The Evolutionary Trade-off between Stem Cell Niche Size, Aging, and Tumorigenesis

Many epithelial tissues within large multicellular organisms are continually replenished by small independent populations of stem cells. These stem cells divide within their niches and differentiate into the constituent cell types of the tissue, and are largely responsible for maintaining tissue homeostasis. Mutations can accumulate in stem cell niches and change the rate of stem cell division and differentiation, contributing to both aging and tumorigenesis. Here, we create a mathematical model of the intestinal stem cell niche, crypt system, and epithelium. We calculate the expected effect of fixed mutations in stem cell niches and their expected effect on tissue homeostasis throughout the intestinal epithelium over the lifetime of an organism. We find that, due to the small population size of stem cell niches, fixed mutations are expected to accumulate via genetic drift and decrease stem cell fitness, leading to niche and tissue attrition, and contributing to organismal aging. We also explore mutation accumulation at various stem cell niche sizes, and demonstrate that an evolutionary trade-off exists between niche size, tissue aging, and the risk of tumorigenesis; where niches exist at a size that minimizes the probability of tumorigenesis, at the expense of accumulating deleterious mutations due to genetic drift. Finally, we show that the probability of tumorigenesis and the extent of aging trade-off differently depending on whether mutational effects confer a selective advantage, or not, in the stem cell niche.

(1) Figure 1: The general architecture of a crypt system. Population names are within the boxes and the rates at which cells accumulate within or are transferred between populations are next to the arrow portraying their transition.
Setting the left-hand sides for each equation in the system (1) to zero, we can solve for the 92 steady state mean of the terminally differentiated population, Z . We find that Z can be 93 expressed in terms of the system's rate parameters and the stem cell niche population size X 1 .

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The steady state Z (Eq. 2) is a function of all of the rate parameters in the system except the 95 TA cell division rate: This function is used in subsequent analyses to calculate how mutations to the various rate 97 parameters change the tissue population size. Of note, a prediction of this model is that a 98 mutation to either division rate or differentiation rate will result in an amplified effect on the 99 proportion of change in steady state post-mitotic cells. That is, if λ 0 is mutated to λ 1 , the 100 proportional difference in the post-mitotic cell population is consist of approximately 3500 cells [Potten and Loeffler, 1990]. The total steady state mean 132 population in our model contributed by 8 crypts to a villus using the parameters in Table 1 is 133 8 × 446 = 3568 cells. 134 We are interested in estimating the effects of mutation accumulation throughout the whole 135 intestinal tract in the mouse, hence we take the median value of certain parameters that vary 136 from proximal small intestine to colon (as described above). We note that if one was interested 137 in the effects of steady state terminally differentiated population size in just the proximal small 138 intestine or colon this analysis may overestimate or underestimate the effects, respectively, 139 since there are more stem cells dividing faster in the colon than the proximal small intestine.

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Additionally, the large intestine TA cells may undergo more rounds of division than the small 141 intestine TA cells [Potten, 1998].

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A new lineage with a division rate relative to the background division rate λ λ0 has probability of eventually replacing the original lineage We use Bayes' theorem to calculate the probability density of the division rate given the mutant lineage fixed, and, redefining Eq. 5 such that f 1 (λ) is equal to the density given the first fixation of a mutant 198 lineage, we calculate the expected value of the division rate of this new lineage: From Eq. 6 we can calculate the expected value of division rate given m mutations, which is 210 the expected value of these probability densities: We model the rate at which new lineages arise and fix in the crypts as being constant over 212 time with rateμ =pµλ 0 X 1 , wherep is the total probability of fixation Thus, we can estimate the total number of crypts with m mutations as the organism ages.

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Calculating the probability that fixed mutations initiate tumorigenesis.
where Z 1 Z 0 is the steady state population size of the post-mitotic cell population after one fixed 247 mutation divided by the healthy population size with zero mutations (Eq. 2). When mutations 248 alter the division rate the fraction and θ = ν λ0 . Thus, Eq. 9 simplifies to When mutations affect the differentiation rate of stem cells, the fraction In this case Eq. 9 simplifies to and the derivative of this function with respect to time is: the inverse of one another.

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Equations 10 and 11 are close approximations of the rate of tissue size change per day. it is likely the mutation conferred a beneficial effect. 308 q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q 0.0 0.   q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q X 1 =20 Mutations Affect Division Rate −− Linear Approximation X 1 =20 Mutations Affect Differentiation Rate −− Linear Approximation X 1 =6 Mutations Affect Division Rate −− Linear Approximation X 1 =6 Mutations Affect Differentiation Rate −− Linear Approximation Figure 5: The expected tissue size change in humans due to mutation accumulation.

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ing mediated by stem cell niche size. 330 We vary initial stem cell population size, along with the total number of crypts in the system, 331 such that the total output of crypts, i.e., healthy tissue, remains constant. This analysis was 332 initially conducted using the crypt parameters described for the mouse in Table 1 Figure 6A). However, at this crypt size, the expected value 338 of the epithelium tissue size is expected to decrease over a lifetime due to the accumulation 339 of deleterious mutations in stem cell niches ( Figure 6B). Furthermore, when mutations affect 340 differentiation rate and fix neutrally, the probability of tumorigenesis is minimized for large 341 stem cell niche sizes ( Figure 6C) and the expected effect on tissue size is invariant to stem cell 342 niche size ( Figure 6D). As the true mutational parameters governing somatic tissue evolution 343 are unknown, we do not wish to analyze the predicted magnitude of tumorigenesis and aging 344 presented here, but rather the trade-off that exists between niche population size, drift, and 345 selection among mutations that confer a selective advantage and those that fix neutrally. The 346 nature of the dynamics presented in Figure 6 holds true for a large range of mutation parameters.

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For instance, we next analyzed this evolutionary trade-off when considering the parameters 348 discussed in Section 2.4 for the human colon (Figure 7). We find that the probability of a  increasing the chance that a fixed mutation was beneficial, leading to higher chances of tumorigenesis. Here, given the specified parameters for mice, we find that the minimum probability of tissue homeostasis. For instance, a mutation that arises within a niche that is ten times the size 387 of another niche has ten times smaller probability of fixation, but ten times higher influence on 388 the total epithelium, meaning that the expected influence of the total amount of accumulated 389 mutations in systems with different stem cell population sizes but consistent total epithelium 390 sizes is invariant.

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It is probable that the mutational target size for mutations that affect the propensity for stem 392 cells to commit to differentiation is smaller than that for mutations that might affect the overall 393 division rate, and therefore selection may not act as strongly to optimize niche size in light on minimizing tumorigenesis caused by failure to differentiate. Furthermore, our model results

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indicate that both the probability of tumorigenesis and the extent of tissue size change are larger 396 for the scenarios where mutations only affect differentiation rate. This is due to mutations in 397 this scenario fixing neutrally and having a larger absolute influence towards tumorigenesis (we model expected mutational effect sizes as a proportion of the rate they govern). We use the 399 same rate of mutation when modeling both mutations that affect division rate and differentiation 400 rate, but given that there is probably a smaller target size for mutations that change the rate 401 at which stem cells commit to differentiation, there will also be a smaller mutation rate.

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The intestinal epithelium population is expected to decline with age through 403 stem cell attrition. When employing distributions of mutational effects commonly found in 404 experiments on whole organisms we find that the total intestinal epithelium size is expected 405 to decrease with age. This attrition is modest in the mouse, with a mouse three years into 406 adulthood having an intestinal epithelium approximately 0.3% -0.5% smaller than a mouse that 407 just entered adulthood. This attrition is potentially much more substantial for humans, given