Patterns of cell division and the risk of cancer

Epidermal and intestinal tissues divide throughout life to replace lost surface cells. These renewing tissues have long-lived basal stem cell lineages that divide many times, each division producing one stem cell and one transit cell. The transit cell divides a limited number of times, producing cells that move up from the basal layer and eventually slough off from the surface. If mutation rates are the same in stem and transit divisions, we show that minimal cancer risk is obtained by using the fewest possible stem divisions subject to the constraints imposed by the need to renew the tissue. In this case, stem cells are a necessary risk imposed by the constraints of tissue architecture. Cairns suggested that stem cells may have lower mutation rates than transit cells do. We develop a mathematical model to study the consequences of different stem and transit mutation rates. Our model shows that stem cell mutation rates two or three orders of magnitude less than transit mutation rates may favor relatively more stem divisions and fewer transit divisions, perhaps explaining how renewing tissues allocate cell divisions between long stem and short transit lineages.     

Role of heterozygous APC mutation in niche succession and initiation of colorectal cancer--a computational study

Mutations in the adenomatous polyposis coli (APC) gene are found in most colorectal cancers. They cause constitutive activation of proliferative pathways when both alleles of the gene are mutated. However studies on individuals with familial adenomatous polyposis (FAP) have shown that a single mutated APC allele can also create changes in the precancerous colon crypt, like increased number of stem cells, increased crypt fission, greater variability of DNA methylation patterns, and higher somatic mutation rates. In this paper, using a computational model of colon crypt dynamics, we evolve and investigate a hypothesis on the effect of heterozygous APC mutation that explains these different observations. Based on previous reports and the results from the computational model we propose the hypothesis that heterozygous APC mutation has the effect of increasing the chances for a stem cell to divide symmetrically, producing two stem cell daughters. We incorporate this hypothesis into the model and perform simulation experiments to investigate the consequences of the hypothesis. Simulations show that this hypothesis links together the changes in FAP crypts observed in previous studies. The simulations also show that an APC(+/-) stem cell gets selective advantages for dominating the crypt and progressing to cancer. This explains why most colon cancers are initiated by APC mutation. The results could have implications for preventing or retarding the onset of colon cancer in people with inherited or acquired mutation of one APC allele. Experimental validation of the hypothesis as well as investigation into the molecular mechanisms of this effect may therefore be worth undertaking.     

Quantification of Colonic Stem Cell Mutations

The ability to measure stem cell mutations is a powerful tool to quantify in a critical cell population if, and to what extent, a chemical can induce mutations that potentially lead to cancer. The use of an enzymatic assay to quantify stem cell mutations in the X-linked glucose-6-phosphate dehydrogenase gene has been previously reported.¹ This method requires the preparation of frozen sections and incubation of the sectioned tissue with a reaction mixture that yields a blue color if the cells produce functional glucose-6-phosphate dehydrogenase (G6PD) enzyme. If not, the cells appear whitish. We have modified the reaction mixture using Optimal Cutting Temperature Compound (OCT) medium in place of polyvinyl alcohol. This facilitates pH measurement, increases solubilization of the G6PD staining components and restricts diffusion of the G6PD enzyme. To demonstrate that a mutation occurred in a stem cell, the entire crypt must lack G6PD enzymatic activity. Only if a stem cell harbors a phenotypic G6PD mutation will all of the progeny in the crypt lack G6PD enzymatic activity. To identify crypts with a stem cell mutation, four consecutive adjacent frozen sections (a level) were cut at 7 µm thicknesses. This approach of making adjacent cuts provides conformation that a crypt was fully mutated since the same mutated crypt will be observed in adjacent sections. Slides with tissue samples that were more than 50 µm apart were prepared to assess a total of >10⁴ crypts per mouse. The mutation frequency is the number of observed mutated (white) crypts ÷ by the number of wild type (blue) crypts in a treatment group.     

Matrix simulation of duodenal crypt cell kinetics. I. The steady state.

