Fundamental Theorems of Evolution

Queller 2017 Fundamental Theorems of Evolution. The American Naturalist. 189: 345-353.

Figure 1 from Queller (2017) illustrating the relationship between the Price equation and four other fundamental equations of evolutionary biology. An arrow from one equation to another indicates that the latter can be derived form the former. Variables and subscripts are explained in the main text.


  Brad Duthie

Conceptual unification of disparate phenomena is a major goal of theory in the natural sciences, and many of the most revolutionary scientific theories are those that have shown how seemingly disparate ideas and observations follow logically from a single unifying framework. The most momentous of these theories include Newton’s unification of gravity and the laws of planetary motion, Darwin’s explanations of adaptation and biodiversity as following from natural selection and descent with modification, respectively, and Einstein’s general relativity unifying gravity, space, and time. In all of these examples, theory has changed how scientists understand the world by revealing a fundamental concept, the consequences of which encompass an entire field of study.

Perhaps to this list of discoveries we should include the unifying equation of George Price, which, in a recent paper in the American Naturalist, David Queller argues to be the most fundamental theorem of evolution. The Price equation as a unifying framework has been a subject of recent interest both within evolutionary biology and across disciplines from mechanics to music. At its core, the Price equation is a unifying framework for understanding any correlated change between any two entities. Queller proposes it to be fundamental because it encompasses all evolutionary forces acting on a population, and because it can be used to derive other less general equations in population and quantitative genetics, all of which require stricter assumptions about the evolutionary forces and environmental conditions affecting entities in the population. The Price equation includes two terms to describe the change in any trait Δz.

The first term isolates how a trait (z) covaries with fitness (w) for entities (i), and encompasses the evolutionary processes of natural selection and drift. The second term encompasses everything else that affects trait change (often called the ‘transmission bias’), such as mutation or background changes in environment. Intepreting the Price equation can be a bit daunting at first, perhaps in part because of how abstract the entities (i) are — representing anything from alleles, to unmeasured genotypes, to organisms, to even groups of organisms as the situation requires. Likewise, traits (z) can be any aspect of phenotype associated with such entities, including fitness (w) itself!

It’s here where Queller’s synthesis really shines, as he carefully walks the reader through how Price’s abstract equation can be used to derive multiple other less fundamental equations in which variables represent something concrete and measureable in empirical populations. These equations include Fischer’s average excess equation describing allele frequency change in population genetics, the Robertson and breeder’s equations of trait change in quantitative genetics, and Fischer’s fundamental theorem of evolution. In all cases, Queller notes the additional assumptions that are required to use these equations, particularly that the second term of the Price equation equals zero meaning that no transmission bias exists.

In our discussion, we reviewed the Price equation and its importance in evolutionary biology. We noted the interesting timeline of the discoveries of the equations; although the Price equation is fundamental in the sense that all of the other equations that Queller cites can be derived from it, it is also the most recent equation to be published. All of the other equations, which serve fundamental roles in population or quantitative genetics, were published decades before Price’s equation and have been used regularly by specialists in these sub-fields. This led us to talk a bit about what we value from theorems in evolutionary biology, and whether all of these theorems are better taught as independent solutions to particular problems in evolutionary biology, or as sub-components of a more fundamental framework grounded in the Price equation. Finally, we discussed the second term of the Price equation, noting that all of the equations that can be derived from the Price equation assume that this term equals zero. This effectively isolates natural selection, or some partition of natural selection, but ignores processes that are known to be important for trait change, particularly changing environment.

An extended discussion of Queller’s paper can be found on Brad’s own website.


Selection and evolution of causally covarying traits

Michael Morrissey. Evolution 68(6): 1748-1761 DOI:10.1111/evo.12385. Selection and evolution of evolution of causally covarying traits

I'm baaaaaaaaaaad at thinking of image captions. Soay sheep, picture courtesy of BBC / Arpat Ozgul.

I’m baaaaaaaaaaad at thinking of image captions. Soay sheep, picture courtesy of BBC / Arpat Ozgul.

Will Pearse

Will Pearse

This was not the paper I was expecting, but it was a paper I enjoyed. Morrissey argues that we can use path analysis to tease apart the evolution of different aspects of species’ traits, and I think he makes a very good case for it. I quite enjoyed reading about Soay sheep again – it was a bit like bumping into an old friend in the street.

