Abstract Formulations Of Darwinian Principles

Though it is widely agreed that the Darwinian principles can be characterized abstractly, without reference to any specific level of biological organization, the literature contains a number of non-equivalent characterizations. For example, some authors distinguish units of selection from levels of selection; others distinguish units of selection from units of evolution; still others recognize neither of these distinctions. Some authors argue that evolution by natural selection requires two types of entities, replicators and interactors, while others offer analyses in terms of a single type of entity. Still others argue that reproduction, rather than replication, is the fundamental notion. To some extent these are questions of terminological preference, though there are substantive issues at stake too. For clarity in what follows, conceptual and terminological uniformity is required.

In a well-known article, Lewontin (1970) identified three principles that he said 'embody the principle of evolution by natural selection', namely phenotypic variation, differential fitness, and heritability; entities possessing these properties he called 'units of selection' (p. 1). Fitness he defined as rate of survival and reproduction, and heritability as parent-offspring correlation. Oddly, Lewontin required that the differences in fitness, rather than the phenotypic differences, should be heritable, that is, the parent—offspring correlation should hold with respect to fitness rather than phenotype. This is odd because if selection is to produce cross-generational phenotypic change, it is the phenotypic differences, not the fitness differences, that must be heritable.3 But leaving this oddity aside, Lewontin's formulation seems to capture the essence of the Darwinian process very neatly. In a similar vein, Maynard Smith (1987a) wrote that evolution by natural selection will operate on

3 Heritability of fitness is required if selection is to lead mean population fitness to increase over generations, as Fisher's (1930) 'fundamental theorem' states. But if by evolutionary change one means change in mean phenotype, rather than mean fitness, as Lewontin does, then it is the phenotypic differences, not the fitness differences, that must be heritable.

any entities that exhibit 'multiplication, variation and heredity', so long as the variation affects the probability of multiplying. Entities satisfying these three criteria he called 'units of evolution' rather than selection; here I stick with Lewontin's terminology.4

Note that both Lewontin and Maynard Smith treat the relation of reproduction or multiplication as primitive; neither offers an account of what it means for one entity to multiply, or to produce an offspring entity. Griesemer (2000) argues that this is a significant lacuna, and offers an analysis of what reproduction amounts to, based on two key ideas. First, there should be 'material overlap' between parent and offspring entities. This means that offspring must contain, as physical parts, objects or structures that used to be physical parts of their parents. Organismic reproduction, cell division, DNA replication, speciation, and 'demic reproduction' all satisfy this criterion, Griesemer argues: in each case, a physical part of the parent becomes a physical part of its offspring. Secondly, the capacity to reproduce is something that entities must acquire; they are not born with it. In effect, this second requirement means that entities capable of reproduction must develop, or have a life cycle.

Griesemer's account is a plausible way of fleshing out the abstract notion of reproduction, and forges an interesting link between developmental and evolutionary processes. But in the interests of maximum generality, I prefer to work with a purely abstract notion. However, Griesemer's account does bring out one important feature of the Lewontin—Maynard Smith characterization that can go unnoticed. This is that reproduction, or multiplication, is generally understood to mean the production of offspring entities that occupy the same level in the biological hierarchy as the parental entity. Thus when organisms reproduce they give rise to offspring organisms, when cells divide they give rise to offspring cells, when colonies reproduce they give rise to offspring colonies, and so on. This is the most intuitive way to think of reproduction, and unless otherwise stated is what I shall mean by the term. But as we shall see, there are contexts where reproduction at one level has been defined in terms of the production of offspring entities at another level.

4 In one respect Maynard Smith's terminology is superior, given that the three criteria really describe necessary and sufficient conditions for there to be an evolutionary response to selection, rather than for selection to occur. However, the label 'unit of evolution' is sometimes used in a quite different sense, as in the title of Ereshefsky's (1992) collection, for example.

