A size-structured model of bacterial growth and reproduction.

We consider a size-structured bacterial population model in which the rate of cell growth is both size- and time-dependent and the average per capita reproduction rate is specified as a model parameter. It is shown that the model admits classical solutions. The population-level and distribution-level behaviours of these solutions are then determined in terms of the model parameters. The distribution-level behaviour is found to be different from that found in similar models of bacterial population dynamics. Rather than convergence to a stable size distribution, we find that size distributions repeat in cycles. This phenomenon is observed in similar models only under special assumptions on the functional form of the size-dependent growth rate factor. Our main results are illustrated with examples, and we also provide an introductory study of the bacterial growth in a chemostat within the framework of our model.

The dynamics of single-substrate continuous cultures: the role of ribosomes.

When a chemostat is perturbed from its steady state, it displays complex dynamics. For instance, if the identity of the growth-limiting substrate is switched abruptly, the substrate concentration and cell density undergo a pronounced excursion from the steady state that can last several days. These dynamics occur because certain physiological variables respond slowly. In the literature, several physiological variables have been postulated as potential sources of the slow response. These include transport enzymes, biosynthetic enzymes, and ribosomes. We have been addressing this problem by systematically exploring the role of these variables. In previous work Shoemaker et al. (J. Theor. Biol., 222 (2003) 307-322), we studied the role of transport enzymes, and we showed that transients starting from low transport enzyme levels could be quantitatively captured by a model taking due account of transport enzyme synthesis. However, there is some experimental data indicating that slow responses occur even if the initial enzyme levels are high. Here, we analyse this data to show that in these cases, the sluggish response is most probably due to slow adjustment of the ribosome levels. To test this hypothesis, we extend our previous model by accounting for the evolution of both the transport enzyme and the ribosomes. Based on a kinetic analysis of the data in the literature, we assume that the specific protein synthesis rate is proportional to the ribosome level, and the specific ribosome synthesis rate is autocatalytic. Simulations of the model show remarkable agreement with experimentally observed steady states and the transients. Specifically, the model predictions are in good agreement with (1) the steady-state profiles of the cell density, substrate concentration, RNA, proteins, and transport enzymes, (2) the instantaneous specific substrate uptake, growth, and respiration rates in response to a continuous-to-batch shift, and (3) the transient profiles of the cell density, substrate concentration, and RNA in response to feed switches and dilution rate shifts. Time-scale analysis of the model reveals that every transient response is a combination of two fundamental (and simpler) dynamics, namely, substrate-sufficient batch dynamics and cell-sufficient fed-batch dynamics. We obtain further insight into the transient response by analysing the equations describing these fundamental dynamics. The analysis reveals that in feed switches or dilution rate shift-ups, the transport enzyme reaches a maximum before RNA achieves its maximum, and in dilution rate shift-downs the cell density reaches a maximum before RNA achieves a minimum.

Global exponential stability of competitive neural networks with different time scales.

The dynamics of cortical cognitive maps developed by self-organization must include the aspects of long and short-term memory. The behavior of such a neural network is characterized by an equation of neural activity as a fast phenomenon and an equation of synaptic modification as a slow part of the neural system. We present a new method of analyzing the dynamics of a biological relevant system with different time scales based on the theory of flow invariance. We are able to show the conditions under which the solutions of such a system are bounded being less restrictive than with the K-monotone theory, singular perturbation theory, or those based on supervised synaptic learning. We prove the existence and the uniqueness of the equilibrium. A strict Lyapunov function for the flow of a competitive neural system with different time scales is given and based on it we are able to prove the global exponential stability of the equilibrium point.

Modeling immune responses with handling time.

Simple predator-prey type models have brought much insight into the dynamics of both nonspecific and antigen-specific immune responses. However, until now most attention has been focused on examining how the dynamics of interactions between the parasite and the immune system depends on the nature of the function describing the rate of activation or proliferation of immune cells in response to the parasite. In this paper we focus on the term describing the killing of the parasite by cell-mediated immune responses. This term has previously been assumed to be a simple mass-action term dependent solely on the product of the densities of the parasite and the immune cells and does not take into account a handling time (which we define as the time of interaction between an immune cell and its target, during which the immune cell cannot interact with and/or destroy additional targets). We show how the handling time (i) can be incorporated into simple models of nonspecific and specific immunity and (ii) how it affects the dynamics of both nonspecific and antigen-specific immune responses, and in particular the ability of the immune response to control the infection.

Models of immune memory: on the role of cross-reactive stimulation, competition, and homeostasis in maintaining immune memory.

There has been much debate on the contribution of processes such as the persistence of antigens, cross-reactive stimulation, homeostasis, competition between different lineages of lymphocytes, and the rate of cell turnover on the duration of immune memory and the maintenance of the immune repertoire. We use simple mathematical models to investigate the contributions of these various processes to the longevity of immune memory (defined as the rate of decline of the population of antigen-specific memory cells). The models we develop incorporate a large repertoire of immune cells, each lineage having distinct antigenic specificities, and describe the dynamics of the individual lineages and total population of cells. Our results suggest that, if homeostatic control regulates the total population of memory cells, then, for a wide range of parameters, immune memory will be long-lived in the absence of persistent antigen (T1/2 > 1 year). We also show that the longevity of memory in this situation will be insensitive to the relative rates of cross-reactive stimulation, the rate of turnover of immune cells, and the functional form of the term for the maintenance of homeostasis.

Modeling T-cell proliferation: an investigation of the consequences of the Hayflick limit.

Somatic cells, including immune cells such as T-cells have a limited capacity for proliferation and can only replicate for a finite number of generations (known as the Hayflick limit) before dying. In this paper we use mathematical models to investigate the consequences of introducing a Hayflick limit on the dynamics of T-cells stimulated with specific antigen. We show that while the Hayflick limit does not alter the dynamics of T-cell response to antigen over the short term, it may have a profound effect on the long-term immune response. In particular we show that over the long term the Hayflick limit may be important in determining whether an immune response can be maintained to a persistent antigen (or parasite). The eventual outcome is determined by the magnitude of the Hayflick limit, the extent to which antigen reduces the input of T-cells from the thymus, and the rate of antigen-induced proliferation of T-cells. Counter to what might be expected we show that the persistence of an immune response (immune memory) requires the density of persistent antigen to be less than a defined threshold value. If the amount of persistent antigen (or parasite) is greater than this threshold value then immune memory will be relatively short lived. The consequences of this threshold for persistent mycobacterial and HIV infections and for the generation of vaccines are discussed.