Deterministic and stochastic regimes of asexual evolution on rugged fitness landscapes

We study the adaptation dynamics of an initially maladapted asexual population with genotypes represented by binary sequences of length L. The population evolves in a maximally rugged fitness landscape with a large number of local optima. We find that whether the evolutionary trajectory is deterministic or stochastic depends on the effective mutational distance d(eff) up to which the population can spread in genotype space. For d(eff) = L, the deterministic quasi-species theory operates while for d(eff) < 1, the evolution is completely stochastic. Between these two limiting cases, the dynamics are described by a local quasi-species theory below a crossover time T(x) while above T(x) the population gets trapped at a local fitness peak and manages to find a better peak via either stochastic tunneling or double mutations. In the stochastic regime d(eff) < 1, we identify two subregimes associated with clonal interference and uphill adaptive walks, respectively. We argue that our findings are relevant to the interpretation of evolution experiments with microbial populations.     

Stochastic tunnels in evolutionary dynamics

We study a situation that arises in the somatic evolution of cancer. Consider a finite population of replicating cells and a sequence of mutations: type 0 can mutate to type 1, which can mutate to type 2. There is no back mutation. We start with a homogeneous population of type 0. Mutants of type 1 emerge and either become extinct or reach fixation. In both cases, they can generate type 2, which also can become extinct or reach fixation. If mutation rates are small compared to the inverse of the population size, then the stochastic dynamics can be described by transitions between homogeneous populations. A "stochastic tunnel" arises, when the population moves from all 0 to all 2 without ever being all 1. We calculate the exact rate of stochastic tunneling for the case when type 1 is as fit as type 0 or less fit. Type 2 has the highest fitness. We discuss implications for the elimination of tumor suppressor genes and the activation of genetic instability. Although our theory is developed for cancer genetics, stochastic tunnels are general phenomena that could arise in many circumstances.     

Expression of lamin A mutated in the carboxyl-terminal tail generates an aberrant nuclear phenotype similar to that observed in cells from patients with Dunnigan-type partial lipodystrophy and Emery-Dreifuss muscular dystrophy

Autosomal dominantly inherited missense mutations in lamins A and C cause familial partial lipodystrophy of the Dunnigan-type (FPLD), and myopathies including Emery-Dreifuss muscular dystrophy (EDMD). While mutations responsible for FPLD are restricted to the carboxyl-terminal tails, those responsible for EDMD are spread throughout the molecules. We observed here the same structural abnormalities in the nuclear envelope and chromatin of fibroblasts from patients with FPLD and EDMD, harboring missense mutations at codons 482 and 453, respectively. Similar nuclear alterations were generated in fibroblasts, myoblasts, and preadipocytes mouse cell lines overexpressing lamin A harboring either of these two mutations. A large variation in sensitivity to lamin A overexpression was observed among the three cell lines, which was correlated with their variable endogenous content in A-type lamins and emerin. The occurrence of nuclear abnormalities was reduced when lamin B1 was coexpressed with mutant lamin A, emphasizing the functional interaction of the two types of lamins. Transfected cells therefore develop similar phenotypes when expressing lamins mutated in the carboxyl-terminal tail at sites responsible for FPLD or EDMD.     