Dawkins (1976, 1982) and Hull (1981) offered a somewhat different characterization of the Darwinian process. In Hull's version, evolution by natural selection occurs when 'environmental interaction' leads to 'differential replication'; it thus involves two types of entity—interactors and replicators. Dawkins spoke of 'vehicles' in place of Hull's 'inter-actors'. Replicators are defined as any entities of which accurate copies are made—they 'pass on their structure intact' from one generation to another and are characterized by their 'copying fidelity' and 'longevity'. Interactors are defined as entities which 'interact as a cohesive whole with their environment' so as to cause the differential transmission of replicators. Dawkins and Hull argued that the expression 'unit of selection', as it appeared in the early literature, was often ambiguous between replicators and interactors, leading to equivocation.

Despite its popularity, there are reasons for doubting that the Dawkins—Hull characterization offers a fully general account of Darwinian evolution, applicable across the board. One is simply that the Lewontin—Maynard Smith characterization does seem fully general, and involves just one type of entity, not two. Gould (2002) argues that treating the replicator—interactor account as fundamental leads to a 'historical paradox', given Darwin's own views on inheritance. If blending rather than particulate inheritance had turned out to be correct, then replicators as defined by Dawkins and Hull would not exist, so replication cannot be essential to the Darwinian process, he argues. Gould's 'paradox' is a dramatic way of making a valid point, namely that what matters for evolution by natural selection is sufficient parent—offspring resemblance, or heritability; the transmission of replicating particles from parent to offspring is not in itself necessary (cf. Godfrey-Smith 2000a).

Another way of appreciating this point is to note that cultural and behavioural, as well as genetic, inheritance can generate the parent—offspring similarity needed for an evolutionary response to selection (Avital and Jablonka 2000; Boyd and Richerson 2005). These inheritance channels do not involve particles bequeathing 'structural copies' of themselves to succeeding generations. So evolutionary changes mediated by cultural and behavioural inheritance cannot be described as the differential transmission of replicators.5 This suggests that the

5 As Avital and Jablonka say, 'the replicator concept is associated with a very specialized type of information transmission, which does not cover all types of inheritance, and therefore cannot be the basis of all evolution.' (2000 p. 359).

replicator—interactor conceptualization is not a fully general account of Darwinian evolution.6 Therefore, I do not employ the Dawkins—Hull framework in what follows; the theoretical work done by the replicat-or/interactor distinction can be captured in other ways, permitting us to remain within the simpler Lewontin-Maynard Smith framework.

Griesemer (2000) raises a quite different objection to the generality of the Dawkins—Hull framework, namely that it characterizes the evolutionary process in terms of features that are themselves the product of evolution. The longevity and copying fidelity of replicators (such as genes) and the cohesiveness of interactors (such as organisms) are highly evolved properties, themselves the product of many rounds of cumulative selection. The earliest replicators must have had extremely poor copying fidelity (Maynard Smith and Szathmary 1995), and the earliest multicelled organisms must have been highly non-cohesive entities, owing to the competition between their constituent cell-lineages (Buss 1987; Michod 1999). If we wish to understand how copying fidelity and cohesiveness evolved in the first place, we cannot build these notions into the very concepts used to describe natural selection.

This is an important consideration, whose implications extend beyond the question of the suitability of the Dawkins—Hull framework. It highlights a subtle transformation in the levels-of-selection question since the discussions of the 1960s and 1970s (Griesemer 2000; Okasha 2006). These early discussions tended to take the existence of the biological hierarchy for granted, as if hierarchical organization were simply an exogenous fact about the living world. But of course the biological hierarchy is itself the product of evolution—entities further up the hierarchy, such as eukaryotic cells and multicelled organisms, obviously have not existed since the beginning of life on earth. So ideally, we would like an evolutionary theory which explains how the biological hierarchy came into existence, rather than treating it as a given. From this perspective, the levels-of-selection question is not simply about identifying the hierarchical levels(s) at which selection now acts, which is how it was traditionally conceived, but about identifying the mechanisms which led the various hierarchical levels to evolve in the first place. Increasingly, evolutionary theorists have turned their attention to this latter question.

