Biomedical subjects

Martin A Nowak

Publications and source records attributed to Martin A Nowak.

At least 37 records · Page 2Linked to original sources

PubMed2006

Evolution of resistance during clonal expansion.

Acquired drug resistance is a major limitation for cancer therapy. Often, one genetic alteration suffices to confer resistance to an otherwise successful therapy. However, little is known about the dynamics of the emergence of resistant tumor cells. In this article, we consider an exponentially growing population starting from one cancer cell that is sensitive to therapy. Sensitive cancer cells can mutate into resistant ones, which have relative fitness alpha prior to therapy. In the special case of no cell death, our model converges to the one investigated by Luria and Delbrück. We calculate the probability of resistance and the mean number of resistant cells once the cancer has reached detection size M. The probability of resistance is an increasing function of the detection size M times the mutation rate u. If Mu << 1, then the expected number of resistant cells in cancers with resistance is independent of the mutation rate u and increases with M in proportion to M(1-1/alpha) for advantageous mutants with relative fitness alpha>1, to l nM for neutral mutants (alpha = 1), but converges to an upper limit for deleterious mutants (alpha<1). Further, the probability of resistance and the average number of resistant cells increase with the number of cell divisions in the history of the tumor. Hence a tumor subject to high rates of apoptosis will show a higher incidence of resistance than expected on its detection size only.

Antineoplastic Agents↗

PubMed2006

Evolution of resistance to cancer therapy.

Acquired drug resistance is a major limitation for successful treatment of cancer. Resistance emerges due to drug exclusion, drug metabolism and alteration of the drug target by mutation or overexpression. Depending on therapy, the type of cancer and its stage, one or several genetic or epigenetic alterations are necessary to confer resistance to treatment. The fundamental question is the following: if a genetically diverse population of replicating cancer cells is subjected to chemotherapy that has the potential to eradicate it, what is the probability of emergence of resistance? Here, we review a general mathematical framework based on multi-type branching processes designed to study the dynamics of escape of replicating organisms from selection pressures. We apply the general model to evolution of resistance of cancer cells and discuss examples for diverse mechanisms of resistance. Our theory shows how to estimate the probability of success for any treatment regimen.

Animals↗

PubMed2005

Stochastic dynamics of metastasis formation.

Tumor metastasis accounts for the majority of deaths in cancer patients. The metastatic behavior of cancer cells is promoted by mutations in many genes, including activation of oncogenes such as RAS and MYC. Here, we develop a mathematical framework to analyse the dynamics of mutations enabling cells to metastasize. We consider situations in which one mutation is necessary to confer metastatic ability to the cell. We study different population sizes of the main tumor and different somatic fitness values of metastatic cells. We compare mutations that are positively selected in the main tumor with those that are neutral or negatively selected, but faster at forming metastases. We study whether metastatic potential is the property of all (or the majority of) cells in the main tumor or only the property of a small subset. Our theory shows how to calculate the expected number of metastases that are formed by a tumor.

Computer Simulation↗

PubMed2005

Evolution of indirect reciprocity.

Natural selection is conventionally assumed to favour the strong and selfish who maximize their own resources at the expense of others. But many biological systems, and especially human societies, are organized around altruistic, cooperative interactions. How can natural selection promote unselfish behaviour? Various mechanisms have been proposed, and a rich analysis of indirect reciprocity has recently emerged: I help you and somebody else helps me. The evolution of cooperation by indirect reciprocity leads to reputation building, morality judgement and complex social interactions with ever-increasing cognitive demands.

Altruism↗

PubMed2005

Synergy and discounting of cooperation in social dilemmas.

The emergence and maintenance of cooperation by natural selection is an enduring conundrum in evolutionary biology, which has been studied using a variety of game theoretical models inspired by different biological situations. The most widely studied games are the Prisoner's Dilemma, the Snowdrift game and by-product mutualism for pairwise interactions, as well as Public Goods games in larger groups of interacting individuals. Here, we present a general framework for cooperation in social dilemmas in which all the traditional scenarios can be recovered as special cases. In social dilemmas, cooperators provide a benefit to the group at some cost, while defectors exploit the group by reaping the benefits without bearing the costs of cooperation. Using the concepts of discounting and synergy for describing how benefits accumulate when more than one cooperator is present in a group of interacting individuals, we recover the four basic scenarios of evolutionary dynamics given by (i) dominating defection, (ii) coexistence of defectors and cooperators, (iii) dominating cooperation and (iv) bi-stability, in which cooperators and defectors cannot invade each other. Generically, for groups of three or more interacting individuals further, more complex, dynamics can occur. Our framework provides the first unifying approach to model cooperation in different kinds of social dilemmas.

