IsoMaTrix: a framework to visualize the isoclines of matrix games and quantify uncertainty in structured populations.

SUMMARY: Evolutionary game theory describes frequency-dependent selection for fixed, heritable strategies in a population of competing individuals using a payoff matrix. We present a software package to aid in the construction, analysis and visualization of three-strategy matrix games. The IsoMaTrix package computes the isoclines (lines of zero growth) of matrix games, and facilitates direct comparison of well-mixed dynamics to structured populations on a lattice grid. IsoMaTrix computes fixed points, phase flow, trajectories, (sub)velocities and uncertainty quantification for stochastic effects in spatial matrix games. We describe a result obtained via IsoMaTrix's spatial games functionality, which shows that the timing of competitive release in a cancer model (under continuous treatment) critically depends on the initial spatial configuration of the tumor. AVAILABILITY AND IMPLEMENTATION: The code is available at: https://github.com/mathonco/isomatrix. SUPPLEMENTARY INFORMATION: Supplementary data are available at Bioinformatics online.

The 2019 mathematical oncology roadmap.

Whether the nom de guerre is Mathematical Oncology, Computational or Systems Biology, Theoretical Biology, Evolutionary Oncology, Bioinformatics, or simply Basic Science, there is no denying that mathematics continues to play an increasingly prominent role in cancer research. Mathematical Oncology-defined here simply as the use of mathematics in cancer research-complements and overlaps with a number of other fields that rely on mathematics as a core methodology. As a result, Mathematical Oncology has a broad scope, ranging from theoretical studies to clinical trials designed with mathematical models. This Roadmap differentiates Mathematical Oncology from related fields and demonstrates specific areas of focus within this unique field of research. The dominant theme of this Roadmap is the personalization of medicine through mathematics, modelling, and simulation. This is achieved through the use of patient-specific clinical data to: develop individualized screening strategies to detect cancer earlier; make predictions of response to therapy; design adaptive, patient-specific treatment plans to overcome therapy resistance; and establish domain-specific standards to share model predictions and to make models and simulations reproducible. The cover art for this Roadmap was chosen as an apt metaphor for the beautiful, strange, and evolving relationship between mathematics and cancer.

Moral externalization may precede, not follow, subjective preferences.

We offer four counterarguments against Stanford's dismissal of moral externalization as an ancestral condition, based on requirements for ancestral states, mismatch between theoretical and empirical games, passively correlated interactions, and social interfaces that prevent agents' knowing game payoffs. The fact that children's externalized phenomenology precedes their discovery of subjectivized phenomenology also suggests that externalized phenomenology is an ancestral condition.

Cancer treatment scheduling and dynamic heterogeneity in social dilemmas of tumour acidity and vasculature.

BACKGROUND: Tumours are diverse ecosystems with persistent heterogeneity in various cancer hallmarks like self-sufficiency of growth factor production for angiogenesis and reprogramming of energy metabolism for aerobic glycolysis. This heterogeneity has consequences for diagnosis, treatment and disease progression. METHODS: We introduce the double goods game to study the dynamics of these traits using evolutionary game theory. We model glycolytic acid production as a public good for all tumour cells and oxygen from vascularisation via vascular endothelial growth factor production as a club good benefiting non-glycolytic tumour cells. This results in three viable phenotypic strategies: glycolytic, angiogenic and aerobic non-angiogenic. RESULTS: We classify the dynamics into three qualitatively distinct regimes: (1) fully glycolytic; (2) fully angiogenic; or (3) polyclonal in all three cell types. The third regime allows for dynamic heterogeneity even with linear goods, something that was not possible in prior public good models that considered glycolysis or growth factor production in isolation. CONCLUSIONS: The cyclic dynamics of the polyclonal regime stress the importance of timing for anti-glycolysis treatments like lonidamine. The existence of qualitatively different dynamic regimes highlights the order effects of treatments. In particular, we consider the potential of vascular normalisation as a neoadjuvant therapy before follow-up with interventions like buffer therapy.

Fidelity drive: a mechanism for chaperone proteins to maintain stable mutation rates in prokaryotes over evolutionary time.