Steady state crypt cell kinetics have been simulated using matrix algebra. The model crypt cell population is distributed through two proliferation compartments (P1 and P2) and a quiescent state (Q). Under steady state conditions half the daughter cells produced on completion of P1 enter G1 of P2 and half enter G1 of P1. Both P2 daughter cells enter Q. Cells in Q are non-dividing but retain the potential to divide. On completion of Q, cells lose the potential to divide and move up onto the villi. The model has been developed by simultaneously simulating the following biological data: (1) the per cent labeled mitosis (PML) curve, (2) the number of labeled cells per crypt as a function of time following an injection of 3H-thymidine, and (3) the total number of cells per crypt.     

Optimality in the development of intestinal crypts

Intestinal crypts in mammals are comprised of long-lived stem cells and shorter-lived progenies. These two populations are maintained in specific proportions during adult life. Here, we investigate the design principles governing the dynamics of these proportions during crypt morphogenesis. Using optimal control theory, we show that a proliferation strategy known as a "bang-bang" control minimizes the time to obtain a mature crypt. This strategy consists of a surge of symmetric stem cell divisions, establishing the entire stem cell pool first, followed by a sharp transition to strictly asymmetric stem cell divisions, producing nonstem cells with a delay. We validate these predictions using lineage tracing and single-molecule fluorescence in?situ hybridization of intestinal crypts in infant mice, uncovering small crypts that are entirely composed of Lgr5-labeled stem cells, which become a minority as crypts continue to grow. Our approach can be used to uncover similar design principles in other developmental systems.     

Patterns of Proliferative Activity in the Colonic Crypt Determine Crypt Stability and Rates of Somatic Evolution

Epithelial cells in the colon are arranged in cylindrical structures called crypts in which cellular proliferation and migration are tightly regulated. We hypothesized that the proliferation patterns of cells may determine the stability of crypts as well as the rates of somatic evolution towards colorectal tumorigenesis. Here, we propose a linear process model of colonic epithelial cells that explicitly takes into account the proliferation kinetics of cells as a function of cell position within the crypt. Our results indicate that proliferation kinetics has significant influence on the speed of cell movement, kinetics of mutation propagation, and sensitivity of the system to selective effects of mutated cells. We found that, of all proliferation curves tested, those with mitotic activities concentrated near the stem cell, including the actual proliferation kinetics determined in in vivo labeling experiments, have a greater ability of delaying the rate of mutation accumulation in colonic stem cells compared to hypothetical proliferation curves with mitotic activities focused near the top of the crypt column. Our model can be used to investigate the dynamics of proliferation and mutation accumulation in spatially arranged tissues.     

The conservation of redundancy in genetic systems: effects of sexual and asexual reproduction

The relationship between probability of survival and the number of deleterious mutations in the genome is investigated using three different models of highly redundant systems that interact with a threatening environment. Model one is a system that counters a potentially lethal infection; it has multiple identical components that act in sequence and in parallel. Model two has many different overlapping components that provide three-fold coverage of a large number of vital functions. The third model is based on statistical decision theory: an ideal detector, following an optimum decision strategy, makes crucial decisions in an uncertain world. The probability of a fatal error is reduced by a redundant sampling system, but the chance of error rises as the system is impaired by deleterious mutations. In all three cases the survival profile shows a synergistic pattern in that the probability of survival falls slowly and then more rapidly. This is different than the multiplicative or independent survival profile that is often used in mathematical models. It is suggested that a synergistic profile is a property of redundant systems. Model one is then used to study the conservation of redundancy during sexual and asexual reproduction. A unicellular haploid organism reproducing asexually retains redundancy when the mutation rate is very low (0001 per cell division), but tends to lose high levels of redundancy if the mutation rate is increased (001 to 01 per cell division). If a similar unicellular haploid organism has a sexual phase then redundancy is retained for mutation rates between 0001 and 01 per cell division. The sexual organism outgrows the asexual organism when the above mutation rates apply. If they compete for finite resources the asexual organism will be extinguished. Variants of the sexual organism with increased redundancy will outgrow those with lower levels of redundancy and the sexual process facilitates the evolution of more complex forms. There is a limit to the extent that complexity can be increased by increasing the size of the genome and in asexual organisms this leads to progressive accumulation of mutations with loss of redundancy and eventual extinction. If complexity is increased by using genes in new combinations, the asexual form can reach a stable equilibrium, although it is associated with some loss of redundancy. The sexual form, by comparison, can survive, with retention of redundancy, even if the mutation rate is above one per generation. The conservation and evolution of redundancy, which is essential for complexity, depends on the sexual process of reproduction.     

