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).

Game theoretic analysis of motility in cancer metastasis

As I am starting to write this post, the last of the 4th of July fireworks are winding down outside. I am currently traveling through New York on my way to Swarmfest 2013 in Orlando, FL and to visit David Basanta and Jacob Scott at the Integrated Mathematical Oncology Department of Moffitt Cancer Research Institute in Tampa, FL. The weather is unbearably hot already, so I imagine I will struggle in Florida, especially since I forgot all my shorts in Montreal!

The more important struggle, however, will be my lack of background in biology and medicine. On Jake’s suggestion, I decided to look at a paper of David’s on an evolutionary game theoretic approach to the emergence of motility in cancer cells. As far as I understand, one of the key steps in going from a benign tumor to a malignant cancer is metastasis or the ability of a cancer to spread from one organ to another non-adjacent organ. To achieve this, a cancer cell has to transition from a simple proliferative cell to a motile one. However, motility usually involves a cost to the organism.

How ethnocentrism evolves: a simulation of evolutionary dynamics

Cooperation is a paradox—it just doesn’t make sense. Why should I help you when there’s no direct benefit for me? Artem, Professor Tom Shultz, and I have been working for quite some time on a paper about cooperation, and we’re psyched to announce that it’s finally been published in The Journal of Artificial Societies and Social Simulation (JASSS). JASSS is an open web journal, so you can view the full text of our article for free on their website. Or you could skip the 8000 or so words and check out this summary post. Read more of this post

Micro-vs-macro evolution is a purely methodological distinction

On the internet, the terms macroevolution and microevolution (especially together) are usually used primarily in creationist rhetoric. As such, it is usually best to avoid them, especially when talking to non-scientists. The main mistake creationist perpetuate when thinking about micro-vs-macro evolution, is that the two are somehow different and distinct physical processes. This is simply not the case, they are both just evolution. The scientific distinction between the terms, comes not from the physical world around us, but from how we choose to talk about it. When a biologist says “microevolution” or “macroevolution” they are actually signaling what kind of questions they are interested in asking, or what sort of tools they plan on using.

Environmental austerity and the anarchist Prince of Mutual Aid

Prince Pyotr Alexeyevich Kropotkin

Any good story starts with a colourful character, a complicated character, and — to be complacent with modern leftist literature — an anarchist intellectual well-versed in (but critical and questioning of) Marxism; enter Pyotr Alexeyevich Kropotkin. Today he is best known as one of the founders and leading theorist of anarcho-communism, but in his time he was better known as an anti-Tsarist revolutionary, zoologist, geographer and explorer. Kropotkin was born to the Prince of Smolensk, a descendant of the Rurik dynasty that ruled and eventually unified many of the Principalities and Duchies of Rus into the Tsardom of Russia. By the 9 December 1842 birth of our protagonist, Russia had been under Romanov rule for over 200 years, but the house of Rurik still held great importance. Even though the young boy renounced his Princely title at age 12, he was well-off and educated in the prestigious Corps of Pages. There he rose to the highest ranks and became the personal page of Tsar Alexander II. Upon graduation this entitled Kropotkin to his choice of post, and our first plot twist.

Analogous to the irresistible pull of critical theory on modern liberal-arts students, the young Kropotkin was seduced by the leftist thought of his day: French encyclopédistes, the rise of Russian liberal-revolutionary literature, and his personal disenfranchisement with and doubt of the Tsar’s “liberal” reputation. Instead of choosing a comfortable position in European Russia, the recent graduate requested to be sent to the newly annexed Siberian provinces and in 1862 was off to Chita. This city has a personal significance to me, it is where my grandfather was stationed over 100 years later and most of my mother’s childhood was spent there. Chita has become a minor place of pilgrimage for modern anarchists, but it (and the other Siberian administrative centre at Irkutsk) did not hold Kropotkin’s attention for long.

Unable to enact substantial change as an administrator, he followed his passion as a naturalist. In 1864, he took command of a geographic survey expedition into Manchuria. Having read Darwin’s On the Origin of Species when it was published 5 years earlier, Kropotkin embarked on a distinctly Siberian variant of the HMS Beagle — sleigh dogs instead of wind to power his way. His hope was to observe the same ‘tooth and claw’ competition as Darwin, but instead he saw primarily cooperation. In the harsh environment of Siberia, it wasn’t a struggle of beast versus beast, but animal against environment.