Stochastic dynamics of cancer initiation

Most human cancer types result from the accumulation of multiple genetic and epigenetic alterations in a single cell. Once the first change (or changes) have arisen, tumorigenesis is initiated and the subsequent emergence of additional alterations drives progression to more aggressive and ultimately invasive phenotypes. Elucidation of the dynamics of cancer initiation is of importance for an understanding of tumor evolution and cancer incidence data. In this paper, we develop a novel mathematical framework to study the processes of cancer initiation. Cells at risk of accumulating oncogenic mutations are organized into small compartments of cells and proliferate according to a stochastic process. During each cell division, an (epi)genetic alteration may arise which leads to a random fitness change, drawn from a probability distribution. Cancer is initiated when a cell gains a fitness sufficiently high to escape from the homeostatic mechanisms of the cell compartment. To investigate cancer initiation during a human lifetime, a 'race' between this fitness process and the aging process of the patient is considered; the latter is modeled as a second stochastic Markov process in an aging dimension. This model allows us to investigate the dynamics of cancer initiation and its dependence on the mutational fitness distribution. Our framework also provides a methodology to assess the effects of different life expectancy distributions on lifetime cancer incidence. We apply this methodology to colorectal tumorigenesis while considering life expectancy data of the US population to inform the dynamics of the aging process. We study how the probability of cancer initiation prior to death, the time until cancer initiation, and the mutational profile of the cancer-initiating cell depends on the shape of the mutational fitness distribution and life expectancy of the population.     

X-ray induction of microsatellite instability at autosomal loci in human lymphoblastoid WTK1 cells

Many models of carcinogenesis posit that multiple genetic events are required for a normal cell to become cancerous. As the mutation rate of a single gene is in the range of 10(-8) to 10(-5) per cell division, a central question remains, how does a single cell acquire multiple mutations? One hypothesis, originally articulated by Loeb [10], proposed that some mutations may not be isolated events, but are associated with a mutator phenotype that leads to the occurrence of additional mutations elsewhere in the cellular genome. To test this hypothesis, we utilized a human lymphoblastoid cell line (WTK1) that is known to be hypermutable at the autosomal thymidine kinase (TK) locus. We isolated 139 independent clones which were selected for new TK mutations that arose either spontaneously or as the result of a single X-ray exposure of 1.5Gy. These clones were examined for second-site alterations in several microsatellite loci scattered throughout the genome using polymerase chain reaction (PCR) amplification followed by both denaturing gel electrophoresis and single-stranded conformational polymorphism (SSCP) analysis. Of these clones, 21 exhibited second-site mutations primarily involving loss of heterozygosity, 17 arose from irradiated cells whereas the remaining four arose from non-irradiated cells. We further examined the 17 clones which exhibited alterations specifically at the D16S265 locus; alterations at this site were associated with an enhanced frequency of mutations at other loci in the same region of chromosome 16q, but were not associated with additional mutations at other sites in the genome. Furthermore, new mutations arose in loci on 16q when these clones were propagated for 6 months in culture. Overall, these results support the hypothesis that radiation can induce a type of genetic instability which may facilitate the occurrence of multiple mutations throughout the genome in a small population of exposed cells. Furthermore, some cells may possess localized regions in the genome which are highly sensitive to the induction of instability.     

Mutation-selection networks of cancer initiation: tumor suppressor genes and chromosomal instability

In this paper, we derive analytic solutions of stochastic mutation-selection networks that describe early events of cancer formation. A main assumption is that cancer is initiated in tissue compartments, where only a relatively small number of cells are at risk of mutating into cells that escape from homeostatic regulation. In this case, the evolutionary dynamics can be approximated by a low-dimensional stochastic process with a linear Kolmogorov forward equation that can be solved analytically. Most of the time, the cell population is homogeneous with respect to relevant mutations. Occasionally, such homogeneous states are connected by 'stochastic tunnels'. We give a precise analysis of the existence of tunnels and calculate the rate of tunneling. Finally, we calculate the conditions for chromosomal instability (CIN) to precede inactivation of the first tumor suppressor gene. In this case, CIN is an early event and a driving force of cancer progression. The techniques developed in this paper can be used to study arbitrarily complex mutation-selection networks of the somatic evolution of cancer.     

The rate of multi-step evolution in Moran and Wright-Fisher populations

Several groups have recently modeled evolutionary transitions from an ancestral allele to a beneficial allele separated by one or more intervening mutants. The beneficial allele can become fixed if a succession of intermediate mutants are fixed or alternatively if successive mutants arise while the previous intermediate mutant is still segregating. This latter process has been termed stochastic tunneling. Previous work has focused on the Moran model of population genetics. I use elementary methods of analyzing stochastic processes to derive the probability of tunneling in the limit of large population size for both Moran and Wright-Fisher populations. I also show how to efficiently obtain numerical results for finite populations. These results show that the probability of stochastic tunneling is twice as large under the Wright-Fisher model as it is under the Moran model.     

