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Biomedical subjects

F B Christiansen

Publications and source records attributed to F B Christiansen.

At least 19 recordsLinked to original sources
PubMed2005

The evolution of n-player cooperation-threshold games and ESS bifurcations.

An evolutionary game of individuals cooperating to obtain a collective benefit is here modelled as an n-player Prisoner's Dilemma game. With reference to biological situations, such as group foraging, we introduce a threshold condition in the number of cooperators required to obtain the collective benefit. In the simplest version, a three-player game, complex behaviour appears as the replicator dynamics exhibits a catastrophic event separating a parameter region allowing for coexistence of cooperators and defectors and a region of pure defection. Cooperation emerges through an ESS bifurcation, and cooperators only thrive beyond a critical point in cost-benefit space. Moreover, a repelling fixed point of the dynamics acts as a barrier to the introduction of cooperation in defecting populations. The results illustrate the qualitative difference between two-player games and multiple player games and thus the limitations to the generality of conclusions from two-player games. We present a procedure to find the evolutionarily stable strategies in any n-player game with cost and benefit depending on the number of cooperators. This was previously done by Motro [1991. Co-operation and defection: playing the field and the ESS. J. Theor. Biol. 151, 145-154] in the special cases of convex and concave benefit functions and constant cost.

Animals↗
PubMed2001

An evolutionary-game model of tumour-cell interactions: possible relevance to gene therapy.

Evolutionary games have been applied as simple mathematical models of populations where interactions between individuals control the dynamics. Recently, it has been proposed to use this type of model to describe the evolution of tumour cell populations with interactions between cells. We extent the analysis to allow for synergistic effects between cells. A mathematical model of a tumour cell population is presented in which population-level synergy is assumed to originate through the interaction of triplets of cells. A threshold of two cooperating cells is assumed to be required to produce a proliferative advantage. The mathematical behaviour of this model is explored. Even this simple synergism (minor clustering effect) is sufficient to generate qualitatively different cell-population dynamics from the models published previously. The most notable feature of the model is the existence of an unstable internal equilibrium separating two stable equilibria. Thus, cells of a malignant phenotype can exist in a stable polymorphism, but may be driven to extinction by relatively modest perturbations of their relative frequency. The proposed model has some features that may be of interest to biological interpretations of gene therapy. Two prototypical strategies for gene therapy are suggested, both of them leading to extinction of the malignant phenotype: one approach would be to reduce the relative proportion of the cooperating malignant cell type below a certain critical value. Another approach would be to increase the critical threshold value without reducing the relative frequency of cells of the malignant phenotype.

Algorithms↗
PubMed2000

The strength of the selection barrier between populations.

Genetic differences among populations exposed to selection form barriers against genetic exchange by mortality among hybrids. The strength of such a selection barrier, with which one (recipient) population reacts against immigration from another (donor) population, may be measured as the cumulative mean fitness of hybrids and their descendants relative to the fitness of the recipient population. Previous work analysed a case of weak selection with pairwise epistatic interactions by assuming small genetic distance between two populations in contact. The present study allows large genetic difference between the donor and recipient populations and considers weak multilocus selection with arbitrary epistatic interactions between two or more linked loci. An approximate analytical expression for the barrier strength is obtained as an expansion in which the strength of selection plays the role of a small parameter. It is shown that allele frequencies and gametic linkage disequilibria contribute in different ways to the strength of the selection barrier.

Models, Genetic↗
PubMed1998

Waiting with and without recombination: the time to production of a double mutant.

