How to Analyze Models of Nonlinear Public Goods

Public goods games often assume that the effect of the public good is a linear function of the number of contributions. In many cases, however, especially in biology, public goods have nonlinear effects, and nonlinear games are known to have dynamics and equilibria that can differ dramatically from linear games. Here I explain how to analyze nonlinear public goods games using the properties of Bernstein polynomials, and how to approximate the equilibria. I use mainly examples from the evolutionary game theory of cancer, but the approach can be used for a wide range of nonlinear public goods games

Codon Usage Bias and Mutation Constraints Reduce the Level of ErrorMinimization of the Genetic Code

Studies on the origin of the genetic code compare measures of the degree of error minimization of the standard code with measures produced by random variant codes but do not take into account codon usage, which was probably highly biased during the origin of the code. Codon usage bias could play an important role in the minimization of the chemical distances between amino acids because the importance of errors depends also on the frequency of the different codons. Here I show that when codon usage is taken into account, the degree of error minimization of the standard code may be dramatically reduced, and shifting to alternative codes often increases the degree of error minimization. This is especially true with a high CG content, which was probably the case during the origin of the code. I also show that the frequency of codes that perform better than the standard code, in terms of relative efficiency, is much higher in the neighborhood of the standard code itself, even when not considering codon usage bias; therefore alternative codes that differ only slightly from the standard code are more likely to evolve than some previous analyses suggested. My conclusions are that the standard genetic code is far from being an optimum with respect to error minimization and must have arisen for reasons other than error minimizatio

Selection on Codon Usage for Error Minimization at the Protein Level

Given the structure of the genetic code, synonymous codons differ in their capacity to minimize the effects of errors due to mutation or mistranslation. I suggest that this may lead, in protein-coding genes, to a preference for codons that minimize the impact of errors at the protein level. I develop a theoretical measure of error minimization for each codon, based on amino acid similarity. This measure is used to calculate the degree of error minimization for 82 genes of Drosophila melanogaster and 432 rodent genes and to study its relationship with CG content, the degree of codon usage bias, and the rate of nucleotide substitution. I show that (i) Drosophila and rodent genes tend to prefer codons that minimize errors; (ii) this cannot be merely the effect of mutation bias; (iii) the degree of error minimization is correlated with the degree of codon usage bias; (iv) the amino acids that contribute more to codon usage bias are the ones for which synonymous codons differ more in the capacity to minimize errors; and (v) the degree of error minimization is correlated with the rate of nonsynonymous substitution. These results suggest that natural selection for error minimization at the protein level plays a role in the evolution of coding sequences in Drosophila and rodent

Heterogeneity for IGF-II production maintained by public goods dynamics in neuroendocrine pancreatic cancer

The extensive intratumor heterogeneity revealed by sequencing cancer genomes is an essential determinant of tumor progression, diagnosis, and treatment. What maintains heterogeneity remains an open question because competition within a tumor leads to a strong selection for the fittest subclone. Cancer cells also cooperate by sharing molecules with paracrine effects, such as growth factors, and heterogeneity can be maintained if subclones depend on each other for survival. Without strict interdependence between subclones, however, nonproducer cells can free-ride on the growth factors produced by neighboring producer cells, a collective action problem known in game theory as the “tragedy of the commons,” which has been observed in microbial cell populations. Here, we report that similar dynamics occur in cancer cell populations. Neuroendocrine pancreatic cancer (insulinoma) cells that do not produce insulin-like growth factor II (IGF-II) grow slowly in pure cultures but have a proliferation advantage in mixed cultures, where they can use the IGF-II provided by producer cells. We show that, as predicted by evolutionary game theory, producer cells do not go extinct because IGF-II acts as a nonlinear public good, creating negative frequency-dependent selection that leads to a stable coexistence of the two cell types. Intratumor cell heterogeneity can therefore be maintained even without strict interdependence between cell subclones. Reducing the amount of growth factors available within a tumor may lead to a reduction in growth followed by a new equilibrium, which may explain relapse in therapies that target growth factors

A test of the coevolution theory of autumn colours: colour preference of Rhopalosiphum padi on Prunus padus

According to the coevolution theory of autumn colours, the bright colours of trees evolved as a warning signal towards parasites colonizing the plant in autumn. We monitored colonization of the aphid Rhopalosiphum padi on individual tress of Prunus padus in autumn and observed a strong preference of aphids for trees with green leaves. This is the first direct observation of a key assumption of the theory, that parasites avoid bright colours. Moreover our observations, compared with previous data gathered on the same species, suggest that aphids colonizing trees with green leaves develop better in spring than aphids colonizing trees with bright autumn colours, which is consistent with the second main assumption of the coevolution theory