Stem cell factor is encoded at the Sl locus of the mouse and is the ligand for the c-kit tyrosine kinase receptor.

We have cloned a partial cDNA encoding murine stem cell factor (SCF) and show that the gene is syntenic with the Sl locus on mouse chromosome 10. Using retroviral vectors to immortalize fetal liver stromal cell lines from mice harboring lethal mutations at the Sl locus (Sl/Sl), we have shown that SCF genomic sequences are deleted in these lines. Furthermore, two other mutations at Sl, Sld and Sl12H, are associated with deletions or alterations of SCF genomic sequences. In vivo administration of SCF can reverse the macrocytic anemia and locally repair the mast cell deficiency of Sl/Sld mice. We have also provided biological and physical evidence that SCF is a ligand for the c-kit receptor.     

Stochastic Drift in Mitochondrial DNA Point Mutations: A Novel Perspective Ex Silico

The mitochondrial free radical theory of aging (mFRTA) implicates Reactive Oxygen Species (ROS)-induced mutations of mitochondrial DNA (mtDNA) as a major cause of aging. However, fifty years after its inception, several of its premises are intensely debated. Much of this uncertainty is due to the large range of values in the reported experimental data, for example on oxidative damage and mutational burden in mtDNA. This is in part due to limitations with available measurement technologies. Here we show that sample preparations in some assays necessitating high dilution of DNA (single molecule level) may introduce significant statistical variability. Adding to this complexity is the intrinsically stochastic nature of cellular processes, which manifests in cells from the same tissue harboring varying mutation load. In conjunction, these random elements make the determination of the underlying mutation dynamics extremely challenging. Our in silico stochastic study reveals the effect of coupling the experimental variability and the intrinsic stochasticity of aging process in some of the reported experimental data. We also show that the stochastic nature of a de novo point mutation generated during embryonic development is a major contributor of different mutation burdens in the individuals of mouse population. Analysis of simulation results leads to several new insights on the relevance of mutation stochasticity in the context of dividing tissues and the plausibility of ROS ”vicious cycle” hypothesis.     

High-throughput trapping of secretory pathway genes in mouse embryonic stem cells

High-throughput gene trapping is a random approach for inducing insertional mutations across the mouse genome. This approach uses gene trap vectors that simultaneously inactivate and report the expression of the trapped gene at the insertion site, and provide a DNA tag for the rapid identification of the disrupted gene. Gene trapping has been used by both public and private institutions to produce libraries of embryonic stem (ES) cells harboring mutations in single genes. Presently, approximately 66% of the protein coding genes in the mouse genome have been disrupted by gene trap insertions. Among these, however, genes encoding signal peptides or transmembrane domains (secretory genes) are underrepresented because they are not susceptible to conventional trapping methods. Here, we describe a high-throughput gene trapping strategy that effectively targets secretory genes. We used this strategy to assemble a library of ES cells harboring mutations in 716 unique secretory genes, of which 61% were not trapped by conventional trapping, indicating that the two strategies are complementary. The trapped ES cell lines, which can be ordered from the International Gene Trap Consortium (http://www.genetrap.org), are freely available to the scientific community.     

