Fixation probability, the probability that the frequency of a newly arising mutation in a population will eventually reach unity, is a fundamental quantity in evolutionary genetics. Here we use a number of models (several versions of the Moran model and the haploid Wright-Fisher model) to examine fixation probabilities for a constant size population where the fitness is a random function of both allelic state and spatial position, despite neither allele being favored on average. The concept of fitness varying with respect to both genotype and environment is important in models of cancer initiation and progression, bacterial dynamics, and drug resistance. Under our model spatial heterogeneity redefines the notion of neutrality for a newly arising mutation, as such mutations fix at a higher rate than that predicted under neutrality. The increased fixation probability appears to be due to rare alleles having an advantage. The magnitude of this effect can be large, and is an increasing function of the spatial variance and skew in fitness. The effect is largest when the fitness values of the mutants and wild types are anti-correlated across environments. We discuss results for both a spatial ring geometry of cells (such as that of a colonic crypt), a 2D lattice and a mass-action (complete graph) arrangement.

Studying the stem cell (SC) niche architecture is a crucial step for investigating the process of oncogenesis and obtaining an effective stem cell therapy for various cancers. Recently, it has been observed that there are two groups of SCs in the SC niche collaborating with each other to maintain tissue homeostasis: border stem cells (BSCs), which are responsible in controlling the number of non-stem cells as well as stem cells, and central stem cells (CeSCs), which regulate the SC niche. Here, we develop a bi-compartmental stochastic model for the SC niche to study the spread of mutants within the niche. The analytic calculations and numeric simulations, which are in perfect agreement, reveal that in order to delay the spread of mutants in the SC niche, a small but non-zero number of SC proliferations must occur in the CeSC compartment. Moreover, the migration of BSCs to CeSCs delays the spread of mutants. Furthermore, the fixation probability of mutants in the SC niche is independent of types of SC division as long as all SCs do not divide fully asymmetrically. Additionally, the progeny of CeSCs have a much higher chance than the progeny of BSCs to take over the entire niche.

The unwelcome evolution of malignancy during cancer progression emerges through a selection process in a complex heterogeneous population structure. In the present work, we investigate evolutionary dynamics in a phenotypically heterogeneous population of stem cells (SCs) and their associated progenitors. The fate of a malignant mutation is determined not only by overall stem cell and non-stem cell growth rates but also differentiation and dedifferentiation rates. We investigate the effect of such a complex population structure on the evolution of malignant mutations. We derive exactly calculated results for the fixation probability of a mutant arising in each of the subpopulations. The exactly calculated results are in almost perfect agreement with the numerical simulations. Moreover, a condition for evolutionary advantage of a mutant cell versus the wild type population is given in the present study. We also show that microenvironment-induced plasticity in invading mutants leads to more aggressive mutants with higher fixation probability. Our model predicts that decreasing polarity between stem and non-stem cells’ turnover would raise the survivability of non-plastic mutants; while it would suppress the development of malignancy for plastic mutants. The derived results are novel and general with potential applications in nature; we discuss our model in the context of colorectal/intestinal cancer (at the epithelium). However, the model clearly needs to be validated through appropriate experimental data. This novel mathematical framework can be applied more generally to a variety of problems concerning selection in heterogeneous populations, in other contexts such as population genetics, and ecology.

Colon and intestinal crypts have been widely chosen to study cell dynamics because of their fairly simple structures. In the colon and intestinal crypts, stem cells (SCs) are located at very bottom of the crypt, fully differentiated cells (FDs) are located in the top of the crypt, and transit-amplifying cells (TAs) are in the middle of the crypt between FDs and SCs. Recently, it has been discovered that there are two types of stem cells in the intestinal crypts: central stem cells (CeSCs) and border stem cells. To investigate dynamics of mutants in colon and intestinal crypts, we develop a four-compartmental stochastic model, which includes two SC compartments, and TAs and FDs compartments. We calculate the probability of the progeny of marked or mutant cells located at each of these compartments taking over the entire crypt or being washed out from the crypt. We found that the progeny of CeSCs will take over the entire crypt with a probability close to one. Interestingly, the progeny of advantageous mutant TAs and FDs will be washed out faster than disadvantageous mutants. Saliently, the model predicts that the time that the progeny of wild-type central stem cells will take over the mouse intestinal crypt is around 60 days, which is in perfect agreement with an experimental observation.