Frontiers in Mathematical Biology

A generation ago, research on the molecular basis of Ufe was confined to a small, though significant group of scientists. Their methods were experimental- and though many were trained in physics, only on the most infrequent of occasions did their investigations demand the exercise of mathematical methods beyond elementary statistics. But the elucidation of the DNA double helix, which was driving their research, had forever transformed biology to a conceptual science, and with that transformation it had set in motion an inevitable mathematical evolution.

In the second half of the twentieth century biology has progressed at breakneck speed. James Watson and Francis Crick in 1953 proposed the now famous double helical structure for DNA. This structure gave a physical model for how one DNA molecule can divide and become two identical molecules. On this point they wrote one of the most famous sentences of science: “It has not escaped our notice that the specific pairing we have postulated immediately suggests a possibly copying mechanism for the genetic material.” And that copying mechanism based on the adenine (A)-thymine (T) and guanine (G)-cytosine (C) base pairing turned out to be correct and is the foundation of molecular genetics. While about 100 years earlier Mendel gave an abstract model of inheritance, Watson and Crick gave a specific molecular model that can be studied and manipulated. The last 50 years of molecular biology has been in large part based on the Watson-Crick discovery. See Lewin (1990) and Alberts et al. (1983) for excellent general accounts of the subject.

How do cells move? There is no shortage of theories, but a definitive answer is still elusive. Here we will present some models for cell motions along with proposals for experiments to address the theories. We will restrict ourselves to the problem of cell protrusion, although the models apply to other motility phenomena as well.

Biological cells are widely regarded as the elementary units of life, insofar as they already comprise the principles of behavior that we associate with living organisms including ourselves: the adaptive ability to survive, i.e. to maintain a certain identity while being exposed to a changing environment. Obviously, nutrient or energy uptake and metabolism are the most important prerequisites for such a (cellular) household, but also important is persistence of other “essential” constituents and functional structures as a “unique” genome, a well-defined boundary and adequate sensory systems.

Embryogenesis depends on a series of processes which generate specific patterns at each stage of development. For example, gastrulation, chondrogenesis, formation of scale, feather and hair primordia all involve major symmetry breaking. These ubiquitous spatial pattern formation requirements depend on specific pattern generation mechanisms which are still unknown. They are the subject of much research both theoretical and experimental. In the case of integumental patterns, for example, we do not in general even know when in development the pattern is actually formed. This was the key question studied by Murray et al. (1990) in a recent theoretical and experimental paper on alligator (Alligator missippiensis) stripes.

This brief essay moves from a personal perspective of research in neurobiology to the identification of some important areas for future research.

In the late 1950’s Norbert Wiener became interested in the spectrum of human brain waves (Wiener 1958, 1961). Along with his medical collaborators, he made high-resolution electroencephalographic recordings from subjects who were awake but resting with their eyes closed. Under these conditions, the electroencephalogram shows conspicuous activity at frequencies around 10 Hz — the so-called alpha rhythm.

There still exists no quantitative understanding of ventricular fibrillation (VF), the main cause of sudden cardiac death among 300,000 Americans annually. It would be useful to have one, not only for engineering design of implantable defibrillators and clinical management of arrhythmias that threaten to become VF, but also for the satisfaction of understanding this aspect of the dynamics of excitable media and of the corresponding partial differential equations.

Jerne (1974) introduced the idea that the immune system is regulated by idiotypic interactions. When an antigen (Ag) is encountered by the immune system an immune response normally results, with the antibodies (Ab) directed against the antigen called Ab₁ or first level antibodies. Jerne suggested that the unique or idiotypic portions of these Ab₁ antibodies would act as antigens and induce an anti-idiotypic immune response resulting in the production of Ab₂ antibodies. These in turn could induce third level antibodies, etc., so that an entire network of interactions might be involved in the response.

About 25 years ago, a symposium, “Mathematical challenges to the neo-Darwinian interpretation of evolution,” was held at the Wistar Institute, and there was a hot debate between the mathematicians and evolutionists about whether or not the neo-Darwinian theory explains the accumulation of genetic information contained in genomes of higher organisms (Moorhead and Kaplan 1967). In particular, it was argued that the development of the amino acid sequence of any protein is improbable, if the sequence has been selected from all possible sequences via a random mutation and selection process. It was further argued that “the human genetic complement comprises about 109 nucleotides or about one nucleotide for each year since life appeared on earth. Because at some time or other there were no nucleotides, the average rate of accrual is about one nucleotide per year”, and that this vast accumulated genetic information can not be explained by the neo-Darwinian model.

