A mathematical description of metabolizing systems I

An integral equation is established applicable to many types of metabolizing systems. The equation is applied to first order chemical reactions and to some biological systems.     

Hierarchical evolutive systems: A mathematical model for complex systems

The notion of an evolutive hierarchical system proposed in this paper is a mathematical model for systems, like organisms, with more or less complex objects. This model, based on category theory, retains the following characteristics of natural systems: they have an internal organization consisting of components with interrelations; they maintain their organization in time though their components are changing; their components are divided into several levels corresponding to the increasing complexity of their own organization, and the system may be studied at any of these levels (e.g. molecular, cellular...). The state of the system at a given instant is modeled by a category whose objects are its components, the state transition by a functor, a complex object by the (direct) limit of a pattern of linked objects (which describes its internal organization). The properties of limits in a category make it possible to ‘measure’ the emergence of properties for a complex object with respect to its components, and to reduce the study of a hierarchical system to that of its components of the lowest degree and their links. Categorical constructions describe the formation of a hierarchical evolutive system stepwise, by means of the operations: absorption of external objects, destruction of some components, formation of new complex objects.     

A mathematical description of metabolizing systems: II

Some of the laboratory procedures available for determining the functions in the integral equation established in part I are discussed. The tracer or tagged molecule technique is shown to be especially promising including the use of “double tracer” molecules. Conversely, the integral equation may be a convenient device for correlating and integrating some of the work now being done with tracer molecules in biological systems.     

A survey of mathematical models of competition with an inhibitor

Mathematical models of the effect of inhibitors on microbial competition are surveyed. The term inhibitor is used in a broad sense and includes toxins, contaminants, allelopathic agents, etc. This includes both detoxification where the inhibitor is viewed as a pollutant and control where the inhibitor is viewed as an aid to controlling a bioreactor. The inhibitor may be supplied externally or may be created as an anti-competitor toxin. This includes plasmid-bearing, plasmid-free competition. The literature is spread across journals in different disciplines and with different notation. The survey attempts to present the mathematical models and the results of the corresponding analysis within a common framework and notation. Detailed mathematical proofs are not given but the methods of proof are indicated, references cited, and the results presented in tables. Open problems are indicated where there is a gap in the theory.     

The role of unequal crossover in alpha-satellite DNA evolution: a computational analysis.

Human DNA consists of a large number of tandem repeat sequences. Such sequences are usually called satellites, with the primary example being the centromeric alpha-satellite DNA. The basic repeat unit of the alpha-satellite DNA is a 171 bp monomer. Arbitrary monomer pairs usually have considerable sequence divergence (20-40%). However, with the exception of peripheral alpha-satellite DNA, monomers can be grouped into blocks of k-monomers (4 < or = k < or = 20) between which the divergence rate is much smaller (e.g., 5%). Perhaps the simplest and best understood mechanism for tandem repeat array evolution is unequal crossover. Although it is possible that alpha-satellite sequences developed as a result of subsequent unequal crossovers only, no formal computational framework seems to have been developed to verify this possibility. In this paper, we develop such a framework and report on experiments which imply that pericentromeric alpha-satellite segments (which are devoid of higher order structure) are evolutionarily distinct from the higher order repeat segments. It is likely that the higher order repeats developed independently in distinct regions of the genome and were carried into their current locations through an unknown mechanism of transposition.     

Temporal properties of some biological systems and their fractal attractors

In this paper we analyse time series data as the growth of organisms using markers such as treerings and otolith deposits (fish). The series studied belong to two tree species (Pinus uncinata, Fagus sylvatica) and one fish species (Dicentrarchus labrax). Spectral analyses of the time series growth show that the main frequencies of fluctuation may be due to variations of the energy input. However, any causal explanation must consider the internal continuous readjustment in the system as reported by the corresponding chaotic properties of the asymptotic decay of the spectra time structure. Since the output of noisy and chaotic systems tend to show similar spectral densities, an attempt to differentiate them has been carried out. The chaotic behaviour has been characterized by the study of the attractors. The dimmensions of these multiple topologies were 3.2 and 3.4 for the tree species and 2.3 for the fish species. Therefore, we are dealing with fractal attractors and the minimum number of variables that can be used to describe the systems are 4 and 3 respectively. It is suggested that some of the variables that most influence growth are those obtained by the response functions in the case of trees.     

