A note on Rashevsky's theorem about point-bases in topological biology

A graphG may have more than one point-baseB _G. In a primordial graphP of Rashevsky's (1954) TransformationT, some of the pointbases may consist of nonspecialized points only, and some other pointbases may contains specialized points. In this case, Rashevsky's Theorem (1955a) on point-bases may not hold. The Theorem is certainly true ifall point-bases ofP consists of nonspecialized points. A rigorous proof is given. Some results are derived for the more general case, when point-bases include both kinds of points. A general expression for the point-base ratio of the transformed graphP(T) is obtained. It is shown that with some biologically plausible assumptions Rashevsky's interpretation of the point-base ratio and his conclusions are still true. A few simple Theorems on point-bases of graphs are included in this work.     

Topological biology: A note on Rashevsky's transformation T.

In the bio-topological transformation between graphs denoted by (T ⁽¹⁾ X) N. Rashevsky (Bull. Math. Biophysics,18, 173–88, 1956) considers the number of fundamental sets which (a) have only one specialized point as source (and no other sources), (b) have no points in common (are “disjoined”); he proves that this number is an invariant of the transformation. In this note we show that Rashevsky's Theorem can be extended as follows:The number of fundamental sets of the first category is an invariant of the transformation. We must, however, count the subsidiary points of the transformed graph as specialized points. We recall that fundamental sets of the first category are those whose sources consist of specialized points only (Trucco,Bull. Math. Biophysics,18, 65–85, 1956). But in this modified version of the Theorem the fundamental sets may have more than one source and need not be disjoined.     

Note on the growth of bacterial populations

In this note the explicit solution is given to an equation, suggested by C. N. Hinshelwood (1946), describing the growth of a bacterial population under the assumption that toxic products are a limiting factor. The behavior of the culture as a function of time and the parameters (initial number, rate of growth, and rate of production of toxic substance) is discussed. Public Health Service Research Fellow of the National Cancer Institute, National Institutes of Health, Federal Security Agency.     

Augmented growth equation for cell wall expansion

The Growth Equation representing the relative rate of irreversible wall expansion is augmented with an elastic expansion component. Some of the utility of this augmented Growth Equation is demonstrated through selected applications.     

A cellular growth model for root tips

Growth of the root tip is modeled using a one-dimensional string of cells. Each cell is characterized by three distinct phases, division, elongation-only or maturity. In this model two hypothetical phytohormones, one produced at the root tip and the other at the shoot, determine the behavior of the cell, and therefore the growth of the entire tip. While the division rate is taken to be a step function of the string coordinate, the growth rate of each cell is assumed to be piecewise linear and composed of linear functions of cell length. Thereafter, suitable operators for the calculation of the velocity and relative growth rate distributions are given. The results of the model are finally compared to measurements of Arabidopsis thaliana, Nicotiana tabacum and Pisum sativum roots.     

A delay differential equation model for tumor growth

We present a competition model of tumor growth that includes the immune system response and a cycle-phase-specific drug. The model considers three populations: Immune system, population of tumor cells during interphase and population of tumor during mitosis. Delay differential equations are used to model the system to take into account the phases of the cell cycle. We analyze the stability of the system and prove a theorem based on the argument principle to determine the stability of a fixed point and show that the stability may depend on the delay. We show theoretically and through numerical simulations that periodic solutions may arise through Hopf Bifurcations.Send offprint requests to:Minaya Villasana     

Simulation experiments with a model for cellular growth

Computer models have been used by various authors to simulate both the growth of normal cellular tissue and the development of cancerous cells within normal tissue. As these models were the result of considerable idealization, the purpose of the present paper is to propose a model in which the degree of simplification is relaxed: the features of simultaneous growth, and cell growth whose rate depends on the free absorbing periphery of the cell are introduced. Simulation experiments have been conducted using the model, and the results are presented. Now in Department of Civil Engineering.     

Mathematical models for cellular systems the von Foerster equation. Part I

P. B. M. Walker (1954) and H. C. Longuet-Higgins (quoted by Walker), as well as O. Scherbaum and G. Rasch (1957), made the first attempts towards a mathematical study of the age distribution in a cellular population. It was H. Von Foerster (1959), however, who derived the complete differential equation for the age density function,n(t, a). His equation is obtained from an analysis of the infinitesimal changes occurring during a time elementdt in a group of cells with ages betweena anda+da. The behavior of the population is determined by a quantity λ which we call the loss function. In this paper a rigorous discussion of the Von Foerster equation is presented, and a solution is given for the special case when λ depends, ont (time) anda (age) but not on other variables (such asn itself). It is also shown that the age density,n(t, a), is completely known only if the birth rate,α(t), and the initial age distribution, β(a), are given as boundary conditions. In Section II the steady state solution and some plausible forms of intrinsic loss functions (depending ona only) are discussed in view of later applications. This work was performed under the auspices of the U.S. Atomic Energy Commission.     

