Yoh Iwasa

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.

THE EVOLUTION OF COSTLY MATE PREFERENCES II. THE "HANDICAP" PRINCIPLE.

We use a general additive quantitative genetic model to study the evolution of costly female mate choice by the “handicap” principle. Two necessary conditions must be satisfied for costly preference to evolve. The conditions are (i) biased mutation pressure on viability and (ii) a direct relationship between the degree of expression of the male mating character and viability. These two conditions explain the success and failure of previous models of the “handicap” principle. Our model also applies to other sources of fitness variation like migration and host‐parasite coevolution, which cause effects equivalent to biased mutation.

Prey distribution as a factor determining the choice of optimal foraging strategy

The optimal-patch-use problem in predation theory is investigated by use of a stochastic discrete model to match experimental situations when deterministic continuous models are inappropriate. We first consider three elementary strategies, differing in when to leave the patch in which the predator has been foraging; namely, (1) a fixed time has passed, (2) a fixed number of prey has been captured, and (3) the interval between two successive catches has exceeded a fixed time. Each of these fixed quantities has same value that optimizes the strategy concerned, given certain conditions. The optimized strategies are compared to determine the most efficient. It is shown that the ranking of the strategies depends critically on the type of prey distribution between patches, other things being equal; e.g., strategy 3, the best when the variance of prey distribution is sufficiently high, tends to be the worst when the variance is minimal. The above strategies do not use full information for estimating the number of prey remaining in the patch; for example, every time interval between two successive catches might be used by a more sophisticated predator. The statistical decision theory reveals, however, that the number of prey already taken (n) and the total time spent foraging (t) in the patch are the minimal sufficient statistics for estimating the number of unexploited prey under the random-search assumption. In other words, under the assumption of random search the estimator r is a function of n and t only at most, and detailed knowledge of the distribution of the time intervals is immaterial. Thus, the sophisticated predator should leave the patch when the estimator r(n,t) falls below a certain value that optimizes the strategy adopted. The estimator r depends on the prey distribution: If prey are distributed contagiously, estimated value of the remaining prey jumps up each time a capture is made, but it steps down if prey are distributed regularly. However, r may be a function of n or t alone, depending on the type of the between-patch distribution of prey. In particular, r is a function of n only for a completely regular distribution, and the best strategy is reduced to elementary strategy 2. If the distribution is random (Poisson), r is a function of t only. Then the best strategy becomes simply elementary strategy 1. It is pointed out that whether the predator under observation actually behaves optimally in accordance with the distribution of prey should be revealed by plotting the number of captures against the length of period for which the predator stayed in each patch.

Influence of nonlinear incidence rates upon the behavior of SIRS epidemiological models.

When the traditional assumption that the incidence rate is proportional to the product of the numbers of infectives and susceptibles is dropped, the SIRS model can exhibit qualitatively different dynamical behaviors, including Hopf bifurcations, saddle-node bifurcations, and homoclinic loop bifurcations. These may be important epidemiologically in that they demonstrate the possibility of infection outbreak and collapse, or autonomous periodic coexistence of disease and host. The possible mechanisms leading to nonlinear incidence rates are discussed. Finally, a modified general criterion for supercritical or subcritical Hopf bifurcation of 2-dimensional systems is presented.

Demographic Theory for an Open Marine Population with Space-Limited Recruitment

We introduce a demographic model for a local population of sessile marine invertebrates that have a pelagic larval phase. The processes in the model are the settling of larvae onto empty space, and the growth and mortality of the settled organisms. The rate of settlement per unit of unoccupied space is assumed to be determined by factors outside of the local system. The model predicts the number of animals of each age in the local system through time. The model is offered in both discrete and continuous—time versions. The principal result is that the growth of the settled organisms is destabilizing. In the model, there is always a state where recruitment balances mortality. However, growth can interfere with recruitment and can destabilize this steady state, provided also that the settlement rate is sufficiently high. The model suggests that two qualitatively distinct pictures of population structure result, depending on the settlement rate. In the high settlement limit, the intertidal landscape is a mosaic of cohorts, punctuated with occasional gaps of vacant substrate. In the low settlement limit, the intertidal landscape has vacant space and organisms of all ages mixed together and spatial variation in abundance is caused by microgeographic variation in settlement and mortality rates.

The evolution of costly mate preferences. I, Fisher and biased mutation

Fishers runaway process is the standard explanation of the evolution of exaggerated female preferences. But mathematical formulations of Fishers process (haploid and additive diploid) show it cannot cause stable exaggeration if female preference carries a cost. At equilibrium female fitness must be maximized. Our analysis shows that evolutionary stable exaggeration of female preference can be achieved if mutation pressure on the male character is biased, that is, mutation has a directional effect. At this equilibrium female fitness is not maximized. We discuss the reasons and evidence for believing that mutation pressure is typically biased. Our analysis highlights the previously unacknowledged importance of biased mutation for sexual selection.

