# Mathematical models of lineage selection

I'm interested in the concept of lineage selection (Aboitiz, 1991) as an explanation for why traits would be selected for that enhance the rate at which evolution can occur, rather than directly enhancing an individual's fitness. (Such traits could include gene duplication; body plans that can easily be adapted to new niches; learning (which enhances evolution due to the Baldwin effect) and so on. Questions about the evolution of such traits are collectively termed "evolution of evolvability.") The term "lineage selection" does not refer to group selection or kin selection, but simply to selection between distinct genetic lineages in a mixed population.

What I would like to know is whether a mathematical theory has been developed along the lines of the Price equation that can capture the concept of lineage selection and make predictions about it. Lineage selection does not appear to be modelled by the Price equation itself, at least not in a straightforward and obvious way, because on the face of it the Price equation only directly accounts for the relation between a trait and the number of offspring in the next generation, rather than the number of surviving descendants several generations later, which is what is required for lineage selection.

The reason I'm looking for a mathematical model is that there are a number of intuitions that many people seem to share about lineage selection and I'd like to understand what assumptions are needed in order for them to be true. In particular, it seems clear that large populations are needed for lineage selection to be effective, and I'd like to understand precisely the relationship between population size and the effectiveness of lineage selection. (My intuition is that with a population size of around a million, lineage selection can select for traits that have a beneficial effect around 20 generations later, since $1,000,000\approx 2^{20}$, but this intuition lacks a formal basis.)

A literature search did not turn up anything along the lines of what I'm looking for, but this is not my main field and I may simply not know the most appropriate search terms to use.

• Question that is a bit related: Empirical evidence for species selection – fileunderwater Mar 25 '15 at 9:49
• I think that there is quite a good agreement among the kin selectionist community that kin selection and group selection is exactly the same thing (see here for example). – Remi.b Mar 25 '15 at 14:35
• – Remi.b Mar 25 '15 at 18:04
• @Remi.b I'm pretty aware of that argument (not sure which side I fall on), but lineage selection in the sense I intend it is not the same as either of them - it's not about adaptations that benefit relatives, it's purely about adaptations that benefit your descendants, several (or many) generations later. – Nathaniel Mar 25 '15 at 22:37
• @sterid I didn't find it in the literature, so I decided to do the research myself. An initial conference paper version of the work should be published soon - I'm revising the manuscript now. I will try to remember to post here once it's available online. – Nathaniel May 28 '17 at 23:28

The writings by Samir Okasha (philosopher of biology/science) could be a good starting point. In his book Evolution and the Levels of Selection, he explicitly uses the Price equation to discuss selection at multiple levels (e.g. chapter 2.3: Price's equation in a hierarchical setting), and also derives a multi-level version of the Price equation:

$$\bar{w}\Delta\bar{z} = Cov(W,Z) + E(Cov_k(w,z))$$

where the first part on the right hand side represents collective-level selection and the second part particle-level selection. This is mostly aimed at aspects of group-level selection though. There is also a chapter on "Species Selection, Clade Selection, and Macroecolution" (ch. 7), but this is not directly tied to the Price equation. A note though; I own the book and have read parts of it casually, but I haven't worked with the Price equation myself, and I haven't checked exactly how the extended version of the Price equation is developed and used in the book.

A couple of references that might be useful are Frank (1998) (and other papers by him) and papers by Vrba. The work presented in this recent conference abstract by Rankin, Fox et al (2014), "Using the Extended Price Equation to Analyze Species Selection in Mammalian Body Size Evolution Across the Paleocene/eocene Thermal Maximum", seems to describe exactly the type of theoretical developments that you are looking for (I didn't attend the conference, and haven't heard the talk though).

It's fairly straightforward to expand the Price equation if you fix the genetic operators. This was done in one CS-oriented paper. Bassett et al. (2004) (preprint): where k iterates over the genetic operators.

But it's not at all straightforward (and I suspect not possible) to do that for an all-encompassing abstract notion of evolvability (of every rule/mechanism that might be evolvable). I mean if the rules/operators are countable, you could write an infinite sum on the right hand side of that equation, but it's not clear how that helps with anything in practice.

If you use the multi-level (= group selection) version of the Price equation (as suggested in the other answer), you still have to decide (a priori) on grouping so basically enumerate over the (potentially evolvable) operators, albeit do that in terms of groups. For example, an operator can be recombination, which is the same as deciding to have groups for sexual and asexual organisms. I think the operator approach may be preferable in some cases because it can be real-valued (i.e. a continuous quantity). Although not using the (above) Price equation explicitly, such research on evolvability of specific (continuous) operators has been carried out, e.g. on the optimal recombination rate (Lobkovsky et al., 2016) etc.

On a different tack, and possibly closer to biology as a general idea, in another CS-oriented paper, Touissant (2003) proposed a (non-trivial) notion of embedding the evolvability of the parameters of interest into the "normal" genotype space.

In simple cases where the genotype space decomposes (strategy parameters) an embedding space of neutral sets is obvious. To generalize to arbitrary neutral sets and arbitrary genotype-phenotype mappings we introduced the σ-embedding, i.e., we embedded neutral sets in the space of probability distributions over the genotype space (exploration distributions). [...]

the question of how variational properties evolve has also been raised in numerous variations (pleiotropy, canalization, epistatis, etc.) in the biology literature. These discussions aim at understanding how evolution can handle to introduce correlations or mutational robustness or functional modularity in phenotypic exploration. Our answer is that variational properties evolve as to approximate the selection distribution. If, for example, certain phenotypic traits are correlated in the selection distribution F, then the Kullback-Leibler divergence decreases if these correlations are also present in the exploration distribution σ.

Alas he doesn't derive anything resembling a Price equation using his approach. (He does have a twin notion of evolution and σ-evolution, but there's no hierarchy/grouping.) Whether this notion is actually biologically useful, probably depends crucially on the embedding.

There are some papers in actual biology research that are closer to this idea of "embedded" evolvability. For example one paper (Lehman and Stanley, 2013) which looks at (actually defined evolvability in terms of) phenotypic variability.

one widely-held conception of evolvability as the capacity of an organism to “generate heritable phenotypic variation” [...], which is also the definition adopted in this paper. While evolvability is also sometimes discussed in relation to adaptation [...], the chosen definition reflects a growing consensus in biology that phenotypic variability in its own right deserves study in the context of evolvability

So this is one kind of concrete embedding. But no Price equation based on this idea that I could find in there...

On the other hand Rice (2008) does propose a stochastic extension of the Price equation, based on this idea that variability matters (or is all that there is to evolvability according to some):

A number of studies have shown that when fitness is treated as a random variable, meaning that each individual has a distribution of possible fitness values, then both the mean and variance of individual fitness distributions contribute to directional evolution. Unfortunately the most general mathematical description of evolution that we have, the Price equation, is derived under the assumption that both fitness and offspring phenotype are fixed values that are known exactly. The Price equation is thus poorly equipped to study an important class of evolutionary processes. [...]

I present a general equation for directional evolutionary change that incorporates both deterministic and stochastic processes and applies to any evolving system. This is essentially a stochastic version of the Price equation [...]

This [new] equation shows that the effects of selection are actually amplified by random variation in fitness.

Finally, (as acknowledged in this latter paper) all variations on Price equation are over immediate successive generations. If you want to iterate forward (to infinity) you need to use a diffusion equation/model. (See chapter 7 in Durrett's book, for instance.)

• Thank you, this is a very helpful answer with some great references, which I will work through in detail. – Nathaniel Mar 28 '19 at 14:45