R.A. Fisher and H.J. Muller argued in the 1930s that a major evolutionary advantage of recombination is that it allows favorable mutations to be combined within an individual even when they first appear in different individuals. This effect is evaluated in a two-locus, two-allele model by calculating the average waiting time until a new genotypic combination first appears in a haploid population. Three approximations are developed and compared with Monte Carlo simulations of the Wright-Fisher process of random genetic drift in a finite population. First, a crude method, based on the deterministic accumulation of single mutants, produces a waiting time of 1/square root of N mu(2) with no recombination and [formula: see text] with recombination between the two loci, where mu is the mutation rate, N is the haploid population size, and R is the recombination rate. Second, the waiting time is calculated as the expected value of a heterogeneous geometric distribution obtained from a branching process approximation. This gives accurate estimates for small values of N mu large. The estimates for small values of N mu are considerably lower than the simulated values. Finally, diffusion analysis of the Wright-Fisher process provides accurate estimates for N mu small, and the time scales of the diffusion process show a difference between R = 0 and for R >> 0 of the same order of magnitude as seen in the deterministic analysis. In the absence of recombination, accurate approximations to the waiting time are obtained by using the branching process for high N mu and the diffusion approximation for low N mu. For low N mu the waiting time is well approximated by 1/the square root of 8N2 mu(3). With R >> 0, the following dependence on N mu is observed: For N mu > 1 the waiting time is virtually independent of recombination and is well described by the branching process approximation. For N mu approximately equal to 1 the waiting time is well described by a simplified diffusion approximation that assumes symmetry in the frequencies of single mutants. For N mu << 1 the waiting time is well described by the diffusion approximation allowing asymmetry in the frequencies of single mutants. Recombination lowers the waiting time until a new genotypic combination first appears, but the effect is small compared to that of the mutation rate and population size. For large N mu, recombination has a negligible effect, and its effect is strongest for small N mu, in which case the waiting time approaches a fixed fraction of the waiting time for R = 0. Free recombination lowers the waiting time to about 45% of the waiting time for absolute linkage for small N mu. Selection has little effect on the importance of recombination in general.

Biological Evolution↗
PubMed1998

Allelic genealogies in sporophytic self-incompatibility systems in plants.

Expectations for the time scale and structure of allelic genealogies in finite populations are formed under three models of sporophytic self-incompatibility. The models differ in the dominance interactions among the alleles that determine the self-incompatibility phenotype: In the SSIcod model, alleles act codominantly in both pollen and style, in the SSIdom model, alleles form a dominance hierarchy, and in SSIdomcod, alleles are codominant in the style and show a dominance hierarchy in the pollen. Coalescence times of alleles rarely differ more than threefold from those under gametophytic self-incompatibility, and transspecific polymorphism is therefore expected to be equally common. The previously reported directional turnover process of alleles in the SSIdomcod model results in coalescence times lower and substitution rates higher than those in the other models. The SSIdom model assumes strong asymmetries in allelic action, and the most recessive extant allele is likely to be the most recent common ancestor. Despite these asymmetries, the expected shape of the allele genealogies does not deviate markedly from the shape of a neutral gene genealogy. The application of the results to sequence surveys of alleles, including interspecific comparisons, is discussed.

Alleles↗
PubMed1998

A codon-based model designed to describe lentiviral evolution.

A codon-based model designed to describe lentiviral evolution is developed. The model incorporates unequal base compositions in the three codon positions and selection against the CpG dinucleotide within codons to account for a deficit of this dinucleotide exhibited by lentiviral genes. The model is, to a large extent, able to account for the pattern of codon usage exhibited by the HIV1 genes gag, pol, and env, in spite of its parameter paucity. The model is extended to a similar model which operates on pentets (codons and their neighboring bases). The results obtained by the pentet model establish the importance of depression of CpGs across codon boundaries as well as within codons. The goodness of fit of the CpG depression model to the observed evolution in pairwise alignments of HIV1 sequences is assessed. The model provides a significantly better description of the observed evolution than the simpler models examined. The parameter estimates indicate that part of the unusually large biases in nucleotide frequencies observed in HIV1 genes is caused by selection against CpGs. We find that the estimates of expected numbers of substitutions, of transitions to transversions, and of synonymous to nonsynonymous substitution rates are robust to CpG depression, whereas the ratio of CpG-generating substitutions to other substitutions is strongly influenced by the choice of model.

Base Composition↗
PubMed1997

Evolutionary dynamics of sporophytic self-incompatibility alleles in plants.

