Aspirin’s effect on kinetic parameters of cells contributes to its role in reducing incidence of advanced colorectal adenomas, shown by a multiscale computational study
Abstract
Aspirin intake has been shown to lead to significant protection against colorectal cancer, for example with an up to twofold reduction in colorectal adenoma incidence rates at higher doses. The mechanisms contributing to protection are not yet fully understood. While aspirin is an antiinflammatory drug and can thus influence the tumor microenvironment, in vitro and in vivo experiments have recently shown that aspirin can also have a direct effect on cellular kinetics and fitness. It reduces the rate of tumor cell division and increases the rate of cell death. The question arises whether such changes in cellular fitness are sufficient to significantly contribute to the epidemiologically observed protection. To investigate this, we constructed a class of mathematical models of in vivo evolution of advanced adenomas, parameterized it with available estimates, and calculated population level incidence. Fitting the predictions to age incidence data revealed that only a model that included colonic crypt competition can account for the observed ageincidence curve. This model was then used to predict modified incidence patterns if cellular kinetics were altered as a result of aspirin treatment. We found that changes in cellular fitness that were within the experimentally observed ranges could reduce advanced adenoma incidence by a sufficient amount to account for age incidence data in aspirintreated patient cohorts. While the mechanisms that contribute to the protective effect of aspirin are likely complex and multifactorial, our study demonstrates that direct aspirininduced changes of tumor cell fitness can significantly contribute to epidemiologically observed reduced incidence patterns.
Editor's evaluation
This work develops a multistage/component mathematical model to analyze advanced colorectal adenomas and the impact that aspirin therapy has on adenoma formation rates. This study will be interesting to the cancer evolution community and in particular those interested in colorectal cancer incidence. While the model is mainly focused on aspirin chemoprevention, the model could be adapted to test other putative preventative agents, and thus could have a broad impact.
https://doi.org/10.7554/eLife.71953.sa0Introduction
Colorectal cancer currently affects about 5% of the population in the USA and is a major cause of cancerrelated deaths (Siegel et al., 2020). Prevention of colorectal cancer is an important goal in the quest to reduce morbidity and mortality. In this respect, longterm aspirin use has been shown to be effective (Chan et al., 2012; Thun et al., 1991). Aspirin is a nonsteroidal antiinflammatory drug (NSAID) and is a cyclooxygenase (COX)–2 inhibitor (Goel et al., 2003). The CAPP2 trial Burn et al., 2011 demonstrated that the intake of 600 mg of aspirin per day for 2 years resulted in a 63% reduction in colorectal cancer incidence in Lynch Syndrome patients. Interestingly, observation of the protective effect of aspirin required a followup time of more than 55 months (Burn et al., 2011). In a range of studies, aspirin has also been shown to reduce incidence of sporadic colorectal cancer (Thun et al., 1991; Tougeron et al., 2014; Friis et al., 2015; Lochhead and Chan, 2016; Chan et al., 2007; Drew et al., 2016; Chan et al., 2005; Rothwell, 2013), and of adenomas (Sandler et al., 2003; Chan et al., 2004), which are a precursor of cancer. This was evident both in observational studies and in randomized controlled trials, which are reviewed for example in Lochhead and Chan, 2016; Drew et al., 2016. These studies report a reduction in cancer or adenoma incidence of the order of 10–50% in aspirintreated compared to placebo groups, depending on the exact dose and frequency of aspirin intake. While some studies failed to detect significant protective effects of aspirin, larger studies with higher aspirin doses and longer treatment times yielded significant results.
The mechanisms underlying the protective effect of aspirin are likely complex and multifactorial. Inflammation is a possible driver of colorectal carcinogenesis (Itzkowitz and Yio, 2004), and aspirin can reduce the extent of inflammation in the cellular microenvironment, which might contribute to a reduced development of disease. Other microenvironmental effects, such as the composition of the colorectal microbiome (Prizment et al., 2020; Brennan et al., 2021; Zhao et al., 2020), have also been shown to determine the degree of protection provided by aspirin. Our previous in vitro and in vivo work, however, has shown that physiologically relevant aspirin concentrations can have a direct effect on tumor cells, reducing their rate of proliferation and increasing their death rate (Shimura et al., 2020; Zumwalt et al., 2017). This not only results in reduced tumor growth, but can also lead to a lower probability that newly generated tumor cells successfully give rise to clonal expansion, thus increasing the likelihood that these initially transformed cells go extinct (Wodarz et al., 2017). This effect might contribute to the reduced incidence of colorectal cancer as a result of aspirin intake.
While these direct effects of aspirin on tumor cell division and death rates have been documented by us in vitro and in vivo (Shimura et al., 2020; Zumwalt et al., 2017), and occurred under physiologically realistic doses, it is unclear to what extent these changes in cellular kinetics can potentially alter disease incidence. To evaluate this quantitatively, a mathematical modeling framework needs to be developed that predicts epidemiological incidence data based on cellular processes. There is a rich history of such approaches in the cancer literature in different contexts (Fisher and Hollomon, 1951; Nordling, 1953; Armitage and Doll, 1954; Luebeck and Moolgavkar, 2002; Meza et al., 2008; Moolgavkar, 1978; Hornsby et al., 2007), which has allowed researchers to gain fundamental insights into carcinogenic processes based on the interpretation of ageincidence data. Here, we describe a mathematical model of advanced adenoma formation and parameterize it by fitting epidemiological predictions to incidence data that document advanced adenoma occurrence as a function of age. We then use this model to test whether aspirinmediated changes in cellular kinetics, as documented by our experiments, can result in reductions in advanced adenoma incidence that are comparable to those observed in aspirintreated patient cohorts. We find that the magnitude of changes in the kinetics of transformed cell populations that we observed experimentally can result in a pronounced reduction of advanced adenoma incidence, and that the epidemiologically observed incidence reductions (between 10% and 50%) can be explained by our model. This indicates that the direct effects of aspirin on dividing cells can in principle explain a significant amount of the chemoprotective effect exerted by this drug. We note, however, that while this is a clear result that emerges from this mathematical modeling effort, other mechanisms of aspirin not included in this model (such as antiinflammatory effects [Sostres et al., 2014] or modulation of the microbiome [Prizment et al., 2020; Brennan et al., 2021; Zhao et al., 2020]) are likely to also contribute to the observed protective effect.
We start by describing a mathematical model of advanced adenoma formation and show that when parameterized with experimentally obtained estimates, it can account for epidemiologically observed ageincidence curves, only as long as intercrypt competition is explicitly included. We then use this model to simulate the effect of aspirin on the incidence of advanced adenomas in human populations, and compare model predictions to epidemiological data.
Results
Computational modeling
In order to quantify the effects of aspirin on colorectal cancer initiation and progression, we have designed a mathematical model that is rooted in the process of multistep carcinogenesis (Luebeck and Moolgavkar, 2002; Hornsby et al., 2007; Fearon, 2011; Ashley, 1969). Its assumptions are similar in principle to those in a recent study (Paterson et al., 2020), with important differences that are discussed below. There are two early molecular events that we postulate (without assuming their temporal order): (1) An inactivation of the APC gene, or a related event that affects the functioning of the betacatenin/WNT signaling pathway, and (2) an activation of the KRAS oncogene (or another gainoffunction mutation). For simplicity, we will be referring to these mutations as APC and KRASmutations, keeping in mind that the model still applies in the presence of a pair of another lossoffunction and a gainoffunction mutation (further discussed below). The inactivation of the APC tumor suppressor gene is a classic example of a lossoffunction mutation, which implies two molecular events, corresponding to the inactivation of the two copies of the gene. The associated mutation rate is therefore assumed to be u = 10^{–7} per cell division. The activation of the KRAS oncogene, on the other hand, is a gainoffunction event, whose mutation rate is about two orders of magnitude lower (μ = 10^{–9} per cell division). The associated selectionmutation diagram is shown in Figure 1 and contains six different cell populations, denoted as types 1 through 6. The populations occupying the top row (types 1–3) are characterized by an unmutated KRAS oncogene; the populations of the bottom row (types 4–6) all have the KRAS mutation activated. Moving from left to right on this diagram, the number of inactivated copies of the APC gene increases from 0 to 2, such that populations of types 1 and 4 are APC+/+, populations of types 2 and 5 are both APC+/, and populations of types 3 and 6 are APC/.
These different populations correspond to different stages in the pathway towards colorectal cancer. The presence of the APC/ genotype has been related to the appearance of early adenomas (type 3) (Armaghany et al., 2012). Cells that have not inactivated APC but are characterized by KRAS activation (types 4 and 5) have been linked to aberrant crypt foci (Pretlow and Pretlow, 2005; Jass, 2006). The combination of both types of mutations (type 6) is thought to correlate with the growth of advanced adenomas (Armaghany et al., 2012). As in the previous study (Paterson et al., 2020), we assume flexibility regarding the order with which the different mutations can occur. Hence, it is assumed that the initial mutation can occur either in APC or in KRAS. It is, however, controversial whether adenoma formation can indeed be initiated by a mutation in KRAS. Some studies indicate that an initial mutation in KRAS leads to the formation of nondysplastic polyps, which could represent an evolutionary dead end for neoplasias (Jen et al., 1994; Chan et al., 2003). On the other hand, it has been suggested that an initial KRAS mutation might be able to drive the initiation of colorectal carcinogenesis (Jass, 2006; Pretlow and Pretlow, 2005), based on mutation frequencies in aberrant crypt foci and adenomas. The assumed flexibility in the evolutionary pathway of the model accommodates these conflicting notions.
While the model assumptions about the pathways to adenoma formation are clearly defined in our model, it is important to point out that there are uncertainties in those assumptions, and that there is heterogeneity in the types of mutations that can lead to colorectal carcinogenesis. For example, it has been reported that among nonhypermutated colorectal tumors, KRAS was mutated in only about 43% of patient samples (Network, 2012), indicating the importance of a variety of evolutionary pathways. Our model, however, does not depend on the identity of particular mutations, but assumes the occurrence of mutation types; these are the inactivation of a tumor suppressor gene (which is a lossoffunction mutation, e.g. APC/), and a gainoffunction mutation, which can be in KRAS or an alternative gene. Our model predictions hold as long as the evolutionary pathway to advanced adenomas involves these two types of mutational events, regardless of their identity. We note that our model does not apply to potential cases of advanced adenomas that might develop via pathways characterized by a different number or different types of initiating events.
We model the population dynamics of the colon by using a colonic crypt as a basic unit, which is similar in concept to recently published work (Paterson et al., 2020). Our model is related to many previous theoretical investigations of the cell population dynamics of crypts (Komarova et al., 2002; Komarova et al., 2003; Nowak et al., 2002; Shahriyari and Komarova, 2013; Shahriyari et al., 2016), where stem cells (SC) were assumed to acquire random mutations in a constantpopulation turnover (birth and death) process, and selection happened at the level of individual stem cells. Once it was discovered that there were very few stem cells per crypt (Nicholson et al., 2018; Humphries and Wright, 2008), it became clear that the evolutionary dynamics can be conveniently described at the level of crypts, because crypts are likely to be homogeneous with respect to the driver mutations. The rate at which a crypt changes its mutational status from i to j, denoted by R_{ij}, depends on the population size (the number of stem cells per crypt), the mutation rate, and the relative fitness of the invading cell type compared to the resident cell type (Komarova et al., 2003; Nowak et al., 2002). The latter can be calculated from the cell displacement data reported in the literature. Cell types APC+/, APC/, and KRAS + all have a selective advantage compared to the wild type, which we assume results in an increase of the SC division rate (see Section 2 of Appendix 1 for details).