I know a lot of people who use aster models to examine different components of fitness, and estimate how traits affect fitness. I’m not really capable of assessing which of aster and path analysis is better, but I do feel that they’re complementary and so I doubt that it’s valid to ask which is best anyway. A thing that always confuses me in these analyses is how we define fitness: fitness isn’t just whether you reproduce that year or survive until the next – it’s inherently multi-faceted. Fitness also changes with temporal scale – the total number of offspring is a great measure, but it matters whether those children reproduce. Indeed, you could keep going on down that road with grandchildren, and great-grandchildren… until no one really knows what’s going on. One of the strengths of a structural equation model (…call it a path analysis if you prefer…) in other settings is that one can have multiple ‘predictor’ variables. I wonder if the same sort of approach could be used in an evolutionary setting, at the very least to explore the ‘decision’ behind whether to reproduce one year or wait in hope of a better season.

I finished the paper wondering about generalities. Figures 2 and 3, which show how what contributes to fitness, look very different to me: for sheep, there are lots of indirect effects and everything maps onto fitness, whereas for the plants not everything maps onto fitness and everything seems less messy. Do we tend to find clustered groupings of traits and environmental conditions that all interact with one-another and nothing else (germination –> flowering time –> other things –> fitness), or does everything interact with everything else with no neat modules (birthdate –> fitness and (birth weight –> fitness and (weight in August –> fitness)))? I wonder if the structure of these relationships can alter the stability of the system too – is a tightly-connected system more stable, or do isolated units have some capacity to compensate for one-another? I feel like path analysis might be the way to find out!

Lynsey McInnes

Lynsey McInnes

My heart sank a little when I opened Will’s paper choice this week. a hard core extension of existing quantitative genetic theory. I can barely get to grips with everyday quantitative genetics, let alone pull apart a new extension. So, I closed the paper and my eyes, then opened them again to give it a go.

The jist of the paper appears to be an extension to the quantification that includes more than just direct effects of traits on fitness. Rather, it tries to include all indirect effects of multiple traits, an idea that can be visualised in a path analysis. To be honest, I have no idea if this is a major advance for the field or not, but it does sound like a good, though perhaps dangerous, idea.

I am only familiar with path analyses from various papers that implement structural equation modelling to infer direct and indirect effects of various biotic and abiotic factors on macroecological patterns. Although I have used them myself, I remain wary of how comprehensive they can be. As Morrissey also indicates here, they rely on the user specifying what can and can’t be a direct or an indirect effect and and are susceptible to missing variables or poorly specified paths. OK, you could argue all models might miss a variable or two and that that is not the point, my worry though is that path analyses in particular create a kind of false confidence that all bases are covered and that the model is bang on. Dangerous.

On the flip side, that these models provide scope for accounting for both direct and indirect effects is good. Both quantitative geneticists and macroecologists then have the capacity to incorporate a broader number of variables and are more or less forced to think about how variables might interact with each other. And the modelling framework itself can to some extent remove erroneously included variables. Heck, you could even throw in a latent variable if you know something else matters, but you can’t measure it or don’t know what it is.

I am a bit dubious despite the intrinsic appeal of drawing parallels between two pretty different fields (and in particular data types), but remain sucked in by frameworks that try to do just that, e.g. Vellend‘s between genetic and ecological diversity. I wonder then how much useful crosstalk could occur between quantitative geneticists on the one hand and macroecologists on the other in terms of how best to set up their paths? Perhaps it is more useful to think more generally about how geneticists and ecologists might talk to each other. What benefits would come from thinking about the ecological background/drivers of trait variation or of selection coefficients or of the genetic variation (available or sought after) for ecological adaptation. OK OK, I don’t want to pretend to reinvent the wheel, lots of such analyses are already occurring, but it is far from the norm. Perhaps another path to add to the analysis?

Resolving the paradox of stasis: models with stabilizing selection explain evolutionary divergence on all timescales



Stasis: A population of mollusks is experiencing stasis, living, dying, and getting fossilized every few hundred thousand years. Little observable evolution seems to be occurring judging from these fossils. From Evolution 101.

Suzanne Estes and Steven J Arnold. The American Naturalist 169: 227-244. Resolving the paradox of stasis: models with stabilizing selection explain evolutionary divergence on all timescales

Lynsey McInnes

Lynsey McInnes


The last time I read this paper was when I was complaining that all the macroevolutionary analyses I was attempting to conduct were kind of crap and far-fetched. Someone recommended this paper to me as a great example of an elegant, meaningful analysis of a heterogeneous dataset with a surprisingly simple outcome. I liked it then, but it made me despair even more about the state of my exclusively macroevolutionary analyses even more.

Now that I’ve jumped ship and am trying to find my way within the field of population genetics (with a lot of exposure to quantitative genetics), I like this paper even more. But enough angst from me. What about the paper itself? The authors quickly assume that stabilising selection is the general explanation for the extensive amounts of stasis observed in temporal datasets of a variety of phenotypes and set about attempting to find what kinds of models of phenotypic evolution can generate observed datasets.

