Rebecca B. Hoyle

When to rely on maternal effects and when on phenotypic plasticity

Existing insight suggests that maternal effects have a substantial impact on evolution, yet these predictions assume that maternal effects themselves are evolutionarily constant. Hence, it is poorly understood how natural selection shapes maternal effects in different ecological circumstances. To overcome this, the current study derives an evolutionary model of maternal effects in a quantitative genetics context. In constant environments, we show that maternal effects evolve to slight negative values that result in a reduction of the phenotypic variance (canalization). By contrast, in populations experiencing abrupt change, maternal effects transiently evolve to positive values for many generations, facilitating the transmission of beneficial maternal phenotypes to offspring. In periodically fluctuating environments, maternal effects evolve according to the autocorrelation between maternal and offspring environments, favoring positive maternal effects when change is slow, and negative maternal effects when change is rapid. Generally, the strongest maternal effects occur for traits that experience very strong selection and for which plasticity is severely constrained. By contrast, for traits experiencing weak selection, phenotypic plasticity enhances the evolutionary scope of maternal effects, although maternal effects attain much smaller values throughout. As weak selection is common, finding substantial maternal influences on offspring phenotypes may be more challenging than anticipated.

The benefits of maternal effects in novel and in stable environments.

Natural selection favours phenotypes that match prevailing ecological conditions. A rapid process of adaptation is therefore required in changing environments. Maternal effects can facilitate such responses, but it is currently poorly understood under which circumstances maternal effects may accelerate or slow down the rate of phenotypic evolution. Here, we use a quantitative genetic model, including phenotypic plasticity and maternal effects, to suggest that the relationship between fitness and phenotypic variance plays an important role. Intuitive expectations that positive maternal effects are beneficial are supported following an extreme environmental shift, but, if too strong, that shift can also generate oscillatory dynamics that overshoot the optimal phenotype. In a stable environment, negative maternal effects that slow phenotypic evolution actually minimize variance around the optimum phenotype and thus maximize population mean fitness.

The fitness costs of adaptation via phenotypic plasticity and maternal effects

Summary 1. Phenotypes are often environmentally dependent, which requires organisms to track environmental change. The challenge for organisms is to construct phenotypes using the most accurate environmental cue. 2. Here, we use a quantitative genetic model of adaptation by additive genetic variance, within- and transgenerational plasticity via linear reaction norms and indirect genetic effects respectively. 3. We show how the relative influence on the eventual phenotype of these components depends on the predictability of environmental change (fast or slow, sinusoidal or stochastic) and the developmental lag s between when the environment is perceived and when selection acts. 4. We then decompose expected mean fitness into three components (variance load, adaptation and fluctuation load) to study the fitness costs of within- and transgenerational plasticity. A strongly negative maternal effect coefficient m minimizes the variance load, but a strongly positive m minimises the fluctuation load. The adaptation term is maximized closer to zero, with positive or negative m preferred under different environmental scenarios. 5. Phenotypic plasticity is higher when s is shorter and when the environment changes frequently between seasonal extremes. Expected mean population fitness is highest away from highest observed levels of phenotypic plasticity. 6. Within- and transgenerational plasticity act in concert to deliver well-adapted phenotypes, which emphasizes the need to study both simultaneously when investigating phenotypic evolution.

League tables and school effectiveness: a mathematical model

‘School performance tables’, an alphabetical list of secondary schools along with aggregates of their pupils’ performances in national tests, have been published in the UK since 1992. Inevitably, the media have responded by publishing ranked ‘league tables’. Despite concern over the potentially divisive effect of such tables, the current government has continued to publish this information in the same form. The effect of this information on standards and on the social make–up of the community has been keenly debated. Since there is no control group available that would allow us to investigate this issue directly, we present here a simple mathematical model. Our results indicate that, while random fluctuations from year to year can cause large distortions in the league–table positions, some schools still establish themselves as ‘desirable’. To our surprise, we found that ‘value–added’ tables were no more accurate than tables based on raw exam scores, while a different method of drawing up the tables, in which exam results are averaged over a period of time, appears to give a much more reliable measure of school performance.