Our model keeps track of crypts of different types (denoted as n_{i} for each type i). Modified crypts of types APC/ and KRAS + have been reported to undergo crypt fission; in other words, while the total population of a single crypt remains constant (even though it is populated by SCs that are fitter than the wildtype SCs), the crypt can undergo a doubling, thus increasing the total number of such modified crypts. The fission rates of different crypt types have been reported in the literature (Paterson et al., 2020; Nicholson et al., 2018; Humphries et al., 2013; Baker et al., 2014) and are denoted by γ_{i}; we further denote by δ the death rate of crypts, see Birtwell et al., 2020 for the role of crypt turnover. We model these dynamics by using the following system of ordinary differential equations:
where on the left hand side we have the rate of change for the population of crypts of each type, and the terms with the carrying capacity (K_{A} for KRAS^{} crypts and K_{R} for KRAS^{+} crypts) represent competition among modified crypts that undergo crypt fission; in reference (Paterson et al., 2020) no crypt competition was included, such that K_{R} = K_{A} = ∞ in that model. The initial conditions for the system above are given by n_{1}(0) = N_{crypt}, n_{i}(0) = 0, 1 < i ≤ 5, that is, initially all N_{crypt} crypts are wild type. Parameter values are presented in Appendix 1—table 2.
The probability to have produced a single crypt of type 6 (the APC^{/}KRAS^{+} phenotype) by time t is denoted by P(t) and is approximated by the following equation Chou and Wang, 2015,
We further assume that crypts of type 6 engage in a fissiondeath dynamics (with the corresponding rates γ_{6} and δ_{6}). At the time of detection, an advanced adenoma is characterized by a certain size, N. If ΔT denotes the expected time for the crypt population, n_{6}, to grow to the size of detection, then the value P(tΔT) calculated above approximates the mathematical expression for the ageincidence curve for advanced adenoma. These approximations were checked against stochastic (Gillespie) simulations recording the incidence of size N colonies of type 6 crypts, yielding excellent agreement (see Section 5 of Appendix 1 for details).
Fitting the adenoma incidence curve
Until recently, most of the parameters associated with cellular dynamics in colonic crypts were unknown, but presently many of the rates have been estimated with a high degree of confidence (Paterson et al., 2020), which makes it possible to parameterize the model and use it to answer questions about the process of crypt transformation and the dynamics of cancer initiation. Using the published data on the mutation rates, the total number of crypts, the number of SCs per crypt, and the relative fitness of different cell types (see Appendix 1—table 2), we first attempted to fit the model in the absence of crypt competition (K_{R} = K_{A} = ∞), by varying the SC division rate within the physiological range and finding the best fitting value for crypt fission rates. The best fitting parameter combinations always corresponded to zero crypt fission rates. Nonzero crypt fission rates resulted in a much steeper rise in the adenoma incidence compared to that reported in Brenner et al., 2014. A similar result was obtained when we used different values for fitness differences (the exhaustive parameter search and a model selection procedure are described in Section 3 of Appendix 1). Finally, using the reported crypt fission rates (Appendix 1—table 2) we were not able to find a SC division rate within the biologically applicable range that would give the correct shape of the advanced adenoma incidence curve. The conclusion is that an unlimited exponential expansion of crypts by fission gives an unrealistically steep rise in incidence. This problem did not occur in reference (Paterson et al., 2020) because fitting of the agespecific incidence of CRC was not attempted, and instead, only the total lifetime risk of CRC was compared to the model prediction.
Including crypt competition in the model has resolved this issue. We fixed the carrying capacity of type 3–5 crypts (parameters K_{R} and K_{A}) to values much smaller than the initial number of healthy crypts, N_{crypt}, to ensure the presence of significant competition among the partially transformed crypts. Using this model, we were able to fit the data for a wide range of the SC division rates, with the nonzero bestfitting crypt fission rates that have the correct order of magnitude. Additionally, fixing the crypt fission rates to their reported values, we were able to find very wellfitting incidence curves for a wide range of SC division rates (r_{1}), with the carrying capacity parameters ranging between about 100 and about 5000.
For the model that includes crypt competition, it was possible to find nearly equally good fits for a range of biologically plausible parameter values, see Figure 2. The amount of data in the advanced adenoma incidence curve does not allow finding unique values for all the parameters, but instead it allows using many of the parameters fixed to their experimentally obtained values, and just finetuning the small number of remaining parameters whose value is unknown (such as K_{A} and K_{R}) or only its range is known (such as the SC division rate). When using the parameterized model to study the role of aspirin, instead of selecting the best fitting parameter set, we included a number of parameter sets from the best fitting parameter ranges, to see how this variability influences the result.
Pathways to adenoma
Next, we asked what is the most likely pathway that leads to the creation of the type 6 (advanced adenoma). It is possible that crypts of type 6 could be created by a KRAS mutation in a crypt of type 3 (we called this ‘APC path’), or by an APC mutation in a crypt of type 5 (‘KRAS path’), see panel (a) of Figure 3. We found, consistent with (Paterson et al., 2020), that the likelihood of each of these two pathways is determined by the crypt fission rates, and not by mutation rates or crypt conversion rates (Figure 3(d)); in addition, it is sensitive of the carrying capacity parameters, K_{A} and K_{R}. This is demonstrated in Figure 3 by examining the best fitting parameter sets of Figure 2(b). They naturally fall into two groups (circled in yellow and green): for the former group, the best fitting carrying capacity satisfy K_{A} >K_{R}, and for the latter group this inequality is reversed (see Figure 3(b)). We consider the former group biologically relevant, not only because it yields a smaller fitting error, but also because it corresponds to the type (APC^{/},KRAS^{}) crypts having a larger carrying capacity, which is consistent with this type being a more advanced stage.
The model allows for the calculation of the probabilities to develop an advanced adenoma through the APC/ and KRAS pathways, functions P_{APC}(t) and P_{KRAS}(t). Panel (c) of Figure 3 plots these quantities as functions of age (t) for the two groups of parameters. We observe that for the biologically relevant group where the carrying capacity associated with type 3 (APC^{/},KRAS^{}) is larger, the pathway through the inactivation of the APC gene is predominant. This is consistent with the conclusions of reference (Paterson et al., 2020). We also generated probability distributions of the numbers of type 3 and type 5 crypts at the time when the first type 6 crypt is generated (Appendix 1—figures 12 and 13). We observe that in the model, the number of type 3 (APC^{/},KRAS^{}) crypts is in the hundreds while type 5 (APC^{/+},KRAS^{+}) crypts are relatively rare. This might further argue against the importance of KRAS as an initiating event in disease evolution.
The effect of aspirin
We asked, given that a variety of parameter values could lead to the same incidence curve, can we still say anything about the possible role of aspirin in cancer prevention/delay? To model the effect of aspirin on the relevant kinetic parameters, we used a variety of sources. One type of data was obtained by us in our earlier studies, where the effect of aspirin was quantified by measuring cells’ kinetic parameters with and without aspirin treatment, in vitro and in xenografts (Shimura et al., 2020; Zumwalt et al., 2017). In other work, it has also been demonstrated that a related nonsteroidal antiinflammatory drug, sulindac, inhibited the fission of APCdeficient crypts and thus reduced adenoma numbers in mice.
It is, however, unclear which exact cell populations aspirin might affect in vivo. Therefore, we implemented the effect of aspirin in the epidemiological model by testing different sets of assumptions: (a) aspirin affects the fitness of cells within crypts (intracrypt dynamics), and it may or may not affect crypt turnover dynamics through crypt fission and death rates (intercrypt dynamics); (b) the fitness of type 6 cells is reduced, and the fitness of type 2–5 cells may or may not be reduced as well. In other words, only the most transformed cell type (that is, the most modified cell type that combines both the APC/ mutation and the KRAS + mutation) is affected by aspirin, or all mutated cells, that is types 2—6, are affected. Different combinations of these assumptions have been explored, as summarized in Figure 4(a).
The effect of aspirin on cellular kinetics is modeled by using results of our xenograft experiments (Shimura et al., 2020), where we documented a dosedependent reduction in the cell division rate (fold difference F_{r} <1) and an increase in the cell death rate (fold difference F_{d} >1), see Table 1. The relationship between the experimentally used aspirin doses in mice (Shimura et al., 2020) and the number of aspirin pills per weeks in humans (Nair and Jacob, 2016) is also given in Table 1. The strongest dose we used in study (Shimura et al., 2020) roughly translates to 1–2 standard aspirin pills a day for humans, which is the second strongest dose considered in Chan et al., 2004 (6–14 pills per week), see Section 6 of Appendix 1. When aspirin is applied in our model, we assumed the following:
Effect on the intracrypt dynamics
For the purposes of our model, it is the combined effect of aspirin on cell division and death rates that changes the cells' relative fitness and decreases the probability of crypt conversion. To translate this information into the fold decrease in SC fitness, we note that, while the foldreduction in division rate could be directly implemented, an increase in death rate is less straightforward. This is because in contrast to cell lines, with SCs, cell removal can occur through a combination of apoptosis and loss through differentiation, which might be the dominant component in the colorectal tissue. Therefore, if the rate of SC apoptosis is increased, say, twofold in the presence of aspirin, this does not translate to a twofold reduction in SC fitness. In the extreme scenario of zero SC death in the absence of aspirin, a twofold increase in this parameter will not lead to a change in SC fitness. To calculate the fitness factor, we assumed that the removal rate of SCs, d, is comprised of 90% differentiation and 10% apoptosis, and that it is the latter that is affected by aspirin. If in the absence of aspirin, cellular fitness is given by the ratio r/d, then in the presence of aspirin this changes to $r/d\times {F}_{r}/(0.9+0.1{F}_{d}),$ which gives the fitness factor in Table 1. This factor enters into the crypt conversion rate, see Section 2 of Appendix 1. In particular, if only type 6 is affected, then rates R_{36} and R_{56} will experience a reduction. If types 2 to –6 are affected, then all conversion rates will be reduced.
Effect of intercrypt dynamicsf
In addition to affecting cellular fitness within the crypts, it is also logical to assume that aspirin reduces crypt fission rates and increases crypt death rates (the bottom row of the table in Figure 4(a)). This is supported by data (Fischer et al., 2014), and the rationale behind this assumption is that crypt fission is ultimately connected with divisions of individual cells, and crypt death is associated with cell death. Therefore, we assume that under aspirin treatment, $\gamma}_{i}\to {F}_{r}\to {\gamma}_{i$ and $\delta \to {F}_{d}\delta$ (that is, the folddifferences apply to the crypt fission and death rates). Again, this could affect the most modified crypts only (type 6), thus reducing the rate γ_{6} and increasing the death rate δ _{6}; alternatively, this could affect to all type 2 to –6 crypts, thus reducing all the crypt fission rates and increasing all the crypt death rates.
The delaying effect of aspirin was studied by using stochastic (Gillespie) simulations, where the models were run according to the schematic in Figure 4(b). In particular, model parameters were switched from their (best fitted, aspirinfree) values to their modified values for the duration of treatment from T_{start} to T_{end}. Simulations were stopped when the colony of type6 crypts grew to its detection size (or when T = 80 was reached). Many simulation runs were aggregated to derive the ageincidence curve for advanced adenoma. Figure 4(c) presents typical simulation results for the models where aspirin affects both intracrypt (conversion) and intercrypt (crypt fission and death) dynamics for all the mutated cell types. The thin black lines represent the incidence curve in the absence of treatment, as was obtained by fitting the advanced adenoma data from Brenner et al., 2014 (black dots). The yellow curves represent the predicted ageincidence for individuals who were undergoing aspirin treatment during the timewindow (of 10 years) in different decades of their lives (see the green shading in each panel representing treatment). Panel (d) of Figure 4 shows the relative risk of advanced adenoma for patients that received aspirin treatment during different decades of their lives. This is calculated at the timepoint that is referred as ‘Relative risk assessment time’ in panel (b), and corresponds to a zero follow up time in this case. The yellow bars correspond to the prediction for individuals treated with a relatively strong dose of aspirin (roughly 6–14 pills a week) for 10 years. We have further performed simulations to obtain model predictions pertaining to lower aspirin doses (Table 1), which we referred to as medium (about 6 pills a week, brown bars in Figure 4(d)), and light (less than two pills a week, blue bars in Figure 4(d)).