From 1890 to 1896, Kropotkin published his Siberian observations as a series of essays in the British monthly literary magazine, Nineteenth Century. Motivated as a response to Huxley’s “The Struggle for Existence”, the essays highlighted cooperation among nonhuman animals, in primitive societies and medieval cities, and in contemporary times. He concluded that not competition, but cooperation, were the most important factors in survival and the evolution of species. Kropotkin assembled the essays into book form, and in 1902 published Mutual Aid: A Factor of Evolution. A magnum opus on cooperation, much like E.O. Wilson Sociobiology of nearly 75 years later, Kropotkin started from the social insects and traced a common thread to the human society around him; he was the first student of cooperation.

Unfortunately, his mechanism for cooperation did not extend beyond group selection. Kropotkin left it to modern researchers to find more basic engines of altruism. Only now are we starting to build mathematical, computational, and living models. To study cooperation in the laboratory, especially when looking at the effect of environmental austerity, Strassmann & Queller (2011) have proposed the social microbe Dictyostelium discoideum or slime mold as the perfect model. These single-cell soil-dwelling amoeba are capable of working together under austere conditions, and even display rudimentary swarm intelligence. A long time expert on slime molds, John Bonner of Princeton University, made a video of them during his undergraduate years at Harvard:

Under plentiful conditions, D. discoideum are solitary predators of bacteria, which they consume by engulfment. If the environment deteriorates and the amoebae begin to starve, then they enter a social stage. Using their quorum-sensing mechanism they check if enough other amoebae are present in the area and then aggregate into a mound. They coat themselves with a slime (that gives them their name) and move together as a unit, until they find a good location to fruit. The slime mold then extends a stalk up from the soil with most cells forming a spore at the top. At a certain height, the spore is released, allowing the amoebae at the top to disperse to greener pastures; the cells in the stalk die. Since all the cells are free-living independent organisms during the non-social stage, this shows the clearest form of altruism: fellow D. discoideum sacrificing their own lives in order to give their brethren a chance at a future.

Sadly, most evolutionary game theory models assume constant population size and no resource variability. In these models, it is difficult to introduce a parameter analogous to environmental austerity. To allow for resource limitations, we need to introduce variable population sizes and thus create an ecological game. I explored this modification for a Hammond & Axelrod-like model back in the summer of 2009 and thought I would share some results here.

The agents inhabit a toroidal lattice, and each round the agents interacts with their 4 adjacent neighbours via the Prisoner’s dilemma. The payoffs are added to their default birth rate, reproduction is asexual and into adjacent empty sites. At each time step, each agent has a fixed (0.25) probability of expiring and vacating its site. The worlds start empty and are gradually filled with agents.

This figure has three graphs; in each figure the line thickness represents standard error from averaging 30 independent runs. The leftmost graph is the proportion of cooperation versus cycle, with two conditions for default birth rate: 0.24 (high austerity; top line) and 0.28 (low austerity; bottom line). The two figures on the right show the total number of cooperators (blue) and defectors (red). The rightmost graph has time flowing from right to left. The left panel is high austerity (def ptr = 0.24) and the right panel is low austerity (def ptr = 0.28).

Above are the results for a Prisoner’s dilemma interaction with $c/b = 0.5$ — a rather competitive environment. Matching Shultz, Hartshorn, & Kaznatcheev (2009) and consistent with Milbiner, Cremer, & Frey (2010), we can see an early spike in the number of cooperators as the world reaches its carrying capacity. After this transient period, the dynamics shift and defection becomes more competitive. The dynamics settle to a stable distribution of cooperators and defectors. The proportion of cooperation depends heavily on the environmental austerity. In a harsh environment with a low default birth rate of 0.24, the agents band together and cooperate and in a plentiful environment with high default birth rate of 0.28, defection dominates. As Kropotkin observed: cooperation is essential to surviving environmental austerity.

Analogous to the results from the Hauert, Homles, & Doebeli (2006) ecological public-goods game the proportion of cooperation tends to bifurcate around default birth rate equal to to the death rate (0.25), although I don’t present the visuals here. The increase in default birth rate results in a slight increase in the world population at saturation, but even by raw number there are more cooperators in the high austerity than the low austerity setting. Thus, it is not simply defectors benefiting more from the decrease in austerity (since defectors go from a regime where clusters are not-self sustaining (def ptr = 0.24) to one where it is (def ptr = 0.28)), but also an effect of defectors out-competing and disproportionately exploiting and crowding out the cooperators.

Each graph is evolutionary cycles versus proportion of cooperation, line thickness is standard error from averaging 30 independent runs. Environmental austerity decreases from the left graph (where default birth rate is equal to death rate) to the right (where their ratio is 1.1). The blue line is the model where agents can discriminate based on arbitrary non-strategy related tag (the green-beard effect/ethnocentrism are possible) and the green line is simulations where no conditional strategy is possible.

