Mathematical Topics in Population Biology, Morphogenesis and Neurosciences

Since life originated in the ancient sea, the interplay between the environmental fluid and organisms has become an inseparable part of physiology, ecology and evolution of organisms. Yet our understanding of the mechanisms and processes in the interaction remains irritatingly poor. An often hostile attitude between biologists and physicists/mathematicians is in part at fault.

When investigating the biological world we can concentrate on different levels of organization. For instance, we can look at individual organisms and try to understand their life cycle and the physiological processes that are essential for their functioning. Or we can look at populations of such organisms and try to understand or even predict how numbers change in the course of time.

I will demonstrate how a simple empirical regularity observed in the topology of real food webs may be used to mathematically deduce a necessary and sufficient constraint operating during system assembly. This differs from the classical bottom-up approach for studying ecosystem structure in that “plausible” assembly rules are not used to induce large scale patterns, but rather, a macroscopic regularity is used to deduce an assembly rule. This is an extension of results reported earlier on patterns in the graphical structure of food webs (Sugihara 1983, 1984).

The problems facing a foraging animal have been studied on the basis of a variety of mathematical models. MacArthur and Pianka (1966) used marginality arguments familiar in economics to study optimal patch selection and optimal prey selection within patches. Fretwell (1972) discussed the equilibrium distribution of foragers among patches. Charnov (1976) characterized the problem of moving from one patch to another in terms of a Marginal Value Theorem, also familiar in economics. Schoener (1971) modelled territorial behavior. The predictions of these models have been tested in numerous experiments, with varying degrees of agreement between theory and observation (Krebs et al 1983).

Models of population dynamics are based on the population size as the complete descriptor of the dynamic state. We can write this central assumption of traditional theory as follows: $$\frac{1}{N}\frac{{dN}}{{dt}} = f(E)$$ where N is the population size; $$\frac{1}{N}\frac{{dN}}{{dt}}$$ is the relative growth rate (average number of surviving offspring per parent per unit of time); f (E) is a function of the environment with the understanding that population size itself might be one of the environmental parameters.

One of the most fascinating challenges in ecology involves the statistical description of movement, and the understanding of population distributions in terms of the behavior of individuals. Knowledge of plant and animal dispersal patterns is fundamental to an understanding of basic and applied issues ranging from the evolution of life history traits to the spread of genetically engineered organisms. The spatial and temporal structure of environment, and its effects upon the movements of individuals, are central issues in ecological and evolutionary theory. Seed and pollen dispersal, together with germination of dormant seeds and released growth of understory plants, play important roles in secondary successional patterns of forest communities following disturbance. Dispersal is key to the maintenance of gene flow among populations, the dynamics of pest outbreaks, the recovery of disturbed areas, and the optimal spatial design of agricultural systems and natural reserves.

Since Fisher’s pioneering work (Fisher 1937), many studies on traveling waves in a growing population have been performed. The model proposed by Fisher consists of a diffusion equation with a logistic growth term: $${u_t} = d{u_{xx}} + \left( {\varepsilon - u} \right)u\quad for\quad x \in \left( { - \infty ,\infty } \right)$$. Here u(x,t) denotes the population density at position x and time t, and d and ε are diffusivity and intrinsic growth rate, respectively. It has been shown from this equation that, starting from a localized distribution, the population evolves into a propagating wave of constant speed, $$ 2\sqrt {\varepsilon d} $$.

In this presentation we discuss methods for inverse or parameter estimation problems which can be employed as quantitative modeling techniques in models for distributed (spatially, age, size, etc.) biological systems. In this context they may be useful in attempts to understand, elaborate on, or further refine details of specific mechanisms for dispersal, growth, interaction, etc. in wide classes of models. We have also used these techniques in a number of biologically related problems [1] such as bioturbation [12], [14], [15] and climatology [19]. In addition to an overview of ideas underlying these techniques, we shall present here brief discussions and some findings on two specific biological problems for which we are currently using them successfully.

Natural environments are heterogeneous in space and time. This heterogeneity favors the evolution of mechanisms such as dispersal of seeds or other propagules, because dispersal allows escape in space from locally unfavorable conditions and, on the average, exploitation of ones temporarily more favorable (Levin 1976, Motro 1982). Similarly, delayed germination may be advantageous because it allows seeds to avoid exposure to unfavorable conditions and, on the average, to exploit more favorable ones (Cohen 1966, Templeton and Levin 1979).

