Outline of a mathematical approach to the cancer problem

The fact that the growth of every organism, as it develops through cell multiplication from the ovum, eventually ceases, makes plausible the hypothesis that each cell of an organism exerts some kind of inhibitory influence upon the growth and multiplication of every other cell through the medium of some factorj. Each cell, however, is considered to possess an amount of a “growth factor”g, the activity of which is inhibited byj. In the course of development, most cells differentiate. A number of them remains relatively undifferentiated, and those are used for repairs of natural tear and wear as well as for healing of wounds. With age either the number of those cells or their “potency” decreases, which accounts for a general decrease in the speed of wound healing. Due to statistical fluctuations, the amount of theg is not uniformly distributed amongst the cells. Some of them have accidentally a greater amount than others. As the number or potency of the undifferentiated cells decreases with age, so does their inhibitory influence upon other cells. Therefore, those cells which do have an accidental dental excess ofg-factor will begin to grow and multiply, producing neoplasms. Putting the above assumptions into mathematical form, it is possible to derive a relation which gives theprobability of cancer incidence as a function of age. The theoretical relation is compared to the rather inadequate empirical data and found in agreement with them. A relation for the speed of wound healing as a function of age is also derived, and also found in general agreement with the rather scant data. Several other conclusions are discussed and possibilities of their practical applications pointed out.     

Outline of a mathematical biology of leadership

Leadership, whether executive, political or any other type, is connected with the achievement of some goal by the social group through an appropriate organization of that group. From this point of view different leadership ranks in a group would be assigned to individuals according to their ability to organize the group for the purpose of reaching a specific goal. The situation is actually complicated by the circumstance that an individual may have the necessary ability but may not like the responsibility connected with the leadership, or vice versa. Also, he may not be interested in the goal. The suggested mathematical approach is to consider that the satisfaction of an individual is a function of his leadership rank, of the goal, and of several other parameters. If each individual tends to adjust his position in society so as to maximize his own satisfaction, this condition gives us the equations which determine the leadership rank of each individual. It is found that, in general, the rank of an individual depends not only on his ability, but on the abilities of all other individuals. The method enables us to calculate the distribution function of abilities among individuals of a given rank, and leads to results which allow, in principle, experimental verification.     

Outline of a possible mathematical approach to the problem of the effects of environment upon the incidence of some psychoneuroses

In previous papers we have studied the distribution of the differences of two desires or inner drives in a group of individuals, with a view to developing a theory of mass behavior. The same concept is now applied to the problem of incidence of neuroses. It is natural to consider that the incidence of neuroses in a group of individuals, which is characterized by a near equality of two opposite drives, will be particularly high. Applying the results of a previous paper (Bull. Math. Biophysics,18, 323–36, 1956) in which the distribution of certain types of opposite drives has been connected with the distribution of incomes, expressions are now derived which give the relative incidences of some types of neuroses for different income groups. The expressions derived hold only for certain types of assumptions and cannot, therefore, be directly applied to actual data. They suggest, however, new possibilities in experimental and observational approaches to the problem.     

Outline of a mathematical biosociology of beliefs and prejudices

Behavior based on acceptance on faith is considered as a result of a special conditioning, and is assumed to be mediated through a special center. Behavior based on rational critical analysis is mediated through a different center. The previously developed theory of two mutually exclusive behaviors is applied to the situation. Making simple assumptions about the development of the two centers with age, it is shown how equations can be derived which describe the percentage of individuals exhibiting one type of behavior or another as a function of time and of different biosociological parameters.     

Outline of a probabilistic approach to animal sociology: III

The probabilities of the emergence of the two kinds of social structure in a 3-bird flock (chain and cycle) are deduced under the assumption of certain biases acting on the social dynamics of the flock. In particular a bias against the reversal of peck order and a bias against encounters of individuals of disparate social rank are considered. Like-wise a distribution of an “inherent” fighting ability is considered which influences the outcomes of encounters. A functional relation is derived between the importance of this ability and the initial probability of a chain structure.     

Outline of a probabilistic approach to animal sociology: II

Under certain assumptions concerning the probabilities of “mutations,” i.e. changes of structure of bird societies, it is shown that the probability distribution for all possible structures of a society ofN individuals approaches a limit independent of the initial probability distribution. A formula for the limiting distribution is derived.     

Outline of a probabilistic approach to animal sociology: I

The non-transitive character of he peck right relation gives rise to different “social structures” in an aggregate of individuals. A method is developed for computing the respective probabilities of occurrence for each type of structure in small aggregates on the basis of random determination of peck right between each pair of individuals.     

