Cooperative success in epithelial public goods games
Introduction
Oncogenesis is a process of somatic evolution. In order to become cancerous there are certain key mutations which cells must obtain, corresponding to the hallmarks of cancer (Hanahan and Weinberg, 2000, Hanahan and Weinberg, 2011). Evolutionary game theory provides a framework for modelling mutations which have a fitness effect beyond the cell itself. For example, certain mutations can be considered cooperative, in that they invoke a cost to the cell which is recuperated as a shared benefit. This is evident when the benefit relies on the production of a diffusible growth factor (Jouanneau et al., 1994, Axelrod et al., 2006), as is the case for a number of the hallmarks of cancer, such as self-sufficiency in growth signalling and sustained angiogenesis. The Warburg effect, whereby tumour cells metabolise through glycolysis even when oxygen is abundant (Warburg, 1956), can also be considered cooperative (Archetti, 2014).
Cooperative mutations benefit the population as a whole; however, it is often the case that defection (e.g. not producing growth factor) results in higher individual fitness. This is because the defector shares in the benefits without paying any fitness costs associated with cooperating. Understanding the conditions under which cooperation can evolve, despite the incentive to defect, has been a topic of extensive study within evolutionary game theory (Nowak, 2006, Ohtsuki et al., 2006a, Allen et al., 2016).
Cooperation is usually considered to be a desirable outcome. For example, within the social sphere or amongst healthy constituent cells of a multicellular organism. Cooperation between cancer cells, however, can drive tumour growth (Marusyk etal., 2014). This is of course detrimental to the patient, and thus, disrupting cooperation between cancer subclones, possibly by exploiting its evolutionary weaknesses, could be an important avenue for treatment (Archetti, 2013a, Zhou et al., 2017).
Applications of evolutionary game theory to model cancer evolution have mainly focussed on two-player games, whereby cells participate in multiple pairwise interactions within the population (Tomlinson, 1997, Basanta et al., 2008a, Hummert et al., 2014). Interactions between cancer cells, however, tend to happen in groups. For example, a cell producing a growth factor will provide a benefit to other cells within its diffusion range. These types of mutations are thus better represented as multiplayer public goods games (PGGs) (Archetti and Scheuring, 2012), played between producer (cooperator) and non-producer (defector) cells. The former produce growth factor at a fixed cost to their fitness. Both producers and non-producers receive a fitness benefit as a function of the frequency of producers in their interaction neighbourhood.
The most common PGG, known as the N-person prisoner’s dilemma (NPD), uses a linear benefit function (Hauert et al., 2002, Santos et al., 2008). However, non-linear benefit functions may be more realistic (Archetti et al., 2015, Archetti et al., 2020), and can lead to much richer dynamics, even for well-mixed populations. An example is the volunteer’s dilemma (VD), which defines the benefit as a Heaviside step function (Bach et al., 2001, Bach et al., 2006, Archetti, 2009a, Archetti, 2009b).
A sigmoid benefit function has been proposed as an appropriate model for growth factor production. Experiments on neuroendocrine pancreatic cancer cells in vitro have found sigmoid dependence of proliferation rates on the concentration of growth factor IGF-II (Archetti et al., 2015). Furthermore, such a function is relatively general, with both the NPD and VD arising as extreme cases (Archetti and Scheuring, 2011).
Most cancers originate in epithelia. These are tissues formed of sheets of cells, which are approximately polygonal on their apical surfaces. It is important to take into account this population structure when modelling the evolutionary dynamics. For both two-player cooperation games (Ohtsuki et al., 2006b, Nowak et al., 2010) and multiplayer PGGs (Peña et al., 2016), cooperators tend to have greater success in structured populations, as compared to well-mixed ones, because they are able to form mutually beneficial clusters.
Evolution on structured populations is usually modelled within the framework of evolutionary graph theory (Lieberman et al., 2005), in which the population is represented as a fixed graph. Epithelial cells tend to have six neighbours on average, and thus can be represented as a hexagonal lattice. Introducing more realistic population structures, with small variation in neighbour number, does not have a significant impact on evolutionary outcomes (Archetti, 2016, Renton and Page, 2019).
The success of cooperation is also dependent on the update dynamics. Within evolutionary graph theory, the population evolves according to an update rule. In general, update rules can be divided into two categories: local and global (Nathanson et al., 2009).