We can compare the predicted relative adenoma risk with the data reported in the literature, see e.g. (Chan et al., 2004; Cole et al., 2009; Cao et al., 2016; Drew et al., 2016). In particular, the dosedependence of colorectal adenoma was studied (Chan et al., 2004), and it was shown that the relative risk for adenoma was 0.80 for women who used 0.5–1.5 standard (325 mg) tablets per week, 0.74 for those who used 2–5 tablets per week, 0.72 for those who used 6–14 tablets per week, and 0.49 for those who used more than 14 tablets per week. Comparing this with the relative advanced adenoma risk plot in Figure 4(d), we can see that the model predictions are consistent with the observed bounds: for the aspirin dose that corresponds to 6–14 tablets per week (referred to as high dose, Figure 4(d)), the predicted relative risk ranges between about 0.35 and 0.80, depending on the age when aspirin was administered. Consistently with (Chan et al., 2004), we found that the effect of aspirin decreases with smaller doses, resulting in the relative risk for intermediate doses varying roughly in the 0.6–0.9 range (brown bars in Figure 4(d)), while for the light dose it varied in the 0.8–0.9 range (blue bars in Figure 4(d)).
Note that in our simulations for different scenarios, we observed remarkable quantitative consistency of results over a very wide range of parameters where uncertainties exist. For example, in Appendix 1—figure 14, we compared two markedly different assumptions on the numbers of type6 crypts that constitute advanced adenoma at detection (Lang et al., 2020; Sun et al., 2014; Kim et al., 2007; Tsai and Strum, 2011; Jones et al., 2008; Kang and Shibata, 2013; Dewanji et al., 2011), with the results remaining very similar.
While in Figure 4(c and d) we assume that aspirin treatment lasts 10 years, in Appendix 1—figure 18 we investigated the effect of a shorter duration of aspirin treatment for the high dose regime and found the relative risk closer to 0.80. One conclusion that follows from this and other simulations (see Appendix 1, section 6) is that the extent of the aspirininduced reduction in the adenoma risk, resulting from the mechanisms studied here, is consistent with the reported risk reduction at least for a subset of the parameter combinations. Therefore, this mechanism cannot be rejected based on the predicted advanced adenoma incidence reduction.
Next, we examined how changing model assumptions about the effect of aspirin alter these results. While the simulations of Figure 4 assume that both inter and intracrypt dynamics are affected by aspirin, Appendix 1—figure 15 only includes aspirin’s effect on intracrypt dynamics. Under this assumption, even the strongest aspirin dose did not result in the magnitude of the effect reported in Chan et al., 2004: the reduction in advanced adenoma risk was within a few percent only. This suggests that including aspirin’s effect on intercrypt (fission/death) dynamics is essential to explain the data, which makes intuitive sense because crypt dynamics are thought to be drivers of disease development.
We also investigated the consequence of the assumption that aspirin only affects the most transformed (type 6) cells (Appendix 1—figure 16), and found that while the effect is reduced compared to the full model of Figure 4, one still observes a significant decrease in advanced adenoma risk. Interestingly, if assessment time follows treatment immediately (zero followup time) then there is almost no difference between the prediction of the model where only type 6 is affected compared to that where crypts 2—6 are affected (Appendix 1—figure 17). Increasing the followup time, however, reveals an increase of the difference between the two model predictions. For example, with a 15 years followup time, the model where all types 2—6 are affected shows the relative risk of about 0.6, while the model with only type 6 affected yields a relative risk of about 0.8. The reason for this is the lagphase that exists between the generation of the first type6 crypt and its growth to detection (which in our simulations takes between about 5 and 11 years).
There are several further patterns that emerge. We observe that risk reduction clearly depends on the age of the patients when aspirin was administered. As we see in Figure 4(d), the relative risk can be as low as about 0.4 for patients that started treatment at age 20 and assessed at age 30, compared to a more modest reduction to relative risk of 0.8 for patients receiving treatment later in life (see additional discussion below). Further, we will mention that aspirininduced risk reduction, as predicted by this model, does not disappear even decades after aspirin treatment stopped (Figure 4(b)).
Finally, we comment on another aspect of our model that is different from several other models used in the field (including Luebeck and Moolgavkar, 2002; Meza et al., 2008; Paterson et al., 2020). When predicting the ageincidence curve that results from the microscopic dynamics of selection and mutations, we explicitly included the growth of the most modified crypt type to its detectable size. While stopping the simulations once the first type6 crypt is produced leads to qualitatively similar results, the inclusion of a relatively slow growth of the adenoma significantly changed numerical values of the fitted parameters. It appears that including this stage in the simulations helps improve the quantitative contribution (rather than a proofofprinciple) of this style of mathematical modeling.
Discussion
We used mathematical modeling approaches to test the hypothesis that the changes in tumor cell kinetics observed during aspirin treatment in vitro and in vivo can translate into a protective effect on a population level that is consistent with epidemiological observations for late adenoma. This was done by first constructing a mathematical model of in vivo carcinogenesis describing evolutionary events leading up to the late adenoma stage. This model was then used to calculate expected population incidence as a function of age. Many of the model parameters have recently been estimated experimentally, which provides a solid basis for this modeling effort. Remaining parameters were estimated by fitting the incidence prediction to epidemiological data on late adenoma detection. A linear model that did not include intercrypt competition was rejected because its best fits corresponded to zero crypt fission rates, and the more (statistically) powerful model was adopted instead, where individual mutated crypts experienced both fission and nonlinear competition dynamics. This parameterized model was used as a basis to explore how changes in the kinetics / fitness of cells, brought about by aspirin, can modify the predicted incidence of late adenomas.
Our previous in vitro and in vivo work (Shimura et al., 2020; Zumwalt et al., 2017) indicated that aspirin reduces the rate of colorectal tumor cell division and increases the rate of tumor cell death in a dosedependent way, by up to twofold. In the current modeling study, three different experimental aspirin doses (converted to human aspirin intake) were explored, for which we previously measured their effect on the kinetic parameters. The mathematical analysis demonstrated that parameter changes of a magnitude that lies within our experimentally observed range can lead to significant reductions in predicted late adenoma incidence, which are consistent with the epidemiologically observed reductions (ranging between 10–50%, Chan et al., 2004). The model identified dose, treatment duration, and the age at which treatment was started as important determinants of protection in this context. We can conclude that the aspirininduced changes in cellular fitness that we observed experimentally can in principle explain a significant portion of the protective effect observed on the population level.
This does of course not preclude alternative mechanisms that can further contribute to the protective effect. It is very likely that a reduction in the level of inflammation within the microenvironment of the cells can reduce the incidence of colorectal cancer, because inflammation has been identified as a driver of this disease (Itzkowitz and Yio, 2004). Moreover, other microenvironmental factors, such as the composition of the colorectal microbiome, have been shown to influence the ability of aspirin to reduce tumor growth (Prizment et al., 2020; Brennan et al., 2021; Zhao et al., 2020). This is therefore also likely to play a role in explaining the epidemiological data. Quantification of these further complexities in future work will allow us to introduce these additional aspects into the modeling framework, which would result in a refinement of predictions.
As with most mathematical modeling studies, there are uncertainties in assumptions that need to be kept in mind. Our experiments (Shimura et al., 2020; Zumwalt et al., 2017) were performed with tumor cell lines, both in vitro and in mouse xenografts. While the xenografts capture a higher degree of biological complexity than in vitro experiments, cellular processes in the human colon are even more complex. Colorectal tissues and tumors are characterized by stronger cell hierarchies than our experimental system, including stem, transit amplifying, and terminally differentiated cells. Our analysis was presented under the assumption that colorectal stem cells initiate and maintain tumor growth. While our experimental system did not specifically focus on stem cells, other studies indicate that the effect of aspirin on the kinetics of stem cells in particular is similar (Chen et al., 2018), thus justifying model assumptions. Another uncertainty concerns the cell type in which the tumor originates, and the exact identity of the cell compartment that maintains tumor growth. While we concentrated our model description around stem cells as the cell of origin that drives disease, the model defines this population as having the ability to selfrenew thus maintaining the expansion of the tumor. Hence, this cell population in the model could also correspond to compartments downstream in the differentiation pathway, such as transit amplifying cells, given the marked plasticity within the intestinal epithelium. The model is thus in principle consistent with hypotheses that colorectal cancer might have a different cell of origin (Huels and Sansom, 2015). Interestingly, it has been shown that aspirin had a negative impact on colon organoids derived from nonneoplastic issues, and that aspirin particularly reduced the rapidly cycling transit amplifying cell population (Devall et al., 2021).
Another point of uncertainty concerns the identity of the cell populations that are affected by aspirin. To address this, we made several assumptions, and results remained robust. We first assumed that aspirin influences all mutated cell populations (type 2–6). Results, however, remained fairly similar in an alternative model, where only type 6 cells (characterized by APC/ and KRAS + mutations) were affected by aspirin, although in this case the effect of aspirin is weaker, which is not surprising given that a smaller cell population loses fitness. More crucial was the assumption that aspirin influences not only the cell dynamics themselves, but also the crypt fission dynamics, reducing the rate at which crypts divide and/or increasing crypt death rate. While this is supported by data (Fischer et al., 2014), further experimental investigation into the exact mechanism by which aspirin affects intercrypt dynamics is needed to back up our modeling assumptions.
An important component of all of this work is the underlying mathematical model of in vivo adenoma formation. The assumptions about the genetic events that occur during adenoma formation are consistent with our current understanding of adenoma evolution (Fearon, 2011), and a similar model that also includes evolutionary events beyond adenomas has recently been published (Paterson et al., 2020). While the model description in our and the previously published study (Paterson et al., 2020) was focused on APC/ and KRAS + mutations as initial events, the same evolutionary dynamics would occur if the identity of mutations were different, as long as the evolutionary pathway involves the inactivation of a tumor suppressor gene and the activation of an oncogene, as specified in the model description.
An important difference between our and the previous model concerns assumptions about crypt fission dynamics. The previous study (Paterson et al., 2020) assumed that crypt fission can occur without densitydependent effects. Using experimentally available parameter estimates, this model could account for the lifetime risk of colorectal cancer. When applying a similar model to late adenoma age incidence data, however, we could not obtain a good fit for the ageincidence curve, and the best fit was in fact obtained in the absence of any crypt fission. In the absence of intercrypt competition (that is, with unlimited crypt fission), the predicted adenoma incidence rose too sharply with age compared to epidemiological data. When introducing densitydependence into the crypt fission process, however, late adenoma age incidence data could be readily fit, and so we used this model assumption to go forward. Indeed, it is likely that densitydependent effects play a role in crypt fission, because this process is probably influenced by signaling factors that become limiting as the number of crypts increases. It would be important to verify this assumption experimentally in future work.