Effect of stem cell turnover rates on protection against cancer and aging

Tissue stem cells are responsible for replenishing and maintaining a population of cells which make up a functioning organ. They divide by asymmetric cell division where one daughter remains a stem cell while the other daughter becomes a transit cell, which divides a defined number of times and differentiates. A fully differentiated cell has a finite life-span. A tissue can be maintained by various strategies. Stem cells can divide often and differentiated cells die often (fast turnover). Alternatively, stem cells can divide infrequently, and the differentiated cells are long lived (slow turnover). Genetic alterations and mutations can interfere with tissue homoeostasis. Mutations can induce senescence and apoptosis, and this can result in a reduction of the number of functioning tissue cells which could correlate with tissue aging. Alternatively, mutations can result in the carcinogenic transformation of cells and the formation of a tumour. Using mathematical models, I find that the cellular turnover rate affects the ability of genetic alterations to induce aging and the development of cancer. If mutations occur as a result of errors during cell division, the model suggests that a low cellular turnover rate protects both against aging and the development of cancer. On the other hand, if mutations occur independent from cell division (e.g. if DNA is hit by damaging agents), I find that a high cellular turnover rate protects against aging, while it promotes the development of cancer. Implications for optimal tissue design are discussed.     

Assessment of the symmetry of stem-cell mitoses.

A model of Paneth-cell renewal in the small intestinal epithelium is used to estimate the probability that epithelial stem-cell mitoses are symmetric in the sense that they produce two cells of the same type. I found that counts of the number of Paneth cells per crypt (Paneth cells are terminally differentiated cells derived from small intestinal epithelial stem cells) support a model in which most, if not all, stem-cell mitoses are symmetric.     

MATRIX SIMULATION OF DUODENAL CRYPT CELL KINETICS

Steady state crypt cell kinetics have been simulated using matrix algebra. The model crypt cell population is distributed through two proliferation compartments (P₁ and P₂) and a quiescent state (Q). Under steady state conditions half the daughter cells produced on completion of P₁ enter G₁ of P₂ and half enter G₁ of P₁. Both P₂ daughter cells enter Q. Cells in Q are non-dividing but retain the potential to divide. On completion of Q, cells lose the potential to divide and move up onto the villi. The model has been developed by simultaneously simulating the following biological data: (1) the per cent labeled mitosis (PML) curve, (2) the number of labeled cells per crypt as a function of time following an injection of ³H-thymidine, and (3) the total number of cells per crypt.     

A stochastic branching model with formation of subunits applied to the growth of intestinal crypts

The intestinal epithelium is one of the most rapidly regenerating tissues in mammals. Cell production takes place in the intestinal crypts which contain about 250 cells. Only a minority of 1-60 proliferating cells are able to maintain a crypt over a long period of time. However, so far attempts to identify these stem cells were unsuccessful. Therefore, little is known about their cellular growth and selfmaintenance properties. On the other hand, the crypts appear to exhibit a life cycle which starts by fission of existing crypts and ends by fission or extinction. Data on these processes have recently become available. Here, we demonstrate how these data on the life cycle of the macroscopic crypt structure can be used to derive a quantitative model of the microscopic process of stem cell growth. The model assumptions are: (1) stem cells undergo a time independent supracritical Markovian branching process (Galton-Watson process); (2) a crypt divides if the number of stem cells exceeds a given threshold and the stem cells are distributed to both daughter crypts according to binomial statistics; (3) the size of the crypt is proportional to the stem cell number. This model combining two different stochastic branching processes describes a new class of processes whose stationary stability and asymptotic behavior are examined. This model should be applicable to various growth processes with formation of subunits (e.g. population growth with formation of colonies in biology, ecology and sociology). Comparison with crypt data shows that intestinal stem cells have a probability of over 0.8 of dividing asymmetrically and that the threshold number should be 8 or larger.     

