A. G. Fredrickson

Statistics and dynamics of procaryotic cell populations

Abstract The formulation of a mathematical theory of a cell population requires the ability to specify quantitatively the physiological state of individual cells of the population. In procaryotic cells (bacteria and blue-green algae), the intracellular structure is of a relatively simple nature, and it is postulated that the physiological state of such a cell is specified by its biochemical composition. If we postulate further that the growth rate of a cell and its fission probability depend only on the cells current physiological state and on the current state of the cells environment, then an equation of change for the distribution of physiological states in a population can be derived. In addition, an equation of change for the state of the cellular environment can be obtained. These equations allow us to predict the statistical and dynamical behavior of a cell population from information obtained by analysis of cellular and subcellular structure and function.

A mathematical theory of age structure in sexual populations: random mating and monogamous marriage models

Abstract Equations governing the size and age structure of a large population composed of two sexes are developed. It is assumed that the “significant” environment of the population —that which governs its mortality, nuptiality, and natality patterns and rates— remains constant. Two different social arrangements respecting reproduction are considered. It is assumed first that reproduction is by random mating; it is then assumed that reproduction is via strict monogamous marriage. The equations describing these two cases are coupled and nonlinear. In the case of the random mating model, the problem can be reduced to a single nonlinear integral equation. In the case of the monogamous marriage model, a system of partial differential-integral equations results. Applications of these equations to a population in stable growth and in transient growth are discussed.

Mathematical Models for Fermentation Processes

Publisher Summary This chapter discusses that models serve to correlate data and so provide a concise way of thinking about a system or process. They help to sharpen thinking about a system or process and can be used to guide ones reasoning in the design of experiments to isolate important parameters and elucidate the nature of the system or process. The scope, the scale, and the complexity of fermentation processes are such that changes in technology will likely occur; indeed, there is an emphasis on investigations on continous culture techniques, fermentation vessels are no longer always of the stirred-tank variety, and mixed cultures are starting to become common. The chapter discusses that mathematical models of microbial growth are generated by a combination of three processes. First, well-established physical, chemical, or biological principles may provide certain parts of the model. Second, some hypotheses of the model may be plausible inferences from existing data. Finally, the remaining parts of the model are guesses, but one hopes, educated ones.

Solutions of population balance models based on a successive generations approach

Abstract Microbial and cell cultures are composed of discrete organisms, each of which goes through a cell cycle that terminates in production of new cells. The internal state of an individual cell changes as the cell progresses through the cell cycle, and randomness in various features of the cell cycle always produces a distribution of cell states in the culture. Rigorous models of this situation lead to the so-called population balance equations, which are integropartial differential equations. These equations are notoriously difficult to solve, and the difficulties increase as the number of parameters needed to describe cell state increases. The cells in a culture are of different generations, and cells of the (k + 1)th generation originate only from divisions of cells of the kth generation. A population balance equation written for the (k + 1)th generation is therefore not an integral equation, although it contains a source term which is an integral over the distribution of states of the kth generation. If competition of coexisting generations for environmental resources does not affect growth and reproduction rates, the population balance equations for the various generations in a culture do not have to be solved simultaneously but rather can be solved successively, and thus, some of the major difficulties of population balance equations written for entire populations are circumvented. In this paper, the successive generations approach to modeling is illustrated by its application to two problems where cell state is described by a single parameter, either cell age or cell mass. It is then applied to a problem where two parameters, namely cell age and cell mass, are used to describe cell state at the same time. Analytical solutions of the population balance equations for the successive generations are found for the cases discussed, and the solutions are used to calculate the evolutions of the distributions of cell states with time for the single parameter cases.

Multistaged corpuscular models of microbial growth: Monte Carlo simulations

A new framework is developed by extending the existing population balance framework for modeling the growth of microbial populations. The new class of multistaged corpuscular models allows further structuring of the microbial life cycle into separate phases or stages and thus facilitates the incorporation of cell cycle phenomena to population models. These multistaged models consist of systems of population balance equations coupled by appropriate boundary conditions. The specific form of the equations depend on the assumed forms for the transition rate functions, the growth rate functions, and the partitioning function, which determines how the biological material is distributed at division. A growth model for ciliated protozoa is formulated to demonstrate the proposed framework. To obtain a solution to the system of the partial integro differential equations that results from such formulation, we adopted a Monte Carlo simulation technique which is very stable, versatile, and insensitive to the complexity of the model. The theory and implementation of the Monte Carlo simulation algorithm is analyzed and results from the simulation of the ciliate growth model are presented. The proposed approach seems to be promising for integrating single-cell mechanisms into population models.

