Constant-sum games as a way from non-cell autonomous processes to constant tumour growth rate

A lot of thinking in cancer biology seems to be focused on cell-autonomous processes. This is the (overly) reductive view that key properties of cells, such as fitness, are intrinsic to the cells themselves and not a function of their interaction with other cells in the tumour. As far as starting points go, this is reasonable. But in many cases, we can start to go beyond this cell-autonomous starting point and consider non-cell-autonomous processes. This is when the key properties of a cell are not a function of just that cell but also its interaction partners. As an evolutionary game theorist, I am clearly partial to this view.

Recently, I was reading yet another preprint that has observed non-cell autonomous fitness in tumours. In this case, Johnson et al. (2019) spotted the Allee effect in the growth kinetics of cancer cells even at extremely low densities (seeding in vitro at <200 cells in a 1 mm^3 well). This is an interesting paper, and although not explicitly game-theoretic in its approach, I think it is worth reading for evolutionary game theorists.

Johnson et al.'s (2019) approach is not explicitly game-theoretic because they consider their in vitro populations as a monomorphic clonal line, and thus don't model interactions between types. Instead, they attribute non-cell autonomous processes to density dependence of the single type on itself. In this setting, they reasonably define the cell-autonomous null-model as constant exponential growth, i.e. \dot{N}_T = w_TN_T for some constant fitness w_T and total tumour size N_T.

It might also be tempting to use the same model to capture cell-autonomous growth in game-theoretic models. But this would be mistaken. For this is only effectively cell-autonomous at the level of the whole tumour, but could hide non-cell-autonomous fitness at the level of the different types that make up the tumour. This apparent cell-autonomous total growth will happen whenever the type interactions are described by constant-sum games.

Given the importance of constant-sum games (more famously known as zero-sum games) to the classical game theory literature, I thought that I would write a quick introductory post about this correspondence between non-cell autonomous constant-sum games and effectively cell-autonomous growth at the level of the whole tumour.

Constant-sum games: from classic to evolutionary

Let’s start with some definitions.

In a two-player game in classic game theory, if player one plays pure strategy i and player two plays pure strategy j then they receive a payoff G_{ij} for player one and H_{ij} for player two. We might summarize this by writing (i,j) \mapsto (G_{ij},H_{ij}). In general, these two matrices G and H are arbitrary.

We call a game zero-sum if the gain of any one player are always balanced by the losses of the other. Mathematically, we say that G,H is zero-sum if G = -H, or equivalently G + H = 0.

From this, it isn’t hard to imagine what constant-sum games mean in classic game theory. It is when G + H = K where K is a matrix that has the same constant k in every entry. We might be even more specific, and say that this game is a k-sum game. Classic game theorists don’t usually distinguish between zero-sum and k-sum games because constant off-sets don’t matter, we could just as easily have given each player a base payoff of k/2 and then played a zero-sum game.

What happens when we move to an evolutionary game theory setting?

If we’re studying a single population of several types then players 1 and 2 get equated. In other words, we demand that our game be symmetric: H = G^T (i.e. H is the transpose of G). Since the two matrices are linked in this way, we usually don’t specify H (since it is not redundant) and just make our mapping (i,j) \mapsto (G_{ij},G_{ji}). If we want this game to be k-sum then our some restriction for symmetric games becomes: G_{ij} + G_{ji} = k. In particular, this implies that each diagonal entry is equal to k/2.

This means that for a two-strategy k-sum symmetric game, there is only 1 parameter a:

G = \begin{pmatrix} R & S \\ T & P \end{pmatrix} = \begin{pmatrix} k/2 & a \\ k - a & k/2 \end{pmatrix}

I’ll use the two-strategy case as an example for simplicity. But it isn’t difficult to generalize the main ideas to more strategies.

Constant-sum games in growing populations

Since I’m interested in hiding games within constant exponential growth, we need to start with an exponential growth model for the two strategies:

\begin{aligned}  \dot{N}_A & = w_A N_A \\  \dot{N}_B & = w_B N_B  \end{aligned}

where our growth rates come from the game as:

\begin{aligned}  w_A & = p_A R + p_B S \\  w_B & = p_A T + p_B P  \end{aligned}

and where p_A = N_A/N_T, p_B = N_B/N_T are the proportions of the two strategies with N_T = N_A + N_B as the total tumour size.