6 Symptomatic of this is the fact that attempts to force all selection processes into the replicator—interactor framework often involve significant departures from the original definitions of'replicator'. This point is noted by Szathmary and Maynard Smith (1997) who credit it to J. Griesemer.

This new 'diachronic' perspective gives the levels-of-selection question a renewed sense of urgency. Some biologists were inclined to dismiss the traditional debate as a storm in a teacup—arguing that in practice, selection on individual organisms is the only important selective force in evolution, other theoretical possibilities notwithstanding. But as Michod (1999) stresses, multicelled organisms did not come from nowhere, and a complete evolutionary theory must surely try to explain how they evolved, rather than just taking their existence for granted. So levels of selection other than that of the individual organism must have existed in the past, whether or not they still operate today. From this expanded point of view, the argument that individual selection is 'all that matters in practice' is clearly unsustainable.

It would be a mistake to make too much of this change in perspective. Griesemer (2000) argues that the problem of explaining the emergence of new levels is 'conceptually prior' to that of explaining the evolution of adaptations at pre-existing levels (p. 70); similarly, Fontana and Buss (1994) say that 'selection cannot set in until there are entities to select' (p. 761). In a sense this is obviously true. But even so, the two explanatory problems are not wholly disjoint. Michod (1999) has recently argued that groups of lower-level entities only count as new individuals themselves, and thus generate a new level in the hierarchy, when they evolve a special type of adaptation, namely policing mechanisms to regulate the selfish tendencies of their members. Prior to this stage the groups are merely loose collections of lower-level entities, not genuine evolutionary individuals. If something like this is correct, then the evolution of new levels in the hierarchy cannot be regarded as entirely prior to the evolution of adaptations at those levels. For what converts the group into a true biological unit is precisely the evolution of a special sort of group-level adaptation (cf. Frank 1995b; Szathmary and Wolpert 2003).

I return to this issue in detail in Chapter 8, when I look at the application of multi-level selection theory to the 'major transitions' in evolution. But for the moment, the important point is this. Since the levels-of-selection debate now encompasses questions about the origin of the biological hierarchy, not just the evolution of adaptations at preexisting hierarchical levels, an abstract characterization of Darwinian principles cannot refer to highly evolved features, of either organisms or genetic systems, on pain of an inevitable loss of generality. Characterizations in terms of 'high-fidelity replication' and 'cohesiveness' fall foul of this constraint; arguably, those which describe evolution in terms of'information transfer' do so too (e.g. Williams 1992; Odling-Smee, Laland, and Feldman 2003).7 As we shall see later, the same constraint tells against certain conceptions of what is required for selection to act at a given hierarchical level, for example, the 'emergent character' requirement. Such requirements mistake a product of Darwinian evolution for a prerequisite of it. This is a consideration in favour of the abstract Lewontin characterization.

The expressions 'unit of selection' and 'level of selection' have engendered certain confusion. The following convention will be observed here: if entities at hierarchical level X are units of selection in the Lewontin sense, I shall say that selection 'operates at level X'. The level of selection is simply the hierarchical level occupied by the entities that are units of selection. Thus we can translate easily between talks of units and levels. Note that this convention contrasts with the usage of Brandon (1988), who uses the unit/level distinction in lieu of the replicator/interactor distinction. It also contrasts with the usage of Reeve and Keller (1999), who regard the 'units of selection' question as stale but the 'levels of selection' question as empirically exciting. By the former, they mean the 'gene versus organism' debate prompted by Dawkins's work; by the latter, they mean questions about evolutionary transitions of the sort discussed above.

To summarize, I favour the original Lewontin characterization of the Darwinian principles as a starting point. A population of entities evolves by natural selection where heritable differences between the entities lead to differences in their reproductive output; reproduction is understood as giving rise to an offspring entity that occupies the same hierarchical level as the parent, unless otherwise stated. Entities satisfying these conditions are units of selection; the level in the hierarchy which the entities occupy is the level of selection. This characterization has the virtues of simplicity and generality, though certain complications will emerge. In Section 1.5 we shall see how to integrate it with an abstract mathematical description of the evolutionary process.