Finally, it is interesting to discuss the results of the ASPREE trial (McNeil et al., 2018; McNeil et al., 2021) in the context of the work presented here. This trial investigated the effect of aspirin treatment in a cohort of older individuals, 70 years or older without cardiovascular disease, dementia, or disability. It was found that cancer incidence was not significantly changed by aspirin, but that the aspirintreated group experienced a higher rate of cancerinduced mortality. The absence of a significant effect of aspirin on cancer incidence in this study is consistent with our model predictions. Our mathematical analysis demonstrated that the effect of aspirin treatment on cancer incidence diminished when treatment was initiated in older ages. Our modeling approach, however, cannot make predictions about cancerinduced mortality, because it describes the evolutionary process up to the stage of advanced adenoma only. Our previous work (Wodarz et al., 2017), however, offers an interpretation of these data. Because of their advanced age, it is likely that a certain fraction of the ASPREE participants already harbored tumors that had not been detected yet due to the absence of overt clinical symptoms. In fact, a previous history of cancer was not an exclusion criterion in the trial. As the established tumors continue to grow during aspirin treatment, they likely do so with altered kinetics (reduced division rates and increased death rates, leading to a higher turnover). This means that by the time the tumor has reached a given size (e.g. at which it becomes clinically detectable), it will have undergone more cell divisions under aspirin treatment compared to the placebo group. Hence, the tumor will on average have accumulated more mutations once this detectable tumor size is reached. This in turn means that the aspirintreated tumor might be more virulent and less responsive to therapies, resulting in more deaths. The theoretically derived notion that upon detection, an aspirintreated tumor is more evolved than a tumor that grows without aspirin (Wodarz et al., 2017) is supported by the ASPREE analysis, which found that aspirintreated patients were more likely to have metastasized cancers and stage 4 cancers compared to the placebo group (McNeil et al., 2018; McNeil et al., 2021).
In conclusion, this modeling analysis suggests that a direct impact of aspirin on the kinetics and fitness of mutated cells can significantly reduce the incidence of colorectal adenomas, with a magnitude that is consistent with epidemiological data. This highlights the importance of investigating this effect of aspirin experimentally in more detail, especially under experimental conditions that approximate cell dynamics in the human colorectal tissue with greater accuracy.
Materials and methods
The adenoma incidence data
Request a detailed protocolIn order to study the incidence of advanced adenoma, we used the data reported in Brenner et al., 2014 for the ageranges 55–59, 60–64, 65–69, 70–74, and 75–79. While this study provides incidence data for nonadvanced adenoma, advanced adenoma, and colorectal cancer (CRC), we focused only on the combined incidence of advanced adenoma and cancer. This assumes that individuals that have developed CRC have most likely already developed an advanced adenoma by the age of testing, and further that nonadvanced adenoma likely refers to fewer mutational steps compared to our type 6, where both the APC gene is fully inactivated and the KRAS gene is mutated. The paper reports data separately for males and females; for our purposes we combined the two values to study the average, since the model is not sufficiently detailed to distinguish between the sexes.
Mathematical modeling
Request a detailed protocolThe mathematical model describes stochastic dynamics of colonic crypts. There are six types of crypts that are included in the model, which differ by their mutational status. The number of crypts of each type is denoted by n_{i}, where i = 1 corresponds to the wildtype crypts, i = 2 to type APC^{+/}, i = 3 to APC^{/}, i = 4 to KRAS^{+}, i = 5 to KRAS^{+}APC^{+/}, and i = 6 to KRAS^{+}APC^{/} (the most modified type associated with advanced adenoma). The model contains the processes of crypt conversion (whereby a mutation in a stem cell can fixate in a given crypt thus changing its mutational status), as well as crypt fission/death processes. Intercrypt competition is included by way of nonlinear (logistic) terms. Given the initial condition (wildtype crypts only) and model parameters, the model outputs the probability to observe, by time t, a specified population of type6 crypts (n_{6} = N), which is assumed to be associated with advanced adenoma detection. This represents a numerically generated ageincidence curve. The expected behavior was described by a system of ordinary differential equations (ODEs), and the prediction was fitted to the advanced adenoma incidence reported in Brenner et al., 2014. While the model was parameterized by using the rates found in literature and describing the kinetics in humans, a subset of parameters are unknown (or only their ranges are known); these parameters were estimated by the fitting procedure.
Using the parameterized model that is consistent with the advanced adenoma incidence, we incorporated the effect of aspirin by adjusting the kinetic parameters of cells (division and death rates of cells, which describes the effect of aspirin on intracrypt dynamics), as well as kinetic rates of crypts (crypt fission and death rates, which describes the effect of aspirin on intercrypt dynamics). This was done by using experimentally measured factors. Fully stochastic (Gillespie) simulations were used to quantify the predicted advanced adenoma incidence curves for patients that used different doses and durations of aspirin treatment. For further details of the modeling, see Appendix 1.
Appendix 1
Mathematical formulation of a cryptbased model of adenoma initiation
Let us enumerate the types in the way presented in Table Appendix 1—table 1.
Then we can denote by n_{i} with $1\le i\le 6$ the number of crypts of type $i$. Suppose ${R}_{ij}$ is the conversion rate from crypt type $i$ to type $j$, and ${\gamma}_{i}$ the growth rate (by crypt fission) of crypts of type $i$. We have the following equations:
with the initial conditions
In the ODEs above, we have ignored the process of stochastic tunneling such that the crypts can only convert one step at a time. It is further possible to ignore the negative (outgoing) rates, which simplifies this linear system to the following:
The probability $P(t)$ that by time $t$ at least one crypt of type 6 has been created (using the meanfield approximation from Cole et al., 2009) is given by the solution of the equation
see also section 5.1. The solution can be obtained exactly and is given by
where the quantities ${S}_{ij}$ correspond to the paths $1\to i\to j\to 6$,
see Appendix 1—figure 1. They can be written down by using the following function:
We have
that is, this function increases monotonically in $x$. We have
Note that expressions ${S}_{ij}$ have a singularity if any of the quantities ${\gamma}_{i}$, ${\gamma}_{j}$ is zero and/or if ${\gamma}_{i}={\gamma}_{j}$. For example, if both growth rates are zero (${\gamma}_{i}={\gamma}_{j}=0$), we have
In our context it is reasonable to assume that ${\gamma}_{2}=0$, that is, APC+/ crypts do not divide, and that ${\gamma}_{4}={\gamma}_{5}$, that is, APC+/+ and APC+/ crypts with an additional KRAS mutation divide at the same rate. In the ${\gamma}_{i}={\gamma}_{j}$ case, taking the limit as ${\gamma}_{j}\to {\gamma}_{i}$, we obtain
It is convenient to view S_{25} as a function of ${\gamma}_{4}$, ${S}_{25}=F({\gamma}_{4})$, where
then ${S}_{23}=F({\gamma}_{3})$, and $F(x)$ is an increasing function of $x$. That is, if $\gamma}_{4}>{\gamma}_{3$ then $S}_{25}>{S}_{23$, and if $\gamma}_{4}<{\gamma}_{3$ then $S}_{25}<{S}_{23$. We further note that ${S}_{45}=\stackrel{~}{F}({\gamma}_{4})$ with
To summarize, we can see that $S}_{45}>{S}_{25$, and S_{23} is greater (smaller) than S_{25} if ${\gamma}_{3}$ is greater (smaller) than ${\gamma}_{4}$.
Model parameters
Here we define parameters that appear in this model. We will make the following assumptions (see also Table Appendix 1—table 2):
The mutation rate between types is given as follows: ${u}_{1\to 2}={u}_{4\to 5}=2u$ (inactivation of the first copy of the APC gene, that is, any of the two copies); ${u}_{2\to 3}={u}_{5\to 6}=u$ (inactivation of the remaining copy of the APC gene); ${u}_{1\to 4}={u}_{2\to 5}={u}_{3\to 6}=\mu $ (activation of the KRAS gene).
The fitness of cells with APC+\−, APC−/−, and KRAS+ phenotypes relative to the wild type cells was determined using the cell replacement data from Komarova et al., 2002. In particular, we assumed that for any phenotype,
${F}_{j}\equiv \frac{{r}_{j}{d}_{1}}{{d}_{j}{r}_{1}}=\frac{Pr(j)}{1Pr(j)},j\in 2,3,4$where $Pr(j)$ is the probability of replacement of the wild type by type $j$ found in Komarova et al., 2002. Using this formula, we obtain the values given in Table Appendix 1—table 2. Then the fitness of other types is multiplicative: ${F}_{5}={F}_{4}{F}_{2},{F}_{6}={F}_{3}{F}_{4}$.
The death rates are assumed the same among the types, such that the division rates of cells satisfy${r}_{i}={F}_{i}{r}_{1},\phantom{\rule{1em}{0ex}}1<i\le 6.$
Wildtype crypts and those with only a single copy of APC gene mutated do not proliferate (${\gamma}_{1}={\gamma}_{2}=0$).
Crypts with a KRAS mutation and APC+/+ and APC+/ phenotypes proliferate at the same rate, ${\gamma}_{4}={\gamma}_{5}$.
All the parameters are specified in Table Appendix 1—table 2.
The conversion rates are given by
where r_{i} is the division of type $i$, ${u}_{i\to j}$ is the mutation rate from type $i$ to type $j$, and ${\rho}_{ij}$ is the probability that one cell of type $j$ becomes fixated in a compartment of size $K$ with the host type $i$. To calculate this probability, let us denote by d_{i} the death rate of type $i$. Then we have
where $\frac{{r}_{i}{d}_{j}}{{r}_{j}{d}_{i}}$ is the inverse of the relative fitness of type $j$ with respect to type $i$.
Fitting the linear model to advanced adenoma incidence data
The linear model of the adenoma incidence is given by equations (13,14,15,16,17,18).
In order to fit this model to the data, we fix parameters ${N}_{crypt},K,u,\mu $, and vary the remaining parameters. This is done in stages. We first fix the fitness parameters ${F}_{2},{F}_{3},{F}_{4}$ to their values in table (Appendix 1—table 2) and vary the remaining parameters ${r}_{1},{\gamma}_{3},{\gamma}_{4}$ to find the global minimum of the error between the model and the data, see Appendix 1—figure 2ac. Then we take other select values for the relative fitness parameters to show that the results remain qualitatively similar (not shown). Next, we describe the results of fitting and the patterns that were observed.
To find the global minimum of the error, we varied parameter r_{1} (the division rate of the wild type SCs) between once a day and once every 20 days (which corresponds to the division rate of 365 yrs^{1} and 18 yrs^{1}). For each value of r_{1}, the error was minimized in the 2dimensional parameter space $({\gamma}_{3},{\gamma}_{4})$, and a unique minimum was always found. The best fits corresponding to a subset of these division rates are plotted in Appendix 1—figure 2(a), with the best fitting values of ${\gamma}_{3}$ and ${\gamma}_{4}$ shown in panel (b) for each r_{1}. We can see that as r_{1} increases, that is, SCs are assumed t divide faster, the best fitting crypt division rates decrease (that is, crypt fission proceeds at a slower rate). For r_{1} greater than about 250 yrs^{1} (that is, divisions once every day and a half or faster), the best fitting crypt fission rates are zero. This parameter combination (division rate of about every 1.5 days and zero crypt fission) corresponds to the minimal error of fitting (panel (c)). This is evident from the shape of the best fitting incidence curves, $P(t)$, corresponding to different r_{1} values (panel (a)). For low rates of SC division, the fission rates are relatively high, resulting in an curve $P(t)$ that has a steep rise, yielding a large fitting error and a qualitatively unrealistic shape of the incidence curve if compared with the data. This is the consequence of a pronounced exponential increase in the number of crypts, which sharply accelerates the generation of type 6 crypts. As r_{1} increases and crypt fission rates decrease, the incidence curves become less steep, until the best fitting crypt fission rate reaches zero. At this stage, the best fitting incidence curve is achieved, because further increase in r_{1} results in an increase in the incidence that happens too early.
So far, in order to fit the model to the adenoma data, we fixed several of the parameters to their measured values and varied the three remaining parameters (${r}_{1},{\gamma}_{3},{\gamma}_{4}$) within realistic ranges to investigated the error landscape and find global minima. To obtain a more comprehensive picture of the model behavior, we have implemented a procedure where five parameters were varied: the division rate of stem cells, r_{1}, two cellular fitness parameters, ${R}_{AP{C}^{+/}}$ and ${R}_{KRAS}$ (with ${F}_{AP{C}^{+/+}}=2{F}_{AP{C}^{+/}}$); and two crypt fission parameters, ${\gamma}_{3}$ and ${\gamma}_{4}$. The rest of the parameters were set to their values in Table Appendix 1—table 2.