Mathematical models for cellular systems the von Foerster equation. Part I

P. B. M. Walker (1954) and H. C. Longuet-Higgins (quoted by Walker), as well as O. Scherbaum and G. Rasch (1957), made the first attempts towards a mathematical study of the age distribution in a cellular population. It was H. Von Foerster (1959), however, who derived the complete differential equation for the age density function,n(t, a). His equation is obtained from an analysis of the infinitesimal changes occurring during a time elementdt in a group of cells with ages betweena anda+da. The behavior of the population is determined by a quantity λ which we call the loss function. In this paper a rigorous discussion of the Von Foerster equation is presented, and a solution is given for the special case when λ depends, ont (time) anda (age) but not on other variables (such asn itself). It is also shown that the age density,n(t, a), is completely known only if the birth rate,α(t), and the initial age distribution, β(a), are given as boundary conditions. In Section II the steady state solution and some plausible forms of intrinsic loss functions (depending ona only) are discussed in view of later applications.     

Effects of insulin on cellular growth and proliferation

Insulin stimulates the growth and proliferation of a variety of cells in culture. The growth-stimulatory effects of insulin are observed in G_o/G_l arrested cells limited for serum growth factors or essential nutrients, and in cells growing in hormone-supplemented serum-free media. Some, but not all, of the effects of insulin on growth require superphysiological concentrations of insulin. The action of insulin on growth is synergistic with the action of other hormones and growth factors, including FGF, PDGF, PGF_2α and vasopressin. This observation, as well as other observations regarding the temporal sequence of action of growth factors, suggests that different growth factors act on different intracellular biochemical events. Several hypotheses have been proposed to explain the effect of insulin on cellular proliferation, including regulation of essential metabolic processes and interaction of insulin with receptors for insulin-like growth factors. Evidence supporting these various hypotheses is reviewed. In addition to the growth-stimulatory effect of insulin observed in cell culture, a number of clinical examples suggest that insulin is an important growth-regulating hormone during fetal development.     

Bilayer agar cell cultures for studying cellular growth factors

The method for biological testing of growth factors (GF) produced by transformed cells is described. The method is suitable for studying the ability of donor cells to release GF that stimulate colony formation of test-cells. Donor cells and test-cells are placed into different semisolid agar layers and separated by intermediate agar layer. The method provides a much more efficient testing of GF biological activity than the use of conditioned cultural fluids of transformed cells. It permits the assessment of GF dialyzation and the role of donor cell proliferation in the production of GF.     

Dynamics of cellular growth.

The assumption that prompted the studies reported in this paper was that the unsatisfactory state of our knowledge on the regulation of cellular growth might derive from the reductionistic approach used to investigate it. Thus an analysis of cellular growth which applied concepts derived from systems dynamics was undertaken. First of all a dynamic model of cellular growth has been constructed. It has the following features: the levels of DNA, ribosomes and proteins are the defining levels; cellular growth is expressed by a close loop in which the level of ribosomes per genome and, indirectly, the level of proteins per genome are stabilized around goal values by the action of negative feed backs. The validity of the model has been tested by its ability to predict the growth kinetics of a real system (exponentially growing Neurospora cells). The simulated growth has been found to reproduce with great accuracy that of Neurospora cells. A slightly modified model, which takes into consideration also the degradation of ribosomes and of proteins, is shown to predict with accuracy the dynamics of growth of both growing and resting fibroblasts. These latter results suggest that the rates of macromolecular turnovers play a central role in the control of proliferation of mammalian cells: the condition of zero growth seems to be achieved when the rate of synthesis and the rate of degradation of proteins are the same. The possibility is discussed that the model indicates a unifying hypothesis of the mode of action of growth controlling conditions (hormones, growth factors, contact inhibition).     

Autophagy in cellular growth control

Cell growth is regulated by two antagonistic processes: TOR signaling and autophagy. These processes integrate signals including growth factors, amino acids, and energy status to ensure that cell growth is appropriate to environmental conditions. Autophagy responds indirectly to the cellular milieu as a downstream inhibitory target of TOR signaling and is also directly controlled by nutrient availability, cellular energy status, and cell stress. The control of cell growth by TOR signaling and autophagy are relevant to disease, as altered regulation of either pathway results in tumorigenesis. Here we give an overview of how TOR signaling and autophagy integrate nutritional status to regulate cell growth, how these pathways are coordinately regulated, and how dysfunction of this regulation might result in tumorigenesis.