We show a mechanism by which chaperone proteins can play a key role in maintaining the long-term evolutionary stability of mutation rates in prokaryotes with perfect genetic linkage. Since chaperones can reduce the phenotypic effects of mutations, higher mutation rate, by affecting chaperones, can increase the phenotypic effects of mutations. This in turn leads to greater mutation effect among the proteins that control mutation repair and DNA replication, resulting in large changes in mutation rate. The converse of this is that when mutation rate is low and chaperones are functioning well, then the rate of change in mutation rate will also be low, leading to low mutation rates being evolutionarily frozen. We show that the strength of this recursion is critical to determining the long-term evolutionary patterns of mutation rate among prokaryotes. If this recursion is weak, then mutation rates can grow without bound, leading to the extinction of the lineage. However, if this recursion is strong, then we can reproduce empirical patterns of prokaryotic mutation rates, where mutation rates remain stable over evolutionary time, and where most mutation rates are low, but with a significant fraction of high mutators.

Limitations of the Dirac formalism as a descriptive framework for cognition.

We highlight methodological and theoretical limitations of the authors' Dirac formalism and suggest the von Neumann open systems approach as a resolution. The open systems framework is a generalization of classical probability and we hope it will allow cognitive scientists to extend quantum probability from perception, categorization, memory, decision making, and similarity judgments to phenomena in learning and development.

The Contribution of Evolutionary Game Theory to Understanding and Treating Cancer.

Evolutionary game theory mathematically conceptualizes and analyzes biological interactions where one's fitness not only depends on one's own traits, but also on the traits of others. Typically, the individuals are not overtly rational and do not select, but rather inherit their traits. Cancer can be framed as such an evolutionary game, as it is composed of cells of heterogeneous types undergoing frequency-dependent selection. In this article, we first summarize existing works where evolutionary game theory has been employed in modeling cancer and improving its treatment. Some of these game-theoretic models suggest how one could anticipate and steer cancer's eco-evolutionary dynamics into states more desirable for the patient via evolutionary therapies. Such therapies offer great promise for increasing patient survival and decreasing drug toxicity, as demonstrated by some recent studies and clinical trials. We discuss clinical relevance of the existing game-theoretic models of cancer and its treatment, and opportunities for future applications. Moreover, we discuss the developments in cancer biology that are needed to better utilize the full potential of game-theoretic models. Ultimately, we demonstrate that viewing tumors with evolutionary game theory has medically useful implications that can inform and create a lockstep between empirical findings and mathematical modeling. We suggest that cancer progression is an evolutionary competition between different cell types and therefore needs to be viewed as an evolutionary game.

Measuring competitive exclusion in non-small cell lung cancer.

In this study, we experimentally measure the frequency-dependent interactions between a gefitinib-resistant non-small cell lung cancer population and its sensitive ancestor via the evolutionary game assay. We show that cost of resistance is insufficient to accurately predict competitive exclusion and that frequency-dependent growth rate measurements are required. Using frequency-dependent growth rate data, we then show that gefitinib treatment results in competitive exclusion of the ancestor, while the absence of treatment results in a likely, but not guaranteed, exclusion of the resistant strain. Then, using simulations, we demonstrate that incorporating ecological growth effects can influence the predicted extinction time. In addition, we show that higher drug concentrations may not lead to the optimal reduction in tumor burden. Together, these results highlight the potential importance of frequency-dependent growth rate data for understanding competing populations, both in the laboratory and as we translate adaptive therapy regimens to the clinic.

Computational Complexity as an Ultimate Constraint on Evolution.

Experiments show that evolutionary fitness landscapes can have a rich combinatorial structure due to epistasis. For some landscapes, this structure can produce a computational constraint that prevents evolution from finding local fitness optima-thus overturning the traditional assumption that local fitness peaks can always be reached quickly if no other evolutionary forces challenge natural selection. Here, I introduce a distinction between easy landscapes of traditional theory where local fitness peaks can be found in a moderate number of steps, and hard landscapes where finding local optima requires an infeasible amount of time. Hard examples exist even among landscapes with no reciprocal sign epistasis; on these semismooth fitness landscapes, strong selection weak mutation dynamics cannot find the unique peak in polynomial time. More generally, on hard rugged fitness landscapes that include reciprocal sign epistasis, no evolutionary dynamics-even ones that do not follow adaptive paths-can find a local fitness optimum quickly. Moreover, on hard landscapes, the fitness advantage of nearby mutants cannot drop off exponentially fast but has to follow a power-law that long-term evolution experiments have associated with unbounded growth in fitness. Thus, the constraint of computational complexity enables open-ended evolution on finite landscapes. Knowing this constraint allows us to use the tools of theoretical computer science and combinatorial optimization to characterize the fitness landscapes that we expect to see in nature. I present candidates for hard landscapes at scales from single genes, to microbes, to complex organisms with costly learning (Baldwin effect) or maintained cooperation (Hankshaw effect). Just how ubiquitous hard landscapes (and the corresponding ultimate constraint on evolution) are in nature becomes an open empirical question.