1.2 PRICE'S EQUATION

Price's equation, first published by George Price (1972), is a simple algebraic result that describes a population's evolution from one generation

7 This is because on the standard accounts of what genetic information is, genes contain information as a result of evolutionary processes (cf. Maynard Smith 2000, Sterelny 2000).

to another. The power of the equation lies in its generality: unlike most formal descriptions of the evolutionary process, it rests on no contingent biological assumptions, so always holds true (cf. Frank 1995a, 1998). Moreover, the equation lays bare the essential components of evolution by natural selection in a highly revealing way.

Price's equation actually has special significance for the levels-of-selection question, for reasons that go beyond its generality.8 For the equation lends itself very naturally to a description of selection at multiple hierarchical levels, as Price himself realized. This theme was developed by Hamilton (1975) in a well-known paper, but it is only recently that its full significance has become apparent. Grafen (1985) reported that he could only find two papers, other than Hamilton's, that made use of Price's methods at any length;9 today those methods are very widely used, particularly by theorists interested in multi-level and hierarchical approaches to selection, for example, Frank (1998), Michod (1997, 1999), Queller (1992b), Damuth and Heisler (1988), Tsuji (1995), Sober and Wilson (1998), Rice (2004), and others. I will argue that the Price formalism provides an ideal framework for addressing philosophical questions about the levels of selection.

A simple derivation of the basic Price equation is given below, in relation to a single level of selection. Application of the formalism to multiple levels is postponed until Chapter 2.

Consider a population containing n entities, called the P-population (for parental). It doesn't matter what the entities are. The entities vary with respect to a measurable phenotypic character z, the evolution of which interests us. We let z; denote the character value of the ith entity, and z the average character value in the whole population, i.e. z = jj zi- for example, if z were height, then z; would be the height of the ith entity and z the average height of the whole population.

If the character z is selectively significant, we might expect the quantity z to change over time. To track this change, we need to take account of fitness. We let w; denote the absolute fitness of the ith entity, defined as the total number of offspring entities it produces. For simplicity

8 The history of Price's equation, and its implications for the group selection question in particular, are discussed by Hamilton (1996), Frank (1995a), Sober and Wilson (1998), and Segerstrale (2000).

9 The papers Grafen cites are Seger (1981) and Wade (1985); he could also have mentioned Arnold and Fristrup (1982).

we will assume that reproduction is asexual.10 Average fitness in the P-population as a whole is w = jj- " w;. The relative fitness of the ith entity is therefore co; = w;/w.

To track the evolution of z, we also need to take account of how the character z is transmitted from parent to offspring. We let z- denote the average character value of the offspring of the ith entity. If transmission is perfect, that is, if each parental entity transmits its character to each of its offspring with no deviation, then zi = z; for each i. However, transmission may not be perfect: offspring may deviate from their parents with respect to z. We define Az; as the difference between the character value of the ith entity and the average for its offspring, that is, Az; = z- — z;. So Az; measures the transmission bias of the ith entity with respect to the character z. The closer that Az; is to zero, the more faithfully the ith entity transmits the character.

If the ith entity leaves no offspring, that is, w; = 0, then by convention we let Az; equal the transmission bias that would have resulted, if it had left offspring. (This convention is innocent, for in the Price equation the term Az; appears multiplied by w;, so if w; = 0, the value of Az; can be arbitrarily chosen. The point of the convention will become clear later.) The average transmission bias in the whole population we will denote by E(Az;), where 'E' stands for expected value. Obviously, E(Az;)= ¿EiAz;.

Now consider another population of entities, called the O-population, which comprises all the offspring of entities from the P-population.n We let z0 denote the average character value in the O-population. So if evolution has taken place, zQ will be different from z. To calculate zo, note that the O-population is in effect made up of n disjoint subpopulations, where each subpopulation contains all the wi offspring of the ith entity (see Figure 1.1). By definition, the average character value of the ith subpopulation is z-. So the average character value in the O-population as a whole is the weighted average of all the zi, the weights determined by subpopulation size w;. Therefore zQ = jj- ^z-.