A stochastic analysis of the growth of competing microbial populations in a continuous biochemical reactor

Abstract The dynamics of a chemostat in which two microbial populations grow and compete for a common substrate is examined. It is shown that the two populations cannot coexist in a spatially uniform environment which is subject to time invariant external influences unless the dilution rate takes on one of a discrete set of special values. The dynamics of the same system are next considered in the stochastic environment created by random fluctuations of the dilution rate about a value that allows coexistence. The information needed for the description of the random process of the state of the chemostat is obtained from the transition probability density function. By modeling the system as a Markov process continuous in time and space, the transition probability density is obtained as solution of the Fokker-Planck equation. Analytical and numerical solutions of this equation show that extinction of either one population or the other will ultimately take place. The time required for extinction, the evolution of the mean composition with time, the steady states of the latter and the dependence of all the above on the intensity of the random noise are also calculated using constants appropriate to the competition of E. coli and Spirillum sp. The question of making predictions as to which population is the more likely to become extinct is treated finally, and the probabilities of extinction are calculated as solutions of the steady state version of the backward Fokker-Planck (Kolmogorov) equation.

A new set of population balance equations for microbial and cell cultures

Abstract Population balance equations for microbial or cell cultures contain a state-dependent fission intensity function which is such that the product of this function, evaluated at a given state, and a differential increment of time is the fraction of cells of the given state that divide in that increment of time. Ways to determine experimentally how the fission intensity function depends on cell state have been proposed but, so far as is known to the authors, no one has yet proposed a model that would predict what the state dependence of the fission intensity function is for any population. If one wants to take account of the existence of cell cycle phases, one will have to write a population balance equation for each phase, a transition intensity function for each phase will have to be introduced, and the problem of making models for these functions will arise again, and in multiplied fashion. In this paper, we describe a new and different approach which circumvents the necessity of having intensity functions for transitions between cell cycle phases, and for which the fission intensity function is state-independent.

Kinetics of Growth of the Ciliate Tetrahymena pyriformis on Escherichia coli

The growth of the ciliate Tetrahymena pyriformis on non-growing Escherichia coli has been studied by following the time courses of population densities and protozoan mean cell volume in batch cultures. Viable, non-encysted protozoa always stopped feeding before the bacterial density was reduced to zero and non-feeding ciliates tended to swim faster than feeding ciliates. In addition, the number of bacteria and other particles of bacterial size consumed in the formation of one new ciliate, when averaged over the lag and reproductive phases of a culture, declined toward a limiting value of about 1.6 x 10(4) particles per ciliate as the initial density of such particles was increased.

On relationships between various distribution functions in balanced unicellular growth

The relationships between various size distributions in balanced exponential growth of a batch culture of microorganisms are presented. Starting from the partial differential integral equations (Eakmanet al., 1966; Fredricksonet al., 1967) derived for the growth of a microbial culture expressions are obtained for the growth rate of organisms of specific size and size range. These expressions were first obtained by Collins and Richmond (1962) by an entirely different method. Also derived are equations which link probability functions, which are basic to the growth of a microbial culture, with other size distributions that can be estimated experimentally.

Some physiological aspects of the autecology of the suspension-feeding protozoanTetrahymena pyriformis.

Feeding, growth, and reproductive responses of the suspension-feeding protozoanTetrahymena pyriformis to shifts up or down of the density of its bacterial food were observed. The rates of feeding, growth, and reproduction were determined by measuring the rates of uptake of viable bacterial cells, of change of mean volume of the protozoan cells, and of change of number of protozoan cells, respectively. The effects of the nutritional status of the protozoans at the time of shifting were observed also. Results are interpreted in terms of the limited polymorphism exhibited in the life cycle of this organism. Responses in all cases seem to reflect a strategy for exploiting a patchy, transient environment, a conclusion already reached by several earlier investigators.