# Warburg effect and evolutionary dynamics of metastasis

On Friday, I introduced you guys to the importance of motility in cancer metastasis. However, motility isn’t the whole story, another important factor is the type of respiration (energy generation) that the cell uses. As I gathered from conversations with Jacob Scott (and please correct me if I am wrong), when a tumor saturates the area it is in too quickly to recruit new blood vessels then the cancer cells can become oxygen deprived and be forced to switch from the full aerobic Kreb cycle to a focus on less efficient but anaerobic glycolysis. This shift is known as the Warburg effect and is used for diagnosis and monitoring of cancer progress. For an evolutionary game theorist, this means that when we study metastasis we need to consider three strategies: autonomous growth (AG; called ‘proliferative’ in the previous post), invasive (INV; called ‘motile’ in the previous post), and glycolytic (GLY; new to this this post).

Basanta, Simon, Hatzikirou, and Deutsch (2008) parametrize this game as:

$\begin{pmatrix} 1 - \frac{c}{2} & 1 - c & 1 - c \\ 1 & \frac{1}{2} & \frac{1}{2} - n \\ 1 - k & \frac{1}{2} + n - k & \frac{1}{2} - k \end{pmatrix}$

where the rows (and columns) are INV, AG, GLY and the payoff are written for the row player (note that the paper uses a different convention). The benefit of taking advantage of the environment completely — for instance when an AG player interacts with an INV player that leaves the resource spot — is set to 1 (and thus fixes our units of measurement). Relative to 1, the other parameters are:

• $c$ — cost of motility, both explicit cost and the implicit opportunity cost of not finding a new empty location.
• $k$ — cost of switching to the less efficient glycolytic metabolism. Note that a GLY agent faces this cost regardless of interactive partner (it is present in every column of the 3rd row), similar to how I define cognitive cost in agent based models.
• $n$ — effect of an acidic environment. This represents both the cost that a normal cell incurs when functioning in an acidic environment, and the benefit that a glycolytic cell experiences when its microenvironment is more acidic. This is only relevant to the AG-GLY interaction, since then a higher environmental acidity is maintained that is not present with either two GLY or a single GLY (that happens when an INV moves to a new spot upon noticing an AG). I don’t understand the biology behind this process and would like it if someone could explain it to me in the comments, since — as we will see in the analysis — a large effect is essential for stable coexistence of an AG-INV-GLY polymorphic population.

The replicator dynamics of a three strategy game is a continuous flow on a 2-simplex, which is usually drawn as a triangle. Since the possible number of qualitatively different flows induced by replicator dynamics is very large for a three strategy game, it is instructive to first study the three 1-simplexes that form the edges of the triangle. These are two-strategy games — one for each strategy we could leave out — and as such only have two (four if we don’t consider strategy permutations or flow reversal) qualitatively different flows that are consistent with linear utilities: either we have one strategy dominate the other and every point on the line flows towards the dominant strategy, or we have a polymorphism. If the polymorphism is stable, then at every point on the line we have a flow toward the fixed-point, else the flow is away from the fixed point and towards the end-points. Let us look at these faces:

1. INV-AG: we already studied thisface on Friday, and noted that
1. if $c \geq \frac{1}{2}$ then the flow is towards all AG, else
2. if $c \leq \frac{1}{2}$ we have a stable polyclonal population with $p_{IA} = \frac{1 - 2c}{1 - c}$ proportion of invasive cells.
2. AG-GLY: the simplified game matrix for this face is (removing INV, and adding constants and renormalizing by a positive constant) $\begin{pmatrix}1 & 1 - n/k \\ n/k & 0\end{pmatrix}$ and thus
1. if $n > k$, then the game is a PD so the population goes to all GLY, else
2. if $n \leq k$, it is a harmony game and the population flows toward the AG endpoint.
3. INV-GLY: this is the most interesting face, since it has two independent parameters we can vary: $\alpha = c/(1 - c + 2k)$ and $\kappa = \frac{2k}{1 - c + 2k}$. In terms of these parameters, the simplified game matrix is $\begin{pmatrix}1 & 1 - \alpha \\ 1 + \alpha - \kappa & 0 \end{pmatrix}$. Since $\alpha \geq 0$, $0 \leq \kappa \leq 1$, this payoff matrix can be moved through three different games:
1. If $\alpha \leq \kappa$ (or $c \leq 2k$ in original parameters) then it is a leadership game and the flow is towards all INV,
2. If $\kappa \leq \alpha$ (or $2k \leq c \leq k + \frac{1}{2}$ in original parameters) then it is a Hawk-Dove game and we have a stable polyclonal population with $p_{IG} = \frac{1 - \alpha}{1 - \kappa} = 1 - \frac{c - 2k}{1 - c}$
3. If $\alpha > 1$ (or $c > k + \frac{1}{2}$ in original parameters) then it is a PD and the population flows toward all GLY.

