions. The problem of finding forms of contract which will result in identical critical (maximizing) values of the variable for both satisfaction functions leads to a functional equation.     

Mathematical theory of motivation interactions of two individuals: II

The behavior of two individuals, consisting of effort which results in output, is considered to be determined by a satisfaction function which depends on remuneration (receiving part of the output) and on the effort expended. The total output of the two individuals is not additive, that is, together they produce in general more than separately. Each individual behaves in a way which he considers will maximize his satisfaction function. Conditions are deduced for a certain relative equilibrium and for the stability of this equilibrium, i.e., conditions under which it will not “pay” the individual to decrease his efforts. In the absence of such conditions “exploitation” occurs which may or may not lead to total parasitism. Some forms of the inverse problem are considered, where the form of behavior is given and forms of the satisfaction function are deduced which lead to it.     

生态位理论研究中的数学方法

各种不同的生态位参数测定公式,从不同的侧面揭示物种之间及物种的个体之间的结构关系;多元统计方法的引用,为H-生态位的分析提供了合适的数学模型;运用集合论方法研究群落生态位,为深入探讨群落中物种间的竞争、共存及互作机制提供更多的信息。     

Principles of a new biological theory: A summary

This is a review of the present state of the author's efforts, extending over many years, to construct a formal biological theory that is entirely compatible with quantum mechanics. We point out that the correspondence between detailed realizations on the molecular level and the characteristics of a class or organisms is exceedingly multivalued with respect to the former. Biological theory deals with the process of selection of actual biological states among the very large manifold of possible ones. This may also be expressed by saying that biological theory deals with (biologically) necessary conditions but not with sufficient ones. Conditions that are both necessary and sufficient pertain to physical science (molecular biology). In conformity with a well-known theorem of von Neumann, the selective process (being partly non-mechanistic) is not assumed controlled by simple mathematical rules. The selection from a very large reservoir of possible states is here described as creativity of the organism. The ordinary process of hereditary reproduction may then be represented as creativity with constraints, where the constraints (partly non-mechanistic but compatible with quantum mechanics) are such that the progeny tends to resemble the progenitors. This scheme differs from the usual view of heredity as purely mechanistic molecular replication with possible small errors. In the new scheme a gene appears as an operative symbol which functions as the releaser of a creative process.     

Co-operative response of chemically excitable membrane: I. Formulation: Unified theory of co-operativity

The lattice-model of Changeux, Thiery, Tung & Kittel (1966) was extended in order to examine the co-operative response of chemically excitable membrane and the exact mathematical correspondence to the Ising model was shown. In this model, two conformational states S and R with different affinities for the ligand are assumed to be accessible to each protomer, which is interacting with the nearest-neighbor protomers. The model is applicable to any kind of symmetrically interacting system consisting of oligomers and lattices and is an extension of previously proposed models of allosteric protein. It includes the model of Monod, Wyman, & Changeux (1965) and that of Koshland, Némethy & Filmer (1966) as the extreme cases of the oligomer. By assuming that a state-transition from S to R in a protomer is accompanied by a unit increase in conductance, the characteristics of dose-response curves of chemically excitable membrane are examined. The Hill's coefficient n_H of dose-response curve, the measure of the co-operativity, is shown to be proportional to the square of the mean fluctuation of the state function, the fraction of protomers in R state.