It has been suggested that, in some circumstances, predation may tend to increase species diversity in competitive communities, which is the so called predator-mediated coexistence hypothesis.

Many biological populations tend to aggregate in response to concentration gradients of a chemoattractant secreted by themselves. The present study is motivated by the aggregation observed in Blattella germanica. At properly high densities, B. germanica individuals grow faster than do isolated ones, and they aggregate so as to maintain such densities (Ishii [1969]). This prominent feature suggests that there is evidently a correlation between the growth rate and population density of individuals.

Most diploid organisms have two sexes, but some have three or more. For instance, among ciliates in which mating occurs by the contact of two diploid cells and the subsequent exchange of haploid genome, cells are grouped into several mating types (or sexes) so that mating occurs only between cells of different sexes (sonneborn, 1939). Stylonychia spp. have as many as 48 sexes (Ammermann, 1965). Other organisms with three or more sexes are fungi, in which mating occurs by the fusion of isogamous haploid gametes (Raper, 1966).

In statistical physics we study macroscopic properties of matter on the basis of constituent particles, and in theoretical population biology we study features of populations on the basis of behaviors of individuals or, more basically and generally, on the basis of properties of self-replicating entities such as genes or chromosomes. Let us refer to any object that we broadly regard as a unit of replication as a ‘replicon’, thereby extending the original meaning used by molecular geneticists. Each replicon has a definite genetic state and undergoes birth and death. Therefore, in addition to ‘attraction and repulsion’, interactions between replicons typically includes ‘attacking and helping’, which affects the birth and death of recipients. The particular mode of interaction depends on a replicon’s state. This state is inherited from its parent replicon, and we can therefore study what type of interaction is prevalent in a population by examining that population’s dynamics. This is simply the evolution of behavior by natural selection.

The neutralist-selectionist controversy persists with respect to the population genetical mechanism of molecular evolution. In this paper, we first show how we can explain, from a selectionist perspective, the fact that molecular evolution rates are smaller than total mutation rates. Next, using a statistical analysis of electrophoretic data on local differentiation, we show that, in most cases, stochastic selection is more responsible for local differentiation than is random drift.

Most predators utilize more than one prey species, and it is known that their diets do not directly reflect population densities of available prey species. We consider two prey species which live in two different patches and one predator which allocates its foraging (including searching and handling) activity between the two prey species according to their relative densities. Murdoch(1969) introduced the term “switching” to refer to the case in which the relative amount of prey in the predator’s diet increases more than proportionally to the relative prey density. Predatory switching has been postulated for predators ranging from protozoa to birds (reviewed by Murdoch & Oaten, 1975), and it may be a simple consequence of the predator’s searching behavior (May, 1977).

The diversity and beauty of the pigmentation patterns on the shells of snails and bivalved molluscs invites us to construct models to understand their formation. The similarity of patterns in unrelated species on the one hand and the diversity in closely related species on the other encourage the assumption that most of them are generated by a common mechanism.

In a plant, growth is localized in some special, often distal areas, called meristems. Cell divisions and cell expansion contribute to the elongation of the axes, and also produce infinitely appendages such as leaves, stipules, and branches which are arranged in regular patterns. We want to give here a possible explanation for pattern inception in meristems on the basis of organized tissue growth.

We present here a new theory of epithelial morphogenesis based on the assumption that the cytoplasmic machinery responsible for epithelial cell motility is fundamentally similar to that of freely migrating mesenchymal cells. The additional constraint imposed on epithelial cells is that their apical borders remain attached. This model is able to mimic all of the foldings and invaginations of the earlier apical constriction model (Odell, et al., 1981). Moreover, it provides a mechanism for epithelial cells to actively change neighbors while maintaining the apical seal. This is an essential feature of epithelial morphogenesis, and the application of the model to neural plate formation is discussed in detail in Jacobson, Odell & Oster (1985) and Jacobson. Odell, Oster & Cheng (1986).

The phenomena observed at each stage of the life cycle of cellular slime molds have been widely investigated as a simple system relevant to the studies of Chemotaxis of cell aggregation, regulatory cell differentiation and regenerative pattern formation. Here we shall discuss the cell distribution pattern in cellular slime molds Dictyostelium discoideum at the migrating slug stage, using a standard model of statistical physics.

