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The Importance of Spatial Randomness in the Evolutionary Dynamics of Mutants

Applications to Cancer

By Mohammad Kohandel and Natalia L. Komarova

Cancer is a devastating disease involving abnormalities in cellular control mechanisms, in which cells are aggressive (grow out of control), invasive (spread and destroy adjacent tissues), and sometimes metastatic (travel to other parts of the body). Cancer progression results from the accumulation of multiple mutations and epigenetic changes, which may give mutated cells a selective advantage over healthy cells. To understand cancer dynamics, we must study the evolutionary processes that take place inside the tissues. 

Mutation and selection are the hallmarks of evolution. Generation and spread of mutants propels the evolutionary process forward; in this context, mutant fixation can be very informative. Fixation is the replacement of an initially heterogeneous population with the offspring of just one individual (i.e., a mutant). The probability of fixation and the average time required for a mutant to take over a population are two fundamental quantities in evolution.

The (Homogeneous) Moran Model

Researchers have used the Moran model extensively to study mutant fixation in tissues, where at each update in a constant population of asexually reproducing agents (cells), a cell is removed and replaced with another cell’s offspring. The cell that goes and the one that reproduces is decided probabilistically, based on their kinetic (or fitness) parameters.

In a population of \(N\) cells, suppose that the fitness of \((N-1)\) wild type (i.e., normal, unmutated) cells is equal to \(1\), and a single mutant cell has a smaller fitness, \(r<1\). Unsurprisingly, the mutant cell will have a smaller fixation probability than any of the wild type cells. In the mass-action problem (where all cells compete with each other regardless of location), the probability of such disadvantageous mutant fixation is exponentially small in terms of population size \(N\). In the (opposite) case of an advantageous mutant (\(r>1\)), the probability of mutant fixation is larger than that of fixation of wild type cells, and is given by \(1-1/r\) for large populations. Similar results also hold in the case of circular geometry (where cells only compete with their immediate neighbours along a circle): the probabilities of disadvantageous and advantageous mutant fixation are given respectively by \(2r^{N-2}(1-r)/(3-r)\) and \(2(r-1)/(3r-1)\) [1]. 

If the mutant’s fitness is exactly the same as that of the rest of the cells (\(r=1\), a neutral mutant), the probability of the whole population eventually being replaced by the offspring of such a mutant is exactly the same as the probability of fixation of any other cell and equals \(\frac{1}{N}\); this is the consequence of the system’s symmetries and holds for mass action and other arrangements of cells, like a circle. In general, if we start with \(i\) mutants in a population of \(N\) cells, mutant fixation probability is given by \(i/N\). In other words, cells’ initial frequency yields mutant fixation probability. The aforementioned simple situations are idealizations because, for example, they do not include the environment’s inhomogeneities.

Tumor Microenvironment

In reality, cancers are characterized by highly complex and heterogeneous microenvironments, which include stroma, necrotic cells, blood vessels, etc. Tumors cannot grow beyond a certain size through simple diffusion of oxygen and other essential nutrients. Angiogenesis, the process by which tumors develop their own blood supply, is an important step in tumor growth. The distribution of oxygen and hypoxic regions is highly non-homogeneous, the nutrients are distributed in a complex fashion, and (in general) no two tumors are the same. In a sense, tumors are like unhealed wounds, in that they produce large amounts of inflammatory mediators (cytokines, chemokines, and growth factors). These molecules attract the so-called tumor infiltrating cells, including macrophages, myeloid-derived suppressor cells, mesenchymal stromal cells, and Tie2-expressing monocytes. Collectively, these populations of non-malignant cells contribute to the formation of a rich and heterogeneous tumor microenvironment. In order to understand selection and mutation dynamics of cancer cell populations in such an environment, restricting the modeling efforts to the classic problems—where all wild type cells are exactly (phenotypically) the same and all mutant cells have the same constant fitness value—is not enough. Instead, we must include differences among various spatial locations.

Our Recent Work

We recently made progress towards a more realistic view of cancer dynamics, where fitness values of different genotypes are subject to microenvironmental variations [2, 3]. We studied the effect of spatial randomness on mutant fixation properties, and assumed that  the division rates for wild type and mutant cells at different locations were drawn from probability distributions (and for a given realization, they did not change in time) (see Figure 1). We studied evolutionary dynamics of mutants — specifically the probability of mutant fixation, averaged over all realization of fitness values, and the mean conditional mutant fixation time (averaged over all fitness configurations). The situation of non-selected mutants is especially interesting. 

Figure 1. A schematic illustrating spatial randomness (complete graph is on the left, and circle is on the right). The fitness of cells is represented by color at different spots. The lower left refers to wild type fitness (high fitness for saturated blue, low fitness for unsaturated blue). The upper right refers to mutant fitness (high fitness for saturated red, low fitness for unsaturated red). For example, a wild type cell at location marked by “X” would have low fitness, but a mutant at the same location would have high fitness.

Suppose that fitness values of mutants are drawn from a probability distribution identical to that of wild type cells. If we consider all possible realizations of fitness configurations in this case, we clearly cannot view the mutants as advantageous (i.e., selected for or possessing an intrinsically higher potential for spread than wild type cells) or disadvantageous (i.e., selected against). Therefore, one would expect them to fixate on average with probability equal to their original frequency. However, we discovered that this is not the case. Instead, mutants behaved as if they were under positive selection — as long as they were originally in a minority. In other words, their mean probability of fixation was significantly larger than their initial frequency. We identified this trend in a variety of models, including the Moran model and the haploid Fisher-Wright process (another popular model in genetics), and showed that it held both in mass action and models with spatial mutant interactions. We further demonstrated that the conditional fixation time was significantly affected by randomness, and that randomness increased the mean fixation time on a circle but decreased it on a complete graph.