The fitness parameters were varied between 1.1 and 3.5 to match the measured range. For each pair (${R}_{AP{C}^{+/}},{R}_{KRAS}$), a fitting procedure identical to that of Appendix 1—figure 2 was performed, see Appendix 1—figure 3. Each graph corresponds to a unique pair (${R}_{AP{C}^{+/}},{R}_{KRAS}$), and the values of these coefficients are indicated on each panel. The horizontal axes of each panel is the stem cell division rate, r_{1}. The green curves show the log_{10}(fitting error), and the black (gray) dots show the best fitting values of ${\gamma}_{3}$ (${\gamma}_{4}$), multiplied by 10 for convenience of presentation. If the best fitting crypt fission parameter was negative, then the fitting error shows corresponded to ${\gamma}_{3}={\gamma}_{4}=0$. We observe that in all the cases, the best fitting parameter combination is reached when the crypt fission rates become zero, reproducing the result of Appendix 1—figure 2, but for a wide parameter range. To observe this more clearly, we also presented the error landscape for the best fitting parameter r_{1}, as a function of ${\gamma}_{3}$ and ${\gamma}_{4}$ (the horizontal and vertical axes in each panel, respectively). The quantity log_{10}(fitting error) is shown as a heat map, with darker colors corresponding to lower error values. We can see that the lowest error corresponds to the corner ${\gamma}_{3}={\gamma}_{4}=0$.
Nonlinear model: competition among crypts
Let us assume that the types of crypts that undergo crypt fission (types 3,4, and 5) are in direct competition with each other, modeled as logistic, as opposed to straight exponential, growth. Let us denote te carrying capacity associated with the growth of the KRAS crypt (type 3) as ${K}_{A}$, and that for the KRAS+ crypts (types 4 and 5) as ${K}_{R}$. Then the nonlinear system with crypt competition becomes
with initial conditions (6) and the probability to create a crypt of type 6 given by system (12). While an analytical solution is no longer available, a procedure similar to that performed for the linear system can be performed numerically. Results can be seen in figure Appendix 1—figure 5. As before, all the parameters were fixed to their values in Table Appendix 1—table 2, except the cell division rate, r_{1}, and the crypt fission rates, ${\gamma}_{3}$ and ${\gamma}_{4}$. Additionally, we assumed a crypt carrying capacity ${K}_{A}={K}_{R}=K={10}^{3}$ and crypt death rate $\delta =0.05$ yrs^{1}. The best fitting values of ${\gamma}_{3}$ and ${\gamma}_{4}$ were found for each value of r_{1}, which was varied in the realistic range. This procedure yielded a range of lowerror fits (panel (c)) that correspond to nonzero values of crypt fission rates in the realistic range (panel (b)), with the corresponding incidence curves given in the inset of panel (a).
Since this procedure yielded a wide range of similarly good fits under a fixed value for the crypt carrying capacity, we next performed a fitting procedure where the crypt fission rates ${\gamma}_{3}$ and ${\gamma}_{4}$ were fixed to those in Table Appendix 1—table 2, and the best carrying capacity, $K$, was determined for each division rate, r_{1}, by fitting to the data. Results (under the assumption that ${K}_{A}={K}_{r}=K$) are presented in Appendix 1—figure 6ac. We can see that for a range of values of the cell division rates, r_{1}, a lowerror fit was found, with the carrying capacity values ranging between about $5\times {10}^{2}$ and $5\times {10}^{3}$. The best fitting values are $K=1318,{r}_{1}=141.1$ (divisions approximately every 2.5 days). Changing the crypt death rate to $\delta =0$ yields very similar results, see Appendix 1—figure 6df.
These results suggest that biologically, the nonlinear model is a more appropriate choice because it produces the best fit for values of crypt fission rates that are within the experimentally observed range, while the linear model requires zero crypt fission rates. From the statistical point of view, the nonlinear model is a significantly more powerful model e.g. by applying the Akaike Infromation Criterion (AIC).
Model predictions and advanced adenoma generation dynamics
Gillespie simulations and the meanfield approximation
Equation 12 approximates the probability $P(t)$ that by time $t$, a single type 6 crypt has been generated. In order to test its validity we have run stochastic Gillespie simulations based on system (LABEL:n1non23), which were stopped when the first type 6 crypt was created. The resulting incidence was compared to that obtained from the deterministic system (LABEL:n1non23, 12), see Appendix 1—figure 7, line (0). There, the blue symbols and the blue solid line correspond to the stochastic simulations and the deterministic approximation respectively. The deterministic approximation showed excellent agreement with Gillespie simulations.
Gillespie simulations described here can be used to simulate stochastic effects associated with crypt dynamics studied here, see e.g. results presented in Section 5.3. These simulations were also used to study the effect of aspirin, Section 6. Finally, this methodology can be extended to study the clonality of abnormal crypts. The ODE model does not keep track of clonality. For example, if type 5 crypt is created multiple times (by conversion) in the system, the variable n_{5} simply gives the total number of type5 crypts. In a stochastic Gillespie model, however, it is possible to keep track of different clones by designating each newly generated crypt as a different “subtype”, which can then clonally expand through crypt fission. This however goes beyond the scope of the current study.
The growth phase of advanced adenoma
So far we have stopped the simulations as soon as the first copy of a type 6 crypt was created. Alternatively, one could explicitly account for the growth phase of the APC/ KRAS+ crypts. Estimates for the doubling time of advanced adenomas vary in the literature. For example, (Jones et al., 2008) suggests that the doubling time of advanced adenoma is 250 days (that is, the net rate of expansion 1.01 yr^{1}), with a similar doubling time of about 1 year quoted in Kang and Shibata, 2013 (the net rate of expansion 0.7 yr^{1}). The number of crypts in an advanced adenoma can be estimated as follows. Given that the minimum advanced adenoma size is 10 mm (see Dewanji et al., 2011 and also Chen et al., 2018), and assuming 10^{9} cells per cm^{3} (Huels and Sansom, 2015), we obtain that each advanced adenoma contains about 10^{6} cells. This comprises about 10^{2} crypts, if we assume that type 6 crypts are 10^{4} cells each, which is somewhat larger that normal colonic crypts that measure about $2\times {10}^{3}$ cells (Devall et al., 2021). A somewhat larger estimate is given in McNeil et al., 2018 where it is noted that a 1 mm^{3} adenoma contains on the order of $5\times {10}^{5}$ cells, which translates into $5\times {10}^{8}$ cells in an adenoma of size 10 mm^{3}, or $5\times {10}^{4}$ crypts. Therefore, we have explored a range of crypt expansion to sizes ${N}_{1}={10}^{2}$ and ${N}_{2}={10}^{5}$ ctypts, see Appendix 1—figure 7. Assuming, for type 6 crypts, ${\gamma}_{6}=1.01$ yr^{1} and the crypt death rate ${\delta}_{6}=0.05$ yr^{1}, we obtain that expansion from 1 type 6 crypt to N_{1} crypts will take $\mathrm{\Delta}{T}_{1}=4.79$ years, and expansion from 1 type 6 crypt to N_{2} crypts will take $\mathrm{\Delta}{T}_{2}=11.97$ years respectively.
Using these calculations, we obtain the deterministic prediction that expansion to 10^{2} crypts will shift the incidence curve by $\mathrm{\Delta}{T}_{1}$ years, and expansion to 10^{5} crypts will shift it by $\mathrm{\Delta}{T}_{2}$ years. Apart from this shift, the probability of crypt nonextinction has to be incorporated. If $P(t)$ is the solution of equation (12) under system (LABEL:n1non23), then we can approximate the probability ${P}_{i}(t)$ that by time $t$, ${N}_{i}$ crypts of type 6 have been generated, by
The corresponding curves are marked as curves 1 and 2 in Appendix 1—figure 7, see the orange and green solid lines respectively. To check this approximation, we have run Gillespie simulations that stopped as soon as ${N}_{1}={10}^{2}$ (orange symbols) and ${N}_{2}={10}^{5}$ (green symbols) crypts were generated. Again, very good agreement between the stochastic simulations and our deterministic approximation was observed.
Next, we will show that similar results are obtained when the model with an explicit type 6 crypt expansion phase is fit to the advanced adenoma incidence data. Fixing a crypt carrying capacity, we found the best fitting fission rates ${\gamma}_{3}$ and ${\gamma}_{4}$ for each value of r_{1}. The procedure was similar to that used in Appendix 1—figure 5, except probabilities to have ${N}_{i}$ type 6 crypts, ${P}_{i}(t)$ (equation (24)) were used to fit the incidence data. Results are shown in Appendix 1—figure 8. The plot of the fitting error again shows regions of low error (panel (a)). In those regions, nonzero values of crypt fission correspond to the best fit (see panels (b) and (c), where regions of low error are shaded red). As in Appendix 1—figure 5(b), the best fitting values of ${\gamma}_{4}$ are lower than those of ${\gamma}_{3}$, and both are similar to those found in the literature, see Table Appendix 1—table 2.
Therefore, to proceed, we will fix the crypt fission rates ${\gamma}_{3}$ and ${\gamma}_{4}$ to their values in Table Appendix 1—table 2, and fit the model to the advanced adenoma incidence curve to find the best rate r_{1} and crypt carrying capacities. This is a procedure similar to that in Appendix 1—figure 6, except type 6 crypt populations grow to a given size. The fitting error for the three cases (that is, growth to size 1, size 10^{2}, and size 10^{5} crypts) is shown in Appendix 1—figure 9(a) as a function of r_{1}. The fitting procedure was performed by using the ODE approximation (equation (24)), and the best fitting parameter set then used in a Gillespie simulation ($5\times {10}^{5}$ simulations are used and the error bars are too small to see). Appendix 1—figure 9(b) shows the result of the Gillespie simulation corresponding to the bestfitting parameters for the case of expansion to 10^{5} crypts. The ODE prediction and the fitted curve for the expansion to 10^{5} crypts look identical.
Compartment dynamics and pathways to advanced adenoma
Relaxing the restriction ${K}_{A}={K}_{R}$ improves the fitting, as shown in Appendix 1—figure 9(c). We will explore compartment dynamics in this more general setting, given that by AIC, the model with carrying capacities ${K}_{A}$ and ${K}_{R}$ separately fitted is more powerful. Appendix 1—figure 10 shows the procedure of fitting in the case where type 6 crypts grow to size 10^{2} before detection. For each value of r_{1}, the best fitting pair $({K}_{A},{K}_{R})$ was determined and the resulting error plotted (panel (a)). Panel (c) shows examples of heatplots of the error as a function of ${K}_{A}$ and ${K}_{R}$ (for a fixed r_{1} value). Depending on the size of the resulting error and the location of the minimum in the $({K}_{A},{K}_{R})$ space, three groups of fits can be distinguished. Of particular interest in the group marked by orange and green points. For the fits in the “orange” group, we have $K}_{A}>{K}_{R$, that is, the best fitting carrying capacity that characterizes type3 crypts is larger than that for type4 and 5 crypts. In contrast to that, the “green” group has $K}_{A}<{K}_{R$. This can be seen in Appendix 1—figure 10(b), where the best fitting pairs $({K}_{A},{K}_{R})$ are shown for each r_{1} value. We reject the “green” group of fits because it corresponds to a larger carrying capacity of type4 and 5 crypts compared to that of type3 crypts, which is inconsistent with observations; in addition, the fitting error is minimized for one of the fits in the “orange” group. The same procedure was also performed for the case where type6 crypts grew to size 10^{5} (see Appendix 1—figure 11), with very similar results.