If agents are allowed to condition their behavior on an arbitrary tag then the ethnocentric population is better able to maintain higher levels of cooperation as environmental austerity decreases. In the tag-based model, it would be interesting to know if there is a parameter range where varying environmental austerity can take us from a regime of humanitarian (unconditional cooperator) dominance, to ethnocentric dominance (cooperate with in-group, defect from out-group), to a selfish (unconditional defection) world. I am also curious to know how the irrational hostility I observed in the tag-based harmony game (Kaznatcheev, 2010) would fare as the environment turns hostile. Will groups overcome their biases against each other, or will they compete even more for the more limited resource? Nearly 150 years after Peter Kropotkin’s Siberian expedition, the curtain is still up and basic questions on mutual aid in austere environments remain!

References

Hauert, C., Holmes, M., & Doebeli, M. (2006). Evolutionary games and population dynamics: maintenance of cooperation in public goods games. Proceedings of the Royal Society B: Biological Sciences, 273(1600): 2565-2571

Kaznatcheev, A. (2010). Robustness of ethnocentrism to changes in inter-personal interactions. Complex Adaptive Systems – AAAI Fall Symposium. [pdf]

Melbinger, A., Cremer, J., & Frey, E. (2010). Evolutionary game theory in growing populations. Physical Review Letters, 105(17): 178101. [arXiv pdf]

Shultz, T. R., Hartshorn, M., & Kaznatcheev, A. (2009). Why is ethnocentrism more common than humanitarianism? In N. A. Taatgen & H. van Rijn (Eds.), Proceedings of the 31st Annual Conference of the Cognitive Science Society (pp. 2100-2105). Austin, TX: Cognitive Science Society. [pdf]

Strassmann, J., & Queller, D. (2011). Evolution of cooperation and control of cheating in a social microbe Proceedings of the National Academy of Sciences, 108 (2), 10855-10862 DOI: 10.1073/pnas.1102451108

Ecological public goods game

As an evolutionary game theorist working on cooperation, I sometimes feel like a minimalist engineer. I spend my time thinking about ways to design the simplest mechanisms possible to promote cooperation. One such mechanism that I accidentally noticed (see bottom left graph of results from summer 2009) is the importance of free space, or — more formally — population dynamics. Of course, I was inadvertently reinventing a wheel that Hauert, Holmes, & Doebeli (2006) started building years earlier, except my version was too crooked to drive places.

One of the standard assumptions in analytic treatments of EGT is fixed population size. We either assume that for every birth there is a death (or vice versa) when working with finite populations, or that all relevant effects are captured by the strategy frequency and independent of actual population size (when working with replicator dynamics). This approach only considers evolutionary effects and ignores population dynamics. Hauert, Holmes, & Doebeli (2006) overcome this by building the ecological public goods game.

The authors track three proportions: cooperators ($x$), defector ($y$), and free space ($z$). Since these are still proportions, they must add up to $1 = x + y + z$. Reproduction is modified from standard replicator dynamics, by being restricted to occur only if a free-space is found to reproduce into. An agents fitness is discounted by a multiplicative factor of $z$ — the probability of finding a free space for child placement in an inviscid population. This defined the dynamic system:

\begin{aligned} \dot{x} & = x(zf_C - d) \\ \dot{y} & = y(zf_D - d) \end{aligned}

Where $f_C$ is the average fitness of cooperators, $f_D$ — defectors, and $d$ is a shared death rate. We don’t need to include $\dot{z}$ since we know that $z = 1 - x - y$. Note that if we pick a death rate such that the population density remains constant (by setting $d = z\frac{xf_C + yf_D}{1 - z}$) then we will recover standard replicator dynamics with the constant $z$ as a time-scale parameter.

For interactions, the authors use the public goods game, with max group size $N$ and benefit multiplier $r$. A group is formed to play the game by sampling $N$ times from the distribution $(x,y,z)$ — with probability $x$ a spot is filled by a cooperator, $y$ — defector, and $z = 1 - x - y$ — left empty. This means that the expected number of agents per group is given by a binomial distribution $B(N,z)$. Thus, the average group size is $S = N(1 - z)$.

Each cooperator invests 1 unit of fitness in the public good, and all units invested are multiplied by a constant factor $r$ and uniformly distributed among the agents playing. Defectors invest nothing, but still receive their fraction of the split. Thus, for an agent interacting with $S - 1$ other agents, the expected fitness of being a defector of cooperator are:

\begin{aligned} f_D & = b + \frac{rp(S - 1)}{S} \\ f_C & = b + \frac{r(p(S - 1) + 1)}{S} - 1 \\ & = b + \frac{r(p(S - 1)} + \frac{r}{S} - 1 \\ & = f_D + \frac{r}{S} - 1 \end{aligned}

Where $b$ is the default birth rate and $p = \frac{x}{x + y}$ is the proportion of cooperators among the agents. For the fitness to make sense, we need $f_C > 0$ and so $b > 1 - \frac{r}{S}$, and the strength of selection is given by $\frac{1}{b}$. Note that, unlike the Prisoner’s dilemma (which is dynamically equivalent to PG in the limit of $S \rightarrow \infty$), it becomes rational to cooperate (and irrational to defect) when $r > S$ — this is the regime of weak altruism.