Among our findings is a comparison of the evolutionary dynamics under different cell interaction networks. We contrasted the mass action system (where all cells competed with each other) with a one-dimensional geometry, where cells interacted only with their immediate neighbours along a circle.  This revealed that environmental randomness affects mutant fixation time in opposite ways for the two cases: randomness delayed fixation of mutants on circles, and accelerated fixation on complete graphs. For one-dimensional-type structures (circles), “dead zones” that form randomly in the presence of environmental influences can significantly delay fixation by blocking the necessary paths. Yet for fully-connected graphs, “lucky paths” that facilitated fixation form at random. The difference between fixation probabilities in circular and complete graphs exemplifies the general phenomenon of well-mixedness and its role in evolutionary mutant dynamics; it is further related to the role of dimensionality in system dynamics.

To put this in a larger context, researchers have shown that inactivation of a tumor-suppressor gene (a two-hit evolutionary process in which the cells must first become less fit before becoming more fit) occurs more quickly in one dimension (a row of cells), than in two dimensions (a layer). In turn, this happens faster than in a fully-mixed system with no spatial constraints [4-7]. By contrast, in two-step processes where the intermediate mutant confers a slight selective advantage, the relationship is the opposite and a non-spatial, fully-mixed environment promotes the fastest pace of evolution [5]. These phenomena seem less surprising if we note how reminiscent they are of other fundamental laws of nature in which space dimensionality changes how things work [8], such as the different fundamental solutions to Poisson's equations in one and two dimensions [2].

Biological Applications

Our research into the evolutionary dynamics of random environments revealed that spatial heterogeneities play a significant role in the process of mutant fixation, meaning that traditional models could be misleading in regard to cancer dynamics.  For example, an important biological application of the one-dimensional circle geometry is the model of a human colonic crypt. There, a relatively small (of the order of ten cells or less) population of stem cells is situated along circular bands, which one can view as cross-sections of a three-dimensional colonic crypt (the tissue structure where colon cancer originates). Active stem cells occupy a narrow circular layer and divide mostly symmetrically. The two division types—self-renewal and differentiation—are mathematically equivalent to divisions and deaths in our models. Previous researchers have shown that stochastic evolutionary dynamics of mutant acquisition in the crypts are largely responsible for observed rate incidence curves [9]. In other words, microscopic details of mutant generation and spread (and their statistical features) are important factors in the epidemiological, population-wide statistics of colon cancer. Our results for mutant dynamics on a circle show that random environmental heterogeneities may have a significant effect on the evolutionary processes in a crypt, and should be considered by scientists studying the origins of colon cancer.

In particular, the very first event in colon cancer progression is normally the inactivation of a copy of a tumor suppressor gene. Researchers generally consider this event to result in the formation of a non-selected mutant, and traditional modeling suggests that such mutants fixate with a probability equal to their original frequency (i.e., \(1/N\)). Our recent work shows that, as such mutants are originally in a minority, their fixation probability could on average be significantly higher than this. In other words, ignoring spatial heterogeneity of the environment may result in an error when quantifying the evolutionary rate of cancer generation. Scientists can use our model for more precise quantification of this important process. 

References 
[1] Komarova, N.L. (2006). Spatial stochastic models for cancer initiation and progression. Bull. Math. Bio., 68(7), 1573-1599.
[2] Darooneh, A.H., Nikbakht, M., Komarova, N.L., & Kohandel, M. (2017). The effect of spatial randomness on the average fixation time of mutants, S Farhang-Sardroodi. PLoS Comp. Bio., 13(11), e1005864.
[3] Mahdipour-Shirayeh, A., Darooneh, A.H., Long, A.D., Komarova, N.L., & Kohandel, M. (2017). Genotype by random environmental interactions gives an advantage to non-favored minor alleles. Sci. Rep., 7(1), 5193.
[4] Komarova, N.K. (2006). Spatial stochastic models for cancer initiation and progression. Bull. Math. Biol. 68(7), 1573-99.
[5] Komarova, N.L. (2014). The impact of environmental fluctuations on evolutionary fitness functions. Proceed. Nat. Acad. Sci., 111, 10789-10795.
[6] Komarova, N.L., Shahriyari, L., & Wodarz, D. (2014). Complex role of space in the crossing of fitness valleys by asexual populations. J. Roy. Soc. Inter., 11(95), 20140014.
[7] Durrett, R.,& Moseley, S. (2015). Spatial Moran models I. Stochastic tunneling in the neutral case. Ann. Appl. Prob., 25(1), 104.
[8] Komarova, N.L. (2015). Cancer: A moving target. Nat., 525(7568), 198-199.
[9] Luebeck, E.G., & Moolgavkar, S.H. (2002). Multistage carcinogenesis and the incidence of colorectal cancer. Proceed. Nat. Acad. Sci., 99(23), 15095-15100.

Mohammad Kohandel is a professor in the Department of Applied Mathematics of the University of Waterloo. He is also a member of the Center for Mathematical Medicine at the Fields Institute for Research in Mathematical Sciences in Toronto, Canada. His research interests include applications of theoretical and computational models to biological sciences, particularly to cancer. Natalia Komarova is Chancellor’s Professor of Mathematics at the University of California, Irvine. Her interests lie in the interface of mathematical and life sciences, including biology, medical and social sciences, and linguistics, with the unifying theme of evolution. 

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