In order to determine the likelihood of type y_{6} produced from cells of type y_{3} or cells of type y_{5}, we looked at the probabilities of the two pathways, ${P}_{APC}$ and ${P}_{KRAS}$, which stand for the probability to create adenoma by first inactivating the APC gene and then adding a gainoffunction mutation in KRAS gene, or by first activating KRAS and then inactivating the APC gene, see panel (a) of Figure 3 of the main text. The two probabilities satisfy the following equations,
The proportion of adenomas that originated through the APC pathway is given by
This function is plotted in Appendix 1—figure 10(d) for the parameters corresponding to both “orange” and “green” groups of fits. We can see that for parameter combinations from the orange group the APCpathway is predominant, while for parameter combinations from the “green” group the KRAS pathway is predominant. The same qualitative results are observed in Appendix 1—figure 11(d).
For further simulations, we used the two best fitting parameter combinations obtained from the fitting procedures in Appendix 1—figure 10 and Appendix 1—figure 11. The parameter values obtained by means of these fitting procedure are summarized in Table Appendix 1—table 3.
System (LABEL:n1non23, 12) does not only allow fitting the model to advanced adenoma incidence data, but also shows the prediction for the dynamics of the crypts of different mutational status leading up to the advanced adenoma formation, as shown in figure Appendix 1—figure 12(a). Using the best fitting parameters of Appendix 1—figure 6(b), we can visualize the mean trajectories for the numbers of crypts in each compartment, n_{i} for $i=1,\mathrm{\dots},5$. The stochastic Gillespie simulations provide additional information on the statistics of crypt numbers. Appendix 1—figure 12(b) shows the numerically obtained probability distributions for the crypt numbers in each compartment, at the time when the first type 6 crypt is created. These simulations of panel (b) correspond to the same parameter set as in panel (a); both assume expansion to 10^{2} crypts and ${K}_{A}={K}_{R}$. Panel (c) presents similar simulations but for the best fitting parameters where expansion to 10^{5} crypts takes place. The mean crypt numbers are given in blue in each histogram; we can see that the orders of magnitude are similar for both cases. Similar results are obtained when the restriction ${K}_{A}={K}_{A}$ is dropped, see Appendix 1—figure 13; as expected, the size of the type3 compartment is somewhat larger and the size of the type5 compartment somewhat smaller than the corresponding values for the ${K}_{A}={K}_{R}$ case. Note that Appendix 1—figure 12 and Appendix 1—figure 13 do not contain information on pathways to advanced adenoma. In other words, shown are just the numbers of crypts of each type that are present in the entire simulated colon, at the time when the first crypt associated with the advanced adenoma is generated, regardless of whether it was generated by a mutation of a type 3 or a type 5 crypt.
Aspirin dosage and modeling aspirin’s effect on the crypt dynamics
In mouse experiments performed in Shimura et al., 2020 we used the doses of 15 mg/kg, 50 mg/kg, and 100 mg/kg aspirin each day. Using the conversion table in Drew et al., 2016, and a mass of a human of 70 kg, we obtain the equivalent daily human doses of 85.1 mg/day, 283.5 mg/day, and 567 mg/day.
To compare this with the weekly intake of the participants in the study of Chan et al., 2004, we note that a standard aspirin tablet contains 325 mg, such that 2 tablets a day (i.e. 14 tablets per week) is 650 mg/day. The patients were grouped by their intake into 0.51.5 tablets per week, 25 tablets per week, 614 tablets per week, and gt_{14} tablets per week. Therefore, the highest dose administered in our mouse experiments (100 mg/kg aspirin per day) most closely matches the intake of the 614 tablets per week group, see Table 1 of the main text.
To explore the effect of aspirin on the incidence of advanced adenoma, we used Gillespie simulations of system (LABEL:n1non23), also including a growth phase of type 6,${\dot{n}}_{6}={R}_{36}{n}_{3}+{R}_{56}{n}_{5}+({\gamma}_{6}{\delta}_{6}){n}_{6}.$
In these simulations, aspirinfree parameter values were used for $0\le t<{T}_{start}$, and then during an interval of aspirin treatment $T}_{start}\le t\le {T}_{end$, parameter values modified by aspirin were used, see Figure 4(b) of the main text. As before, simulations were stopped at $t=80$ yrs or when the target number of type 6 crypts were generated, whichever happened first. Incidence curves and relative risk were constructed by processing a large number of such simulations. $2.5\times {10}^{5}$ independent simulations per condition were used unless otherwise noted. The error bars for the age incidence curves or the relative risk are too small to be seen.
Figure Appendix 1—figure 14: Relative incidence of advanced adenoma, where (a) growth to 10^{2} type 6 crypts is assumed and parameter sets #1 and #2 (Table Appendix 1—table 3) are compared, and (b) growth to 10^{5} type 6 crypts is assumed and parameter sets #3 and #4 are compared. Aspirin affects type 2–6 cells, both through conversion rates and crypt fission/death. Treatment is applied for different decades (as marked under the pairs of bars), and the relative risk is evaluated at the end of the treatment decade. $2.5\times {10}^{5}$ independent simulations are used for each condition.
Appendix 1—figure 15 explores the individual contributions of the two possible mechanisms by which the effect of aspirin could be manifested. The yellow bars correspond to the effect on both conversion rates and crypt fission/death rates, and the blue bars only include the effect on conversion rates. It is clear that the effect on conversion rates is not as strong compared to the effect on the crypt fission/death rates.
Appendix 1—figure 16 is similar to main text Figure 4(a), except is contains additional incidence functions that correspond to aspirin only affecting type 6 cells (blue lines). Not surprisingly, the resulting change in adenoma risk is smaller, that is, the blue line (effect on type 6 cells only) is closer to the thin black line (noaspirin adenoma incidence) than the yellow line (where aspirin affects all types 2–6). It is interesting that the difference becomes larger with time. In Appendix 1—figure 17 we change the length of the followup (see Figure 4(b) of the main text), and the difference between the two assumptions (crypts 2–6 affected vs only type 6 crypts are affected) increases as time goes by.
Appendix 1—figure 18 shows the effect of a shorter aspirin duration time
Data availability
Data and some relevant code are available on Dryad under DOI 10.7280/D1M11M. Code for Figures 2, 3 and 4 is uploaded in Source Code files 13.

Dryad Digital RepositoryData files for simulated advanced adenoma age incidence, under aspirin treatment.https://doi.org/10.7280/D1M11M
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Decision letter

Christopher S WilliamsReviewing Editor; Vanderbilt University, United States

Eduardo FrancoSenior Editor; McGill University, Canada

E Georg LuebeckReviewer; Fred Hutchinson Cancer Research Center, United States

Andrew ChanReviewer
Our editorial process produces two outputs: (i) public reviews designed to be posted alongside the preprint for the benefit of readers; (ii) feedback on the manuscript for the authors, including requests for revisions, shown below. We also include an acceptance summary that explains what the editors found interesting or important about the work.
Decision letter after peer review:
Thank you for submitting your article "The protective effect of aspirin in colorectal carcinogenesis: a multiscale computational study from mutant evolution to age incidence" for consideration by eLife. Your article has been reviewed by 3 peer reviewers, and the evaluation has been overseen by a Reviewing Editor and a Senior Editor. The following individuals involved in review of your submission have agreed to reveal their identity: E Georg Luebeck (Reviewer #2); Andrew Chan (Reviewer #3).
As is customary in eLife, the reviewers have discussed their critiques with one another. What follows below is the Reviewing Editor's edited compilation of the essential and ancillary points provided by reviewers in their critiques and in their interaction postreview. Please submit a revised version that addresses these concerns directly. Although we expect that you will address these comments in your response letter, we also need to see the corresponding revision clearly marked in the text of the manuscript. Some of the reviewers' comments may seem to be simple queries or challenges that do not prompt revisions to the text. Please keep in mind, however, that readers may have the same perspective as the reviewers. Therefore, it is essential that you attempt to amend or expand the text to clarify the narrative accordingly.
Essential revisions:
1)) Overall, the manuscript would benefit from a more precise explanation of the assumptions used in the models presented. This would include a more clear discussion/rationalization of advanced adenoma, adenoma classification, and how aspirins effect was implemented at the crypt level (see first reviewers comments)
2) There needs to be increased justification (or modification of the model) for why the assumption of zero crypt death/fusion.
3) Several reviewers mention limitations/concerns with the reliance on mutant KRAS (i.e. lack of determining KRAS status in adenomas/cancers, lack of APC/KRAS mutational status in predicting aspirin response and that the effect estimates are based on preclinical work using relatively high doses of aspirin. Please address these concerns in the manuscript and in response.
4) Addressing the concern from reviewer #3 regarding the assumption that the cell of origin for CRC is an ISC as opposed to more recent theories suggesting alternative origins and suggestions on expanding the discussion to include more recent literature on agedifferences in aspirins effects on CRC.
Reviewer #1 (Recommendations for the authors):
In addition to the points raised in the public review, I have the following comments:
It is unclear to me why the effect of reducing the fitness of type 6 is done through reducing the rates of mutation to type 6 (R36 and R56). What is the justification for that?
For the scenario when aspirin reduces gamma3 and gamma4, why does it also not reduce gamma5?
It would also be important to discuss more precisely (i.e. by referring to specific mathematical models) how the findings that aspirin changes division and death rates in cell culture, where there is no tissue hierarchy, translates to the in vivo setting, where the effects of aspirin may be felt by crypts, stem cells or progenitor cells.
Reviewer #3 (Recommendations for the authors):
I read the manuscript with great interest and was pleased to see that the authors took care to acknowledge the limitations and clearly explain the base assumptions used in their approach. The manuscript is wellwritten and there are only minimal additions that may improve the manuscript.
– Unless I have misunderstood, the basic model allows us to understand the probability of developing APC and/or KRAS mutant adenomas (and using this as a relative measure of the 'advanced' nature of the in silico 'neoplasm'). I appreciated the discussion relative to the formation of aberrant crypts vs. adenomas vs. more advanced precancers. However, it seems that everything operates on the basic premise that an intestinal stem cell must be the tumor cell of origin and that aspirin is having specific effects on these cells. This assumption may be too much an oversimplification to allow the model to have broad reaching applicability. For example, recent work has begun to describe that the tumor cell of origin may not be the classical intestinal stem cell in all CRC cases, especially with advancing age or under different dietary stressors, and separately in parallel, that aspirin may have effects on cell differentiation/states (e.g. Devall et al. Cancer Prev Res 2021), mechanism may be cellcontext specific, or be significantly impacted by epithelial cell extrinsic factors not included in the model (e.g. gut microbiome, see A. Prizment et al. Aliment Pharmacol Ther 2020; C. Brennan et al. mBio 2020; R Zhao et al. Gastro 2020) While the authors do describe that these assumptions may be limiting, I think prudent for the authors to discuss the specific impacts of the assumption that aspirin has a direct effect on intestinal stem cells being tumor cell of origin has on model interpretation. How would estimates be potentially influenced if intestinal stem cells were not the target cell or aspirin only had effects in specific cell types or by cell extrinsic factors? What do these assumptions have on the broader generalizability of this model? Can the authors expand on how this may be expected to be accounted for in the future? What additional information or type of data is needed from clinical and preclinical experiments to allow for more accurate biological modeling of these complex interplays? The last question is particularly important to understand the broader impact of these findings and if the models have to potential to more directly inform future research.
– Similarly, I think more discussion could be owed to the emerging literature around the intersection of agedifferences and aspirin mechanism, especially in light of the recent results from the ASPREE trial that described an increase in cancer death as a result of aspirin intervention among adults over age 70. Is it possible to model the timing of aspirin intervention using this model, particularly in view of the probability differences arising from differential mutational priming outcomes (APC/ vs. APC /+ vs. KRAS+, etc.). The ASPREE results demonstrated that the increase in cancer mortality was not driven by a change in cancer incidence and I wonder if the authors can try to model these effects or at least discuss how the model findings should be interpreted in view of these recent results from trials.
