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Genotype Patterns in Growing Solid Tumors bioRxiv preprint doi: https://doi.org/10.1101/390385; this version posted August 12, 2018. The copyright holder for this preprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made available under aCC-BY-NC-ND 4.0 International license. Genotype patterns in growing solid tumors Mridu Nandaa and Rick Durrettb∗ Harvard U and Duke U August 12, 2018 Abstract Over the past decade, the theory of tumor evolution has largely focused on the selective sweeps model. According to this theory, tumors evolve by a succession of clonal expansions that are initiated by driver mutations. In a 2015 analysis of colon cancer data, Sottoriva et al [34] proposed an alternative theory of tumor evolution, the so-called Big Bang model, in which one or more driver mutations are acquired by the founder gland, and the evolutionary dynamics within the expanding population are predominantly neutral. In this paper we will describe a simple mathematical model that reproduces qualitative features of the observed paatterns of genetic variability and makes quantitative predictions. ∗Both authors are partially supported by NSF grant DMS 1614978 from the math biology program. 1 bioRxiv preprint doi: https://doi.org/10.1101/390385; this version posted August 12, 2018. The copyright holder for this preprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made available under aCC-BY-NC-ND 4.0 International license. 1 Introduction The central region of the plate exhibits a dense speck- led pattern reminiscent of the initial mixed popu- In the multistage theory of carcinogenesis, it is thought lation. From this ring toward the boundary of the that the sequence of “driver” mutations produces a colony, the population segregates into single colored series of selective sweeps. This theory has been con- sectors with boundaries that fluctuate. [20] and [26] firmed by whole genome sequencing of cancer cells. have developed and analyzed models for the develop- Ding et al [10] have identified the clonal structure ment of sectors in the system. of eight relapsed acute myeloid leukemia (AML) pa- In previous work [13], we used the biased voter tients. In one patient the founding clone 1 accounted model [40, 7, 6] to model the microscopic tumor dy- for 12.74% of the tumor at the time of diagnosis. The namics. The genealogies in that model are difficult additional mutations in clones 2 and 3 may have re- to study, so we used ideas of Hallatschek and Nelson sulted in growth or survival advantages because they [19] to build an approximate model of the movement were 53.12% and 29.04% of the tumor respectively. of genealogies. That attempt failed to reproduce ob- Only 5.10% of the cells were in clone 4 indicating that served patterns because it predicted extensive het- it may have arisen last. However, the relapse evolved erogeneity at small scales. We attribute that failure from clone 4 with the resultant clone 5 having 78 new to continual turnover of cells in the interior of the somatic mutations compared to the sampling at day tumor, so here we adopt a new approach in which 170. tumor growth only occurs at the boundary. We will A mathematical theory has been developed for the introduce four models: two for two-dimensional tu- clonal expansion model to make predictions about the mors and two for three dimensional tumors. In the level of intratumor heterogeniety [4, 14], the num- terminology of interacting particle systems, two will ber of passenger (neutral) mutations in cancer cells use voter model dynamics, and two will use contact [2, 39], and more practical questions such the evolu- process dynamics. We describe the models and out tion of resistance to treatment [21, 27, 38], and the results in the next four sections. A final section de- effectiveness of combination therapy [3, 25, 28]. See scribes our conclusions. [12] for an introduction to the mathematics underly- ing many of these applications. The picture of successive selective sweeps has been 2 Voter model d = 1 + 1 confirmed by comparing primary tumors and their metasases [42], and by regional sequencing of breast This model can be thought of as taking place in 1 + 1 cancer [29], glioblastoma [33], and renal carcinoma dimensions, i.e., one space and one time. To formu- 2 [17]. Thus it was surprising when Sottoriva et al late the model