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## Spatial evolutionary games of interaction among generic cancer cells (2003)

Venue: | Computational and Mathematical Methods in Medicine 5 |

Citations: | 4 - 0 self |

### BibTeX

@INPROCEEDINGS{Bach03spatialevolutionary,

author = {L A Bach and D J T Sumpter and J Alsner and V Loeschcke},

title = {Spatial evolutionary games of interaction among generic cancer cells},

booktitle = {Computational and Mathematical Methods in Medicine 5},

year = {2003},

pages = {47--58}

}

### OpenURL

### Abstract

Evolutionary game models of cellular interactions have shown that heterogeneity in the cellular genotypic composition is maintained through evolution to stable coexistence of growth-promoting and non-promoting cell types. We generalise these mean-field models and relax the assumption of perfect mixing of cells by instead implementing an individual-based model that includes the stochastic and spatial effects likely to occur in tumours. The scope for coexistence of genotypic strategies changed with the inclusion of explicit space and stochasticity. The spatial models show some interesting deviations from their mean-field counterparts, for example the possibility of altruistic (paracrine) cell strategies to thrive. Such effects can however, be highly sensitive to model implementation and the more realistic models with semi-synchronous and stochastic updating do not show evolution of altruism. We do find some important and consistent differences between the spatial and mean-field models, in particular that the parameter regime for coexistence of growth-promoting and nonpromoting cell types is narrowed. For certain parameters in the model a selective collapse of a generic growth promoter occurs, hence the evolutionary dynamics mimics observable in vivo tumour phenomena such as (therapy induced) relapse behaviour. Our modelling approach differs from many of those previously applied in understanding growth of cancerous tumours in that it attempts to account for natural selection at a cellular level. This study thus points a new direction towards more plausible spatial tumour modelling and the understanding of cancerous growth. Keywords: Tumour cells; Evolutionary game theory; Individual-based model; Frequency-dependent interaction; Payoff matrices; Cellular interaction INTRODUCTION While there have been many mathematical models of cancerous tumours which attempt to characterise their growth (see for example Cancer cells in tumours typically show a variety of biochemically mediated mutual influences on cell proliferation. Therefore, local selection regimes, and hence the evolutionary dynamics can strongly depend on the nature of such cellular interactions Interaction metaphors such as the Dilemma games, the Hawk-Dove game, and the Ultimatum game have been widely used to model evolution of co-operation and frequency-dependent selection in evolutionary ecology In this paper, we make a step towards spatial models of cancerous growth that include selective forces. We extend Tomlinson and Bodmer's (TB) model to allow for a greater range of cell -cell interactions and thus encompass many of the types of interactions seen in real tumours. We discuss how these interactions affect cell proliferation in the general context of two-player games. We then employ a range of spatial individual-based, or cellular automata type, models to investigate the effect of spatially local interactions between cells. The effect of stochasticity and the importance of local random drift of the cell genotype frequencies are also explored. Bearing in mind the fixed positions of cells in solid tissue tumours, we believe that the significance of spatial effects in tumour cell interactions is importunately relevant for further progress in tumour modelling. The models, we present, are at this stage far from being predictive medical tools but we see our contribution as a "growth promoting factor" in the further development of mathematical models to understand tumour growth. MEAN-FIELD EVOLUTIONARY GAME MODELS In order to determine the evolutionary dynamics of tumour cells, the different genotypic strategies adopted by the cells can be expressed as a two-player game theory model. One such scenario, presented by The payoff matrix for TB model is given in and can be shown to have a single stable equilibrium at growth promoting cells. The TB model thus predicts, in the well mixed case at least, that growth promoter and non-promoter cells will coexist The TB model is in fact a specific instance of a general set of two-player games which can be expressed in terms The payoffs for such games can be normalised so that R ¼ 1 and P ¼ 0. As a result, each of these two player games may be categorised in terms of the values of S and T (Hauert, 2001, and references therein). The replicator equation for these games is then Table Ib gives S and T for the TB model. In terms of the S -T classification of two-player games, the TB game lies on the boundary between the Leader game (where S . 1 and T . 1; see Since S ¼ 1 in the TB game, it cannot be considered a particularly general model of growth promoting interactions. Indeed, if we again return to the biological basis for the model and consider how different cell genotypes may promote and inhibit growth we can produce a more general description of the interaction process. Specifically, we can introduce the additional biological observation that growth promoting cells are likely to obtain an additional benefit from interacting with another growth promoting cells, over and above the autocrine self-benefit, b. We will denote this extra benefit as e $ 0 (note that e ¼ 0 gives the original TB model). In defining a more general model, it is also useful to think of the benefit to non-promoting cells as being a parameter d . 