The effects of background mortality on optimal reproduction in a seasonal environment

We consider optimal annual routines of reproductive behaviour in a seasonal environment. In our model the condition of the organism is adversely affected by hard work, but can recover during easy periods. Our analysis concentrates on the effects of background mortality (i.e., mortality that cannot be avoided) on the optimal strategy and how often an organism following this strategy breeds. In particular, we are concerned with whether reproduction occurs at specific times of year (entrained to the annual cycle), and if so then how many reproductive bouts occur per year. We find that an increase in background mortality can have various effects. If the animal is entrained to the annual cycle and has one breeding attempt per year, then breeding tends to occur earlier and there may be two breeding attempts per season. Another possible outcome is that breeding is no longer entrained. If the animal is entrained but sometimes skips reproduction so that it does not breed every year, then an increase in mortality may make it more likely that the animal breeds every year. We show that as background mortality increases the resultant increase in the frequency of breeding contributes to the increase in annual mortality. We also explore the effects of mortality on the timing of reproduction within a year, highlighting the tension between the interests of the parent and that of the young.     

Effect of Dedifferentiation on Time to Mutation Acquisition in Stem Cell-Driven Cancers

Accumulating evidence suggests that many tumors have a hierarchical organization, with the bulk of the tumor composed of relatively differentiated short-lived progenitor cells that are maintained by a small population of undifferentiated long-lived cancer stem cells. It is unclear, however, whether cancer stem cells originate from normal stem cells or from dedifferentiated progenitor cells. To address this, we mathematically modeled the effect of dedifferentiation on carcinogenesis. We considered a hybrid stochastic-deterministic model of mutation accumulation in both stem cells and progenitors, including dedifferentiation of progenitor cells to a stem cell-like state. We performed exact computer simulations of the emergence of tumor subpopulations with two mutations, and we derived semi-analytical estimates for the waiting time distribution to fixation. Our results suggest that dedifferentiation may play an important role in carcinogenesis, depending on how stem cell homeostasis is maintained. If the stem cell population size is held strictly constant (due to all divisions being asymmetric), we found that dedifferentiation acts like a positive selective force in the stem cell population and thus speeds carcinogenesis. If the stem cell population size is allowed to vary stochastically with density-dependent reproduction rates (allowing both symmetric and asymmetric divisions), we found that dedifferentiation beyond a critical threshold leads to exponential growth of the stem cell population. Thus, dedifferentiation may play a crucial role, the common modeling assumption of constant stem cell population size may not be adequate, and further progress in understanding carcinogenesis demands a more detailed mechanistic understanding of stem cell homeostasis.     

Intestinal crypt homeostasis results from neutral competition between symmetrically dividing Lgr5 stem cells

Intestinal stem cells, characterized by high Lgr5 expression, reside between Paneth cells at the small intestinal crypt base and divide every day. We have carried out fate mapping of individual stem cells by generating a multicolor Cre-reporter. As a population, Lgr5(hi) stem cells persist life-long, yet crypts drift toward clonality within a period of 1-6 months. We have collected short- and long-term clonal tracing data of individual Lgr5(hi) cells. These reveal that most Lgr5(hi) cell divisions occur symmetrically and do not support a model in which two daughter cells resulting from an Lgr5(hi) cell division adopt divergent fates (i.e., one Lgr5(hi) cell and one transit-amplifying [TA] cell per division). The cellular dynamics are consistent with a model in which the resident stem cells double their numbers each day and stochastically adopt stem or TA fates. Quantitative analysis shows that stem cell turnover follows a pattern of neutral drift dynamics.     