Game Theory of Tumor–Stroma Interactions in Multiple Myeloma: Effect of Nonlinear Benefits

Cancer cells and stromal cells often exchange growth factors with paracrine effects that promote cell growth: a form of cooperation that can be studied by evolutionary game theory. Previous models have assumed that interactions between cells are pairwise or that the benefit of a growth factor is a linear function of its concentration. Diffusible factors, however, affect multiple cells and generally have nonlinear effects, and these differences are known to have important consequences for evolutionary dynamics. Here, we study tumor–stroma paracrine signaling using a model with multiplayer collective interactions in which growth factors have nonlinear effects. We use multiple myeloma as an example, modelling interactions between malignant plasma cells, osteoblasts, and osteoclasts. Nonlinear benefits can lead to results not observed in linear models, including internal mixed stable equilibria and cyclical dynamics. Models with linear effects, therefore, do not lead to a meaningful characterization of the dynamics of tumor–stroma interactions. To understand the dynamics and the effect of therapies it is necessary to estimate the shape of the benefit functions experimentally and parametrize models based on these functions

Stable Heterogeneity for the Production of Diffusible Factors in Cell Populations

The production of diffusible molecules that promote survival and growth is common in bacterial and eukaryotic cell populations, and can be considered a form of cooperation between cells. While evolutionary game theory shows that producers and non-producers can coexist in well-mixed populations, there is no consensus on the possibility of a stable polymorphism in spatially structured populations where the effect of the diffusible molecule extends beyond one-step neighbours. I study the dynamics of biological public goods using an evolutionary game on a lattice, taking into account two assumptions that have not been considered simultaneously in existing models: that the benefit of the diffusible molecule is a non-linear function of its concentration, and that the molecule diffuses according to a decreasing gradient. Stable coexistence of producers and non-producers is observed when the benefit of the molecule is a sigmoid function of its concentration, while strictly diminishing returns lead to coexistence only for very specific parameters and linear benefits never lead to coexistence. The shape of the diffusion gradient is largely irrelevant and can be approximated by a step function. Since the effect of a biological molecule is generally a sigmoid function of its concentration (as described by the Hill equation), linear benefits or strictly diminishing returns are not an appropriate approximations for the study of biological public goods. A stable polymorphism of producers and non-producers is in line with the predictions of evolutionary game theory and likely to be common in cell populations

Predicting Climate Change Impacts on the Amount and Duration of Autumn Colors in a New England Forest

Climate change affects the phenology of many species. As temperature and precipitation are thought to control autumn color change in temperate deciduous trees, it is possible that climate change might also affect the phenology of autumn colors. Using long-term data for eight tree species in a New England hardwood forest, we show that the timing and cumulative amount of autumn color are correlated with variation in temperature and precipitation at specific times of the year. A phenological model driven by accumulated cold degree-days and photoperiod reproduces most of the interspecific and interannual variability in the timing of autumn colors. We use this process-oriented model to predict changes in the phenology of autumn colors to 2099, showing that, while responses vary among species, climate change under standard IPCC projections will lead to an overall increase in the amount of autumn colors for most species.Organismic and Evolutionary Biolog

Cooperation among cancer cells as public goods games on Voronoi networks

Cancer cells produce growth factors that diffuse and sustain tumor proliferation, a form of cooperation among cancer cells that can be studied using mathematical models of public goods in the framework of evolutionary game theory. Cell populations, however, form heterogeneous networks that cannot be described by regular lattices or scale-free networks, the types of graphs generally used in the study of cooperation. To describe the dynamics of growth factor production in populations of cancer cells, I study public goods games on Voronoi networks, using a range of non-linear benefits that account for the known properties of growth factors, and different types of diffusion gradients. e results are surprisingly similar to those obtained on regular graphs and different from results on scale-free networks, revealing that network heterogeneity per se does not promote cooperation when public goods diffuse beyond one-step neighbours. e exact shape of the diffusion gradient is not crucial, however, whereas the type of non-linear benefit is an essential determinant of the dynamics. Public goods games on Voronoi networks can shed light on intra-tumor heterogeneity, the evolution of resistance to therapies that target growth factors, and new types of cell therapy