0 which may be changed TABLE II Payoff matrix for the general model Gr þ / C and Gr 2 / D are growth promoter/cooporator and non-promoter/defector, respectively (a) Biologically defined interactions in a population consisting of a generic growth promoter, which can obtain the benefits b, and e and experience the cost c. The nonpromoter will obtain the benefit d from interacting with a growth promoter, whereas interaction with another non-promoter only yields a payoff of unity, i.e. base line proliferation. (b) Identical to (a) but rearranged following Hauert (2002) such that R ¼ 1 and P ¼ 0. SPATIAL EVOLUTIONARY GAMES 49 independently of the costs and benefits to the growth promoting cells. This gives payoff We can now use the payoff The equilibrium proportion of growth promoters in the case where b . c and d . b þ e 2 c is It appears, therefore, that the additional benefit, e does not change the general conclusion that the production of costly angiogenic factors is selected for amongst the cells. However, the equilibrium number of growth promoting cells is increased as the extra benefit increases (MaynardSmith, 1982). Other models of cell interactions are encompassed by the model in SPATIAL EVOLUTIONARY GAME MODELS Spatial patterns undoubtedly arise in the growth of cancerous tumours, for the simple reason that offspring cells grow adjacent to the parent cell. It is important, therefore, that spatial effects are accounted for when attempting to determine the growth and changing genotypic composition of tumours. Indeed, the fact that the Prisoner's Dilemma naturally arises out of cell interactions points to the intriguing possibility that spatial arrangements of cells may allow for truly altruistic effects on proliferation. That is, we may see the evolution of purely paracrine cells. The results of We now propose a variety of simple two dimensional lattice simulation models to determine the behaviour of such cell interaction dynamics Synchronous, Asynchronous or Semi-synchronous Updating Synchronous updating means that all the cells die simultaneously, and they are replaced dependent on the strategy of their neighbours before dying. Although synchronous updating is often the choice of implementation in spatial evolutionary game models it presents some problems. Synchronous updating assumes a global controller of the system, which ensures that all sites are updated exactly once in each iteration. This assumption, which gives a very coarse temporal granularity may be violated in a range of natural situations, particularly in the case of cell populations of a considerable size Asynchronous updating of cells means that on each generation a single cell, chosen at random, dies and is replaced. Asynchronous updating presents its own problems, however. The fact that at most one cellular site is updated in any given iteration means that any two adjacent sites can never be updated simultaneously since updating is strictly sequential. This in turn makes the disappearance of small clusters of cells impossible. Phenomena such as local episodes of low oxygen tension giving rise to apoptotic or necrotic areas with several adjacent cells dying necessitates the consideration of how a relaxation of the assumption of strict sequentiality will affect the system. We therefore, adopt an additional Semi-synchronous updating rule, which does not assume strict sequentiality in the turnover of cells and thus does not suffer the problems of such assumptions. Semi-synchronous updating represents an intermediate and more realistic temporal granularity between the extreme synchronous and extreme asynchronous updating. In the semi-synchronous scenarios presented here individual cellular mortality is 0.1, i.e. one tenth of the cells die and are replaced on each generation. Thus, with a probability of 0.01 any two adjacent cells will be updated concurrently. This method of updating allows for the biologically realistic situation that occasionally more than one local site is available for exploitation by proliferating neighbour cells. Neighbourhood Size A strong criticism of spatial game theory models is that results can be strongly dependent on the type of neighbourhood rules adopted. Indeed, the thorough investigation of Hauert Deterministic or Probabilistic Updating Two different schemes of competition for local reproduction are used in order to disclose possible effects of determinism vs stochasticity in the local competition for reproduction. The deterministic updating corresponds to the competitive situation where the "winner takes it all". The score of each individual is compared to all of its neighbours and only the cells with the highest local maximum score are allowed to reproduce. In case of a tie between two competitors a random cell is chosen. This means that whether or not a local cluster configuration of growth promoters will expand or diminish is governed by deterministic rules. In order to avoid the determinism and discreteness of the "winner takes it all" updating method an alternative implementation was designed. The individual's probability of reproduction is in the probabilistic updating defined in terms of its relative local payoff score. The probability of reproduction is given by the scaled value of own score divided by total score in the neighbourhood. Such local competition allows cellular strategies with lower fitness to have a chance of reproducing, especially