Theoretical population genetics is surely a most unusual subject. At times it appears to have little connection with the parent subject on which it must depend, namely observational and experimental genetics, living an almost inbred life of its own. It cannot claim a status analogous to that of theoretical physics to justify doing this: the latter subject depends on very precise observational data and very precise models, so that if at times it proceeds ahead of observational physics on its own, it is on the basis of quite firm foundations. Theoretical population genetics, on the other hand, rests on often vague and sometimes contradictory foundations and too inbred a development of the theory soon leads to irrelevancy. What should be the methods of progress in population genetics theory, and how should these methods change with changes in mainstream genetics?

Recombination, including chromosomal segregation, shuffles together the genetic material carried by different members of a sexual species. This genetic mixing unties the evolutionary fate of alleles at one locus from the fate of alleles at neighboring loci and can increase the amount of genetic variation found within a population. In the process, however, recombination separates advantageous gene combinations, the very gene combinations that enabled the parents to survive and reproduce. Whether or not the adaptation of a population to an environment is more rapid in the presence of recombination, that is, whether or not recombination speeds up the evolutionary process, depends critically on the ways in which this process is modelled. As we shall see, the effect of recombination depends on the population size, the initial population composition, and the selection regime under consideration.

For most of the twentieth century, techniques for the biometric analysis of organic form fell into one of two incompatible styles. In the first, more indigenous style, a direct extension of techniques introduced into statistics by Galton, Pearson, and their heirs, conventional multivariate techniques were applied to a diverse roster of measures of single forms. The only algebraic structures involved were those of multivariate statistics, limited mainly to covariance matrices; no aspect of the geometric organization of the measures, or their biological rationale, was reflected in the method. Analyses of this mode led at best to path diagrams, not to sketches of typical organisms expressing the developmental or functional import of the coefficients computed.

Behavioral ecology means different things to different people (or even to the same people at different times). Often it is defined as “the survival value of behavior”. Alternatively, a focus on inclusive fitness often leads to the use of optimization methods. “Behavioural ecologists often preface their studies with statements such as ‘individuals are expected to behave so as to maximize their reproductive success’” (Krebs and Davies 1991, pg 3). It is, in fact, extremely unfortunate that the early emphasis in behavioral ecology on optimization deflected discussion of more important concepts (Mangel and Ludwig 1992). In particular, the focus became one of “testing the optimization model” (in which case any variation of behavior disproves the theory) rather than considering fitness consequences of different behaviors (see Mangel and Clark 1988, Mangel and Ludwig 1992).

Stochastic demography deals with populations whose vital rates change randomly over time, and is an extension of classical demography. Cohen (1979) and Tuljapurkar (1990) review the subject. Here I discuss key problems and prospects in applications of stochastic demography, focusing on life history questions and the role of singularities. My discussion is deliberately thin on references; you can trace them starting with the citations, especially the related paper by Orzack (1992), or write me.

A population can be thought of as having, at a particular time, a definite structure. Here “structure” is defined as a frequency distribution of characteristics (here and below we are not referring to relative but to absolute frequencies). Usually the structure changes, both on a population dynamical time scale and over evolutionary time. One of the things we want to understand is how the patterns of change are related to mechanisms. These mechanisms necessarily act on the individual level (i-level). Their reduction to population dynamical essentials amounts to a description of the way in which growth, reproduction, probability of dying and influence on the environment are determined by the state of the individual organism (i-state) and the condition of the environment (E-condition). Here the “environment” is thought to encompass all relevant abiotic factors as well as the local abundances of organisms of the same and other kinds (ranging from food to competitors, predators, and parasites). In this view, the natural representation of density dependence is as a feedback through the environment.

Population dynamics attempts to account for changes in the sizes of biological populations. This is a fundamental problem in biology that has occupied scientists’ attention since at least Aristotle. It is a problem of not only intrinsic interest, but of fundamental importance in other biological investigations as well. Models of population dynamics form the bases of models in ecology, genetics, theories of evolution, cell dynamics, epidemiology, resource management, bioeconomics, ecotoxicology, sociobiology and many other disciplines of the biological, medical, and environmental sciences. Given a biological population’s natural propensity for exponential growth and the finiteness of our natural world, at the heart of this problem is the fundamental problem of how population numbers are “regulated”, i.e. how they are kept from growing without bound.

It is hard to find animals in nature that do not aggregate for one reason or another. The details of such aggregations are important because they influence numerous fundamental processes like mate-finding, prey-detection, predator avoidance, and disease transmission. Yet, despite the near universality of aggregation and its profound consequences, biologists have only recently begun to probe its underlying mechanisms. In this chapter we review theoretical approaches to animal aggregation, concentrating on aggregations which are caused by social interactions. We emphasize methods and limitations, and suggest what we think are the most promising avenues for future research. For an earlier review article on dynamical aspects of animal grouping consult Okubo (1986).

