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.

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.

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.

Different population dynamics of human T cell lymphotropic virus type II in intravenous drug users compared with endemically infected tribes

The phylogeny of human T cell lymphotropic virus type II (HTLV-II) was investigated by using strains isolated from Amerindian and Pygmy tribes, in which the virus is maintained primarily through mother-to-child transmission via breast-feeding, and strains from intravenous drug users (IDUs), in which spread is mainly blood-borne via needle sharing. Molecular clock analysis showed that HTLV-II has two different evolutionary rates with the molecular clock for the virus in IDUs ticking 150–350 times faster than the one in endemically infected tribes: 2.7 × 10−4 compared with 1.71/7.31 × 10−7 nucleotide substitutions per site per year in the long terminal repeat region. This dramatic acceleration of the evolutionary rate seems to be related with the mode of transmission. Mathematical models showed the correlation of these two molecular clocks with an endemic spread of HTLV-II in infected tribes compared with the epidemic spread in IDUs. We also noted a sharp increase in the population size of the virus among IDUs during the last decades probably caused by the worldwide increase in intravenous drug use.

Dynamics of maize endosperm development and DNA endoreduplication.

Endosperm development in Zea mays is characterized by a period of intense mitotic activity followed by a period in which mitosis is essentially eliminated and the cell cycle becomes one of alternating S and G phases, leading to endoreduplication of the nuclear DNA. The endosperm represents a significant contribution to the grain yield of maize; thus, methods that facilitate the study of cellular kinetics may be useful in discerning cellular and molecular components of grain yield. Two mathematical models have been developed to describe the kinetics of endosperm growth. The first describes the kinetics of mitosis during endosperm development; the second describes the kinetics of DNA endoreduplication during endosperm development. The mitotic model is a modification of standard growth curves. The endoreduplication model is composed of six differential equations that represent the progression of nuclei from one DNA content to another during the endoreduplication process. Total nuclei number per endosperm and the number of 3C, 6C, 12C, 24C, 48C, and 96C nuclei per endosperm (C is the haploid DNA content per nucleus) for inbred W64A from 8 to 18 days after pollination were determined by flow cytometry. The results indicate that the change in number of nuclei expressed as a function of the number of days after pollination is the same from one yearly crop to another. These data were used in the model to determine the endosperm growth rate, the maximum nuclei number per endosperm, and transition rates from one C value to the next higher C value. The kinetics of endosperm development are reasonably well represented by the models. Thus, the models provide a means to quantify the complex pattern of endosperm development.