The evolution of two mutations during clonal expansion

Knudson's two-hit hypothesis proposes that two genetic changes in the RB1 gene are the rate-limiting steps of retinoblastoma. In the inherited form of this childhood eye cancer, only one mutation emerges during somatic cell divisions while in sporadic cases, both alleles of RB1 are inactivated in the growing retina. Sporadic retinoblastoma serves as an example of a situation in which two mutations are accumulated during clonal expansion of a cell population. Other examples include evolution of resistance against anticancer combination therapy and inactivation of both alleles of a metastasis-suppressor gene during tumor growth. In this article, we consider an exponentially growing population of cells that must evolve two mutations to (i) evade treatment, (ii) make a step toward (invasive) cancer, or (iii) display a disease phenotype. We calculate the probability that the population has evolved both mutations before it reaches a certain size. This probability depends on the rates at which the two mutations arise; the growth and death rates of cells carrying none, one, or both mutations; and the size the cell population reaches. Further, we develop a formula for the expected number of cells carrying both mutations when the final population size is reached. Our theory establishes an understanding of the dynamics of two mutations during clonal expansion.     

Analysis of a stochastic predator–prey model with applications to intrahost HIV genetic diversity

During an infection, HIV experiences strong selection by immune system T cells. Recent experimental work has shown that MHC escape mutations form an important pathway for HIV to avoid such selection. In this paper, we study a model of MHC escape mutation. The model is a predator–prey model with two prey, composed of two HIV variants, and one predator, the immune system CD8 cells. We assume that one HIV variant is visible to CD8 cells and one is not. The model takes the form of a system of stochastic differential equations. Motivated by well-known results concerning the short life-cycle of HIV intrahost, we assume that HIV population dynamics occur on a faster time scale then CD8 population dynamics. This separation of time scales allows us to analyze our model using an asymptotic approach. Using this model we study the impact of an MHC escape mutation on the population dynamics and genetic evolution of the intrahost HIV population. From the perspective of population dynamics, we show that the competition between the visible and invisible HIV variants can reach steady states in which either a single variant exists or in which coexistence occurs depending on the parameter regime. We show that in some parameter regimes the end state of the system is stochastic. From a genetics perspective, we study the impact of the population dynamics on the lineages of an HIV sample taken after an escape mutation occurs. We show that the lineages go through severe bottlenecks and that in certain parameter regimes the lineage distribution can be characterized by a Kingman coalescent. Our results depend on methods from diffusion theory and coalescent theory.     

Time versus replication dependence of EMS-induced delayed mutation in Chinese hamster cells

We have previously observed in Chinese hamster cells that ethyl methane sulfonate (EMS) induces mutations which are distributed over at least 10-14 cell divisions following treatment. This delayed appearance of mutations could be explained by EMS-induced lesions which remain in DNA and have a probability that is significantly less than 1.0 of producing base mispairing errors during successive replication cycles (replication-dependent). Alternatively, delayed mutation may be a time-dependent process in which a slow acting or damage inducible error-prone repair process removes persistent DNA lesions and replaces them with an incorrect base during the course of 7-10 days of colony growth following EMS exposure. To address this question, the distribution of HGPRT delayed mutation events (fifth division or later) in cells plated immediately for exponential growth after EMS treatment was compared with the distribution in cells which remained under confluent growth conditions for 8 days and then were replated. Both the distribution and rate of accumulation of delayed mutations (mutations/cell division) were similar in the two culture conditions. In contrast, the frequency of early mutations (before the fifth division) in the confluent population was reduced more than 2-fold compared to dividing cells. A comparison of the frequency of EMS-induced DNA lesions in the two populations revealed that the density inhibited population contained one third the DNA lesions of the exponential population. These results argue against a time-dependent process since, if this mechanism applies, one would expect an increase in early mutant events and a decrease in delayed events in the confluent population. The results, however, are consistent with a replication model in which potential early mutant lesions are preferentially removed in the density inhibited culture during the 8 days of incubation while lesions producing late mutants are not removed.     

