From cell death to embryo arrest: Mathematical models of human preimplantation embryo development

Human preimplantation embryos exhibit high levels of apoptotic cells and high rates of developmental arrest during the first week in vitro. The relation between the two is unclear and difficult to determine by conventional experimental approaches, partly because of limited numbers of embryos. We apply a mixture of experiment and mathematical modeling to show that observed levels of cell death can be reconciled with the high levels of embryo arrest seen in the human only if the developmental competence of embryos is already established at the zygote stage, and environmental factors merely modulate this. This suggests that research on improving in vitro fertilization success rates should move from its current concentration on optimizing culture media to focus more on the generation of a healthy zygote and on understanding the mechanisms that cause chromosomal and other abnormalities during early cleavage stages.

Failure of programmed cell death and differentiation as causes of tumors: some simple mathematical models.

Most models of tumorigenesis assume that the tumor grows by increased cell division. In these models, it is generally supposed that daughter cells behave as do their parents, and cell numbers have clear potential for exponential growth. We have constructed simple mathematical models of tumorigenesis through failure of programmed cell death (PCD) or differentiation. These models do not assume that descendant cells behave as their parents do. The models predict that exponential growth in cell numbers does sometimes occur, usually when stem cells fail to die or differentiate. At other times, exponential growth does not occur: instead, the number of cells in the population reaches a new, higher equilibrium. This behavior is predicted when fully differentiated cells fail to undergo PCD. When cells of intermediate differentiation fail to die or to differentiate further, the values of growth parameters determine whether growth is exponential or leads to a new equilibrium. The predictions of the model are sensitive to small differences in growth parameters. Failure of PCD and differentiation, leading to a new equilibrium number of cells, may explain many aspects of tumor behavior--for example, early premalignant lesions such as cervical intraepithelial neoplasia, the fact that some tumors very rarely become malignant, the observation of plateaux in the growth of some solid tumors, and, finally, long lag phases of growth until mutations arise that eventually result in exponential growth.

Variation in per capita interaction strength: Thresholds due to nonlinear dynamics and nonequilibrium conditions

I measured the strength of interaction between a marine herbivore and its growing resource over a realistic range of absolute and relative abundances. The herbivores (hermit crabs: Pagurus spp.) have slow and/or weak functional and numerical responses to epiphytic diatoms (Isthmia nervosa), which show logistic growth in the absence of consumers. By isolating this interaction in containers in the field, I mimicked many of the physical and biological variables characteristic of the intertidal while controlling the densities of focal species. The per capita effects of consumers on the population dynamics of their resource (i.e., interaction strength) were defined by using the relationship between hermit crab density and proportional change in the resource. When this relationship is fit by a Weibull function, a single parameter distinguishes constant interaction strength from one that varies as a function of density. Constant interaction strength causes the proportion of diatoms to fall linearly or proportionally as hermit crab density increases whereas per capita effects that increase with density cause an accelerating decline. Although many mathematical models of species interactions assume linear dynamics and invariant parameters, at least near equilibrium, the per capita effects of hermit crabs on diatoms varied substantially, apparently crossing a threshold from weak to strong when consumption exceeded resource production. This threshold separates a domain of coexistence from one of local extinction of the resource. Such thresholds may help explain trophic cascades, resource compensation, and context-dependent interaction strengths, while indicating a way to predict trophic effects, despite nonlinearities, as a function of vital rates.

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.

Cytotoxic T lymphocytes and viral turnover in HIV type 1 infection

To understand the role of the immune system in limiting HIV type 1 replication, it is critical to know to what extent the rapid turnover of productively infected cells is caused by viral cytopathicity or by immune-mediated lysis. We show that uncultured peripheral blood mononuclear cells of many patients contain cytotoxic T lymphocytes (CTL) that lyse target cells—at plausible peripheral blood mononuclear cell-to-target ratios—with half-lives of less than 1 day. In 23 patients with CD4 counts ranging from 10 to 900 per μl, the average rate of CTL-mediated lysis corresponds to a target cell half-life of 0.7 day. We develop mathematical models to calculate the turnover rate of infected cells subjected to immune-mediated lysis and viral cytopathicity and to estimate the fraction of cells that are killed by CTL as opposed to virus. The models provide new interpretations of drug treatment dynamics and explain why the observed rate of virus decline is roughly constant for different patients. We conclude that in HIV type 1 infection, CTL-mediated lysis can reduce virus load by limiting virus production, with small effects on the half-life of infected cells.

Older plasma lipoproteins are more susceptible to oxidation: a linking mechanism for the lipid and oxidation theories of atherosclerotic cardiovascular disease.

Increases in plasma cholesterol are associated with progressive increases in the risk of atherosclerotic cardiovascular disease. In humans plasma cholesterol is contained primarily in apolipoprotein B-based low density lipoprotein (LDL). Cells stop making the high-affinity receptor responsible for LDL removal as they become cholesterol replete; this slows removal of LDL from plasma and elevates plasma LDL. As a result of this delayed uptake, hypercholesterolemic individuals not only have more LDL but have significantly older LDL. Oxidative modification of LDL enhances their atherogenicity. This study sought to determine whether increased time spent in circulation, or aging, by lipoprotein particles altered their susceptibility to oxidative modification. Controlled synchronous production of distinctive apolipoprotein B lipoproteins (yolk-specific very low density lipoproteins; VLDLy) with a single estrogen injection into young turkeys was used to model LDL aging in vivo. VLDLy remained in circulation for at least 10 days. Susceptibility to oxidation in vitro was highly dependent on lipoprotein age in vivo. Oxidation, measured as hexanal release from n-6 fatty acids in VLDLy, increased from 13.3 +/- 5.5 nmol of 2-day-old VLDLy per ml, to 108 +/- 17 nmol of 7-day-old VLDLy per ml. Oxidative instability was not due to tocopherol depletion or conversion to a more unsaturated fatty acid composition. These findings establish mathematically describable linkages between the variables of LDL concentration and LDL oxidation. The proposed mathematical models suggest a unified investigative approach to determine the mechanisms for acceleration of atherosclerotic cardiovascular disease risk as plasma cholesterol rises.