A local update involves a spatial relationship between birth and death events. Evolutionary graph theory usually requires a local update rule in order to maintain the fixed graph structure. Two commonly used local update rules are defined as follows:
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birth–death: a cell is selected to divide with probability proportional to fitness; one of its neighbours is chosen to die uniformly at random.
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death-birth: a cell is chosen to die uniformly at random; one of its neighbours is selected to divide with probability proportional to fitness
In both cases the offspring of the dividing cell occupies the empty site left by the dead cell (Zukewich et al., 2013). The choice between these update rules has a substantive effect on evolutionary outcomes. For example, consider a two-player prisoner’s dilemma game and a population represented by a regular graph. Cooperation can be favoured for a death-birth update rule, so long as the benefit is high enough. For the birth–death update, however, as is the case with a well-mixed population, cooperation is only favoured for an infinitely large benefit (Ohtsuki et al., 2006b).
These update rules are sometimes referred to as BD-B (birth–death with selection on birth) and DB-B (death-birth with selection on birth) to emphasise that selection is acting on birth. Alternative update rules, for which selection acts on death, can then be referred to as BD-D and DB-D (Masuda, 2009). In this paper, we limit ourselves to the case where selection acts on birth, thus we do not use this notation to differentiate the two cases.
Under a global update rule there is no spatial dependence between birth and death events; thus, cells are selected to reproduce and die from the population as a whole. Global updating is generally seen for well-mixed populations, or when populations are organised in phenotype space (Antal et al., 2009) or by sets (Tarnita et al., 2009b).
Within evolutionary graph theory the shift update rule is an example of global updating. In this case a cell is chosen to divide with probability proportional to fitness, and a second cell is chosen to die uniformly at random. A path is then selected on the graph which connects the two. Cells are shifted along this path until there is an empty node next to the dividing cell for its progeny to occupy. This kind of update works well on a one-dimensional lattice (Allen and Nowak, 2012), and promotes cooperation, even compared to the death-birth update. However, it becomes more complex in two-dimensions (Pavlogiannis et al., 2015), because division causes cellular rearrangement at a distance from the event.
Evolutionary graph theory has several shortcomings for modelling invasion processes in epithelia. Firstly, it assumes that the population can be represented by a static graph, whereas epithelia are dynamic structures. Secondly, as we have discussed, introducing global update rules into evolutionary graph theory presents challenges to the modelling framework (Pavlogiannis et al., 2015).
The question then arises as to which update rule is most realistic for an epithelium. This will depend on the extent to which death and division processes are spatially coupled. For homeostatic tissues it is likely that contact inhibition, the phenomenon whereby cells stop proliferating at high density, plays an important role in maintaining the population size (Mesa et al., 2018). The death-birth update rule could be an appropriate model when contact inhibition is very strong, as tissue density is likely to be low near a recent death. Conversely, a global update rule is likely to be more realistic when contact inhibition is weaker and thus there is less spatial dependence between death and division.
The death-birth and decoupled update rules represent extreme cases of spatial coupling between division and death. In this paper we focus on global updating, as the death-birth update rule, along with other local update rules, has been extensively studied within evolutionary graph theory (Ohtsuki et al., 2006b, Maciejewski et al., 2014, Peña et al., 2016, Allen et al., 2016). In future work, we will consider the spectrum of spatial coupling that can arise in a tissue due to contact inhibition, and how this affects the evolution of cooperation.
In line with our previous work (Renton and Page, 2019), we use the Voronoi tessellation (VT) model (Meineke et al., 2001, Van Leeuwen et al., 2009) to represent epithelial dynamics. Unlike traditional evolutionary graph theory models, the tissue structure is dynamic and cells are able to divide and die independently. It is thus straightforward to spatially decouple birth and death, and we are able to introduce a global form of updating, we call the decoupled update rule. In Renton and Page (2019), we used this framework to analyse the two-player prisoner’s dilemma game, finding that cooperation was more successful for the decoupled update rule, than for a death-birth update rule. The present paper extends these results to a wide range of multiplayer public goods games, as well as deriving general results for global update rules.