Realizing that this is a correct formula for zQ is the key to understanding the Price equation. As Frank (1998) has stressed, the formula's peculiarity lies in the fact that although zQ denotes a property of the

10 The assumption of asexual reproduction is made for expository convenience only; the formalism does not require it.

11 For simplicity it helps to think of generations as non-overlapping, i.e. assume that the P-population goes out of existence as soon as the O-population comes into existence. But the formalism does not depend on this assumption.

P-population

O-population

Figure 1.1. Relation between the P- and O-populations; n = 7

P-population

O-population

Figure 1.1. Relation between the P- and O-populations; n = 7

O-population, namely its average character value, the indices on the RHS of the formula refer to the P-population. In effect, we calculate average character value in the O-population by choosing an entity in the P-population, seeing what fraction of the O-population it is responsible for producing, and multiplying this fraction by the average character value of its offspring; we repeat this calculation for each member of the P-population, then take the summation. Figure 1.1 is a heuristic aid to seeing that this is a correct way of calculating zQ.

The quantity we ultimately are interested in is Az, the change in average character value from one generation to another, where Az = z0 — z. Intuitively, it seems that Az should depend somehow on the fitness differences in the P-population, and the fidelity with which the character z is transmitted. The Price equation captures this dependence precisely, by expressing Az as the sum of two other quantities, as follows:

Note that the quantity of interest, Az, appears on the LHS of equation (1.1) multiplied by average fitness w, which is simply a normalizing constant. The first term on the RHS, Cov (wi, zi), is the covariance between fitness wi and character zi. The second term on the RHS, E(wiAzi) is the average, or expected value, of the quantity w;Az;, which is fitness x transmission bias. For ease of reading, we shall drop the indices wherever possible, so equation (1.1) can be rewritten:

See Box 1.1 for the full derivation of equation (1.1).

See Box 1.1 for the full derivation of equation (1.1).

A useful re-formulation of the Price equation results when we divide both sides by w:

where Cov (w, z) is the covariance between z; and relative fitness rather than absolute fitness w;; and Ew(Az) is the fitness-weighted average of the quantity Az;, rather than the simple average of the quantity w; Az;, as in equation (1.1). We shall make use of both the absolute fitness and relative fitness formulations of Price's equation in what follows; obviously, it is easy to translate from one to the other.

1.3 INTERPRETATION OF PRICE'S EQUATION

What exactly does Price's equation mean? As we can see from equation (1.2), it expresses the total change in z, between parent and offspring generations, as the sum of two other quantities. The first quantity, Cov (w, z), measures the statistical association between the character z and fitness. If entities with a high character value tend to be fitter than average, then Cov (w, z) will be positive; if such entities tend to be less fit than average, then Cov (w, z) will be negative. If character value and fitness are completely unassociated, or if neither shows any variation at all, then Cov (w, z) = 0. The covariance term is therefore a measure of the extent to which the character z is subject to natural selection; it is sometimes called the 'selection differential' on z.

The second quantity, Ew(Az), is a measure of the overall transmission bias in the population, weighted by fitness. To understand it, recall that each of the n entities in the P-population has a Azi term associated with it. Ew(Az) is the average of these n Azi terms, weighted by fitness. If each entity transmits its z-value perfectly, then Azi = 0 for each i, so Ew(Az) = 0. However, if offspring deviate from their parents with respect to the character z, whether systematically or simply as a result of 'noise' during transmission, then Ew(Az) may be non-zero. Note that Ew(Az) is a fitness-weighted expectation: it takes into account not just how much the offspring of the ith entity deviate from it in character, but also how many offspring there are.

