Edge effects in game theoretic dynamics of spatially structured tumours.

Edge effects in game theoretic dynamics of spatially structured tumours. Artem Kaznatcheev1† , Jacob G. Scott2,3† & David Basanta2∗

arXiv:1307.6914v2 [q-bio.PE] 10 Jan 2014

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School of Computer Science and Department of Psychology, McGill University, Montreal, QC, Canada 2 Integrated Mathematical Oncology, H. Lee Moffitt Cancer Center and Research Institute, Tampa, FL, USA 3 Centre for Mathematical Biology, Mathematical Institute, University of Oxford, Oxford, UK Abstract Evolutionary game theory has been used to model many situations in ecology where different species, or different phenotypes within one species, compete against one another. Of late, this has extended to cancer biology to understand the dynamics of disease progression as a game between competing cellular phenotypes. A major assumption in this modeling paradigm is that the population is inviscid: the probability of a player with a given phenotypic strategy interacting with another depends exclusively on the respective abundance of those strategies in the population. While there are scenarios where this assumption might be useful, in solid tumours, where the populations have spatial structure, this assumption can yield misleading results. In this study we use a recently developed mathematical tool, the Ohtsuki-Nowak transform, to study the effect of interaction neighborhood size on a canonical evolutionary cancer game: go vs. grow. We show that spatial structure promotes 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 in solid tumours. This edge effect allows a tumour with no invasive phenotypes expressed in the bulk to have a polyclonal boundary with both invasive and non-invasive cells. We focus on evolutionary game dynamics between competing cancer cells, but we anticipate that our approach can be extended to games between other types of players where interaction neighborhoods change at a static system boundary.

keywords: edge effects — evolutionary game theory — cancer — heterogeneity

† ∗

Contributed equally [email protected], [email protected], [email protected]

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Signifigance statement Cancer is a highly heterogeneous disase, and its progression depends on competition between different cell types which is governed by Darwinian evolution. Evolutionary game theory (EGT) provides tools for understanding how intercell interactions shape selection. Unfortunately, traditional EGT in mathematical oncology ignores spatial structure in tumours. We introduce the Ohtsuki-Nowak transform into cancer modeling to study how selective pressures differ in-bulk versus at a static edge of a tumour such as against a blood vessel. We observe that smaller neighbourhoods at an edge select for an invasive phenotype; even if a tumour is not invasive in-bulk, it could be at the boundary, where it matters most. Our technique could also prove useful for studying edge effects in other ecological settings.

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Introduction

Evolutionary Game Theory (EGT) is a mathematical approach to modeling frequencydependent selection where players interact strategically not by choosing from a set of strategies but using a fixed strategy given by their phenotype. This tool has been used to model many ecological scenarios including yeast population dynamics [1], invasive species [2], community dynamics in plants [3] and many more. Recently, because of the evolutionary nature of cancer [4, 5], it has been applied to study how the interactions between different types of cells in a polyclonal tumour could drive the evolutionary dynamics of a given cancer [6]. The first use of EGT in the field of cancer was done by Tomlinson and Bodmer [7, 8]: in their models, EGT is used to analyse the circumstances that lead to coexistence of two phenotypes. Subsequent research carried out by Bach and colleagues extended this idea to interactions between three players in the angiogenesis problem [9]. Gatenby and Vincent adopted a game theory model heavily influenced by population dynamics to investigate the influence of the tumour-host interface in colorectal carcinogenesis [10, 11] and suggested therapeutic strategies [12]. Our own work, as well as that of others, has shown that EGT can be used to study the conditions that select for more aggressive tumour phenotypes in gliomas [13, 14], colorectal cancer [10, 11], multiple myeloma [15] and prostate cancer [16]. Furthermore, EGT has been used to investigate the impact of treatment on cancer progression [17, 18]. For an in depth overview of the game theoretical approach to cancer, see Basanta & Deutsch [6]. A standard assumption in evolutionary game theory is a perfectly mixed (inviscid) population, in which every cell in the population is equally likely to interact with every other cell [19, 20]. This may be a reasonable assumption in liquid tumours, but in solid tumours (or any situation being modeled) in which spatial structure is important, the validity of this assumption should be questioned [21]. The current solution to this is to map analytic EGT cancer models onto a lattice and run in-silico experiments to simulate the resulting Cellular Automaton [22, 23, 24, 25]. Unfortunately, such transformations sacrifice the analytical power and full theoretical understanding of pure EGT approaches. To analytically model how spatial structure effects evolutionary games in the limit of 2

large populations and weak selection, Ohtsuki & Nowak [26] derived a simple rule for taking a first-order approximation of spatial structure. This approach is exact only for infinite Beth-lattices, but highly accurate for any static structure where higher-order terms – like the correlations between neighbours of neighbours – are negligible. While spatially structured biological populations, such as solid tumours, can have non-negligible higher-order interactions, this first-order approximation is an improvement over the common inviscid assumption while still allowing us to explore an analytic model. Given a game matrix A, one can compute the Ohtsuki-Nowak (ON) transform A0 = ONk (A) and can recover the dynamics of the spatially structured game A by simply looking at the inviscid replicator equation of A0 . Here, we present the transform in a form that stresses its important qualitative aspects: ONk (A) = A +