Early studies of the dynamics of biological populations tended to look for stable equilibrium points in spatially homogeneous contexts. Although such models are occasionally ridiculed today (usually for transparently careerist reasons), they represent a sensible first step toward understanding how populations interact, and how they respond to natural or human-created changes. And greater dynamical complexity has been hinted at from the earliest studies: witness Lotka and Volterra’s [1] neutrally stable cycles, or Skellam’s [2] work on what today would be called competitive coexistence of metapopulations.

Food webs should be more closely integrated with other descriptions of community ecology. Recent technology should be exploited to observe food webs better. The effects of human interventions on food webs should be more carefully studied and better understood.

In 1880, Lorenzo Camerano, then a 24-year-old assistant in the laboratory of the Royal Zoological Museum of Torino, Italy, published a paper “On the equilibrium of living beings by means of reciprocal destruction” in the Acts of the Royal Academy of Sciences of Torino. This pioneering paper contains an early, perhaps the first, graphical representation of a food web as a network of groups of species linked by feeding relations. It also contains a vivid dynamic model of the consequences of trophic links for population dynamics. The model is based on the propagation of sound waves in an organ pipe. An English translation of Camerano’s paper follows this introductory essay.

The recent history of Zoology, starting some time before the middle of this century and still continuing, is principally characterized by many researches on both the utility and the damages to man due to animals. Applied zoology, in a word, was born in this period of development of the zoological sciences.

There is, in biology, a convenient if imperfect hierarchy, reflected in the organization of this volume: molecules to cells, cells to tissues, tissues to organisms, organisms to populations, populations to communities. The ecosystem concept (Tansley 1935) breaks the pattern. The ecosystem is not another step in the hierarchy, but rather the broadening of the community concept beyond the biological parts to include the physical environment, and especially the flows of materials and energy.

At the level of ecological populations and communities, modelers have usually followed the state variable approach, in which a separate state variable equation is written for each species population. This approach has been extended to account for age and size structure within populations by dividing a population into classes, each repesented by a density. Similarly, to account for the spatial patchiness of populations, populations have been modeled as sets of subpopulations connected by migration, with a separate variable to describe each subpopulation. Still, the basic nature of even these more complex models involves variables representing highly aggregated components of populations.

Modern theoretical ecology owes much to the modeling legacies of Vito Volterra, Alfred Lotka, and Georgii F. Gause. In 1949, Hutchinson and Deevey (1949) acknowledged their contribution by writing “Perhaps the most important theoretical development in general ecology has been the application of the logistic by Volterra, Gause, and Lotka to 2 species cases.” To this group we should add the name of C. S. Holling who, in the late 1950s (Holling, 1959), made important contributions to quantifying the rate at which consumers exploit resource populations in two species interactions.

The most pervasive image of an ecological system, both in popular and in scientific imagination, surely is that of a highly interacting collective of many parts, each influencing the others in strange, wonderful, and somewhat mysterious ways — an image not absurdly distant from the truth. Some of the most pressing practical problems faced by ecological science require an understanding of exactly how human intervention affects these complex systems, hence of how the systems themselves, the whole systems including all interactions, work. One need seek no further for examples than this morning’s newspaper, which chronicled the scientifically “baffling” collapse of the northwest Atlantic cod fishery, and the not-so-baffling decline of the St. Lawrence River beluga whale population, whose habitat could serve as a catalog of contemporary toxins (The Globe and Mail, Toronto, June 6, 1992).

The transmission of diseases, genetic characteristics, or cultural traits is influenced by many factors including the contact/social structure of the interacting subpopulation, that is, the social environment. Classical demography (see MacKendrick, 1926; Lotka, 1922; and Leslie, 1945) ignores social dynamics and usually concentrates on the birth and death processes of female populations under the assumption that they have reached a stable age distribution. They usually ignore the specific mating/contact structure of the population. The incorporation of mating structures or marriage functions, as they are commonly referred to in human demography, was pioneered by Kendall (1949) and Keyfitz (1949). However, despite the fact that their work was extended by Parlett (1972), Predrickson (1971), McFarland (1972), and Pollard (1973) two decades ago, their impact on demography, epidemiology, and population biology has been minimal.

One of the most important challenges facing ecologists over the next decades is to help with the conservation of endangered species and ecosystems. As has been recognized increasingly within ecology in general, and in conservation in particular, meeting these challenges will require including the role of spatial structure in the models that are used.

