Quantifying evolutionary potential

Procedure

If traits have the capability to respond to natural selection, then they have evolutionary potential. The trait may or may not experience selection, yet traits have the potential to evolve if they vary and if the trait is inherited with some fidelity from parent to offspring. The two components of variation and heritability together define the evolutionary potential. Without variation in a trait, there can be no evolution, because without phenotypic variation, there can be no variation in fitness. If heritability is zero, then it means that there is no degree of similarity between parents and offspring. Then, likewise, there can be no evolution by natural selection no matter the strength of selection or the amount of phenotypic variation. This is because, with zero heritability, offspring are free to take any form, and therefore, the change in frequencies of forms relative to their fitnesses is suppressed by the inability for forms to propagate across generations. A trait with heritability maintains its phenotype over time, much like a thrown snowball that maintains its identity as it flies through the air. A trait without heritability changes its phenotype overtime unpredictably, just as throwing a handful of sand spreads through the air. As soon as it leaves your hand it spreads out, increasingly losing its coherence the further it flies from your hand.

*Multivariate Price’s theorem*. The evolutionary logic above forms the basis of quantitative genetics and breeding by artificial selection. These fields have given us powerful mathematical tools to understand just how evolution proceeds given patterns of selection, variation, and heritability (*22*–*25*). These tools derive part of their power because they are not reductionists; they deal with phenotypic evolution at the phenotypic level. There is no need to dig lower into genetic levels of explanation for them to work. Thus, the methods do not offer a complete understanding of the mechanism of evolutionary change at all levels, nevertheless their success in agriculture and evolutionary biology underline their utility. For our purposes, we want to know how hierarchically organized *Stylopoma* colonies evolve, given the simultaneous proliferation of asexually and sexually produced zooids.

To see why this works, let us use a formality of Price’s theorem. Price’s theorem describes the evolutionary response to selection of a trait or set of covarying traits, given the structure of variation and heritability of those traits. Price’s theorem defines the evolutionary potential of traits in terms of two matrices: **C**, the heritability matrix, which measures the similarity of traits between parents and offspring. The diagonal values within the heritability matrix can be calculated as the variation in the offspring phenotype multiplied by the linear regression of parent phenotype onto offspring phenotype. The off-diagonal elements measure the co-heritability between two traits, for example, how similar egg size in a bird is to the egg number that her chicks are able to produce. These interactions can be strong if, as in some birds, egg size and egg numbers have a strong inverse relationship that persists over generations. The phenotypic covariance matrix, **P**, describes the amount of variation of all traits and the covariance between them. The product of **C** and the inverse of **P** define the evolutionary potential of traits. Selection, β_{w,ϕ}, is defined as the linear regression of fitness (*w*) on phenotypes (ϕ). The response to selection is measured as the change in the average phenotype, denoted $\mathrm{\Delta}\overline{\mathrm{\varphi}}$, and is determined by the product of evolutionary potential and selection (*25*)$$\mathrm{\Delta}\overline{\mathrm{\varphi}}={\text{CP}}^{-1}{\mathrm{\beta}}_{w,\mathrm{\varphi}}$$()1

Because this equation is a simple product of three terms, if any of the three factors selection (β_{w,ϕ}), variation (P), or heritability (C) is equal to zero, then there will be no response to selection.

*Hierarchical expansion*. The equation above is very simple and it is another way to write the Breeder’s equation. For solitary organisms such as cattle, chickens, or beans, it summarizes the evolutionary processes involved adequately. However, it, as written, only gets half the story for colonial organisms like *Stylopoma*. It either partially describes colony-level evolution or it partially describes zooid level evolution. We want both. In the general form of Price’s theorem, there is an additional term, $\overline{\mathrm{\delta}}$, which represents the expected change in the mean phenotype due to processes within the parts$$\mathrm{\Delta}\overline{\mathrm{\varphi}}={\text{CP}}^{-1}{\mathrm{\beta}}_{w,\mathrm{\varphi}}+\overline{\mathrm{\delta}}$$ ()2

It may help to use a paleontological example to think about this term (*43*–*45*). Species can evolve over time. New species may also change phenotypes during speciation so that they differ more or less from their ancestor. There is even a process of selection at the species level that acts by differential rates of extinction and speciation. The term, $\overline{\mathrm{\delta}}$, is the average amount of evolution within all species—and in colonies, it is the change in phenotype due to biased changes among zooid members. Hamilton (*46*) and Price (*47*) were among the first to realize that Price’s theorem can be hierarchically expanded such that this last additive term is equivalent to a lower level of selection. The way this works is to notice that the change in the average traits within an entity has the same units as the change in the average traits among entities$${\overline{\mathrm{\delta}}}_{\text{whole}}=\mathrm{\Delta}{\overline{\mathrm{\varphi}}}_{\text{parts}}$$()3

If we rewrite Eq. 2 with this recursive evolutionary level in mind, then we can combine the evolution of wholes with the evolution of constituent parts$$\mathrm{\Delta}{\overline{\mathrm{\varphi}}}_{\text{whole}}={\mathrm{C}}_{\text{whole}}{\mathrm{P}}_{\text{whole}}^{-1}{\mathrm{\beta}}_{{\mathrm{w}}_{\text{whole},\mathrm{\varphi}\text{whole}}}+{\mathrm{C}}_{\text{parts}}{\mathrm{P}}_{\text{parts}}^{-1}{\mathrm{\beta}}_{{\mathrm{w}}_{\text{parts},\mathrm{\varphi}\text{parts}}}$$()4

This looks more complex than it is because of the notation keeping track of whole colonies and their zooid parts. However, in words, Eq. 4 shows that the evolution of whole colonies is due to the product of the evolutionary potential of traits and selection at the colony level and the product of the evolutionary potential of traits and selection at the zooid level.

Unlike previous hierarchical expansions of Price’s theorem (*26*, *46*–*48*), we do not assume that fitness at the colony level is a direct function of fitness at the zooid level. The standard assumption would be that average fitness of zooids is equal the average fitness of colonies. However, we know, from observed life-history patterns of *Stylopoma* and other bryozoans, that colony fitness is not a simple function of zooid fitness (*49*). As discussed in the context of the natural history of *Stylopoma*, the rate of production of ovicells in a colony is not related to the growth rate of the colony. Ovicells can be rare in fast-growing large colonies and common in small slow-growing colonies just as often as ovicells can be common in fast growing colonies (*20*, *50*, *51*).

Therefore, selection may occur at both the colony level and the zooid level as colonies beget colonies sexually and zooids beget zooids asexually. Given the importance of sexual and asexual modes of reproduction in these colonies, we should assume that selection is rampant at both levels. This fact brings the importance of evolutionary potential at each level into stark focus. What is the pattern of and cause of evolutionary potential at both the colony and the zooid level? If colony traits are variable and heritable, then they can respond to colony-level selection. Likewise, if zooid traits are variable and heritable, then they too have the potential to evolve by natural selection. There may be a conflict between these two levels of selection or they may be aligned.

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