1 1 ~ ~T ~ ~ T (∆1 − 1∆ ) + (A − AT ) . k − 2 (k + 1)(k − 2) {z } | | {z } local dispersal

(1)

finite sampling from death-birth updating

~ is the diagonal of the game matrix A = [aij ], i.e. Where ~1 is the all ones vector and ∆ ~ i = aii ; thus, ∆ ~ ~1T (~1∆ ~ T ) is a matrix with diagonal elements repeated to fill each row ∆ (column). The first summand is the original payoff matrix. The second summand accounts for the more common same-strain interactions that are a consequence of local dispersal. The type of perturbation in the third summand was shown by Hilbe [27] to result from finite sampling of interaction partners. Effects of indirect interaction from more distant cells could be introduced as further corrections to equation 1, but would require more assumptions about the tumour microstructure [28]. Since the local neighbour relation is emperically well studied [29] we focus this article on incorporating this first-order structure into EGT models of cancer dynamics. In this study, we introduce spatial structure into the canonical ‘go versus grow’ game [24, 30] in which proliferation and motility compete within a tumour. We use our spatial treatment to consider a familiar scenario for conservation biology – the edge effect of an ecological system (usually a forest in landscape ecology) at a static boundary [31, 32, 33]. In tumour progression, this is analogous to a cancer cell surrounded entirely by other cancer cells as opposed to constrained by a physical boundary, such as a basement membrane, organ capsule or blood vessel (see figure 1). The evolutionary dynamics that occur at such boundaries governs progression past key cancer stages: the change from in situ to invasive; locally contained to regional advanced growth; and the dramatic shift from local to metastatic disease. The former situation occurs early in the progression of most epithelial tumours, and it is commonly believed that it is at this point when the Warburg effect occurs, pushing cells toward the glycolytic (acid producing) phenotype which promotes invasion and motility - the so called ‘acid mediated invasion’ hypothesis [34, 35, 36, 13]. The latter situation occurs when tumours grow up against blood vessels and is likely the first opportunity for hematogenous dissemination, the first step in the metastatic process [37] We show a striking change in the evolutionary game dynamics that occurs at the tumour’s physical boundary (Fig. 1). This study represents, to our knowledge, the first attempt to understand the effects of changing neighborhood structure on evolutionary game dynamics of tumours. 3

Figure 1: A tumour (blue cells) has several different scenarios that govern the architecture of its neighborhood geometry. Structures like the blood vessels and organ capsules represent static boundaries which we consider in this paper. Most cells in a solid tumour experience a largely static neighbourhood architecture. However, cells in the bulk of the tumour (turquoise) have many more neighbors than cells against static boundaries (pink) like an organ capsule (left close-up) or blood vessel (right close-up). This change in relative number of neighbours affects evolutionary game dynamics. The boundary between the tumour and healthy cells (yellow) is a dynamic edge and not considered in this article. The figure is intended as an intuition, and cells are not represented with numeric accuracy. We would expect a biologal tumour to be 3 dimensional, have many more cells, and a more intricate neighbourhood microstructure. However, the key feature of cells in bulk versus boundary having more neighbours will remain.

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2

Inviscid game for motility

Giese and colleagues have shown that in Gliomas there is a relationship between proliferative and invasive behaviours [38]. Specifically it was shown that migration and proliferation were mutually exclusive processes and motile cells are unable to proliferate while they are moving. This is known as the Go or Grow dichotomy [39]. Furthermore, Godlewski and colleagues have recently provided experimental evidence suggesting that metabolic stress is responsible for this behaviour in gliomas[40]. To mathematically model this we considered the situation in which the tumour is made of a population of rapidly proliferating cells capable of autonomous growth (AG). In this population a mutation can confer motility/invasiveness (INV) to tumour cells. The game has two parameters: c, represents the direct and indirect costs of motility incurred by cells with the INV phenotype resulting from the reduced proliferation rate of motile/invasive cells [39] Parameter b is the maximum fitness a tumour cell will have under ideal circumstances when it does not have to share space or nutrients with other cells. As the units of measure are arbitrary, it is usually just the ratio c/b that matters for game theoretic dynamics. With INV as strategy 1 and AG as strategy 2, the game’s payoff matrix is:

