John D. Nagy

Dynamics of a delay differential equation model of hepatitis B virus infection

We formulate and systematically study the global dynamics of a simple model of hepatitis B virus in terms of delay differential equations. This model has two important and novel features compared to the well-known basic virus model in the literature. Specifically, it makes use of the more realistic standard incidence function and explicitly incorporates a time delay in virus production. As a result, the infection reproduction number is no longer dependent on the patient liver size (number of initial healthy liver cells). For this model, the existence and the component values of the endemic steady state are explicitly dependent on the time delay. In certain biologically interesting limiting scenarios, a globally attractive endemic equilibrium can exist regardless of the time delay length.

Biological Stoichiometry in Human Cancer

Background A growing tumor in the body can be considered a complex ecological and evolutionary system. A new eco-evolutionary hypothesis (the “Growth Rate Hypothesis”, GRH) proposes that tumors have elevated phosphorus (P) demands due to increased allocation to P-rich nucleic acids, especially ribosomal RNA, to meet the protein synthesis demands of accelerated proliferation. Methodology/Principal Findings We determined the elemental (C, N, P) and nucleic acid contents of paired malignant and normal tissues from colon, lung, liver, or kidney for 121 patients. Consistent with the GRH, lung and colon tumors were significantly higher (by approximately two-fold) in P content (fraction of dry weight) and RNA content and lower in nitrogen (N):P ratio than paired normal tissue, and P in RNA contributed a significantly larger fraction of total biomass P in malignant relative to normal tissues. Furthermore, patient-specific differences for %P between malignant and normal tissues were positively correlated with such differences for %RNA, both for the overall data and within three of the four organ sites. However, significant differences in %P and %RNA between malignant and normal tissues were not seen in liver and kidney and, overall, RNA contributed only ∼11% of total tissue P content. Conclusions/Significance Data for lung and colon tumors provide support for the GRH in human cancer. The two-fold amplification of P content in colon and lung tumors may set the stage for potential P-limitation of their proliferation, as such differences often do for rapidly growing biota in ecosystems. However, data for kidney and liver do not support the GRH. To account for these conflicting observations, we suggest that local environments in some organs select for neoplastic cells bearing mutations increasing cell division rate (“r-selected,” as in colon and lung) while conditions elsewhere may select for reduced mortality rate (“K-selected,” as in liver and kidney).

A clinical data validated mathematical model of prostate cancer growth under intermittent androgen suppression therapy

Prostate cancer is commonly treated by a form of hormone therapy called androgen suppression. This form of treatment, while successful at reducing the cancercell population, adversely affects quality of life and typically leads to a recurrence of the cancer in an androgen-independent form. Intermittent androgen suppression aims to alleviate some of these adverse affects by cycling the patient on and off treatment. Clinical studies have suggested that intermittent therapy is capable of maintaining androgen dependence over multiple treatment cycles while increasing quality of life during off-treatment periods. This paper presents a mathematical model of prostate cancer to study the dynamics of androgen suppression therapy and the production of prostate-specific antigen (PSA), a clinical marker for prostate cancer. Preliminary models were based on the assumption of an androgen-independent (AI) cell population with constant net growth rate. These models gave poor accuracy when fitting clinical data during simulation. The final model presented hypothesizes an AI population with increased sensitivity to low levels of androgen. It also hypothesizes that PSA production is heavily dependent on androgen. The high level of accuracy in fitting clinical data with this model appears to confirm these hypotheses, which are also consistent with biological evidence.

The evolutionary impact of androgen levels on prostate cancer in a multi-scale mathematical model