Animals↗

PubMed2005

Why are phenotypic mutation rates much higher than genotypic mutation rates?

The evolution of genotypic mutation rates has been investigated in numerous theoretical and experimental studies. Mutations, however, occur not only when copying DNA, but also when building the phenotype, especially when translating and transcribing DNA to RNA and protein. Here we study the effect of such phenotypic mutations. We find a maximum phenotypic mutation rate, umax, that is compatible with maintaining a certain function of the organism. This may be called a phenotypic error threshold. In particular, we find a minimum phenotypic mutation rate, umin, with the property that there is (nearly) no selection pressure to reduce the rate of phenotypic mutations below this value. If there is a cost for lowering the phenotypic mutation rate, then umin is close to the optimum phenotypic mutation rate that maximizes the fitness of the organism. In our model, there is selective pressure to decrease the rate of genotypic mutations to zero, but to decrease the rate of phenotypic mutations only to a positive value. Despite its simplicity, our model can explain part of the huge difference between genotypic and phenotypic mutation rates that is observed in nature. The relevant data are summarized.

Animals↗

PubMed2005

Stochastic evolutionary dynamics on two levels.

We consider a population that is subdivided into groups. Individuals reproduce proportional to their fitness. When a group reaches a certain size it has a probability to split into two groups while another group is eliminated. In this stochastic process, the number of groups is constant, while the total population size fluctuates between well-defined bounds. We calculate the fixation probability of newly introduced mutants under constant selection. We show that the described population structure acts as a suppressor of selection compared to an unstructured population of the same size. The maximum suppression of selection is obtained, when the number of groups equals the number of individuals per group. We also study opposing selection on two or more levels by analysing the evolutionary dynamics of hierarchically embedded Moran processes.

Animals↗

PubMed2005

Evolutionary cycles of cooperation and defection.

The main obstacle for the evolution of cooperation is that natural selection favors defection in most settings. In the repeated prisoner's dilemma, two individuals interact several times, and, in each round, they have a choice between cooperation and defection. We analyze the evolutionary dynamics of three simple strategies for the repeated prisoner's dilemma: always defect (ALLD), always cooperate (ALLC), and tit-for-tat (TFT). We study mutation-selection dynamics in finite populations. Despite ALLD being the only strict Nash equilibrium, we observe evolutionary oscillations among all three strategies. The population cycles from ALLD to TFT to ALLC and back to ALLD. Most surprisingly, the time average of these oscillations can be entirely concentrated on TFT. In contrast to the classical expectation, which is informed by deterministic evolutionary game theory of infinitely large populations, stochastic evolution of finite populations need not choose the strict Nash equilibrium and can therefore favor cooperation over defection.

Journal Article↗

PubMed2005

Dynamics of chronic myeloid leukaemia.

The clinical success of the ABL tyrosine kinase inhibitor imatinib in chronic myeloid leukaemia (CML) serves as a model for molecularly targeted therapy of cancer, but at least two critical questions remain. Can imatinib eradicate leukaemic stem cells? What are the dynamics of relapse due to imatinib resistance, which is caused by mutations in the ABL kinase domain? The precise understanding of how imatinib exerts its therapeutic effect in CML and the ability to measure disease burden by quantitative polymerase chain reaction provide an opportunity to develop a mathematical approach. We find that a four-compartment model, based on the known biology of haematopoietic differentiation, can explain the kinetics of the molecular response to imatinib in a 169-patient data set. Successful therapy leads to a biphasic exponential decline of leukaemic cells. The first slope of 0.05 per day represents the turnover rate of differentiated leukaemic cells, while the second slope of 0.008 per day represents the turnover rate of leukaemic progenitors. The model suggests that imatinib is a potent inhibitor of the production of differentiated leukaemic cells, but does not deplete leukaemic stem cells. We calculate the probability of developing imatinib resistance mutations and estimate the time until detection of resistance. Our model provides the first quantitative insights into the in vivo kinetics of a human cancer.