INV AG

INV AG  1 b + 2 (b − c) b − c 2 1 b b 2

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(2)

To understand the inviscid model [24], imagine two cells meeting at random in a resource spot; in contrast, for a structured population, the probabily of meeting again will be higher. If both are motile (INV) then one cell stays in the resource spot and consumes the resources b, and the other pays a cost, c, to move and find a new empty site with resources b. If after a move, a motile cell only finds a new empty spot with probability r, then the expected move payoff is rb − c. This can be captured as an indirect cost by adjusting the cost to be c0 = c + (1 − r)b without introducing an extra parameter. As the cell that has to move is chosen randomly from the two, the expected payoff for each cell is the average of the no-move (b) and move (b − c) payoffs. On the other hand, if an INV cell meets an AG cell then the INV cell will move, incurring the cost, c, before receiving the benefit, b, for a total fitness (b − c). The AG cell, however, will stay in this case and consume all the resources (b). Finally, if two non-motile cells (AG) are in the same resource spot then they simply share the resources, for a payoff of b/2.

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1 Figure 2: Level of viscosity k−2 versus relative cost of motility cb . The parameter space is divided into three regions that correspond to qualitatively different dynamics. In the red, the population evolves toward all INV; in the yellow – toward a polyclonal tumor of INV and AG 1 = 0 (k → ∞) we recover cells; and in the green the tumor remains benign (all AG). When k−2 1 the standard inviscid replicator dynamics of our previous work [24]. For k−2 = 1 (k = 3, the top edge of the plot), we have the environment with the smallest local neighbourhood to which the ON-transform applies. The first horizontal dotted line marks k = 5 (pink cell in fig. 1) and the bottom line is k = 8 (teal cell in fig. 1). The left vertical dotted line is at c/b = 0.23, and shows that it is possible to go from a polyclonal tumor in the bulk to a completely invasive population at a static edge. The right vertical dotted lines shows that is possible to see a qualitative shift from all AG to a polyclonal tumour in dynamics with the 1 from 1/6 to 1/3) at game fixed at c/b = 0.53 by decreasing k from 8 and 5 (increasing k−2 the tumour boundary. Example dynamics are shown in the insets, where proportion of INV (p) is plotted versus time (t). The left inset corresponds to c/b = 0.23, an initial proportion of invasive agents of p0 = 0.93, k = 5 (tumour edge) for the red line, and k = 8 (tumour bulk) for the yellow. The right inset correspond stp c/b = 0.53, an initial proportion of invasive agents of p0 = 0.04, k = 5 (tumour edge) for the yellow line, and k = 8 (tumor bulk) for the green.

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3

Spatial effects and boundaries

Whenever b > c, the game in eq.2 is a social dilemma with invasive cells as the cooperators, and AG cells as defectors. Rules of thumb from EGT [41] suggest that cooperators benefit from the structure of small interaction neighbourhoods, in agreement with our biological intuition that, in this game, having the ability for conditional motion is of more use in a more structured and viscous environment than in one where all cells are already stochastically moving around and interacting at random. We look at this formally by explicitly considering spatial effects on the previously inviscid model. Applying the transform from equation 1 to the game in equation 2 yields:  2k − 3 − c 2k 2 − k − 1 1 b + 1 (b − c) b 2 2 2(k − 2) 2(k − 2)(k + 1)   2k − 5 k + 3 1 b +c 2b 2(k − 2) 2(k − 2)(k + 1) 

This game has three qualitatively different regimes that depend on the value of

(3) c b

and k:

c 1. If kk+1 2 +1 ≥ b then there is a single stable fixed point with all cells invasive. All polyclonal tumors evolve toward this fixed point. For inviscid populations (k → ∞) this condition is satisfied only if motility is cost-free (c = 0) and hence the possibility of an all invasive tumour was not noticed in previous non-spatial analysis [24]. c k+1 2. If kk+1 2 +1 < b < 2k+1 then the game has Hawk-Dove dynamics and there is a stable polyclonal equilibrium with a proportion p of INV cells:

p=

1 1 2c b − 2c + − b−c k − 2 (k + 1)(k − 2) (b − c)

(4)