Dynamics of cancer progression

Evolutionary concepts such as mutation and selection can be best described when formulated as mathematical equations. Cancer arises as a consequence of somatic evolution. Therefore, a mathematical approach can be used to understand the process of cancer initiation and progression. But what are the fundamental principles that govern the dynamics of activating oncogenes and inactivating tumour-suppressor genes in populations of reproducing cells? Also, how does a quantitative theory of somatic mutation and selection help us to evaluate the role of genetic instability?

Theory of oviposition strategy of parasitoids. I. Effect of mortality and limited egg number

The optimal oviposition strategies of parasitoids, the host range, and the number of eggs laid per host which result in the maximum lifetime performance of reproduction, are investigated. To study the effects of parasitoid mortality and of limiting total number of eggs laid by a parasitoid, a standard criterion used in previous theories of optimal diet and optimal patch use, the maximization of the foraging rate, is no longer suitable. The model is solved analytically by using dynamic programming. The results are as follows: The host preference of solitary parasitoids depends on the mortality during handling times; i.e., the forager tends to avoid hosts with high risk of foraging mortality. If the total number of eggs produced by a parasitoid is limited, and if the mortality during handling is negligible, the host range is wider when a larger number of eggs remains in the parasitoids body. In general, however, the mortality-cost of forager and the egg-cost interplay, because the loss of future reproduction by mortality increases with the number of available eggs. In an example with two host types, host range is widest with an intermediate number of eggs available in the body. The optimal number of eggs per host laid by a gregarious parasitoid is also affected by the differential mortality of the forager, and by the number of available eggs.

THE EVOLUTION OF MATE PREFERENCES FOR MULTIPLE SEXUAL ORNAMENTS.

Males of many species use multiple sexual ornaments in their courtship display. We investigate the evolution of female sexual preferences for more than a single male trait by the handicap process. The handicap process assumes that ornaments are indicators of male quality, and a female benefits from mate choice by her offspring inheriting “good genes” that increase survival chances. A new handicap model is developed that allows equilibria to be given in terms of selection pressures, independent of genetic parameters. Multiple sexual preferences evolve if the overall cost of choice is not greatly increased by a female using additional male traits in her assessment of potential mates. However, only a single preference is evolutionarily stable if assessment of additional male traits greatly increases the overall cost of choice (more than expected by combining the cost of each preference independently). Any single preference can evolve, the outcome being determined by initial conditions. The evolution of one preference effectively blocks the evolution of others, even for traits that are better indicators of male quality. Comparison is made with sexual selection caused by Fishers runaway process in which male traits are purely attractive characters. This shows that sexual preferences for multiple Fisher traits are likely to evolve alongside preference for a single handicap trait that indicates male quality. This is a general difference in the evolutionary outcome of these two causes of sexual selection.

Optimal size of storage for recovery after unpredictable disturbances

Terrestrial plants often live in environments in which above-ground photosynthetic organs (production parts) are suddenly removed by unpredictable disturbances, such as fire, frost, desiccation, pathogen attack, breakage by wind and trampling, or herbivory by insects and mammals. We study the optimal growth schedule for a plant having a below-ground storage organ that is used for recovery (or regrowth) of photosynthetic organs after disturbances. We assume the following: (1) the daily production rate increases with the production part size, but saturates for large size due to shading and local resource depletion, (2) disturbances occur randomly and remove all the aerial parts, (3) plants are finally killed by fatal disturbances that also occur randomly and (4) the plant chooses the pattern of growth, reproduction, storage and recovery after disturbances by reallocation of stored material to maximize the total lifetime reproductive success. The model is analysed by stochastic dynamic programming. The results are as follows: (1) the ratio of storage size to production part size (S/F ratio) is large if the longevity is large and if the disturbance rate is large but a little smaller than the productivity coefficient, (2) the S/F ratio is larger for mature plants than for small immature plants, (3) after disturbances, the above-ground production part recovers relatively quickly, but reproductive activity is depressed until storage size recovers and (4) the variations over time and between habitats differing in disturbance frequency are larger for storage size and for reproductive activity than for production part size. These tendencies are more pronounced for a linear production function (with initial linear increase followed by a sudden stop), but less so for a hyperbolic production function (with a gradually decreasing slope). We also discuss the growth and regrowth behaviour of plants adapted to a disturbance frequency growing under one different disturbance frequency.