A mathematical model for resolution enhancement in layered sensory systems

Heiligenberg (1987) recently proposed a model to explain how the representation of a stimulus variable through an ordered array of broadly tuned receptors could allow a degree of stimulus resolution greatly exceeding the resolution of the individual receptors which make up the array. In his model, this hyperacuity is achieved by connecting the receptors to a higher level pool interneuron according to a linear synaptic weighting function. We have extended this model to the general case of arbitrary polynomial synaptic weighting functions, and showed that the response function of this higher level interneuron is a polynomial of the same order as the weighting function. We also proved that Hermite polynomials are eigen-functions of the system. Further, by allowing multiple interneurons in the higher level pool, each of which is connected to the receptors according to a different orthogonal weighting function, we demonstrated that extended stimulus functions can be represented with enhanced precision, rather than just the value of individual point stimuli. Finally, we suggest a solution to the problem of edge effect errors arising near the ends of finite receptor arrays.     

A mathematical model for the mechanical properties of scallop shells

The mechanical behaviour of one or both shell valves of five species of pectinacean and one anomiacean was determined using strain gauge rosettes attached to the inner surfaces of the valves. A multiple linear regression analysis accounted for over 64% of the total vanance in tensile shell compliance. The important shell architectural variables were thickness, corrugation and, much less important, convexity and shell length. Calculation showed that in general corrugation of the shell was easily the cheapest way of producing stiffness, in terms of the amount of shell material required. Suggestions are offered as to why not all scallops are scalloped.     

A mathematical model of two-trophic-level aquatic systems with two complementary nutrients

A model of phytoplankton, zooplankton, and two complementary nutrients, nitrogen and phosphorus, that simulates the dynamics of their interactions in lakes or oceans is presented. The paper studies a mathematical model, based on a modified Michaelis-Menten kinetics, for the phytoplankton uptake of the two nutrients, and a herbivore-density dependent grazing rate. Given the parameters of the system, we answer the basic questions about persistence, extinction, and limiting behavior of the microorganisms. The analytical studies demonstrate the role of the total system nutrients in governing system behavior and in turn in governing water quality control. Features of the model system include behavior determined solely by the total system nutrients, nitrogen and phosphorus. A conclusion robust for the model is the coupling of nitrogen and phosphorus cycles in more realistic dynamics that increase the effectiveness of water quality-control programs.     

A mathematical model for water and nutrient uptake by plant root systems

This article deals with modelling the simultaneous uptake of water and highly buffered nutrient, such as phosphate, by root branching structures from partially saturated soil. We use the simultaneous water and nutrient uptake model to investigate the effect that water movement has on nutrient uptake. With the aid of this model we are also able to show that the previous models by Barber and Tinker and Nye systematically underestimated the phosphate uptake, due to the oversimplified approach in dealing with root branching structure. In this article we show how this discrepancy can be remedied and the root branching structure included in the models of plant nutrient uptake. We will also discuss the differences in the results for continuous and spot fertilization combined with variable rainfall.     

A mathematical analysis of focal adhesion lifetimes and their effect on cell motility

By analyzing the distributions of focal adhesion (FA) lifetimes from different cell types, we found that a gamma distribution best matched the experimental distributions. In all but one case, it was a unimodal, non-symmetric gamma distribution. We used a mathematical model of cell motion to help understand the mechanics and data behind the FA lifetime distributions. The model uses a detach-rate function to determine how long an FA will persist before it detaches. The detach-rate function that produced distributions with a best-fit gamma curve that closely matched that of the data was both force and time dependent. Using the data gathered from the matching simulations, we calculated both the cell speed and mean FA lifetime and compared them. Where available, we also compared this relationship to that of the experimental data and found that the simulation reasonably matches it in most cases. In both the simulations and experimental data, the cell speed and mean FA lifetime are related, with longer mean lifetimes being indicative of slower speeds. We suspect that one of the main predictors of cell speed for migrating cells is the distribution of the FA lifetimes.     

A field course survey of three English river systems

A hydrobiological field course for undergraduates in the Department of Biology, University of Salford has proved useful in investigating river pollution, and parts of the course may be suitable for upper school studies. The course compares the Lancashire rivers Lune, Ribble and Irwell, but could be adapted for still waters.

The water quality was assessed by simple chemical methods and the bacteriological quality by multiple tube and membrane filtration techniques. These assessments were then related to the benthic inverte-brates in the rivers—dipteran larvae, oligochaete worms, leeches, molluscs, stoneflies and mayflies. The Irwell was found to be grossly polluted, the Ribble mildly polluted and the Lune relatively unpolluted.