All polyclonal populations will converge toward this proportion of INV cells. In the unstructured limit as k → ∞ we have perfect agreement with our previous results [24] and recover the condition cb ≤ 12 that was assumed for the inviscid equilibrium to exist and the exact numeric value of b−2c for the equilibrium proportion of INV agents. For b−c any finite k the proportion of invasive cells is strictly higher. k+1 3. If 2k+1 ≤ cb then the game has Prisoner’s dilemma dynamics any polyclonal population converges toward all AG and the tumor remains non-invasive. 1 The above three regimes are plotted in figure 2. When k−2 = 0, we have the previously 1 studied inviscid game [24] and for k−2 = 1 we have the most structured regime possible with small neighbourhoods (k = 3). The red region corresponds to a completely INV tumour, the yellow to a polyclonal tumour, and the green to all AG. As we make the tumor more structured and reduce the number of neighbours, it becomes easier for the INV cells to be expressed in the tumor. In particular, consider the case where cb = 0.53 and as in figure 1 in the tumor away from boundary (the teal cell) there are k = 8 neighbours and the dynamics favour all AG, so no INV cells will be present at equilibrium. However, if the solid tumor is

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pressed up against a static boundary then cells at the edge have fewer neighbours (k = 5, the purple cell) and the dynamics at the boundary favour a polyclonal population with about 8% of the cells INV. The edge effect causes a polyclonal boundary in a tumor with a homogeneous all AG body.

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Discussion

A standard assumption in evolutionary game theory is that all players interact with all other players: the population is inviscid. There are a number of biological scenarios in which such an assumption could be misleading, and we consider such a scenario in the form several key aspects of solid tumour progression. The role of spatial heterogeneity has been explored before in mathematical models studying evolutionary processes in cancer [42, 43, 44]. Those models show that different environments produce different selective pressures and that phenotypic heterogeneity results from, and drives, the spatial one. Our approach tackles a different question related to the nature of physical edges on cancer evolution for which we have applied the Ohtsuki-Nowak transform [26] to the standard go-grow game of mathematical oncology [24]. We have shown a quantitative effect: spatial-structured tumours promote the invasive phenotype compared to inviscid tumours with the same ratio of cost of motility to benefit of resources c/b. We also considered the decrease in neighbourhood size experienced by cells at the boundary of a tumour, compared to cells within the tumour bulk. This could represent a number of feasible and very relevant scenarios, including a tumour against a blood vessel or organ capsule, or when an in situ neoplasm reaches the basement membrane. We have shown that this change in neighbourhood for tumour cells can, independently from any other parameters, significantly effect the evolutionary stable strategies: in this case a promotion of the INV phenotype. The edge effect allows a tumour that internally has no invasive phenotypes expressed to have a polyclonal boundary with both invasive and non-invasive cells. The results of our mathematical model could have significant translational implications. Genetic heterogeneity has recently become recognized as the rule in cancer [45], but as long as physicians have had microscopes, we have realised that spatial organisational heterogeneity was an equally defining factor. The Gleason score [46] is a classic example of greater heterogeneity predicting worse survival in prostate cancer. We have shown that a specific change in neighbourhood size at a static boundary can dramatically alter game dynamics, and select for novel phenotypes. Our work offers a plausible non-genetic explanation for some of these poorer tumour outcomes. We have shown that the local spatial structure of a tumour can strongly affect the evolutionary pressures on its constituent cells, even if all other factors are held constant. This can add yet another source of sampling bias to tissue biopsies and suggests that the architectural, not just the molecular, context is important. For instance, consider an idealized fine-needle aspiration biopsy [47], assuming the standard 0.7 mm needle samples a perfect column around 20 cells in diameter of tumour cells right next to a critical boundary such as a capillary. In our running example of c/b = 0.53 (see figures 1 and 2) this would result in the sample containing only about 0.4% invasive cells 8

since 19 out of every 20 cells are not at the boundary, and so contain no invasive phenotypes. This is below the detection levels of the state of the art medical practice [48, 49]. However, the critical 1 out of every 20 cells at the boundary, would have a dangerous 8% invasive cells. Thus, an oncologists performing a diagnostic fine-needle aspiration biopsy would misdiagnose a malignant tumour as benign because they are using a technique that destroys the structure of the tumour and mixes the cells in the critical tumour boundary with the (in this case) irrelevant tumour body. Our model was motivated by the study of cancer, but it is based on spatial edge effects in games of interacting players which could represent any number of other scenarios. While we have focused on the specifics of metastasis and cancer invasion, this method could yield insights into many other interesting problems ranging from ecology to medicine, and highlights the importance of neighborhood geometry when studying the evolutionary dynamics of competing biological agents.

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