This paper is a beautiful example of an attempt to cut to the chase of a bunch of models floating around in the literature using a set-up that makes just the right amount of simplifying assumptions for a tractable answer to emerge. Estes & Arnold find that the best model of the evolution of phenotypic means (where ‘stasis’ appears to be the norm) is one of tracking a fitness optimum that can move within fixed limits. They do this by seeing what quantitative genetics model fits best to a dataset of phenotypic mean changes across one to over a million generations (so, anagenetic rather than cladogenetic/splitting evolution). As an aside, I love that their analysis could be distilled as – does our elegant QG model generate points that fit within an ellipse around our data, or not. Genius!

Their set-up allows them to dismiss the common Brownian motion model (see Will’s post below) as well as the punctuational peak shift model in favour of a model that fits nicely with Simpson’s model of adaptive zones. Phew. This is a pleasing outcome for me as it sits comfortably with a lot of macro-scale analyses (using totally different data) that often find reasonably-sized clades filling up niche space to a certain point and then not really increasing in disparity or diversity until they jump over to new empty niche space (of course, there are counter examples left, right and centre). The matching results are convincing and underline further how naïve models of trait evolution are really quite unhelpful.

The data here consists of phenotypic means through time rather than across lineages at one time point (the typical format for macroevolutionary trait evolution datasets). I wonder how you could conduct a similar meta-analysis on such data? (Related tests have been done on individual traits like body size using the Ornstein-Uhlenbeck model of bounded evolution). I wonder if the signal Este & Arnold obtain is because they include phenotypic change across time-scales (from a single to millions of generations). Their best fitting model fits the amount of change observable at these vastly different time scales (i.e., massive change on a short-time scale that irons out into ‘stasis’ at macroevolutionary time-scales). Is it possible and/or interesting to attempt this kind of analysis across lineages? What do I even mean by this?

Taking a possibly more useful track – how can this result influence how we set up and test our cross lineage trait evolution studies? Can it be used to create more useful null models?

Most of my interest in thinking of stasis in phenotypic evolution comes from thinking about and observing phylogenetic niche conservatism (really just the narrow-sense niche encompassing abiotic environmental variables). The literature is replete with purported examples of strong evidence of PNC, but pretty bare on the process of keeping a niche axis conserved. I like this paper as it demonstrates to us how stabilising selection can generate the right amount of evolution observed at different time-scales. My favoured next step would be to add in some ecology to find out the mechanisms that prevent a lineage’s niche (or elements/axes within) from wandering amok?

Apologies for the rambling nature of this point. I’d be very keen to hear what others thought of it and how this result could be used to inform future analyses, particularly at the macro scale.

Will Pearse

Will Pearse

Too few papers draw links between models of evolution among and within species (phylogenetics vs. quantitative genetics to my mind). Lynsey is doing just that, so I’m not surprised she picked this paper this week! I liked it, if only (but not just) for its excellent summary of a lot of quantitative genetic ground.

The authors make reference to how, under a Brownian motion model, noise increases through time. This is a good point that’s often missed – I’ve brought this up to comparative biologists in the past, and they often retort that the signal of Brownian motion is never lost. This is very true, but if the noise  is so large that is swamps the signal (look at figure 1 in this), then what’s the point? Drawing broad generalisations, I think this reflects how most biologists are taught statistics; we’re taught that bias is always a bad thing (beware a biased predictor! bad!) whereas machine learning people are fine absorbing a little bit of bias if the precision is sufficiently increased (intro chapter of this excellent book). Yes, under Brownian motion the central tendency doesn’t change, but the precision of your prediction is tiny because so much error is introduced given sufficient time. Thus we can still make inferences about the deep past, but sometimes we might do better asking a different question.

Which brings me to the different models that were tested, many of which are ~two decades old, which is awesome in every sense of the word. A lot of people are scrambling to build ever-more complicated models that incorporate more and more detail, and yet more are turning to methods like Approximate Bayesian Computation as the only way to fit such complex models. This paper shows that might not be needed: they/Lande simplify by taking polynomial approximations of difficult equations, and then work with those. I’m a huge fan of non-linear interactions, but even these can (under certain conditions) be linearised and approximated to draw inferences about biology. The authors go to some pains to talk about whether some of these models could be fitted to phylogenetic data (some already are); were we to make such simplifications I really can’t see why these, and even more complex models, couldn’t be.

In passing, it’s interesting that they view models of DNA evolution in phylogenetics as successfully integrated and all fine and dandy. I really don’t – I think we need to start taking into account geography, and I occasionally see someone talk about ways to integrate directional selection into phylogenetics which sounds fantastic. I shudder when I consider how many phylogenies are built using loci under incredibly strong directional selection, like rbcL and matK (I do it!), and in so doing violate so many of the assumptions phylogenetics is based on.

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