Mathematical models have become important tools in analyzing the spread and control of infectious diseases. Although chronic diseases such as cancer and heart disease receive more attention in developed countries, infectious diseases are the most important causes of suffering and mortality in developing countries. Recently, the human immunodeficiency virus (HIV), which is the etiological agent for acquired immunodeficiency syndrome (AIDS), has become an important sexually-transmitted disease throughout the world. Tuberculosis is again becoming a problem because drug-resistant strains have evolved. Understanding the transmission characteristics of infectious diseases in communities, regions and countries can lead to better approaches to decreasing the transmission of these diseases. Mathematical models are useful in building and testing theories, and in comparing, planning, implementing and evaluating various detection, prevention, therapy and control programs. See Hethcote and Van Ark [30] for a discussion of the purposes and limitations of epidemiological modeling.

Fisheries management provides an important example of the application of theoretical ideas to practical biological problems. The interest of mathematicians in the subject goes back at least as far as the fundamental work of Volterra (1928), and there is now a voluminous literature. In spite of this long history, certain crucial aspects of the problem have not been understood as well as they might. These involve statistical issues and the relationship between models and data. Two of these ideas can be stated in the form of paradoxes: 1.Management for sustained yield cannot be optimal.2.Effective management models cannot be realistic.

While there have been established guidelines in the United States for projecting the effects of a chemical toxicant on humans for many years, only recently have efforts been made to develop á set of guidelines for ecological risk assessment. Human risk assessment addresses the most completely studied species in existence. Homo sapiens generally lives in a relatively clean environment because the species is able to control its environment to a large extent. Antipodally, ecological risk assessment must be concerned with numerous (for practical purposes, essentially infinitely many) species, most of which have been only sparsely investigated, and which generally live in heterogeneous, polluted environments over which they have little control. The problems of ecological risk assessment are more difficult by many orders of magnitude than those of human risk assessment, which are certainly nontrivial.

The subject of health information is not primarily biomathematical. Medical and sociological facts together with computing and traditional descriptive statistics play a predominant role. It is, however, becoming increasingly clear that even under the conditions of developing countries, and in fact precisely because of them, modern mathematical-statistical methods, in particular epidemiologic ones, are going to be one of the main pillars of health information systems.

In the early 1950s a Soviet biochemist, Boris P. Belousov, was trying to develop a simple chemical model of the oxidation of organic molecules in living cells. Central to these pathways is the Krebs cycle, whereby organic acids are oxidized to CO₂ and H₂O. In aerobic organisms, oxygen is the oxidizing agent, and the reactions are catalyzed by enzymes and electron-transport proteins, many of which rely on iron ions (Fe2+/Fe3+) to move electrons around. In his testtube version of metabolism, Belousov used citric acid (one of the intermediates of the Krebs cycle) as an organic substrate, bromate ions (BrO3-) as oxidizing agent, and cerium ions as catalyst. Any chemist would expect the reaction to proceed monotonically to equilibrium, perhaps showing one visible sign of progress by changing from a colorless solution (cerium in the reduced state, Ce3+) to pale yellow (the oxidized state, Ce4+). So we can imagine Belousov’s surprise when his reaction mixture turned yellow then colorless, then yellow again and colorless, oscillating dozens of times between oxidized and reduced states (Fig. 1).

Our procedures for analyzing differential equations models in mathematical biology used to be straightforward, directed by simple, clear expectations. We set the right side equal to zero and solved to find the equilibria, not necessarily an easy job when the equations were nonlinear. We then linearized at each equilibrium and computed the eigenvalues. When all had negative real parts the equilibrium was stable, attracting solutions beginning at nearby points. Some systems, like predator-prey models, exhibited stable nonequilbrium behavior like limit cycles but the reasons for such apparently exceptional behavior were usually clear.

The problem of modelling a biological system should be revisited as an inverse problem. Namely:

given a set of observed properties exhibited by the system, the problem is the identification of a dynamical system which implies such properties.

The research literature on topics related to population models in which size structure is a dominant feature has experienced substantial growth during the past decade. This is due in part to the ever increasing ability of scientists to compute with models that describe fine detail of complex mechanisms. The wide availability of high performance computers (including vector and parallel machines) has encouraged development in a number of areas: modeling concepts, computational methods (including estimation techniques for nonconstant model parameters), and statistical methods for model evaluation and validation. Recent contributions involving parallel computations offer the potential for major advances in both theoretical modeling and applications to experimental design and data collection in the near future. The purpose of this note is to outline some of these recent developments and contributions and to give some indications as to their possible role in future research directions. Our discussions here will entail a somewhat personal perspective in that we shall not attempt to give a survey of all the recent literature. We cite literature (some of it still “in preparation” or “to appear”) as needed to illustrate the points we wish to make.