Benzamides↗

PubMed2005

Mutation in evolutionary games can increase average fitness at equilibrium.

We study game dynamical interactions between two strategies, A and B, and analyse whether the average fitness of the population at equilibrium can be increased by adding mutation from A to B. Classifying all two by two games with payoff matrix [(a,b),(c,d)], we show that mutation from A to B enhances the average fitness of the whole population (i) if both a and d are less than (b + c)/2 and (ii) if c is less than b. Furthermore, we study conditions for maximizing the productivity of strategy A, and we analyse the effect of mutations in both directions. Depending on the biological system, a mutation in an evolutionary game can be interpreted as a genetic alteration, a cellular differentiation, a change in gene expression, an accidental or deliberate modification in cultural transmission, or a learning error. In a cultural context, our results indicate that the equilibrium payoff of the population can be increased if players sometimes choose the strategy with lower payoff. In a genetic context, we have shown that for frequency-dependent selection mutation can enhance the average fitness of the population at equilibrium.

Animals↗

PubMed2005

Evolutionary dynamics on graphs.

Evolutionary dynamics have been traditionally studied in the context of homogeneous or spatially extended populations. Here we generalize population structure by arranging individuals on a graph. Each vertex represents an individual. The weighted edges denote reproductive rates which govern how often individuals place offspring into adjacent vertices. The homogeneous population, described by the Moran process, is the special case of a fully connected graph with evenly weighted edges. Spatial structures are described by graphs where vertices are connected with their nearest neighbours. We also explore evolution on random and scale-free networks. We determine the fixation probability of mutants, and characterize those graphs for which fixation behaviour is identical to that of a homogeneous population. Furthermore, some graphs act as suppressors and others as amplifiers of selection. It is even possible to find graphs that guarantee the fixation of any advantageous mutant. We also study frequency-dependent selection and show that the outcome of evolutionary games can depend entirely on the structure of the underlying graph. Evolutionary graph theory has many fascinating applications ranging from ecology to multi-cellular organization and economics.

Biological Evolution↗

PubMed2005

Virus evolution within patients increases pathogenicity.

Viruses like the human immunodeficiency virus (HIV), the hepatitis B virus (HBV), the hepatitis C virus (HCV) and many others undergo numerous rounds of inaccurate reproduction within an infected host. The resulting viral quasispecies is heterogeneous and sensitive to any selection pressure. Here we extend earlier work by showing that for a wide class of models describing the interaction between the virus population and the immune system, virus evolution has a well-defined direction toward increased pathogenicity. In particular, we study virus-induced impairment of the immune response and certain cross-reactive stimulation of specific immune responses. For eight different mathematical models, we show that virus evolution reduces the equilibrium abundance of uninfected cells and increases the rate at which uninfected cells are infected. Thus, in general, virus evolution makes things worse. An idea for combating HIV infection, however, is constructing a virus mutant that could outcompete the existing infection without being pathogenic itself.

Antigens, Viral↗

PubMed2005

Can chromosomal instability initiate tumorigenesis?

Cancers result from the accumulation of inherited and somatic mutations in oncogenes and tumor suppressor genes. These genes encode proteins that function in growth regulatory and differentiation pathways. Mutations in those genes increase the net reproductive rate of cells. Chromosomal instability (CIN) is a feature of most human cancers. Mutations in CIN genes increase the rate at which whole chromosomes or large parts of chromosomes are lost or gained during cell division. CIN causes an imbalance in chromosome number (aneuploidy) and an enhanced rate of loss of heterozygosity, which is an important mechanism of inactivating tumor suppressor genes. A crucial question of cancer biology is whether CIN is an early event and thus a driving force of tumorigenesis. Here we discuss mathematical models of situations where inactivation of one or two tumor suppressor genes is required for tumorigenesis. If two tumor suppressor genes have to be inactivated in rate-limiting steps, then CIN is likely to emerge before the inactivation of the first tumor suppressor gene.

Animals↗

PubMed2005

Dynamics of colorectal cancer.