The stationary frequency distribution and allelic dynamics in finite populations are analyzed through stochastic simulations in three models of single-locus, multi-allelic sporophytic self-incompatibility. The models differ in the dominance relationships among alleles. In one model, alleles act codominantly in both pollen and style (SSIcod), in the second, alleles form a dominance hierarchy in pollen and style (SSIdom). In the third model, alleles interact codominantly in the style and form a dominance hierarchy in the pollen (SSIdomcod). The SSIcod model behaves similarly to the model of gametophytic self-incompatibility, but the selection intensity is stronger. With dominance, dominant alleles invade the population more easily than recessive alleles and have a lower frequency at equilibrium. In the SSIdom model, recessive alleles have both a higher allele frequency and higher expected life span. In the SSIdomcod model, however, loss due to drift occurs more easily for pollen-recessive than for pollen-dominant alleles, and therefore, dominant alleles have a higher expected life span than the more recessive alleles. The process of allelic turnover in the SSIdomcod and SSIdom models is closely approximated by a random walk on a dominance ladder. Implications of the results for experimental studies of sporophytic self-incompatibility in natural populations are discussed.

Alleles↗
PubMed1996

Population genetic perspectives on the evolution of recombination.

Optimality arguments and modifier theory are reviewed as paradigms for the study of the evolution of recombination. Optimality criteria (such as maximization of mean fitness) may agree with results from models developed in terms of the evolution of recombination at modifier loci. Modifier models demonstrate, however, that equilibrium mean fitness can decrease during the evolution of recombination rates and is not always maximized. Therefore, optimality arguments do not successfully predict the conditions under which increased or decreased recombination will evolve. The results from modifier models indicate that decreased recombination rates are usually favored when the population is initially near a polymorphic equilibrium with linkage disequilibrium. When the population is subject to directional selection or to deleterious mutations, increased recombination may be favored under certain conditions, provided that there is negative epistasis among alleles.

Biological Evolution↗
PubMed1995

Slow coevolution of a viral pathogen and its diploid host.

We study a population exposed to a lethal infectious disease. Host response is carried at one locus with two alleles while the pathogen occurs in two variants. Based on an SI-type epidemic model we derive explicit equations for the dynamics of each genotype. By assuming small variations in both host and disease, we obtain a separation in time scales between epidemic and evolutionary processes. This allows us to describe explicitly the changes in host and disease gene frequencies. The resulting model has a rich behaviour including multiple stable states and oscillations. However, in the oscillatory situation the model is degenerate excluding the possibility of limit cycles. We show that the degeneracy can only be removed by frequency dependent selection in the pathogen, for example by including direct interaction of virus in a free-living stage. The qualitative conclusions extend to an SIR-type epidemic model, where recovery with immunity from the disease is possible.

Biological Evolution↗
PubMed1995

Sex determination in a symmetric autosomal multi-locus model.

A model of determination of sex by an individual's genotype at n loci is discussed. The parameters that determine the probability that an individual is male (or female) depend only on the loci at which the individual is heterozygous, extending the two-gene theory of Feldman et al. (1991, Genetics 129, 297-312). The analysis uses a set indexation technique that allows a compact expression of the recombination process, and a transformation of the chromosome frequencies that induces a natural subgroup structure on hyperplanes in the frequency simplex. It is shown that the polymorphic equilibria correspond to these subgroups. Conditions are determined under which equilibria that exhibit the even sex ratio are stable. How recombination affects the stability of other equilibria, including those with linkage disequilibrium, are also explored.

Animals↗
PubMed1994

Evolution of recombination among multiple selected loci: a generalized reduction principle.

Conditions for invasion by a new allele that controls the recombination pattern among an arbitrary number of genes under viability selection are studied. The recombination pattern may include interference. The new allele increases if its appropriately averaged marginal fitness is greater than the mean fitness prior to its introduction. Under weak additive-by-additive epistatic selection, this condition involves a weighted average of the changes in pairwise recombination rates relative to those prior to the introduction of the modifier. The weights here are positive functions of the epistatic selection components. In particular, the modifier allele may succeed even if it increases recombination among some pairs of loci, provided the overall average effect is one of reduction.

Alleles↗
PubMed1993

Disease-induced natural selection in a diploid host.

Using for each genotype an SIR-type model of disease transmission dynamics, we describe natural selection in a continuously breeding diploid host whose disease susceptibility and resistance are carried at one locus with two alleles. The system is transformed into variables that for each disease class describe the number of individuals, the gene frequency, and the deviation from Hardy-Weinberg proportions as measured by Wright's fixation index. An assumption of small variation in disease response among genotypes (slow selection) separates the system to first-order into three blocks. One block describes the population-wide disease dynamics, the second considers the fixation index in each class, and the third block provides the change in gene frequencies. The first two blocks settle to equilibrium at a rate determined by the population turnover time while the last block after a while is dominated by a slowly changing variable, the average gene frequency. The dynamics of the gene frequency take the usual form for a continuous time slow selection model, and this provides explicit, epidemiologically justified expressions for the genotypic fitnesses. We apply the method to other disease transmission patterns (SEI and SIS) and discuss how suitable time averages extend our results to diseases with temporally varying incidence.