Fibroblasts and alectinib switch the evolutionary games played by non-small cell lung cancer.

Heterogeneity in strategies for survival and proliferation among the cells that constitute a tumour is a driving force behind the evolution of resistance to cancer therapy. The rules mapping the tumour's strategy distribution to the fitness of individual strategies can be represented as an evolutionary game. We develop a game assay to measure effective evolutionary games in co-cultures of non-small cell lung cancer cells that are sensitive and resistant to the anaplastic lymphoma kinase inhibitor alectinib. The games are not only quantitatively different between different environments, but targeted therapy and cancer-associated fibroblasts qualitatively switch the type of game being played by the in vitro population from Leader to Deadlock. This observation provides empirical confirmation of a central theoretical postulate of evolutionary game theory in oncology: we can treat not only the player, but also the game. Although we concentrate on measuring games played by cancer cells, the measurement methodology we develop can be used to advance the study of games in other microscopic systems by providing a quantitative description of non-cell-autonomous effects.

Fractionated follow-up chemotherapy delays the onset of resistance in bone metastatic prostate cancer.

Prostate cancer to bone metastases are almost always lethal. This results from the ability of metastatic prostate cancer cells to co-opt bone remodeling leading to what is known as the vicious cycle. Understanding how tumor cells can disrupt bone homeostasis through their interactions with the stroma and how metastatic tumors respond to treatment is key to the development of new treatments for what remains an incurable disease. Here we describe an evolutionary game theoretical model of both the homeostatic bone remodeling and its co-option by prostate cancer metastases. This model extends past the evolutionary aspects typically considered in game theoretical models by also including ecological factors such as the physical microenvironment of the bone. Our model recapitulates the current paradigm of the "vicious cycle" driving tumor growth and sheds light on the interactions of heterogeneous tumor cells with the bone microenvironment and treatment response. Our results show that resistant populations naturally become dominant in the metastases under conventional cytotoxic treatment and that novel schedules could be used to better control the tumor and the associated bone disease compared to the current standard of care. Specifically, we introduce fractionated follow up therapy - chemotherapy where dosage is administered initially in one solid block followed by alternating smaller doeses and holidays - and argue that it is better than either a continuous application or a periodic one. Furthermore, we also show that different regimens of chemotherapy can lead to different amounts of pathological bone that are known to correlate with poor quality of life for bone metastatic prostate cancer patients.

Edge effects in game-theoretic dynamics of spatially structured tumours.

Cancer dynamics are an evolutionary game between cellular phenotypes. A typical assumption in this modelling paradigm is that the probability of a given phenotypic strategy interacting with another depends exclusively on the abundance of those strategies without regard for local neighbourhood structure. We address this limitation by using the Ohtsuki-Nowak transform to introduce spatial structure to the go versus grow game. We show that spatial structure can promote the invasive (go) strategy. By considering the change in neighbourhood size at a static boundary--such as a blood vessel, organ capsule or basement membrane--we show an edge effect that allows a tumour without invasive phenotypes in the bulk to have a polyclonal boundary with invasive cells. We present an example of this promotion of invasive (epithelial-mesenchymal transition-positive) cells in a metastatic colony of prostate adenocarcinoma in bone marrow. Our results caution that pathologic analyses that do not distinguish between cells in the bulk and cells at a static edge of a tumour can underestimate the number of invasive cells. Although we concentrate on applications in mathematical oncology, we expect our approach to extend to other evolutionary game models where interaction neighbourhoods change at fixed system boundaries.