From here, it is straightforward to write down the equation for the tumour growth rate:

\begin{aligned}  \dot{N}_T & = \dot{N}_A + \dot{N}_B \\  & = w_A N_A + w_B N_B \\  & = N_T(p_Aw_A + p_Bw_B) \\  & = \langle w \rangle N_T  \end{aligned}

where \langle w \rangle = p_Aw_A + p_Bw_B is the average fitness of the population. And if we let p = p_A then we can also write down the complete system dynamics as:

\begin{aligned}  \dot{N}_T & = \langle w \rangle N_T \\  \dot{p} & = p(1 - p)(w_A - w_B)  \end{aligned}

Note that this decouples the proportion dynamics from the population growth. This is why it can be perfectly reasonable to use replicator dynamics in growing populations (at least if you have a reason to believe that w_A, w_B are functions of proportions and not densities.

However, in the above equations, the population growth can still depend on the proportion dynamics. This is due to the proportions appearing within \langle w \rangle. But if we’re dealing with constant-sum games then this dependence will disappear, as \langle w \rangle will be a constant.

There are several ways to see that \langle w \rangle will be constant for constant-sum games. The easiest is by noting that the average is just a normalized sum of all the interactions and in each interaction the two payoffs sum to k and thus the average will be k regardless of which pairs meet.

But if we insist on seeing the arithmetic then we get:

\begin{aligned}  \langle w \rangle & = p_A w_A + p_B w_B \\  & = p_A(p_A R + p_B S) + p_B(p_A T + p_B P) \\  & = \sum_{i \in \{A, B\}} \sum_{j \in \{A, B\}} p_i p_j G_{ij} \\  & = \sum_{i \in \{A, B\}} p_i^2 G_{ii} + \sum_{j > i} 2 p_i p_j \frac{G_{ij} + G_{ji}}{2}  \end{aligned}

At this point, we haven’t used that G is constant-sum. We’ve also written the equations in such a form that if we replaced the summation to be over a set of n strategies instead of just the two in {A,B} then nothing would change.

Let’s now use that G is constant-sum: i.e. that G_{ij} + G_{ji} = k for all i,j. Then we can continue with:

\begin{aligned}  \langle w \rangle & = \frac{k}{2} \sum_{i \in \{A, B\}} p_i^2 + \sum_{j > i} 2 p_i p_j \\  & = \frac{k}{2} \sum_{i \in \{A, B\}} p_i (\sum_{j \in \{A, B\}} p_j ) \\  & = \frac{k}{2}  \end{aligned}

From this, our complete dynamic system (for the two strategy case) simplifies to:

\begin{aligned}  \dot{N}_T & = \frac{k}{2}N_T \\  \dot{p} & = p(1 - p)(w_A - w_B)  \end{aligned}

With the dynamics for the total tumour size and the dynamics for the frequencies as completely decoupled. More improtantly, if we were looking at just the total tumour burden then it would look like it has cell-autonomous growth even though there is non-cell-autonomous growth at the level of the subpopulations.

What does this mean? Not much. Especially for Johnson et al. (2019), my observation is irrelevant since they already rule out the constant growth null-model. If one was to level objections about their definitions, it would be about if simply the use of an abiotic resource should count as non-cell-autonomous or not. But I’ve already engaged with this point when I talked about EGT without interactions.

Mostly, I wanted to share this as another cute observation to add to the list of fun discrepancies possible between the reductive and effective views of population dynamics (for more on that, see Kaznatcheev, 2017).

References

Johnson, K. E., Howard, G., Mo, W., Strasser, M. K., Lima, E. A., Huang, S., & Brock, A. (2019). Cancer cell population growth kinetics at low densities deviate from the exponential growth model and suggest an Allee effect. BioRxiv, 585216.

Kaznatcheev, A. (2017). Two conceptions of evolutionary games: reductive vs effective. BioRxiv, 231993.

About Artem Kaznatcheev
From the Department of Computer Science at Oxford University and Department of Translational Hematology & Oncology Research at Cleveland Clinic, I marvel at the world through algorithmic lenses. My mind is drawn to evolutionary dynamics, theoretical computer science, mathematical oncology, computational learning theory, and philosophy of science. Previously I was at the Department of Integrated Mathematical Oncology at Moffitt Cancer Center, and the School of Computer Science and Department of Psychology at McGill University. In a past life, I worried about quantum queries at the Institute for Quantum Computing and Department of Combinatorics & Optimization at University of Waterloo and as a visitor to the Centre for Quantum Technologies at National University of Singapore. Meander with me on Google+ and Twitter.

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