Colorectal cancer results from an accumulation of mutations in tumor suppressor genes and oncogenes. An additional defining characteristic of colorectal cancer is its genetic instability. Two main types of genetic instability have been identified. Microsatellite instability leads to an increased point mutation rate, whereas chromosomal instability refers to an enhanced rate of accumulating gross chromosomal aberrations. All colon cancer cell lines are genetically unstable. An interesting question is whether genetic instability arises early in tumorigenesis. An early emergence of genetic instability could drive most of the somatic evolution of cancer. Here, we review mathematical models of colorectal tumorigenesis and discuss the role of genetic instability.

Colorectal Neoplasms↗

PubMed2004

Win-stay, lose-shift in language learning from peers.

Traditional language learning theory explores an idealized interaction between a teacher and a learner. The teacher provides sentences from a language, while the learner has to infer the underlying grammar. Here, we study a new approach by considering a population of individuals that learn from each other. There is no designated teacher. We are inspired by the observation that children grow up to speak the language of their peers, not of their parents. Our goal is to characterize learning strategies that generate "linguistic coherence," which means that most individuals use the same language. We model the resulting learning dynamics as a random walk of a population on a graph. Each vertex represents a candidate language. We find that a simple strategy using a certain aspiration level with the principle of win-stay, lose-shift does extremely well: stay with your current language, if at least three others use that language; otherwise, shift to an adjacent language on the graph. This strategy guarantees linguistic coherence on all nearly regular graphs, in the relevant limit where the number of candidate languages is much greater than the population size. Moreover, for many graphs, it is sufficient to have an aspiration level demanding only two other individuals to use the same language.

Algorithms↗

PubMed2004

Population genetics of tumor suppressor genes.

Cancer emerges when a single cell receives multiple mutations. For example, the inactivation of both alleles of a tumor suppressor gene (TSG) can imply a net reproductive advantage of the cell and might lead to clonal expansion. In this paper, we calculate the probability as a function of time that a population of cells has generated at least one cell with two inactivated alleles of a TSG. Different kinetic laws hold for small and large populations. The inactivation of the first allele can either be neutral or lead to a selective advantage or disadvantage. The inactivation of the first and of the second allele can occur at equal or different rates. Our calculations provide insights into basic aspects of population genetics determining cancer initiation and progression.

Alleles↗

PubMed2004

Determinants of human immunodeficiency virus type 1 escape from the primary CD8+ cytotoxic T lymphocyte response.

CD8+ cytotoxic T lymphocytes (CTLs) play an important role in containment of virus replication in primary human immunodeficiency virus (HIV) infection. HIV's ability to mutate to escape from CTL pressure is increasingly recognized; but comprehensive studies of escape from the CD8 T cell response in primary HIV infection are currently lacking. Here, we have fully characterized the primary CTL response to autologous virus Env, Gag, and Tat proteins in three patients, and investigated the extent, kinetics, and mechanisms of viral escape from epitope-specific components of the response. In all three individuals, we observed variation beginning within weeks of infection at epitope-containing sites in the viral quasispecies, which conferred escape by mechanisms including altered peptide presentation/recognition and altered antigen processing. The number of epitope-containing regions exhibiting evidence of early CTL escape ranged from 1 out of 21 in a subject who controlled viral replication effectively to 5 out of 7 in a subject who did not. Evaluation of the extent and kinetics of HIV-1 escape from >40 different epitope-specific CD8 T cell responses enabled analysis of factors determining escape and suggested that escape is restricted by costs to intrinsic viral fitness and by broad, codominant distribution of CTL-mediated pressure on viral replication.

Amino Acid Sequence↗

PubMed2004

Evolutionary dynamics of tumor suppressor gene inactivation.

Tumor suppressor genes (TSGs) are important gatekeepers that protect against somatic evolution of cancer. Losing both alleles of a TSG in a single cell represents a step toward cancer. We study how the kinetics of TSG inactivation depends on the population size of cells and the mutation rates for the first and second hit. We calculate the probability as function of time that at least one cell has been generated with two inactivated alleles of a TSG. We find three different kinetic laws: in small, intermediate, and large populations, it takes, respectively, two, one, and zero rate-limiting steps to inactivate a TSG. We also study the effect of chromosomal and other genetic instabilities. Small lesions without genetic instability can take a very long time to inactivate the next TSG, whereas the same lesions with genetic instability pose a much greater risk for cancer progression.

Alleles↗