We aim to extend the range of applicability of quasi-analytical methods from evolutionary game theory to more realistic tissue models. We have chosen to use the VT model, because it uses a very simple force law and, as a cell-centre model, naturally provides the graph structure needed for evolutionary games (Meineke et al., 2001). Furthermore, unlike cellular automata models, cell division leads only to local topological changes. The VT model has been used to represent cellular dynamics in colonic and intestinal crypts, including for models of invasion (Mirams et al., 2012, Romijn et al., 2020). Other tissue models, such as the vertex model (Farhadifar et al., 2007), could also be appropriate for our purposes.
The version of the VT model we use represents a simple epithelium1 as a two-dimensional structure. Thus our results are mostly relevant to the early stages of tumorigenesis or field cancerization (Curtius et al., 2017) in simple epithelia. While models of later stage tumour evolution would be more appropriately modelled in three dimensions, two-dimensional models can still be useful in the first instance.
For stochastic evolutionary games without mutation, we can compare the success of different strategies by calculating fixation probabilities. Here we consider the dynamics of two cell types: A and B. The fixation probability is then defined as the probability that a single initial mutant X will eventually take over the entire population. We consider two measures for the success of an A mutant (Zukewich et al., 2013, Maciejewski et al., 2014):
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A is a beneficial mutation when . Here is the fixation probability for a neutral mutant and Z is the population size.
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A is favoured by selection, or has an evolutionary advantage, when . This is equivalent to the condition that the equilibrium frequency of A is greater than a half when mutation is allowed (A is the dominant strategy).
In general, these conditions are not equivalent, thus it is possible for a mutation to be beneficial but not favoured, or vice versa. One or the other condition might be more relevant to quantifying mutant success depending on the circumstances. Furthermore, under certain circumstances these two conditions are equivalent (Maciejewski et al., 2014).
The remainder of this paper explores conditions under which a mutation is beneficial and/or favoured. We begin in Section 2 by setting out the mathematical formalism for multiplayer evolutionary games, focussing particularly on PGGs played between cooperators and defectors. Section 2.1 then introduces the -rule, which is used to determine whether a strategy is favoured. We outline several known results on graphs with local update rules, as well as deriving results for a birth–death and shift update rule on a cycle. We then derive the conditions for favourability on a general population structure with global updating. In Section 2.2 we derive a similar rule, but for a strategy to be beneficial. In Section 3 we apply this theory to consider conditions for cooperator success in an epithelium, using spatial statistics calculated through simulation of the Voronoi tessellation model. Finally, in Section 4, we discuss the implications of our work for the evolution of cooperative public goods in epithelia and make some remarks on the different significance of beneficial and favourable mutants.
Section snippets
Evolutionary dynamics of multiplayer games
We consider an arbitrary multiplayer game with two strategies, A and B. Players interact in groups of size , and obtain payoffs and respectively, where j is the number of A co-players and k is the total number of co-players. For a graph-structured population, the co-players are direct neighbours. The fitness of each individual is then defined as or , where is the selection strength parameter.
The population evolves according to a Moran process (Moran, 1958), i.e.
Public goods games in an epithelium
A number of studies have considered the evolutionary dynamics of sigmoid PGGs in epithelia, representing the tissue either as a well-mixed population (Archetti, 2013b), or a fixed graph structure with various local update rules (Archetti, 2013c, Archetti, 2016). Here, we use the framework introduced in Renton and Page (2019) to incorporate explicit tissue dynamics, using the Voronoi tessellation (VT) model, with a spatially decoupled (global) update rule. This means that when the population is
Discussion
There is an extensive literature on cancer modelling, which goes way beyond evolutionary game theory. For a review, see for example (Altrock et al., 2015). However, evolutionary game theory is increasingly used in cancer modelling (Rockne et al., 2019, Archetti and Pienta, 2019, Wölfl et al., 2020) both to elucidate tumorigenesis (Tomlinson and Bodmer, 1997, Bach et al., 2003, Basanta et al., 2008b, Basanta et al., 2008c, Archetti, 2016) and to inform potential treatment strategies (Basanta et
Data accessibility
The code and data can be accessed at https://github.com/jessierenton/pgg-epithelium.
CRediT authorship contribution statement
Jessie Renton: Conceptualization, Methodology, Software, Formal analysis, Writing - original draft. Karen M. Page: Conceptualization, Formal analysis, Writing - review & editing, Supervision.
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgments
This research was funded by an EPSRC studentship held by JR.
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