In many developing multicellular systems the location of a cell within the aggregate determines its developmental fate. In such systems the orderly specialization of cell structure and function and the arrangement of cells into tissues and organs require mechanisms for the spatial and temporal control of cellular activity. The concept of positional information [22], which is based on the supposition that a cell in a developing system must know where it is relative to other cells in order to follow the appropriate developmental pathway, has provided the framework for many analyses of developing systems. An alternative hypothesis to such direct linkage between the location of a cell and its subsequent fate is to suppose that cellular differentiation occurs independently of the spatial location of the cell, and that spatial patterns of cell type result from other processes such as cell sorting which occur after cell differentiation has occurred. That sorting of different cell types in vitro can produce nonuniform spatial distributions has long been known, but until recently there has been little evidence that cells in vivo can be determined at random locations in a developing tissue. However, recent experimental evidence [15,16,19] indicates that this can occur in the slug stage of the cellular slime mold Dictyostelium discoideum. A mathematical model based on these recent observations has been developed and analyzed elsewhere [14], and here we shall discuss some of the predictions of that model and compare them with recent experimental evidence.

Pattern formation is one of the central issues in the study of developmental biology. Although its implications are diverse, it is related mainly to cell differentiation and morphogenesis. In this respect, cellular slime molds seem to be quite handy as a model for the aggregation process, various typical morphogenetic changes, and the formation and regulation of slug pattern, to name a few (Bonner, 1967; Loomis, 1982). This note treats the slug stage of the cellular slime mold Dictyostelium discoideum and proposes a simple mathematical model to explain the prestalk/prespore pattern formation through density dependent cell movement and differentiation. The model will have good regulation capability and generate size independent typical slug patterns.

The periodic generation of cyclic AMP (cAMP) pulses during aggregation of the slime mold Dictyostelium discoideum is one of the best-known examples of periodic behavior at the cellular level (Gerisch and Wick, 1975; Goldbeter and Caplan, 1976). We have recently analyzed a model for this phenomenon based on receptor modification (Goldbeter, Martiel and Decroly, 1984; Martiel and Goldbeter, 1984). In the course of this analysis, we found that in addition to simple periodic oscillations, the model is capable of more complex modes of oscillatory behavior, namely, bursting and chaos (Martiel and Goldbeter, 1985) as well as birhythmicity (Goldbeter and Martiel, 1985). The latter phenomenon refers to the coexistence between two simultaneously stable periodic regimes (Decroly and Goldbeter, 1982). It is the rhythmic counterpart of the more common mode of bistability in which two stable steady states coexist in the same conditions.

Theoretical studies of the brain have concentrated mainly on microscopic phenomena at the membrane and cellular levels. At the macroscopic level, there are “gross” brain phenomena, e.g., some kinds of epilepsy, that occur on space scales which are large compared to cell size and on time scales which are long compared to time constants associated with molecular and cellular events. Because of the large number of nerve and glial cells involved (on the order of 1011 (Hubel, 1979)), it is difficult to construct a mathematical model of such phenomena by starting with individual cells, connecting them into small circuits, and then combining these circuits into a coherent model for a particular brain structure.

Burst activity is characterized by slowly alternating phases of near steady state behavior and trains of rapid spike-like oscillations; examples of bursting patterns are shown in Fig. 2. These two phases have been called the silent and active phases respectively [2], In the case of electrical activity of biological membrane systems the slow time scale of bursting is on the order of tens of seconds while the spikes have millisecond time scales. In our study of several specific models for burst activity we have identified a number of different mechanisms for burst generation (which are characteristic of classes of models). We will describe qualitatively some of these mechanisms by way of the schematic diagrams in Fig. 1.

The brain can self-organize its structure based on environmental information. More specifically, when a set of signals is applied repeatedly to a nerve system, for each signal in the set, the system forms by self-organization sets of representative cells that are excited in response to particular signals but are not excited by any other signals. Such a representation may be regarded as a model of the outer world formed in the nervous system. This is one simple aspect of self-organization taking place in the brain. Physiologists have so far found hypercolumnar and microcolumnar structures in the primary visual cortex in which orientation-detecting cells, i.e., cells representative of lines of various orientations, are formed and fixed by self-organization. Moreover, hypercolumns are arranged retinotopically, and orientation-detecting cells are arranged in each hypercolumn in the order of preferable orientations. Physiologists have also found in various parts of the cerebrum cells which are responsive to specific shapes of objects, specific types of motions, and, in particular, faces of men or monkeys.