This results in an interesting feedback between the population size ($1 - z$) and the proportion of cooperators. As there are more cooperators in the population, the average fitness becomes higher and the population grows, $\dot{z} < 0$, As the population increased we get $S = (1 - z)N > r$ and then defectors fare better than cooperators, causing the proportion of cooperators (relative to defectors) to decrease $\dot{p} 0$). When the population is small enough ($1 - z > \frac{r}{N}$), then it becomes rational to cooperate ($f_C > f_D$) and the proportion of cooperators starts to grown $\dot{p} > 0$ — restarting the cycle.

Hauert, Holmes, & Doebeli (2006) and later Hauert, Wakano, & Doebeli (2008) carefully analyzed these dynamics. They focus on 4 qualitatively distinct regimes (depending on parameter values). In all settings, if the initial population is too small or has too few cooperators then it will go extinct, however in two of the regimes it is possible to maintain the population leading either to the co-existence of cooperators and defectors or even cooperate dominance.

Four possible phase profiles for the ecological public goods game in groups of at most N = 8. Individuals. The figure plots proportion of cooperation $p = \frac{x}{x + y}$ versus total population density $x + y$. The left hand side is extenction and the top is cooperator dominance. Stable fixed points are in solid black, while unstable fixed points are not coloured in. The four figures differ in their values of (r,d) with (a: 3, 0.5), (b: 5, 1.6), (c: 2.7, 0.5), (d: 2.1, 0.5). Figure 2 in Hauert, Holmes, & Doebeli (2006).

As $r$ is increased, the interaction becomes less competitive and easier for cooperators; Hauert, Wakano, & Doebeli (2008) discuss this dependence on $r$ in detail. The population goes from

1. regime of total extinction (since we assume $b < d$; figure d), to
2. extinction from an unstable fixed point, to
3. oscillations around an unstable focus that lead to extinction (figure c), to
4. Hopf bifurcation resulting in the stable focus, resulting in co-existance of cooperator and defectors through oscillations, to
5. stable fixed point with static distribution of cooperator and defectors co-existing (figure a), to
6. cooperator dominance (figure b)

This means that cooperation in the public goods is possible with just free-space added to the model. Unfortunately, the results do not hold for the inviscid Prisoner’s dilemma. However, Zhang & Hui (2011) showed that in a viscous population, similar dynamics are possible for the Prisoner’s dilemma. The ecological public goods has also been extended to the spatial setting (Wakano, Nowak, & Hauert, 2009; Wakano, & Hauert, 2011), but we will discuss that extension in a future post.

Although cooperation in the ecological public goods game emerges for $N > r$, I don’t think the cooperation can be called strong altruism. The emergence depends on the population density being occasionally low enough that the effective group size $S < r$, which puts us in the weak altruism range. The authors showed the evolution of cooperation driven by weak altruism. The underlying mechanism is similar to Killingback, Bieri, and Flatt (2006) result that we read in EGT Reading group 6, except the ecological variant uses free-space where Killingback et al. uses group structure.

References

Hauert, C., Holmes, M., & Doebeli, M. (2006). Evolutionary games and population dynamics: maintenance of cooperation in public goods games Proceedings of the Royal Society B: Biological Sciences, 273 (1600), 2565-2571 DOI: 10.1098/rspb.2006.3600

Hauert, C., Wakano, J. Y., & Doebeli, M. (2008). Ecological public goods games: cooperation and bifurcation. Theoretical Population Biology, 73(2), 257.

Killingback, T., Bieri, J., & Flatt, T. (2006). Evolution in group-structured populations can resolve the tragedy of the commons. Proceedings of the Royal Society B: Biological Sciences, 273(1593), 1477-1481.

Wakano, J. Y., Nowak, M. A., & Hauert, C. (2009). Spatial dynamics of ecological public goods. Proceedings of the National Academy of Sciences, 106(19), 7910-7914.

Wakano, J. Y., & Hauert, C. (2011). Pattern formation and chaos in spatial ecological public goods games. Journal of Theoretical Biology, 268(1), 30-38.

Zhang, F., & Hui, C. (2011) Eco-evolutionary feedback and the invasion of cooperation in the prisoner’s dilemma games. PLoS One, 6(11): e27523.