With these interpretations in mind, we see that the Price equation becomes highly intuitive. That Az depends on Cov (to, z) simply reflects the common-sense idea of natural selection—if taller organisms are fitter than shorter ones, that is, if height covaries positively with fitness, we expect average height in the population to increase over time. That Az depends on Ew(Az) reflects the fact that transmission fidelity is important too—unless height is transmitted from parent to offspring with sufficient fidelity, then even if taller entities leave more offspring, average height will not necessarily increase. (Intuitively, this means that the magnitude of Ew(Az) should be related somehow to the heritability of z; see Section 1.5 below.) So Price's equation partitions the total change in z into two components, each of which has a natural biological interpretation.

If all fitness differences between entities stem from differences in survival, rather than fecundity, then the two components of Price's equation can be given a temporal interpretation. Viability selection leads the value of z to change within the P-population, between times ti and t2; the magnitude of this change is given by Cov (w, z). The surviving entities then reproduce, leading to a further change between times t2 and t3, of magnitude Ew(Az). So under pure viability selection, Cov (co, z) equals the within-generation change in z, while E^Az) equals the subsequent change that happens during the process of reproduction. But if there is a component of fecundity selection, this interpretation fails. Price's equation still holds true, of course, but the Cov and Exp components do not correspond to sequential periods of change.

A number of important points about Price's equation should be noted. First, the equation is simply a mathematical tautology whose truth follows from the definition of the terms. Nothing is assumed about the nature of the 'entities', their mode of reproduction, the mechanisms of inheritance, the genetic basis of the character, or anything else. It was Price's view that a properly general formalization of natural selection should abstract away from such contingent details (Price 1972, 1995; Frank 1995a). Rice (2004) observes that parental and offspring entities do not even have to be of the same type, so long as the character z is measurable on both, for example, parents could be groups and offspring organisms, or parents could be organisms and offspring gametes.12

Secondly, the character variable z can be defined however we please. To model the evolution of a 'discrete' rather than a 'continuous' character, for example, we simply need to define z appropriately. Suppose we are interested in the proportion of blue entities in our population, for some reason. We then define Zi = 1 if the ith entity is blue, z; = 0 otherwise. Obviously, z then equals the proportion of blue entities in the P-population, and Az the change in this proportion between the P- and O-populations. So Price's equation applies as usual. Similarly, z could be defined as the frequency of a particular allele at a given locus in an organism (= 1, V2, or 0 for diploids); z would then equal the overall frequency of the allele in the population, and Az the

12 Indeed, the entities in the P- and O-populations do not need to be related as parents and offspring at all; as Price (1972) pointed out, his equation requires only an abstract correspondence between the two sets of entities.

change in frequency across one generation.13 So Price's equation fits naturally with definitions of evolution such as change in gene frequency, or change in relative frequency of different types.

Where z denotes a continuous character, one might question whether all evolutionary change can be compressed down to Az, the change in the mean. For even if Az equals zero, the character distribution may nonetheless have changed, for example, its variance may be different. Indeed in textbook cases of'stabilizing selection', where extreme character values are selected against, the mean character remains the same from one generation to another but the variance is reduced; so only tracking the mean will create the illusion that no evolution has occurred. Though valid, this point compromises the generality of Price's equation less than it may seem. For if we wish, we can define Zi as the squared deviation in character of the ith entity from the population mean—which can be thought of as a relational property of the ith entity; z is then the variance of the character, and Az the change in the variance, so Price's equation applies as usual. If z is suitably defined, the evolution of higher moments of the character distribution can be similarly captured.

Thirdly, note that Price's equation is statistical not causal. If Cov (w, z) is non-zero, this means that differences in character value are correlated with differences in fitness, but the correlation need not reflect a direct causal link.14 It is possible that z itself has no effect on fitness, for example, but is closely correlated with another character which does affect fitness. Where a non-zero value of Cov (w, z) is due to a direct causal link between the character z and fitness, we shall say that z is directly selected; where Cov (w, z) is non-zero for some other reason, then z is indirectly selected. The distinction between direct and indirect selection corresponds closely to Sober's (1984) distinction between 'selection of' and 'selection for'.

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