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.

Rich dynamics of a hepatitis B viral infection model with logistic hepatocyte growth.

Chronic hepatitis B virus (HBV) infection is a major cause of human suffering, and a number of mathematical models have examined within-host dynamics of the disease. Most previous HBV infection models have assumed that: (a) hepatocytes regenerate at a constant rate from a source outside the liver; and/or (b) the infection takes place via a mass action process. Assumption (a) contradicts experimental data showing that healthy hepatocytes proliferate at a rate that depends on current liver size relative to some equilibrium mass, while assumption (b) produces a problematic basic reproduction number. Here we replace the constant infusion of healthy hepatocytes with a logistic growth term and the mass action infection term by a standard incidence function; these modifications enrich the dynamics of a well-studied model of HBV pathogenesis. In particular, in addition to disease free and endemic steady states, the system also allows a stable periodic orbit and a steady state at the origin. Since the system is not differentiable at the origin, we use a ratio-dependent transformation to show that there is a region in parameter space where the origin is globally stable. When the basic reproduction number, R (0), is less than 1, the disease free steady state is stable. When R (0) > 1 the system can either converge to the chronic steady state, experience sustained oscillations, or approach the origin. We characterize parameter regions for all three situations, identify a Hopf and a homoclinic bifurcation point, and show how they depend on the basic reproduction number and the intrinsic growth rate of hepatocytes.

Prostate-specific antigen dynamics predict individual responses to intermittent androgen deprivation.

Intermittent androgen deprivation therapy (IADT) is an attractive treatment for biochemically recurrent prostate cancer (PCa), whereby cycling treatment on and off can reduce cumulative dose and limit toxicities. We simulate prostate-specific antigen (PSA) dynamics, with enrichment of PCa stem-like cell (PCaSC) during treatment as a plausible mechanism of resistance evolution. Simulated PCaSC proliferation patterns correlate with longitudinal serum PSA measurements in 70 PCa patients. Learning dynamics from each treatment cycle in a leave-one-out study, model simulations predict patient-specific evolution of resistance with an overall accuracy of 89% (sensitivity = 73%, specificity = 91%). Previous studies have shown a benefit of concurrent therapies with ADT in both low- and high-volume metastatic hormone-sensitive PCa. Model simulations based on response dynamics from the first IADT cycle identify patients who would benefit from concurrent docetaxel, demonstrating the feasibility and potential value of adaptive clinical trials guided by patient-specific mathematical models of intratumoral evolutionary dynamics.

Dynamic simulations of single-molecule enzyme networks.

Along with the growth of technologies allowing accurate visualization of biochemical reactions to the scale of individual molecules has arisen an appreciation of the role of statistical fluctuations in intracellular biochemistry. The stochastic nature of metabolism can no longer be ignored. It can be probed empirically, and theoretical studies have established its importance. Traditional methods for modeling stochastic biochemistry are derived from an elegant and physically satisfying theory developed by Gillespie. However, although Gillespie's algorithm and its derivatives efficiently model small-scale systems, complex networks are harder to manage on easily available computer systems. Here we present a novel method of simulating stochastic biochemical networks using discrete events simulation techniques borrowed from manufacturing production systems. The method is very general and can be mapped to an arbitrarily complex network. As an illustration, we apply the technique to the glucose phosphorylation steps of the Embden-Meyerhof-Parnas pathway in E. coli . We show that a deterministic version of the discrete event simulation reproduces the behavior of an analogous deterministic differential equation model. The stochastic version of the same model predicts that catastrophic bottlenecks in the system are more likely than one would expect from deterministic theory.

Modeling the population dynamics of lemon sharks.

BACKGROUND: Long-lived marine megavertebrates (e.g. sharks, turtles, mammals, and seabirds) are inherently vulnerable to anthropogenic mortality. Although some mathematical models have been applied successfully to manage these animals, more detailed treatments are often needed to assess potential drivers of population dynamics. In particular, factors such as age-structure, density-dependent feedbacks on reproduction, and demographic stochasticity are important for understanding population trends, but are often difficult to assess. Lemon sharks (Negaprion brevirostris) have a pelagic adult phase that makes them logistically difficult to study. However, juveniles use coastal nursery areas where their densities can be high. RESULTS: We use a stage-structured, Markov-chain stochastic model to describe lemon shark population dynamics from a 17-year longitudinal dataset at a coastal nursery area at Bimini, Bahamas. We found that the interaction between delayed breeding, density-dependence, and demographic stochasticity accounts for 33 to 49% of the variance in population size. CONCLUSIONS: Demographic stochasticity contributed all random effects in this model, suggesting that the existence of unmodeled environmental factors may be driving the majority of interannual population fluctuations. In addition, we are able to use our model to estimate the natural mortality rate of older age classes of lemon sharks that are difficult to study. Further, we use our model to examine what effect the length of a time series plays on deciphering ecological patterns. We find that-even with a relatively long time series-our sampling still misses important rare events. Our approach can be used more broadly to infer population dynamics of other large vertebrates in which age structure and demographic stochasticity are important. REVIEWERS: This article was reviewed by Yang Kuang, Christine Jacob, and Ollivier Hyrien.

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 Barrett's esophagus predicts progression to malignancy.

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.

Evolution of uncontrolled proliferation and the angiogenic switch in cancer.