A Kinetic Model Describing the Processivity of Myosin-V

The precise details of how myosin-V coordinates the biochemical reactions and mechanical motions of its two head elements to engineer effective processive molecular motion along actin filaments remain unresolved. We compare a quantitative kinetic model of the myosin-V walk, consisting of five basic states augmented by two further states to allow for futile hydrolysis and detachments, with experimental results for run lengths, velocities, and dwell times and their dependence on bulk nucleotide concentrations and external loads in both directions. The model reveals how myosin-V can use the internal strain in the molecule to synchronize the motion of the head elements. Estimates for the rate constants in the reaction cycle and the internal strain energy are obtained by a computational comparison scheme involving an extensive exploration of the large parameter space. This scheme exploits the fact that we have obtained analytic results for our reaction network, e.g., for the velocity but also the run length, diffusion constant, and fraction of backward steps. The agreement with experiment is often reasonable but some open problems are highlighted, in particular the inability of such a general model to reproduce the reported dependence of run length on ADP concentration. The novel way that our approach explores parameter space means that any confirmed discrepancies should give new insights into the reaction network model.

Two-species continuum model for aeolian sand ripples

We formulate a continuum model for aeolian sand ripples consisting of two species of grains: a lower layer of relatively immobile clusters, with an upper layer of highly mobile grains moving on top. We predict analytically the ripple wavelength, initial ripple growth rate and threshold saltation flux for ripple formation. Numerical simulations show the evolution of realistic ripple profiles from initial surface roughness via ripple growth and merger.

Spatial period-multiplying instabilities of hexagonal Faraday waves

Abstract A recent Faraday wave experiment with two-frequency forcing reports two types of ‘superlattice’ patterns that display periodic spatial structures having two separate scales [Physica D 123 (1998) 99]. These patterns both arise as secondary states once the primary hexagonal pattern becomes unstable. In one of these patterns (so-called ‘superlattice-two’) the original hexagonal symmetry is broken in a subharmonic instability to form a striped pattern with a spatial scale increased by a factor of 2 3 from the original scale of the hexagons. In contrast, the time-averaged pattern is periodic on a hexagonal lattice with an intermediate spatial scale ( 3 larger than the original scale) and apparently has 60° rotation symmetry. We present a symmetry-based approach to the analysis of this bifurcation. Taking as our starting point only the observed instantaneous symmetry of the superlattice-two pattern presented in [Physica D 123 (1998) 99] and the subharmonic nature of the secondary instability, we show: (a) that a pattern with the same instantaneous symmetries as the superlattice-two pattern can bifurcate stably from standing hexagons; (b) that the pattern has a spatio-temporal symmetry not reported in [Physica D 123 (1998) 99]; and (c) that this spatio-temporal symmetry accounts for the intermediate spatial scale and hexagonal periodicity of the time-averaged pattern, but not for the apparent 60° rotation symmetry. The approach is based on general techniques that are readily applied to other secondary instabilities of symmetric patterns, and does not rely on the primary pattern having small amplitude.

Long wavelength instabilities of square patterns

The long wavelength instabilities of square and rectangular planforms are studied analytically and numerically, using amplitude equations which describe the general interaction of two orthogonal coupled roll patterns. The zigzag and two-dimensional Eckhaus instabilities are found, and in addition it is discovered that the three-dimensional equivalent of the Eckhaus instability splits into two variants. The square Eckhaus instability is the direct equivalent of the two-dimensional case, whereas the rectangular Eckhaus instability is truly three-dimensional in character. In the case of square patterns, nonlinear phase diffusion equations are derived close to the onset of the instabilities. A short wavelength cross square mode is also discussed briefly.

Pattern selection with anisotropy during directional solidification

The effects of surface-tension anisotropy on interface morphology during the directional solidification of a binary alloy are studied. The long-wave evolution equation derived by Brattkus & Davis to describe growth near the absolute stability limit is generalized to include the effects of a surface tension with cubic anisotropy. The special cases of growth in the [001], [011] and [111] directions are considered. The resulting evolution equations are derived, and amplitude equations governing roll/rectangle and roll/hexagon competition are obtained. The coefficients of the amplitude equations depend on the surface-tension anisotropy, and determine how pattern selection is influenced by the presence of geometrically preferred directions. Anisotropy leads to changes in the existence and stability criteria for each pattern, to imperfect bifurcations, and to loss of degeneracy in bifurcations.

Modelling pattern formation in CO+O2 on Pt{100}

We extend a detailed kinetic model for CO + O2 on Pt{1 0 0} to describe pattern formation. The model includes: (i) a non-linear power law to describe the phase transition, (ii) trapping and untrapping processes explicitly considered, and (iii) experimentally determined coverage-dependent sticking probabilities and rate constants. This model is extended to include diffusion and gas global coupling. Diffusion is included through a mass-balance equation which couples the migration of CO with the phase transition. Gas global coupling is introduced considering realistic values of the pumping flow, the reactor volume and the size of the crystal