A mathematical approach to cytoskeletal assembly

The cytoskeleton is a fundamental and important part of cell's structure, and is known to play a large role in controlling the shape, function, division, and motility of the cell. In recent years, the traditional biological and biophysical experimental work on the cytoskeleton has been enhanced by a variety of theoretical, physical and mathematical approaches. Many of these approaches have been developed in the traditional frameworks of physico-chemical and statistical mechanics or equilibrium thermodynamic principles. An alternative is to use kinetic modelling and couch the analysis in terms of differential equations which describe mean field properties of cytoskeletal networks or assemblies. This paper describes two such recent efforts. In the first part of the paper, a summary of work on the kinetics of polymerization, fragmentation, and dynamics of actin and polymers in the presence of gelsolin (which nulceates, fragments, and caps the filaments) is given. In the second part, some of the kinetic models aimed at elucidating the spatio-angular density distribution of actin filaments interacting via crosslinks is described. This model given insight into effects that govern the formation of clusters and bundles of actin filaments, and their spatial distribution. Received: 7 January 1998 / Revised version: 4 March 1998 / Accepted: 7 March 1998     

Mapping baroreceptor function to genome: a mathematical modeling approach

To gain information about the genetic basis of a complex disease such as hypertension, blood pressure averages are often obtained and used as phenotypes in genetic mapping studies. In contrast, direct measurements of physiological regulatory mechanisms are not often obtained, due in large part to the time and expense required. As a result, little information about the genetic basis of physiological controlling mechanisms is available. Such information is important for disease diagnosis and treatment. In this article, we use a mathematical model of blood pressure to derive phenotypes related to the baroreceptor reflex, a short-term controller of blood pressure. The phenotypes are then used in a quantitative trait loci (QTL) mapping study to identify a potential genetic basis of this controller.     

Some suggestions for a possible approach to a mathematical biophysics of mitosis

A tentative suggestion is made to interpret the phenomenon of mitosis as being due to coiling and uncoiling of long chain molecules, which are assumed to constitute the spindle. A possible quantitative approach is briefly outlined.     

A mathematical approach to multiple genetic relationships

A fundamental concept in the treatment of genetic relationships is that of gene identity which first was introduced by Cotterman (1940). Based on this notion several measures of relationship evolved such as the inbreeding coefficient, the coefficient of kinship, and the identity coefficients; by means of these quantities joint and conditional phenotype probabilities could be derived. This paper is an attempt at a general mathematical treatment of genetic relationships: Identity states are defined for any number of individuals, a method is given for the calculation of the corresponding identity coefficients by means of generalized coefficients of kinship, and applications are emphasized.     

Test-and-treat approach to HIV/AIDS: a primer for mathematical modeling

The public benefit of test-and-treat has induced a need to justify goodness for the public, and mathematical modeling studies have played a key role in designing and evaluating the test-and-treat strategy for controlling HIV/AIDS. Here we briefly and comprehensively review the essence of contemporary understanding of the test-and-treat policy through mathematical modeling approaches and identify key pitfalls that have been identified to date. While the decrease in HIV incidence is achieved with certain coverages of diagnosis, care and continued treatment, HIV prevalence is not necessarily decreased and sometimes the test-and-treat is accompanied by increased long-term cost of antiretroviral therapy (ART). To confront with the complexity of assessment on this policy, the elimination threshold or the effective reproduction number has been proposed for its use in determining the overall success to anticipate the eventual elimination. Since the publication of original model in 2009, key issues of test-and-treat modeling studies have been identified, including theoretical problems surrounding the sexual partnership network, heterogeneities in the transmission dynamics, and realistic issues of achieving and maintaining high treatment coverage in the most hard-to-reach populations. To explicitly design country-specific control policy, quantitative modeling approaches to each single setting with differing epidemiological context would require multi-disciplinary collaborations among clinicians, public health practitioners, laboratory technologists, epidemiologists and mathematical modelers.     

Contribution of a mathematical modelling approach to the understanding of the ovarian function

The biological meaning of folliculogenesis is to free fertilisable oocytes at the time of ovulation. We approached the study of the control of follicular development at the level of follicular granulosa cells, on the experimental as well as mathematical modelling grounds. We built a mathematical model allowing for the processes of proliferation, differentiation and apoptosis. State variables correspond to the numbers of cells undergoing these different processes, while control variables correspond to the cellular transition rates. The model results raised the notion of proliferative resources, which leads to consider the optimal management of these resources and has motivated the settling of an experiment investigating the changes in the growth fraction within the granulosa throughout terminal development. We are now investigating the way gonadotrophins, and especially FSH, operate on granulosa cells, in order to account for the hormonal control of the divergent commitment of granulosa cells towards either proliferation, differentiation or apoptosis. We are thus focusing on the dynamics of cAMP production, which appears to be a keypoint in FSH signal transduction.