BackgroundAndrogens bind to the androgen receptor (AR) in prostate cells and are essential survival factors for healthy prostate epithelium. Most untreated prostate cancers retain some dependence upon the AR and respond, at least transiently, to androgen ablation therapy. However, the relationship between endogenous androgen levels and cancer etiology is unclear. High levels of androgens have traditionally been viewed as driving abnormal proliferation leading to cancer, but it has also been suggested that low levels of androgen could induce selective pressure for abnormal cells. We formulate a mathematical model of androgen regulated prostate growth to study the effects of abnormal androgen levels on selection for pre-malignant phenotypes in early prostate cancer development.ResultsWe find that cell turnover rate increases with decreasing androgen levels, which may increase the rate of mutation and malignant evolution. We model the evolution of a heterogeneous prostate cell population using a continuous state-transition model. Using this model we study selection for AR expression under different androgen levels and find that low androgen environments, caused either by low serum testosterone or by reduced 5α-reductase activity, select more strongly for elevated AR expression than do normal environments. High androgen actually slightly reduces selective pressure for AR upregulation. Moreover, our results suggest that an aberrant androgen environment may delay progression to a malignant phenotype, but result in a more dangerous cancer should one arise.ConclusionsThe model represents a useful initial framework for understanding the role of androgens in prostate cancer etiology, and it suggests that low androgen levels can increase selection for phenotypes resistant to hormonal therapy that may also be more aggressive. Moreover, clinical treatment with 5α-reductase inhibitors such as finasteride may increase the incidence of therapy resistant cancers.ReviewersThis article was reviewed by Ariosto S. Silva (nominated by Marek Kimmel) and Marek Kimmel.

Biological Stoichiometry: An Ecological Perspective on Tumor Dynamics

Abstract Biological stoichiometry is the study of the balance of energy and multiple chemical elements in biological systems. A key idea in biological stoichiometry is the growth rate hypothesis, which states that variation in the carbon:nitrogen:phosphorus stoichiometry of living things is associated with growth rate because of the elevated demands for phosphorus-rich ribosomal RNA, a requirement for rapid growth. In this article, we synthesize studies in the cancer literature to test the growth rate hypothesis; consistent with its predictions, rapidly growing tumors have elevated ribosome content, key oncogenes are closely affiliated with regulation of ribosome biogenesis, and tumor development has physiological impacts on patient phosphate metabolism. We also describe a new eco-evolutionary model of tumor dynamics that incorporates stoichiometric mechanisms. Since biological stoichiometry is fundamental in its approach, other areas of biology in which multiple key resources are involved in generating important tradeoffs may also benefit from the new tools provided by stoichiometric theory.

Evolution of Complex Density-Dependent Dispersal Strategies

The question of how dispersal behavior is adaptive and how it responds to changes in selection pressure is more relevant than ever, as anthropogenic habitat alteration and climate change accelerate around the world. In metapopulation models where local populations are large, and thus local population size is measured in densities, density-dependent dispersal is expected to evolve to a single-threshold strategy, in which individuals stay in patches with local population density smaller than a threshold value and move immediately away from patches with local population density larger than the threshold. Fragmentation tends to convert continuous populations into metapopulations and also to decrease local population sizes. Therefore we analyze a metapopulation model, where each patch can support only a relatively small local population and thus experience demographic stochasticity. We investigated the evolution of density-dependent dispersal, emigration and immigration, in two scenarios: adult and natal dispersal. We show that density-dependent emigration can also evolve to a nonmonotone, “triple-threshold” strategy. This interesting phenomenon results from an interplay between the direct and indirect benefits of dispersal and the costs of dispersal. We also found that, compared to juveniles, dispersing adults may benefit more from density-dependent vs. density-independent dispersal strategies.

The roles of predator maturation delay and functional response in determining the periodicity of predator–prey cycles

Population cycles in small mammals have attracted the attention of several generations of theoretical and experimental biologists and continue to generate controversy. Top-down and bottom-up trophic regulations are two recent competing hypotheses. The principal purpose of this paper is to explore the relative contributions of a variety of ecological factors to predator-prey population cycles. Here we suggest that for some species - collared lemmings, snowshoe hares and moose in particular - maturation delay of predators and the functional response of predation appear to be the primary determinants. Our study suggests that maturation delay alone almost completely determines the cycle period, whereas the functional response greatly affects its amplitude and even its existence. These results are obtained from sensitivity analysis of all parameters in a mathematical model of the lemming-stoat delayed system, which is an extension of Gilgs model. Our result may also explain why lemmings have a 4-year cycle whereas snowshoe hares have a 10-year cycle. Our parameterized model supports and extends Mays assertion that time delay impacts cycle period and amplitude. Furthermore, if maturation periods of predators are too short or too long, or the functional response resembles Holling Type I, then population cycles do not appear; however, suitable intermediate predator maturation periods and suitable functional responses can generate population cycles for both prey and predators. These results seem to explain why some populations are cyclic whereas others are not. Finally, we find parameterizations of our model that generate a 38-year population cycle consistent with the putative cycles of the moose-wolf interactions on Isle Royale, Michigan.