Estimates of the number of clonogenic cells in crypts of murine small intestine

There is a proliferative cell hierarchy in the mouse intestinal crypt with ancestral stem cells which can regenerate all components of the lineage after injury (clonogenic cells). The number of these clonogenic or regenerative cells per crypt can be estimated from radiobiological experiments where doses of radiation are used to kill cells and ablate crypts. Various approaches can be adopted which provide different estimates of this number of cells. One of the conventional approaches used in the past provided estimates of about 70-80 clonogenic cells per crypt (i.e. about 50% of the proliferative or 30% of all crypt cells). A technically simpler approach has recently been suggested. This has been used here to provide many independent estimates of the number of crypt clonogenic cells. These suggest about 32 clonogenic cells exist per crypt i.e. about half the previous estimate and about twice the number of putative "functional" stem cells (those which operate as stem cells in the normal steady-state crypt). The reasons for the differences are discussed. The new estimates are compatible with the hypothesis that the crypt contains a ring of about 16 functional stem cells which are expected to be clonogenic, besides which there is a second ring of 16 clonogenic cells which represent early transit cells (the immediate daughters of the stem cells) which can act as clonogenic cells if required after radiation injury.     

Mathematical model for the cancer stem cell hypothesis

Recent research on the origin of brain cancer has implicated a subpopulation of self-renewing brain cancer stem cells for malignant tumour growth. Various genes that regulate self-renewal in normal stem cells are also found in cancer stem cells. This implies that cancers can occur because of mutations in normal stem cells and early progenitor cells. A predictive mathematical model based on the cell compartment method is presented here to pose and validate non-intuitive scenarios proposed through the neural cancer stem cell hypothesis. The growths of abnormal (stem and early progenitor) cells from their normal counterparts are ascribed with separate mutation probabilities. Stem cell mutations are found to be more significant for the development of cancer than a similar mutation in the early progenitor cells. The model also predicts that, as previously hypothesized, repeated insult to mature cells increases the formation of abnormal progeny, and hence the risk of cancer.     

Stochastic model of the formation of cancer metastases via cancer stem cells

The author presents Monte Carlo simulations of the temporal kinetics of the formation of cancer metastases with emphasis on cancer stem cells. The model used implies the existence of premetastatic niches. The population of cancer stem cells located outside tumors and inducing the formation of new tumors in niches is considered to be heterogeneous. If the niches are equivalent with respect to the formation of metastases, the kinetics are predicted to exhibit an induction period and then rapid growth of the number of metastases. If the niches are heterogeneous, the kinetics are found to be more gradual.     

Defining the role of cooperation in early tumor progression

Competition among cells has long been recognized as an important part of carcinogenesis. However, the role of cellular cooperation in cancer has been largely ignored. In this work, we investigated the role of cooperation in early tumor progression using a mathematical and agent-based modeling approach. We hoped to learn how the introduction of cooperative cells into a cell population would affect the dynamics of the system. We modeled the stem cell compartment of tissue using a spatial structure organized into cell patches, with stem cells able to replicate or leave the stem cell compartment through apoptosis or differentiation. The cells could also acquire mutations in three oncogenes and two tumor suppressor genes. Cooperative cells in our model provided a cooperative signal that increased the fitness of their immediate neighbors, but did not affect their own fitness. Running simulations of the model, we found that if cooperative cells are introduced into a cell population, they steadily proliferate and confer a growth advantage to the entire population. This leads us to conclude that providing a cooperative signal is likely to be under positive selective pressure. When cooperative ability and mutation are concurrently present in the same cells, the overall cell population experiences a significant growth advantage, much greater than with cooperation or mutation alone. This growth advantage is diminished if cells with only oncogene/tumor suppressor mutations are also present in the population, suggesting that the optimal scenario for tumor growth would be for cooperative cells to take over a cell population, and then for mutations in oncogenes and tumor suppressors to arise on a background of cooperation. We predict that cooperation is particularly important in the very early stages of carcinogenesis, when tissue is morphologically and histologically normal. Our results have implications for the screening and early diagnosis of cancer.