Since we can set the parameters to achieve any combination listed above, this allows for 12 possible boundary conditions on our simplex represented below. When combined, conditions 1.2 and 3.3 result in a constraint of $k \leq 0$ which is not physically meaningful, so two of the conditions (on the bottom right) are eliminated, leaving us with 10 simplexes to consider.

The 12 simplexes with distinct boundary conditions. The ones in the green region have no internal fixed point and result in the same qualitative dynamics: no GLY in the population. The two simplexes in the bottom right are not physically realizable, since they require k < 2. The top left corner is a rock-paper-scissors dynamics and an internal fixed point always exists. In the remaining 4 simplexes, a fixed point exist if a certain condition is satisfied and thus each one of those simplexes can describe two qualitatively different dynamics depending on if the fixed point exists.

Of course, we are primarily interested in the dynamics inside the 2-simplex and that is not fully specified by the boundaries. The missing ingredient is the internal fixed point which we can find by solving $\vec{1} = G\vec{x}$ where $G$ is the payoff matrix that opened this post. If such an $\vec{x}$ exists and $\vec{x} \geq 0$ (i.e. if every entry of $x$ is positive) then our fixed-point is $\vec{p} = \vec{x}/(\vec{1}\cdot\vec{x})$. In this case, the vector is:

$(1 - \frac{n}{k}, \frac{(1 - c)k + n(2k - c)}{2n^2}, \frac{k + c(n - k)}{2n^2})$

If any component is less than zero then the internal fixed point does not exist. Looking at the first component, we see that there is no fixed points when condition 2.2 is met, thus the 5 realizable simplexes in the green box have no internal fixed point and any initial condition will have a flow to the INV-AG face where it will result in all AG in condition 1.1 and a polyclonal population with no GLY in condition 1.2. On the other hand, in the top left corner (when conditions 1.1, 2.1, and 3.1 are met), a fixed point always exists and we have rock-paper-scissors dynamics. Depending on if the fixed point is stable or not, we either spiral in towards it with oscillating levels of INV, AG, GLY or spiral away from it until two of the strategies go extinct. In the remaining 4 cases, a fixed-point can exist or not depending on the parameters.

Of the remaining cases, my favourite is the top right corner (when conditions 1.1, 2.1, and 3.3 are met) and you have an unstable fixed point (if $\frac{k + c(n - k)}{2n^2} \leq \frac{n}{k}$) then you get the dynamics on the right. An interesting feature here is that your end state can depend very heavily on your initial proportion of cell types. Even though glycolytic cells can outcompete standard proliferate cells (AG), if there is a certain concentration of invasive cells then it can lead to long term behavior that tends towards all AG.

Finally, the flow diagrams are exhaustive and let us make rigorous qualitative conclusions about tumours. If we want tumors to remain benign (not invasive) and not glycolytic then there is no way expect to have $c \geq \frac{1}{2}$ and $n \leq k$. However, if we just want to have a chance of avoiding invasive genotypes then we can replace the $n \leq k$ restriction by $2k \leq c$ and find non-invasive but potentially glycolytic populations. The two rightmost simplexes in the upper row must be studied more closely to make specific treatment recommendations. Of course, this isn’t the end of the story since all results in this post are for randomly interacting cells and we might want to consider introducing some spatial structure.

Basanta, D., Simon, M., Hatzikirou, H., & Deutsch, A. (2008). Evolutionary game theory elucidates the role of glycolysis in glioma progression and invasion. Cell proliferation, 41 (6), 980-7 PMID: 19040573