– The manuscript primarily discusses the CAPP2 trial as the evidence supporting aspirin chemoprevention of colorectal cancers. Although this obviously has clear implications for placing the results in the context of prevention of CRCs in Lynch syndrome, these tumors are neither sporadic, nor arise via the pathways included in the model. The authors could broaden the background to include the preponderance of evidence for the preventive effects in sporadic cases (or even FAP patients which are known to have APC mutant cancers) where aspirin has had less of a potent chemopreventive effect than in Lynch syndrome. However, these data are relevant to the most extreme phenotype in their model.
[Editors' note: further revisions were suggested prior to acceptance, as described below.]
Thank you for resubmitting your work entitled "Aspirin's effect on kinetic parameters of cells contributes to its role in reducing the incidence of advanced colorectal adenomas, shown by a multiscale computational study" for further consideration by eLife. Your revised article has been evaluated by the reviewers and by the Editors.
The manuscript has been improved but there are some remaining issues that need to be addressed, as outlined below:
Reviewer #1 (Recommendations for the authors):
The authors have carefully addressed previous concerns, and I am satisfied with the revision.
Reviewer #2 (Recommendations for the authors):
The revised manuscript has gained considerably in strength and, by in large, clarifies the main points raised by the reviewers. I appreciate the extra work that went into refining/extending the model analysis, in particular the addition of a growth phase for the type 6 (advanced) adenoma.
Two lingering points. I hope they can be addressed.
1. Please clarify whether the size distribution provided for type 3 (APC/) adenoma in Figure Appendix 1 Figure 12 and 13 refer to the particular type 3 adenoma (clone) in which the advanced adenoma first developed or to the entire population of type 3 adenomas in the colon. I think the authors should point out that their ODE model does NOT distinguish individual clones of abnormal crypts (ie individual adenomas). This is a limitation since detectable adenoma number of any kind (other than hyperplastics) is an important clinical factor.
2. I understand the authors' point about postulating other potential gainoffunction mutations, similar to KRAS, such as BRAF. However, BRAF is a poor example as it is associated strongly with mismatch repair deficiency and SSAs, leading frequently to hypermutated cancers. While there may be other yet unidentified gainoffunction drivers for the advanced adenoma, there may also simply none required given the epigenetic plasticity and adaptive epigenetic changes as adenomas sojourn for years. In any case, the mention of BRAF is somewhat misleading in the context defined by the authors.
Reviewer #3 (Recommendations for the authors):
Thank you for a very complete response. The manuscript is excellent!
https://doi.org/10.7554/eLife.71953.sa1Author response
Essential revisions:
1)) Overall, the manuscript would benefit from a more precise explanation of the assumptions used in the models presented. This would include a more clear discussion/rationalization of advanced adenoma, adenoma classification, and how aspirins effect was implemented at the crypt level (see first reviewers comments)
Following the referees’ insightful comments, we have rewritten and extended the parts of the manuscript that describe the model, both in the main text and in the Appendix. The models and the underlying assumptions are now described in greater detail both in the main text and in Materials and methods. This includes a discussion about adenoma classification, and the nature of the mutations. We have significantly expended Section 2 of Appendix 1 that describes all the model parameters. The part of the Results section in the main text that describes the effect of aspirin is now significantly rewritten (starting p13), including the new Figure 4 and Table 1, and we have also added a new Section 6 in Appendix 1, where further details are provided. In particular, we distinguish the effect that aspirin may have on the intra and intercrypt dynamics (that is, its role in modifying conversion rates and fission/crypt death rates, respectively). We have now explicitly included three different doses of aspirin, see new Table 1 for aspirin dosage in mice, the human equivalent, and the resulting fold differences in the division and death rates of cells that were previously measured by us and implemented in this study. The timing of aspirin administration is explained in the schematic of the new Figure 4(b) of the main text.
2) There needs to be increased justification (or modification of the model) for why the assumption of zero crypt death/fusion.
Spontaneous crypt loss is part of the model; it is incorporated in the crypt death rate, d, in the system of equations presented in the main text (as well as system (1923) in Appendix 1). All the simulations in the main text (e.g. Figures24) include a nonzero crypt death rate. We have also researched the effect of crypt death rate, by performing simulations with both zero and nonzero values of d (see Appendix 1 Figure 6). Since the new version of the model also takes into account the expansion phase of type 6 crypts, a nonzero crypt death rate associated with type 6 crypts is also included. It plays a role in the stochastic model, as sometimes, a newly generated crypt may spontaneously disappear. It is also assumed to be affected by aspirin, when we consider the intercrypt dynamics.
3) Several reviewers mention limitations/concerns with the reliance on mutant KRAS (i.e. lack of determining KRAS status in adenomas/cancers, lack of APC/KRAS mutational status in predicting aspirin response and that the effect estimates are based on preclinical work using relatively high doses of aspirin. Please address these concerns in the manuscript and in response.
We now discuss these issues at length. As pointed out by the reviewers, it has been reported that among nonhypermutated colorectal tumors, KRAS was mutated in only about 43% of patient samples, indicating the importance of alternative evolutionary pathways. Our model, however, does not depend on the identity of particular mutations, but assumes the occurrence of mutation types, which are the inactivation of a tumor suppressor gene (which is a lossoffunction mutation, e.g. APC/), and a gainoffunction mutation, which can be in KRAS or an alternative gene, such as BRAF. Our model predictions hold as long as the evolutionary pathway to advanced adenomas involves a lossoffunction mutation and a gain of function mutation, regardless of their identity.
Regarding aspiring dose, we now provide a table relating doses used in our in vivo experiments to the number of aspirin pills used in reference [14]. Further, we now show simulation results for a strong, intermediate, and light aspirin dose, based on the different dosing regimes in our experiments, and relate our predictions to the epidemiological observations in reference [14]. This is described in detail in the Result section, and the accompanying graphs are shown in Figure 4 (as well as Section 6 of Appendix 1).
4) Addressing the concern from reviewer #3 regarding the assumption that the cell of origin for CRC is an ISC as opposed to more recent theories suggesting alternative origins and suggestions on expanding the discussion to include more recent literature on agedifferences in aspirins effects on CRC.
We have now addressed these issues in detail. Regarding the cell of origin issue, the Discussion section now includes the following: “While we concentrated our model description around stem cells as the cell of origin that drives disease, the model defines this population as having the ability to selfrenew thus maintaining the expansion of the tumor. Hence, this cell population in the model could also correspond to compartments downstream in the differentiation pathway, such as transit amplifying cells, given the marked plasticity within the intestinal epithelium. The model is thus in principle consistent with hypotheses that colorectal cancer might have a different cell of origin.”
Regarding the effect of aspirin in older ages, we now provide an extensive discussion of the ASPREE trial (second to last paragraph, Discussion section), and describe how these observations can be interpreted in the light of evolutionary models.
Reviewer #1 (Recommendations for the authors):
In addition to the points raised in the public review, I have the following comments:
It is unclear to me why the effect of reducing the fitness of type 6 is done through reducing the rates of mutation to type 6 (R36 and R56). What is the justification for that?
We thank the referee for pointing this out. This statement was not precise. Changes in the relative fitness of different types (including type 6) are implemented by modifying the parameters r_{i} (relative fitness of cells), and this in turn changes (in a nonlinear fashion) the relevant conversion rates. This is now explained in the new Section 2 of Appendix 1, and also in the heavily revised part of the main text (Results) where we describe modeling the effects of aspirin (see also the new Section 6 of Appendix 1 for details).
For the scenario when aspirin reduces gamma3 and gamma4, why does it also not reduce gamma5?
In our model, we assume that gamma5=gamma4, and therefore both are affected in the model. This was unclear in the original version and is now stated explicitly.
It would also be important to discuss more precisely (i.e. by referring to specific mathematical models) how the findings that aspirin changes division and death rates in cell culture, where there is no tissue hierarchy, translates to the in vivo setting, where the effects of aspirin may be felt by crypts, stem cells or progenitor cells.
We have now done this in the revised manuscript.
Regarding stem cells: There are data in the literature showing that stem cells in particular are also affected by aspirin, which we cite. To describe how the effect on stem cells was implemented, we included the following text: “For the purposes of our model, it is the combined effect of aspirin on cell division and death rates that changes the cells' relative fitness and decreases the probability of crypt conversion. To translate this information into the fold decrease in SC fitness, we note that, while the foldreduction in division rate could be directly implemented, an increase in death rate is less straightforward. This is because in contrast to cell lines, with SCs, cell removal can occur through a combination of apoptosis and loss through differentiation, which might be the dominant component in the colorectal tissue. Therefore, if the rate of SC apoptosis is increased, say, twofold in the presence of aspirin, this does not translate to a twofold reduction in SC fitness. In the extreme scenario of zero SC death in the absence of aspirin, a twofold increase in this parameter will not lead to a change in SC fitness. To calculate the fitness factor, we assumed that the removal rate of SCs, d, is comprised of 90% differentiation and 10% apoptosis, and that it is the latter that is affected by aspirin. If in the absence of aspirin, cellular fitness is given by the ratio r/d, then in the presence of aspirin this changes to r/d x F_{r}/(0.9+0.1F_{d}), which gives the fitness factor in Table 1. This factor enters into the crypt conversion rate, see Section 2 of Appendix 1. In particular, if only type 6 is affected, then rates R_{36} and R_{56} will experience a reduction. If types 26 are affected, then all conversion rates will be reduced.”
Regarding intercrypt dynamics, the following text was included: “In addition to affecting cellular fitness within the crypts, it is also logical to assume that aspirin reduces crypt fission rates and increases crypt death rates (the bottom row of the table in Figure 4(a)). This is supported by data [57], and the rationale behind this assumption is that crypt fission is ultimately connected with divisions of individual cells, and crypt death is associated with cell death. Therefore, we assume that under aspirin treatment, γ_{i} →F_{r} γ_{i} and $\delta \to {F}_{d}\delta$ (that is, the folddifferences apply to the crypt fission and death rates). Again, this could affect the most modified crypts only (type 6), thus reducing the rate g_{6} and increasing the death rate d _{6}; alternatively, this could affect to all type 26 crypts, thus reducing all the crypt fission rates and increasing all the crypt death rates.“
Regarding intercrypt dynamics, the following text was included: “In addition to affecting cellular fitness within the crypts, it is also logical to assume that aspirin reduces crypt fission rates and increases crypt death rates (the bottom row of the table in Figure 4(a)). This is supported by data [57], and the rationale behind this assumption is that crypt fission is ultimately connected with divisions of individual cells, and crypt death is associated with cell death. Therefore, we assume that under aspirin treatment, γ_{i} →F_{r} γ_{i} and $\delta \to {F}_{d}\delta$ (that is, the folddifferences apply to the crypt fission and death rates). Again, this could affect the most modified crypts only (type 6), thus reducing the rate g_{6} and increasing the death rate d _{6}; alternatively, this could affect to all type 26 crypts, thus reducing all the crypt fission rates and increasing all the crypt death rates.“
Reviewer #3 (Recommendations for the authors):
I read the manuscript with great interest and was pleased to see that the authors took care to acknowledge the limitations and clearly explain the base assumptions used in their approach. The manuscript is wellwritten and there are only minimal additions that may improve the manuscript.