The major goal of evolutionary oncology is to explain how malignant traits evolve to become cancer ``hallmarks." One such hallmark---the angiogenic switch---is difficult to explain for the same reason altruism is difficult to explain. An angiogenic clone is vulnerable to ``cheater" lineages that shunt energy from angiogenesis to proliferation, allowing the cheater to outcompete cooperative phenotypes in the environment built by the cooperators. Here we show that cell- or clone-level selection is sufficient to explain the angiogenic switch, but not because of direct selection on angiogenesis factor secretion---angiogenic potential evolves only as a pleiotropic afterthought. We study a multiscale mathematical model that includes an energy management system in an evolving angiogenic tumor. The energy management model makes the counterintuitive prediction that ATP concentration in resting cells increases with increasing ATP hydrolysis, as seen in other theoretical and empirical studies. As a result, increasing ATP hydrolysis for angiogenesis can increase proliferative potential, which is the trait directly under selection. Intriguingly, this energy dynamic allows an evolutionary stable angiogenesis strategy, but this strategy is an evolutionary repeller, leading to runaway selection for extreme vascular hypo- or hyperplasia. The former case yields a tumor-on-a-tumor, or hypertumor, as predicted in other studies, and the latter case may explain vascular hyperplasia evident in certain tumor types.

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

BACKGROUND: Androgens 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. RESULTS: We 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 5alpha-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. CONCLUSIONS: The 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 5alpha-reductase inhibitors such as finasteride may increase the incidence of therapy resistant cancers.

The dynamics of a delay model of hepatitis B virus infection with logistic hepatocyte growth.

Chronic HBV affects 350 million people and can lead to death through cirrhosis-induced liver failure or hepatocellular carcinoma. We analyze the dynamics of a model considering logistic hepatocyte growth and a standard incidence function governing viral infection. This model also considers an explicit time delay in virus production. With this model formulation all model parameters can be estimated from biological data; we also simulate a course of lamivudine therapy and find that the model gives good agreement with clinical data. Previous models considering constant hepatocyte growth have permitted only two dynamical possibilities: convergence to a virus free or a chronic steady state. Our model admits a third possibility of sustained oscillations. We show that when the basic reproductive number is greater than 1 there exists a biologically meaningful chronic steady state, and the stability of this steady state is dependent upon both the rate of hepatocyte regeneration and the virulence of the disease. When the chronic steady state is unstable, simulations show the existence of an attracting periodic orbit. Minimum hepatocyte populations are very small in the periodic orbit, and such a state likely represents acute liver failure. Therefore, the often sudden onset of liver failure in chronic HBV patients can be explained as a switch in stability caused by the gradual evolution of parameters representing the disease state.

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 Gilg's 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 May's 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.

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.

Why don't all whales have cancer? A novel hypothesis resolving Peto's paradox.

Larger organisms have more potentially carcinogenic cells, tend to live longer and require more ontogenic cell divisions. Therefore, intuitively one might expect cancer incidence to scale with body size. Evidence from mammals, however, suggests that the cancer risk does not correlate with body size. This observation defines "Peto's paradox." Here, we propose a novel hypothesis to resolve Peto's paradox. We suggest that malignant tumors are disadvantaged in larger hosts. In particular, we hypothesize that natural selection acting on competing phenotypes among the cancer cell population will tend to favor aggressive "cheaters" that then grow as a tumor on their parent tumor, creating a hypertumor that damages or destroys the original neoplasm. In larger organisms, tumors need more time to reach lethal size, so hypertumors have more time to evolve. So, in large organisms, cancer may be more common and less lethal. We illustrate this hypothesis in silico using a previously published hypertumor model. Results from the model predict that malignant neoplasms in larger organisms should be disproportionately necrotic, aggressive, and vascularized than deadly tumors in small mammals. These predictions may serve as the basis on which to test the hypothesis, but to our knowledge, no one has yet performed a systematic investigation of comparative necrosis, histopathology, or vascularization among mammalian cancers.

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 approximately 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).

The ecology and evolutionary biology of cancer: a review of mathematical models of necrosis and tumor cell diversity.

Recent evidence elucidating the relationship between parenchyma cells and otherwise "healthy" cells in malignant neoplasms is forcing cancer biologists to expand beyond the genome-centered, "one-renegade-cell" theory of cancer. As it becomes more and more clear that malignant transformation is context dependent, the usefulness of an evolutionary ecology-based theory of malignant neoplasia becomes increasingly clear. This review attempts to synthesize various theoretical structures built by mathematical oncologists into potential explanations of necrosis and cellular diversity, including both total cell diversity within a tumor and cellular pleomorphism within the parenchyma. The role of natural selection in necrosis and pleomorphism is also examined. The major hypotheses suggested as explanations of these phenomena are outlined in the conclusions section of this review. In every case, mathematical oncologists have built potentially valuable models that yield insight into the causes of necrosis, cell diversity, and nearly every other aspect of malignancy; most make predictions ultimately testable in the lab or clinic. Unfortunately, these advances have gone largely unexploited by the empirical community. Possible reasons why are considered.

Competition and natural selection in a mathematical model of cancer.

A malignant tumor is a dynamic amalgamation of various cell phenotypes, both cancerous (parenchyma) and healthy (stroma). These diverse cells compete over resources as well as cooperate to maintain tumor viability. Therefore, tumors are both an ecological community and an integrated tissue. An understanding of how natural selection operates in this unique ecological context should expose unappreciated vulnerabilities shared by all cancers. In this study I address natural selection's role in tumor evolution by developing and exploring a mathematical model of a heterogenous primary neoplasm. The model is a system of nonlinear ordinary differential equations tracking the mass of up to two different parenchyma cell types, the mass of vascular endothelial cells from which new tumor blood vessels are built and the total length of tumor microvessels. Results predict the possibility of a hypertumor-a focus of aggressively reproducing parenchyma cells that invade and destroy part or all of the tumor, perhaps before it becomes a clinical entity. If this phenomenon occurs, then we should see examples of tumors that develop an aggressive histology but are paradoxically prone to extinction. Neuroblastoma, a common childhood cancer, may sometimes fit this pattern. In addition, this model suggests that parenchyma cell diversity can be maintained by a tissue-like integration of cells specialized to provide different services.