let L = {(k, `) ∈ Z : ` ≥ 0, k + [34] introduced and validated a ‘Big Bang’ model in ` is even}. It would be more natural to have a grow- which all driver mutations were present at the time ing circular tumor, but that is awkward for computa- of tumor initiation. They collected genetic data of tions. The approach we take here should be thought various types from 349 individual tumor glands sam- of as studying a tumor that is large enough so that pled from the opposite sides of 15 colorectal tumors when we view the dynamics at the level of individual and large adenomas. Data presented in Figure 3 of cells the curvature of the boundary can be ignored. their paper shows that adenomas were characterized To study the dynamics we suppose that at time 0 by mutations and copy number aberrations (CNA) each cell has its own type, which we set equal to its that segregated between tumor sides. In contrast the first (spatial) coordinate. We add a new layer of cells majority of carcinomas exhibited the same private at each time step. The cell at (k, `) is the type of CNA in individual glands from different sides of the the cell at (k − 1, ` − 1) with probability 1/2 and the tumor. Following up on this observation, Ryser et al type of the cell at (k + 1,` − 1) with probability 1/2, [31] found evidence of early abnormal cell movement with the choices for different cells being independent. in 8 of 15 invasive colorectal carcinomas (“born to be The use of L may look odd but it has the desirable bad”) but not in four benign adenomas. For more property that the set of sites occupied by one type is about recent developments see the review article by always an interval Sun, Hu, and Curtis [36]. In making pictures of simulations the fact that are The mutation patterns found in [34] are similar no cells at (k, `) with k+` odd is annoying so we adopt to those found by Hallatschek et al [18] in a remark- the mathematically equivalent approach that at even able experimental paper. Two fluorescently labeled times (k,`) will imitate (k, ` − 1) or (k + 1,` − 1) strains of E. coli were mixed and placed at the cen- with equal probability, while at odd times (k, `) will ter of an agar plate containing a rich growth medium. imitate (k, `−1) or (k−1, `−1) with equal probability. 2 bioRxiv preprint doi: https://doi.org/10.1101/390385; this version posted August 12, 2018. The copyright holder for this preprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made available under aCC-BY-NC-ND 4.0 International license. Figure 1: Simulation of voter model. Figure 2: Genealogies in the voter model Figure 1 shows a simulation. be the genealogy of another site (j, n). If T` = S` This system is easy to analyze because the bound- then the two will agree at all later times. In words, aries between types are random walks. In proving the when the random walks hit they coalesce to 1. It fol- next result we will consider the model on L. lows from results of Bramson and Griffeath [5] that the density of coalescing random walks and hence the 0 −1/2 Lemma 1. Let Nn be the number of individuals at number types at time n decays line cn . 0 P∞ 0 time n whose ancestor at time 0 is at 0. Then Nn is Let N0 = n=0 Nn be the total number of de- a lazy random walk with an absorbing state at 0 and scendents of the cell at 0 at time 0. Combining (1) transition probability p(k, k + 1) = p(k, k − 1) = 1/4, with the fact that the typical distance between bound- p(k, k) = 1/2 for k > 0. aries is 1 over the number of surviving lineages, we see that Proof. Suppose that points at −2j, −2j + 2,... − 2, 0 are red at time n. If j ≥ 1 then the red region will 1/2 −1/2 P (N0 > n · n)= O(n ) (2) surely contain −2j + 1,... − 1. It will contain 1 with probability 1/2, and will contain −2j − 1 with prob- Changing variables n = k2/3 suggests ability 1/2, so if the size k = j + 1 ≥ 2 the size goes −2/3 from k + 1 to k, k + 1 and k + 2 with probabilities Theorem 1. As k → ∞, P (N0 > k) ∼ ck . 1/4, 1/2, and 1/4. Suppose now that j = 0, so the The proof, which is hidden away in an appendix, gives size k = 1. By looking at the four cases for the choice a precise value for the constant and involves some of parent by −1 and by 1, then the size goes to 2, 1, interesting facts for Brownian motion. and 0 with probabilities 1/4, 1/2, and 1/4. 0 Let T0 = min{n : Nn = 0}.
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