Mathematical modelling of hypoxic glioma stem cell niches suggests rapid selective evolutionary dynamics

<p>Poster given at National Center for Regenerative Medicine first annual Stem Cells in Cancer workshop. 7/2014, Cleveland, OH</p> <p> </p> <p>http://www.ncrm.us/events/csc/landing.cfm</p

Anti-angiogenesis therapy changes the benefit function.

<p>In fact, anti-angiogenesis therapy can change <i>b</i>_<i>max</i>, <i>s</i>, and inflection point <i>a</i>. In this figure inflection point <i>a</i> has been changed to <i>a</i>′. Dashed curve shows new benefit function.</p

Evolutionary emergence of angiogenesis in avascular tumors using a spatial public goods game

<div><p>Natural selection in cancer often results in the emergence of increasingly malignant tumor cells that display many if not all of the hallmarks of cancer. One of the most important traits acquired during cancer progression is angiogenesis. Tumor cells capable of secreting pro-angiogenic factors can be seen as cooperators where the improved oxygenation, nutrient delivery and waste disposal resulting from angiogenesis could be seen as a public good. Under this view, the relatively costly secretion of molecular signals required to orchestrate angiogenesis would be undertaken exclusively by cooperating tumor cells but the benefits of angiogenesis would be felt by neighboring tumor cells regardless of their contribution to the process. In this work we detail a mathematical model to better understand how clones capable of secreting pro-angiogenic factors can emerge in a tumor made of non-cooperative tumor cells. Given the importance of the spatial configuration of the tumor in determining the efficacy of the secretion of pro-angiogenic factors as well as the benefits of angiogenesis we have developed a spatial game theoretic approach where interactions and public good diffusion are described by graphs. The results show that structure of the population affects the evolutionary dynamics of the pro-angiogenic clone. Specifically, when the benefit of angiogenesis is represented by sigmoid function with regards to the number of pro-angiogenic clones then the probability of the coexistence of pro-angiogenic and angiogenesis-neutral clones increases. Our results demonstrate that pro-angiogenic clone equilibrates into clusters that appear from surrounding vascular tissues towards the center of tumor. These clusters appear notably less dense after anti-angiogenic therapy.</p></div

The process of angiogenesis.

<p>When tumor’s diameter excessed a critical range (2 <i>mm</i>) (a), angiogenic clones start to secrete pro-angiogenic signals (b). The signals degrade capillary vessel wall (c) and finally results in migration of endothelial cells and formation new tubes with a central lumen (d).</p

Types of evolutionary dynamics with regards to the model’s parameters.

<p>Each panel plots the fitness of two clones based on the relative frequency of the pro-angiogenic. The meeting point of two fitness indicates stable equilibrium of the game (red and black circles show unstable and stable points of the game in each panel). The game has five types of the evolutionary dynamics. 1- Panel A: only pro-angiogenic clone (<i>x</i> = 1) is stable 2- Panel B: only free-rider clone (<i>x</i> = 0) is stable 3- Panel C: both pro-angiogenic and free-rider clones are stable 4- Panel D: only a polymorphic population (<i>x</i> = <i>x</i>_<i>s</i> where 0 ≤ <i>x</i>_<i>s</i> ≤ 1) is is stable 5- Panel E: both polymorphic population and pro-angiogenic clone are stable.</p

Three examples for evolution of cooperation on irregular graph (blue and red nodes show pro-angiogenic and free-rider clones, respectively).

<p>Panel (A) represents an examples of spreading pro-angiogenic clone in the population. In this example, coexistence between pro-angiogenic and free-rider clones arises over time. While panel (B) demonstrates how pro-angiogenesis clone can go extinct in a dense graph. In this example, free-riders lonely spread in the population and act as a stable strategy. We assume that all nodes in the graph have free-rider strategy (red nodes) and then a small fraction of them are mutated to pro-angiogenic clone (blue nodes). Left column shows initial populations including the mutant nodes, middle column represents final topology that has been shaped by natural selection forces over time, and right column indicates fraction of pro-angiogenic clone in the population over time. As shown, both free-rider and pro-angiogenic clones can coexist on irregular graphs. Simulation codes are available in <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0175063#pone.0175063.s001" target="_blank">S1 File</a>.</p

Anti-angiogenesis therapy can restore tumor vessels towards a more normal behavior.

<p>In this figure, there are several examples of tumors that have became dramatically less hyperemic after treatment. These examples (from left to right) are from Willett et al. [<a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0175063#pone.0175063.ref051" target="_blank">51</a>] and Jain [<a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0175063#pone.0175063.ref052" target="_blank">52</a>], respectively, with permission. A: Tumors before tratment. B: Tumor after anti-angiogenesis treatment. As shown in the right panels, our simulations can predict such normal state. In our simulations, after binding inhibitors to the receptors, the number of pro-angiogenic clone that are representative of the tumor blood vessels decreases. Simulation codes are available in <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0175063#pone.0175063.s001" target="_blank">S1 File</a>.</p

Each cell on the graph plays <i>k</i> + 1 PGGs with <i>k</i> neighbors.

<p>Node <i>p</i> participates in six PGGs, simultaneously. Panels (A)-(F) show all these games.</p

Characterization of different levels of angiogenic switch in the parameter space of <i>c</i> and <i>b</i> (right panel), and <i>c</i>, <i>k</i> (left panel).

<p>The red region corresponds to cells that are free-riders (weak angiogenesis), the dark blue region demonstrates cells that strongly produce pro-angiogenic factors (strong angiogenesis), and the pale blue represents a mixed stable strategy between pro-angiogenic and free-rider clones.</p

A sample of human-prostate cancer tissue seen from the proteinatlas station (http://www.proteinatlas.org/learn/dictionary/cancer/prostate+cancer+3).

<p>Panel B indicates tumor cells that have been shown in panel A. As shown, each tumor cell interacts with all neighbors living in a distance <i>r</i> from this cell, by pro-angiogenic factors diffusion.</p