when they are locally superior in numbers. In contrast to the deterministic reproduction this implies an effect of local density as to which strategy succeeds in proliferating. A strategy, which in numbers dominates the local neighbourhood can, albeit being inferior in terms of individual payoff, have a higher collective probability of occupying an empty site than, e.g. a single representative of the superior strategy. Inferentially, this effect becomes more pronounced the closer the payoff values of the competing strategies are. RESULTS Semi-synchronous simulations were run for 20,000 generation whereas asynchronous realisations were run SPATIAL EVOLUTIONARY GAMES 51 for 200,000 generations in order to ensure that a stable genotype distribution had been reached. The results shown are averaged over 2,000 and 20,000 generations for the semi-synchronous and the asynchronous scenarios, respectively. The Tomlinson and Bodmer Scenario The simulations based on the TB scenario Deterministic and probabilistic updating also differ in terms of effect of neighbourhood size. In the deterministic case the Moore and extended Moore neighbourhoods consistently showed lower equilibria of growth promoters for the entire parameter space. This is not the case in simulations applying probabilistic updating. Here, all spatial scenarios gave equlilibria in some cases above and in other cases below the mean-field prediction. In terms of the parameter values for which transitions in the dynamic equilibria occur, these reflect transitions in the local growth dynamics. For example the population level transition for b ¼ 2 as seen from The Hawk -Dove Scenario When a growth promoter interacting with another growth promoter is allowed to obtain the additional benefit (e) the scenario becomes identical to a Hawk -Dove game. By letting e ¼ 1; ðb 2 cÞ ¼ 0:5; and varying d, only the payoff of a non-promoter encountering a promoter (D -C interaction) is allowed to vary. The thin solid lines on SPATIAL EVOLUTIONARY GAMES 53 istic pattern of drastic transitions in the state of the quasi equilibria. Also, in this case there is no effect of temporal synchronousity on the dynamics when reproduction is deterministic (therefore, only the semi-synchronous realisations are shown in The Prisoner's Dilemma Scenario The parameter setting describing growth promotion as a truly altruistic act, hence the Prisoner's Dilemma scenario, showed, in contrast to the mean-field result, that clusters of growth promoters, i.e. co-operators, indeed can persist when c . b: In order to ascertain the robustness of these results similar simulations were conducted with the following parameter values: e ¼ 1:4; ðb 2 cÞ ¼ 20:1 and e ¼ 1:25; ðb 2 cÞ ¼ 20:25: The same pattern emerged as for the results shown in In contrast to the deterministic cases, when updating was probabilistic, growth promoters always went entirely extinct regardless of the initial conditions and the temporal granularity (results not shown). Hence, both the sequentially asynchronous and semi-synchronous realisations of probabilistic reproduction gave populations exclusively consisting of non-promoters. Local stochasticity as in the probabilistic competition for reproduction allows no possibility for persistence of clusters of "altruistic" growth promoters regardless of the level of synchronicity. In the probabilistic scenario clusters of growth promoters will be invaded by non-promoters. Hence with local stochasticity clusters of growth promoters are no longer immune to exploitation. Therefore, given enough time, such clusters will eventually vanish due to local exploitation in spite of spatial correlations. DISCUSSION AND CONCLUSION As might be expected from previous studies, the spatially extended interaction models showed markedly different results compared to their mean-field counterparts. Space seemed to amplify the proportion of growth promoters in many but not in all of our simulations. However, the spatial simulations were highly sensitive to the specific implementation, which in turn limits the scope for general statements about spatial effects. Because, we consider the probabilistic competition for reproduction and the semisynchronous updating more realistic than the corresponding deterministic and asynchronous representations, the conclusion from previous evolutionary models that space generally favours co-operative strategies does in our view not necessarily apply for populations of cells There are in general, however, significant deviations from the mean-field predictions in our spatial models. These seem to be exacerbated by small neighbourhood sizes: both the deterministic and probabilistic updating scenarios show larger deviations in simulations assuming the von Neumann neighbourhood. Moreover, when probabilistic updating is considered, deviations from the mean-field predictions are highly dependent on whether or not concurrent updates of neighbours are permitted, i.e. semi-synchronous or synchronous updating. Our findings do not fully comply with The biologically motivated probabilistic semi-synchronous scenario showed a pronounced deviation from the mean-field result, in particular in the TB scenario. Indeed, the "window of coexistence" between growth promoting and non-promoting cells is smaller for semi-synchronous than synchronous updating (seen by comparing Figs. 2b -c and 3b -c). Such effects were again more pronounced in the smaller von Neumann neighbourhood. Hence, the degree and level of operation of stochasticity play a non-trivial role concerning the deviation from the mean-field results (see for another example