The complex patterns of electroencephalograms(EEGs) have made it very difficult for physiologists and pathologists to interpret the relationships to brain functions. A possible way to approach this problem is to study the dynamic behaviors of a mathematical model of large scale neuron network based on the fundamental electrical properties of neurons and their interrelationships, as done by J.D. Cowan and his co-workers (Cowan 1974; Ermentrout and Cowan 1979, 1980). They proposed two coupled non-linear integro-differential equations, comprising two types of neurons (excitatory and inhibitory), and analysed them by bifurcation theory. Because of the statistical nature of their treatment, all complexities of neurobiological details, such as the plasticities of neurons and synapses, may be taken into consideration, but their results still are very difficult to compare with real EEG patterns, especially those of conscious human beings, which display highly irregular, quasi-random forms. These kinds of behaviors may also be found in general dynamic systems and are called chaotic. This paper is an investigation of the problems concerning the relations between chaotic behaviors of neural network dynamics and EEG patterns, i.e., brain functions.

The term “self-organization” is one of the most popular concepts of modern science. According to the synergetic approach, self-organizing systems are systems which can acquire macroscopic spatial, temporal, or spatio-temporal structures by means of internal processes (Haken 1985). Since dynamic behaviour of hierarchically arranged structures might be interpreted in terms of “self-organization”, the nervous system can be considered as a prototype of self-organizing systems both conceptually (Szentágothai 1984, 1985) and mathematically (Amari 1983).

A major endeavor in modern biology is the effort to unravel the molecular mechanisms of sensory signalling. Here I shall briefly review some of the work I have been doing in this field together with several collaborators. The research to be described has centered in two areas, models for the control of neurotransmitter release and a new molecular model for adaptation at the receptor level.

There are problems in population biology and in neurobiology that require the evaluation of first-passage-time probability density functions (p.d.f.) for diffusion or for Gaussian processes. For instance, the fixation of a gene may be viewed as the process leading the frequency of that gene to attain the value one for the first time; the extinction time of a population can be interpreted as the time when first the population size shrinks to some preassigned small level; the firing of a neuron can at times be modeled as the first crossing through a critical threshold value by the random process depicting the time course of the membrane potential. The literature on this subject is so vast that it cannot and need not be referenced here. It may suffice to mention that an explicit reference to these problems can be found in Ricciardi (1977) and in the literature quoted therein. Instead, we would like to stress that the determination of first-passage-time densities and of its features is in general a formidable task. It is fair to claim that so far only computational methods have been employed, which require the use of large scale computers, are extremely time consuming and hence not quite suitable to carry out the systematic analyses imposed by the nature of the biologically interesting problems that usually involve numerous parameters to be specified by best fitting procedures (Favella et al. 1982) Ricciardi and Sato, 1983a; Ricciardi et al., 1983) .

The purpose of the present paper is to demonstrate the utility of what we call an active rotator model. We will study some dynamical processes for which the interaction of infinitely many degrees of freedom is essential. The processes with which we are concerned take place in a spatially distributed system that is more or less homogeneous in its constitution; the system consists of many similar units, and each unit is assumed to be functionally active in some sense. More specifically, it is assumed that each element (which will ordinarily be called a cell) represents either a limit-cycle oscillator or an excitable functional unit. Since, as is well known, oscillation and excitability are dynamical modes commonly shared by cellular membrane and biochemical reactions, the present subject is relevant to many physiological processes in living organisms. Examples of what are considered to be populations of such active elements include the population of pacemaker cells in the sino-atrial node of a mammal’s heart and neural nets in the cerebral cortex. Our system need not be restricted to discrete populations of well-defined units; even continuous or almost continuous tissues such as the smooth muscle of the small intestine or other digestive organs may also comprise possible examples.

We discuss three problems related to the contraction of striated muscle. These problems involve the matching of the force to the load, the longitudinal stability of a muscle fibre, and the harnessing of the force developed by the contractile elements.