Alleles↗
PubMed1991

Lewontin and Kojima meet Fisher: linkage in a symmetric model of sex determination.

The effect of linkage and epistasis on the evolution of the sex-ratio is studied in a symmetric two-locus model of autosomal sex determination closely related to the symmetric viability model of R. C. Lewontin and K. Kojima. R. A. Fisher's expectation of an even sex ratio for autosomal sex determination by a single gene governs the dynamics when the loci are tightly linked. However, recombination may preclude optimization of the sex ratio just as occurs in viability selection models. Many of the evolutionary phenomena known for the symmetric viability model also occur here. In addition, we exhibit a series of new phenomena related to the presence of surfaces of even sex ratio.

Animals↗
PubMed1990

The generalized multiplicative model for viability selection at multiple loci.

Selection due to differential viability is studied in an n-locus two-allele model using a set indexation that allows the simplicity of the one-locus two-allele model to be carried to multi-locus models. The existence condition is analyzed for polymorphic equilibria with linkage equilibrium: Robbins' equilibria. The local stability condition is given for the Robbins' equilibria on the boundaries in the generalized non-epistatic selection regimes of Karlin and Liberman (1979). These generalized non-epistatic regimes include the additive selection model, the multiplicative selection model and the multiplicative interaction model, and their symmetric versions cover all the symmetric viability models.

Alleles↗
PubMed1989

Persistence of an infectious disease in a subdivided population.

The transmission dynamics of a communicable disease in a subdivided population where the spread among groups follows the proportionate mixing model while the within-group transmission can correspond to preferred mixing, proportionate mixing among subgroups, or mixing between social and nonsocial subgroups, is analyzed. It is shown that the threshold condition for the disease to persist is that either (i) the disease can persist within at least one group through intragroup contacts, or--if (i) does not hold--(ii) the intergroup transmission is sufficiently high. The among-group transmission is computed as an average where each subgroup's reproductive number is weighted according to its intragroup activity level squared and the total number of cases that one infectious individual will cause through intragroup contacts. The model thus allows for a study of the relative importance of communitywide disease transmission and of disease transmission within geographically or socially separate groups.

Communicable Diseases↗
PubMed1988

Frequency dependence and competition.

Intraspecific competition implies interaction among the individuals of a population, so natural selection on genotypic variation in characters related to the competition will necessarily be frequency dependent. Intraspecific antagonistic competition exhibits properties similar to other behavioural interactions between individuals. In exploitative intraspecific competition the interactions among individuals are less direct. Exploitation modifies the abundance of the various limiting resources according to the use of these resources by the individual members of the population. The amount of resource available to an individual is therefore a function of the phenotypes present in the population, through their density and frequency.

Animals↗
PubMed1988

Epistasis in the multiple locus symmetric viability model.

The n-locus two-allele symmetric viability model is considered in terms of the parameters measuring the additive epistasis in fitness. The dynamics is analysed using a simple linear transformation of the gametic frequencies, and then the recurrence equations depend on the epistatic parameters and Geiringer's recombination distribution only. The model exhibits an equilibrium, the central equilibrium, where the 2n gametes are equally frequent. The transformation simplifies the stability analysis of the central point, and provides the stability conditions in terms of the existence conditions of other equilibria. For total negative epistasis (all epistatic parameters are negative) the central point is stable for all recombination distributions. For free recombination either a central point (segregating one, two, ... or n loci) or the n-locus fixation states are stable. For no recombination and some epistatic parameters positive the central point is unstable and several boundary equilibria may be locally stable. The sign structure of the additive epistasis is therefore an important determinant of the dynamics of the n-locus symmetric viability model. The non-symmetric multiple locus models previously analysed are dynamically related, and they all have an epistatic sign structure that resembles that of the multiplicative viability model. A non-symmetric model with total negative epistasis which share dynamical properties with the similar symmetric model is suggested.

Alleles↗