– Unless I have misunderstood, the basic model allows us to understand the probability of developing APC and/or KRAS mutant adenomas (and using this as a relative measure of the 'advanced' nature of the in silico 'neoplasm'). I appreciated the discussion relative to the formation of aberrant crypts vs. adenomas vs. more advanced precancers. However, it seems that everything operates on the basic premise that an intestinal stem cell must be the tumor cell of origin and that aspirin is having specific effects on these cells. This assumption may be too much an oversimplification to allow the model to have broad reaching applicability. For example, recent work has begun to describe that the tumor cell of origin may not be the classical intestinal stem cell in all CRC cases, especially with advancing age or under different dietary stressors, and separately in parallel, that aspirin may have effects on cell differentiation/states (e.g. Devall et al. Cancer Prev Res 2021), mechanism may be cellcontext specific, or be significantly impacted by epithelial cell extrinsic factors not included in the model (e.g. gut microbiome, see A. Prizment et al. Aliment Pharmacol Ther 2020; C. Brennan et al. mBio 2020; R Zhao et al. Gastro 2020) While the authors do describe that these assumptions may be limiting, I think prudent for the authors to discuss the specific impacts of the assumption that aspirin has a direct effect on intestinal stem cells being tumor cell of origin has on model interpretation. How would estimates be potentially influenced if intestinal stem cells were not the target cell or aspirin only had effects in specific cell types or by cell extrinsic factors? What do these assumptions have on the broader generalizability of this model? Can the authors expand on how this may be expected to be accounted for in the future? What additional information or type of data is needed from clinical and preclinical experiments to allow for more accurate biological modeling of these complex interplays? The last question is particularly important to understand the broader impact of these findings and if the models have to potential to more directly inform future research.
These are important points to address in the manuscript. Regarding the cell or origin issue, we did write the text under the premise that colorectal cancer stem cells are driving tumor development and growth, due to the large emphasis on this in the literature. The model, however, is more general in that it assumes a dividing population of cells that initiates and drives tumor growth, without specifying their identity. This could correspond not only to stem cells, but also to cell populations further downstream in the differentiation pathway (such as transit amplifying cells), and could include initial mutant acquisition, followed by dedifferentiation. In the revised manuscript, we now take this broader perspective rather than concentrating solely on stem cells. This is provided in the Discussion section as follows: “Another uncertainty concerns the cell type in which the tumor originates, and the exact identity of the cell compartment that maintains tumor growth. While we concentrated our model description around stem cells as the cell of origin that drives disease, the model defines this population as having the ability to selfrenew thus maintaining the expansion of the tumor. Hence, this cell population in the model could also correspond to compartments downstream in the differentiation pathway, such as transit amplifying cells, given the marked plasticity within the intestinal epithelium. The model is thus in principle consistent with hypotheses that colorectal cancer might have a different cell of origin [69].”
Regarding the effects of other forces driving the response to aspirin in addition to the dynamics described in our paper, we agree. In the original manuscript, we stated that although our measured aspirininduced changes in cell kinetics can in principle account for the epidemiologically observed protective effect, it is likely that other forces, not included in the model, are also important. We mentioned the reduction of an inflammatory microenvironment by aspirin as an example. As the reviewer points out, the colorectal microbiome can also determine the level of protection provided by aspirin, and we have now added a discussion of this. This also points to the importance of quantifying the effect of the microbiome on cellular growth kinetics with and without aspirin in future work, using a combination of experimental and mathematical approaches. This would allow the inclusion of this additional complexity to our model of colorectal carcinogenesis. In the revised manuscript, this has been briefly pointed out in the Introduction, and is brought up more in depth in the Discussion sections by adding the following text:
“Moreover, other microenvironmental factors, such as the composition of the colorectal microbiome, have been shown to influence the ability of aspirin to reduce tumor growth [1618]. This is therefore also likely to play a role in explaining the epidemiological data. Quantification of these further complexities in future work will allow us to introduce these additional aspects into the modeling framework, which would result in a refinement of predictions.”
– Similarly, I think more discussion could be owed to the emerging literature around the intersection of agedifferences and aspirin mechanism, especially in light of the recent results from the ASPREE trial that described an increase in cancer death as a result of aspirin intervention among adults over age 70. Is it possible to model the timing of aspirin intervention using this model, particularly in view of the probability differences arising from differential mutational priming outcomes (APC/ vs. APC /+ vs. KRAS+, etc.). The ASPREE results demonstrated that the increase in cancer mortality was not driven by a change in cancer incidence and I wonder if the authors can try to model these effects or at least discuss how the model findings should be interpreted in view of these recent results from trials.
It is indeed very interesting to discuss the results of the ASPREE trial in the context of our modeling efforts, and in the more general context of evolutionary modeling. (i) In the revised manuscript, we expanded our analysis to show that the effect of aspirin on the predicted cancer incidence diminishes with the age at which aspirin treatment is initiated, being lowest if treatment is initiated at 70 years or older. This fits well with the lack of an effect of aspirin on cancer incidence in the ASPREE trial. (ii) While our modeling approach cannot make predictions about mortality (because we model evolution only up to the advanced adenoma stage), our previously published work on evolutionary dynamics might offer insights into the reasons for the observed increased mortality seen in the APSREE trial. These aspects are now discussed by adding the following paragraph to the end of the Discussion section:
“Finally, it is interesting to discuss the results of the ASPREE trial [71,72] in the context of the work presented here. This trial investigated the effect of aspirin treatment in a cohort of older individuals, 70 years or older without cardiovascular disease, dementia, or disability. It was found that cancer incidence was not significantly changed by aspirin, but that the aspirintreated group experienced a higher rate of cancerinduced mortality. The absence of a significant effect of aspirin on cancer incidence in this study is consistent with our model predictions. Our mathematical analysis demonstrated that the effect of aspirin treatment on cancer incidence diminished when treatment was initiated in older ages. Our modeling approach, however, cannot make predictions about cancerinduced mortality, because it describes the evolutionary process up to the stage of advanced adenoma only. Our previous work [21], however, offers an interpretation of these data. Because of their advanced age, it is likely that a certain fraction of the ASPREE participants already harbored tumors that had not been detected yet due to the absence of overt clinical symptoms. In fact, a previous history of cancer was not an exclusion criterium in the trial. As the established tumors continue to grow during aspirin treatment, they likely do so with altered kinetics (reduced division rates and increased death rates, leading to a higher turnover). This means that by the time the tumor has reached a given size (e.g. at which it becomes clinically detectable), it will have undergone more cell divisions under aspirin treatment compared to the placebo group. Hence, the tumor will on average have accumulated more mutations once this detectable tumor size is reached. This in turn means that the aspirintreated tumor might be more virulent and less responsive to therapies, resulting in more deaths. The theoretically derived notion that upon detection, an aspirintreated tumor is more evolved than a tumor that grows without aspirin [21] is supported by the ASPREE analysis, which found that aspirintreated patients were more likely to have metastasized cancers and stage 4 cancers compared to the placebo group [71,72].“
– The manuscript primarily discusses the CAPP2 trial as the evidence supporting aspirin chemoprevention of colorectal cancers. Although this obviously has clear implications for placing the results in the context of prevention of CRCs in Lynch syndrome, these tumors are neither sporadic, nor arise via the pathways included in the model. The authors could broaden the background to include the preponderance of evidence for the preventive effects in sporadic cases (or even FAP patients which are known to have APC mutant cancers) where aspirin has had less of a potent chemopreventive effect than in Lynch syndrome. However, these data are relevant to the most extreme phenotype in their model.
We agree with this point and have broadened the background accordingly in the introduction.
[Editors' note: further revisions were suggested prior to acceptance, as described below.]
The manuscript has been improved but there are some remaining issues that need to be addressed, as outlined below:
Reviewer #2 (Recommendations for the authors):
The revised manuscript has gained considerably in strength and, by in large, clarifies the main points raised by the reviewers. I appreciate the extra work that went into refining/extending the model analysis, in particular the addition of a growth phase for the type 6 (advanced) adenoma.
Thank you for this positive assessment of our revisions.
Two lingering points. I hope they can be addressed.
1. Please clarify whether the size distribution provided for type 3 (APC/) adenoma in Figure Appendix 1 Figure 12 and 13 refer to the particular type 3 adenoma (clone) in which the advanced adenoma first developed or to the entire population of type 3 adenomas in the colon. I think the authors should point out that their ODE model does NOT distinguish individual clones of abnormal crypts (ie individual adenomas). This is a limitation since detectable adenoma number of any kind (other than hyperplastics) is an important clinical factor.
Figures 12 and 13 of Appendix 1 show probability distributions of the number of crypts of each type (1 through 5) obtained in a fullystochastic model, at the time when he first type 6 crypt is detected. This figure does not contain information on pathways to advanced adenoma. In other words, presented are just the numbers of crypts of each type that are present in the entire simulated colon, at the time when the first crypt associated with the advanced adenoma is generated, regardless of whether it was generated by a mutation of a type 3 or a type 5 crypt. We have included these clarifications in the revised text of the manuscript (Appendix 1, Section 5.3).
We also pointed out that our ODE model does not keep track of the clonality of abnormal crypts. For example, if type 5 crypt is created multiple times (by conversion) in the system, the variable n_{5} simply gives the total number of type5 crypts. In a stochastic Gillespie model, however, it is possible to keep track of different clones by designating each newly generated crypt as a different “subtype”, which can then clonally expand through crypt fission. This however goes beyond the scope of the current study. Text has been added to explain these points (Appendix 1, Section 5.2).
2. I understand the authors' point about postulating other potential gainoffunction mutations, similar to KRAS, such as BRAF. However, BRAF is a poor example as it is associated strongly with mismatch repair deficiency and SSAs, leading frequently to hypermutated cancers. While there may be other yet unidentified gainoffunction drivers for the advanced adenoma, there may also simply none required given the epigenetic plasticity and adaptive epigenetic changes as adenomas sojourn for years. In any case, the mention of BRAF is somewhat misleading in the context defined by the authors.
We agree. We have taken out the reference to BRAF. Further, we stated more explicitly the types of genetic events that are assumed in our model, in more general terms than in the previous version of the manuscript. That is, the model assumes that the pathway to advanced adenoma involves the inactivation of a tumor suppressor gene (e.g. APC), and a gain of function mutation (which can be KRAS or potentially an alternative). This pathway assumed by our model is supported by data available so far. We further pointed out that the model does not hold for evolutionary pathways that deviate from these assumptions. This is expressed in the following text in the manuscript (page 7, first paragraph):
“Our model, however, does not depend on the identity of particular mutations, but assumes the occurrence of mutation types; these are the inactivation of a tumor suppressor gene (which is a lossoffunction mutation, e.g. APC/), and a gainoffunction mutation, which can be in KRAS or an alternative gene. Our model predictions hold as long as the evolutionary pathway to advanced adenomas involves these two types of mutational events, regardless of their identity. We note that our model does not apply to potential cases of advanced adenomas that might develop via pathways characterized by a different number or different types of initiating events.”
https://doi.org/10.7554/eLife.71953.sa2Article and author information
Author details
Funding
National Cancer Institute (NIH 1 U01 CA18795601)
 C Richard Boland
 Ajay Goel
 Dominik Wodarz
 Natalia L Komarova
National Science Foundation (NSFSimons Center for Multiscale Cell Fate Research)
 Yifan Wang
 Natalia L Komarova
MIDAS (AWD00000238)
 Natalia L Komarova
 Yifan Wang
The funders had no role in study design, data collection and interpretation, or the decision to submit the work for publication.
Acknowledgements
We thank the reviewers of this paper for very valuable comments that have influenced to the current form of the manuscript.
Support of the following grants is gratefully acknowledged: NIH 1 U01 CA18795601 (AG, RB, NK, DW); NSFSimons Center for Multiscale Cell Fate Research (NK, YW); NIH/NCI U54CA217378 (NK, DW, YW).
Senior Editor
 Eduardo Franco, McGill University, Canada
Reviewing Editor
 Christopher S Williams, Vanderbilt University, United States
Reviewers
 E Georg Luebeck, Fred Hutchinson Cancer Research Center, United States
 Andrew Chan
Publication history
 Preprint posted: May 12, 2021 (view preprint)
 Received: July 5, 2021
 Accepted: March 15, 2022
 Version of Record published: April 13, 2022 (version 1)
Copyright
© 2022, Wang et al.
This article is distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use and redistribution provided that the original author and source are credited.
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