Mechanisms of Resistance to Intermittent Androgen Deprivation in Patients with Prostate Cancer Identified by a Novel Computational Method

For progressive prostate cancer, intermittent androgen deprivation (IAD) is one of the most common and effective treatments. Although this treatment is usually initially effective at regressing tumors, most patients eventually develop castration-resistant prostate cancer (CRPC), for which there is no effective treatment and is generally fatal. Although several biologic mechanisms leading to CRPC development and their relative frequencies have been identified, it is difficult to determine which mechanisms of resistance are developing in a given patient. Personalized therapy that identifies and targets specific mechanisms of resistance developing in individual patients is likely one of the most promising methods of future cancer therapy. Prostate-specific antigen (PSA) is a biomarker for monitoring tumor progression. We incorporated a cell death rate (CDR) function into a previous dynamical PSA model that was highly accurate at fitting clinical PSA data for 7 patients. The mechanism of action of IAD is largely induction of apoptosis, and each mechanism of resistance varies in its CDR dynamics. Thus, we analyze the CDR levels and their time-dependent oscillations to identify mechanisms of resistance to IAD developing in individual patients.

Population resilience in an American pika (Ochotona princeps) metapopulation

Population resilience in a metapopulation of American pikas (Ochotona princeps) at Bodie, California, was investigated with a series of 18 detailed occupancy surveys conducted between 1989 and 2010. These were compared with earlier 1972 and 1977 censuses and earlier historical records of pikas at Bodie. There is concern that American pikas may be increasingly vulnerable to warm temperatures due to climate change, and this investigation represents the longest study of the species in a relatively low-elevation (warm) environment. The Bodie pika population represents one of the best mammalian examples of a classic metapopulation system. Annual number of observed patch extinctions (total = 114) and recolonizations (109) varied greatly among the 18 census intervals. There has been no decline in percent of patches occupied in the northern half of the study area since 1972, and the number of documented pikas in the north in recent surveys exceeded the numbers found in 1972 and 1977. In contrast, the southern half of the metapopulation collapsed during our study, apparently the result of stochasticity of metapopulation dynamics; no southern patches were occupied after 2006. The potential impact of temperature on metapopulation dynamics was examined using long-term chronic (average summer monthly maximum) and acute threshold (number of days ≥ 25°C and ≥ 28°C within a year) temperatures. There is no evidence that warming temperatures have directly and negatively affected pika persistence at Bodie. Neither warm chronic nor acute temperatures increased the frequency of extinctions of populations on patches, and relatively cooler chronic or acute temperatures did not lead to an increase in the frequency of recolonization events. Warm temperatures, however, could have impeded the dispersal of colonists moving from north to south, thus contributing to the failure of the southern region to become repopulated.

Evolution of Proliferation and the Angiogenic Switch in Tumors with High Clonal Diversity

Natural selection among tumor cell clones is thought to produce hallmark properties of malignancy. Efforts to understand evolution of one such hallmark—the angiogenic switch—has suggested that selection for angiogenesis can “run away” and generate a hypertumor, a form of evolutionary suicide by extreme vascular hypo- or hyperplasia. This phenomenon is predicted by models of tumor angiogenesis studied with the techniques of adaptive dynamics. These techniques also predict that selection drives tumor proliferative potential towards an evolutionarily stable strategy (ESS) that is also convergence-stable. However, adaptive dynamics are predicated on two key assumptions: (i) no more than two distinct clones or evolutionary strategies can exist in the tumor at any given time; and (ii) mutations cause small phenotypic changes. Here we show, using a stochastic simulation, that relaxation of these assumptions has no effect on the predictions of adaptive dynamics in this case. In particular, selection drives proliferative potential towards, and angiogenic potential away from, their respective ESSs. However, these simulations also show that tumor behavior is highly contingent on mutational history, particularly for angiogenesis. Individual tumors frequently grow to lethal size before the evolutionary endpoint is approached. In fact, most tumor dynamics are predicted to be in the evolutionarily transient regime throughout their natural history, so that clinically, the ESS is often largely irrelevant. In addition, we show that clonal diversity as measured by the Shannon Information Index correlates with the speed of approach to the evolutionary endpoint. This observation dovetails with results showing that clonal diversity in Barretts esophagus predicts progression to malignancy.