Multimodel inference in ecology and evolution: challenges and solutions

Information theoretic approaches and model averaging are increasing in popularity, but this approach can be difficult to apply to the realistic, complex models that typify many ecological and evolutionary analyses. This is especially true for those researchers without a formal background in information theory. Here, we highlight a number of practical obstacles to model averaging complex models. Although not meant to be an exhaustive review, we identify several important issues with tentative solutions where they exist (e.g. dealing with collinearity amongst predictors; how to compute model-averaged parameters) and highlight areas for future research where solutions are not clear (e.g. when to use random intercepts or slopes; which information criteria to use when random factors are involved). We also provide a worked example of a mixed model analysis of inbreeding depression in a wild population. By providing an overview of these issues, we hope that this approach will become more accessible to those investigating any process where multiple variables impact an evolutionary or ecological response.     

The heritability of multiple male mating in a promiscuous mammal

The tendency of females to mate with multiple males is often explained by direct and indirect benefits that could outweigh the many potential costs of multiple mating. However, behaviour can only evolve in response to costs and benefits if there is sufficient genetic variation on which selection can act. We followed 108 mating chases of 85 North American red squirrels (Tamiasciurus hudsonicus) during 4 years, to measure each female's degree of multiple male mating (MMM), and used an animal model analysis of our multi-generational pedigree to provide what we believe is the first estimate of the heritability of MMM in the wild. Female red squirrels were highly polyandrous, mating with an average of 7.0 ± 0.2 males on their day of oestrus. Although we found evidence for moderate levels of additive genetic variation (CV(A) = 5.1), environmental variation was very high (CV(E) = 32.3), which resulted in a very low heritability estimate (h(2) < 0.01). So, while there is genetic variation in this trait, the large environmental variation suggests that any costs or benefits associated with differences among females in MMM are primarily owing to environmental and not genetic differences, which could constrain the evolutionary response to natural selection on this trait.     

Integration of molecular and physiological models to explain time of anthesis in wheat

Background and Aims

A model to predict anthesis time of a wheat plant from environmental and genetic information requires integration of current concepts in physiological and molecular biology. This paper describes the structure of an integrated model and quantifies its response mechanisms.

Methods

Literature was reviewed to formulate the components of the model. Detailed re-analysis of physiological observations are utilized from a previous publication by the second two authors. In this approach measurements of leaf number and leaf and primordia appearance of near isogenic lines of spring and winter wheat grown for different durations in different temperature and photoperiod conditions are used to quantify mechanisms and parameters to predict time of anthesis.

Key Results

The model predicts the time of anthesis from the length of sequential phases: 1, embryo development; 2, dormant; 3, imbibed/emerging; 4, vegetative; 5, early reproductive; 6, pseudo-stem extension; and 7, ear development. Phase 4 ends with vernalization saturation (VS), Phase 5 with terminal spikelet (TS) and Phase 6 with flag leaf ligule appearance (FL). The durations of Phases 4 and 5 are linked to the expression of Vrn genes and are calculated in relation to change in Haun stage (HS) to account for the effects of temperature per se. Vrn1 must be expressed to sufficient levels for VS to occur. Vrn1 expression occurs at a base rate of 0·08/HS in winter ‘Batten’ and 0·17/HS in spring ‘Batten’ during Phases 1, 3 and 4. Low temperatures promote expression of Vrn1 and accelerate progress toward VS. Our hypothesis is that a repressor, Vrn4, must first be downregulated for this to occur. Rates of Vrn4 downregulation and Vrn1 upregulation have the same exponential response to temperature, but Vrn4 is quickly upregulated again at high temperatures, meaning short exposure to low temperature has no impact on the time of VS. VS occurs when Vrn1 reaches a relative expression of 0·76 and Vrn3 expression begins. However, Vrn2 represses Vrn3 expression so Vrn1 must be further upregulated to repress Vrn2 and enable Vrn3 expression. As a result, the target for Vrn1 to trigger VS was 0·76 in 8-h photoperiods (Pp) and increased at 0·026/HS under 16-h Pp as levels of Vrn2 increased. This provides a mechanism to model short-day vernalization. Vrn3 is expressed in Phase 5 (following VS), and apparent rates of Vrn3 expression increased from 0·15/HS at 8-h Pp to 0·33/HS at 16-h Pp. The final number of leaves is calculated as a function of the HS at which TS occurred (TS^HS): 2·86 + 1·1 × TS^HS. The duration of Phase 6 is then dependent on the number of leaves left to emerge and how quickly they emerge.

Conclusions

The analysis integrates molecular biology and crop physiology concepts into a model framework that links different developmental genes to quantitative predictions of wheat anthesis time in different field situations.