Mutation as a cause of genetic disease

Mutational changes can be conveniently classified into two sorts: those that appear to involve single genes and are generally referred to as gene mutations, and those that involve chromosomal segments containing many genes, or even whole chromosomes, and are referred to as chromosomal mutations. Both of these kinds of mutation occur in germ-cell lineages and contribute substantially to inherited disease, or pre-disposition to disease, and both also occur in somatic cells and contribute to acquired disease. The mutation rates for inherited disease ascribed to mutation in a single gene differ for different genes and are age-dependent. Moreover, a single disease entity, such as haemophilia B, may be the result of any one of a number of different alterations within the gene responsible for the disease. The mutation rate for inherited chromosomal mutation is also age-dependent, particularly so in the case of mutations involving alterations in chromosome number. Studies in experimental animals demonstrate that exposure to physical or chemical mutagens results in increasing the incidence of inherited gene and chromosomal mutations. However, such increases have not been unequivocally demonstrated in human populations exposed to known mutagens. Studies on mutation in human lymphoid or epithelial somatic cells clearly demonstrate an increased frequency in cells taken from people exposed to ionizing radiations or chemical mutagens or in cells exposed in vitro. The consequences of such mutations will depend upon their nature and the origins and functions of the cells in which they occur. Of particular importance are mutations influencing cell growth and proliferation, and both gene and chromosomal mutations are implicated as causal factors in the development of human cancers.     

A stochastic model of cell replicative senescence based on telomere shortening, oxidative stress, and somatic mutations in nuclear and mitochondrial DNA.

Human diploid fibroblast cells can divide for only a limited number of times in vitro, a phenomenon known as replicative senescence or the Hayflick limit. Variability in doubling potential is observed within a clone of cells, and between two sister cells arising from a single mitotic division. This strongly suggests that the process by which cells become senescent is intrinsically stochastic. Among the various biochemical mechanisms that have been proposed to explain replicative senescence, particular interest has been focussed on the role of telomere reduction. In the absence of telomerase--an enzyme switched off in normal diploid fibro-blasts-cells lose telomeric DNA at each cell division. According to the telomere hypothesis of cell senescence, cells eventually reach a critically short telomere length and cell cycle arrest follows. In support of this concept, forced expression of telomerase in normal fibroblasts appears to prevent cell senescence. Nevertheless, the telomere hypothesis in its basic form has some difficulty in explaining the marked stochastic variations seen in the replicative lifespans of individual cells within a culture, and there is strong empirical and theoretical support for the concept that other kinds of damage may contribute to cellular ageing. We describe a stochastic network model of cell senescence in which a primary role is played by telomere reduction but in which other mechanisms (oxidative stress linked particularly to mitochondrial damage, and nuclear somatic mutations) also contribute. The model gives simulation results that are in good agreement with published data on intra-clonal variability in cell doubling potential and permits an analysis of how the various elements of the stochastic network interact. Such integrative models may aid in developing new experimental approaches aimed at unravelling the intrinsic complexity of the mechanisms contributing to human cell ageing.     

The frequency of shifts between alternative equilibria

We derive a formula giving the frequency with which random drift shifts a population between alternative equilibria. This formula is valid when such shifts are rare (Ns much greater than 1), and applies over a wide range of mutation rates. When the number of mutations entering the population is low (4 N mu much less than 1), the rate of stochastic shifts reduces to the product of the mutation rate and the probability of fixation of a single mutation. However, when many mutations enter the population in each generation (4 N mu much greater than 1), the rate is higher than would be expected if mutations were established independently, and converges to that given by a gaussian approximation. We apply recent results on bistable systems to extend this formula to the general multidimensional case. This gives an explicit expression for the frequency of stochastic shifts, which depends only on the equilibrium probability distribution near the saddle point separating the alternative stable states. The plausibility of theories of speciation through random drift are discussed in the light of these results.     

Tractable Cre-lox system for stochastic alteration of genes in mice

We developed a cell division-activated Cre-lox system for stochastic recombination of loxP-flanked loci in mice. Cre activation by frameshift reversion is modulated by DNA mismatch-repair status and occurs in individual cells surrounded by normal tissue, mimicking spontaneous cancer-causing mutations. This system should be particularly useful for delineating pathways of neoplasia, and determining the developmental and aging consequences of specific gene alterations.