Major and minor groups in evolution

Abstract

Kerr and Godfrey-Smith argue that two mathematically equivalent, alternative formal representations drawn from population genetics, the contextualist and collectivist formalisms, may be equally good for quantifying the dynamics of some natural systems, despite important differences between the formalisms. I draw on constraints on causal representation from Woodward (Making things happen, Oxford University Press, New York, 2003) and Eberhardt and Scheines (Philos Sci 74(5):981–995, 2006) to argue that one or the other formalism will be superior for arbitrary natural systems in which individuals form different types of groups.

Appendices

Appendix 1

Below are equations appropriate for a haplo-diplont model (Scudo 1967), suitable for a system featuring genetic effects on viability at both the gamete and zygote lifecycle phases, along with coefficient k quantifying meiosis. The haplo-diplont model contains three separate sets parameters, v’s, w’s, and k, which quantify differential genetic effects on gametes, zygotes, and meiosis, respectively. The genotypic selection model is a determination of the haplo-diplont model in which v ₁ and v ₂ are set to the same value and k = ½.

• Pre-selection A₁ gametes = p

• Pre-selection A₂ gametes = q = (1 − p)

• Post-selection A₁ gametes = P = ν ₁ p

• Post-selection A₂ gametes = Q = ν ₂ q

• Average gamete fitness = V = P + Q

• Mature A₁ gametes = $P^{\prime } = \frac{P}{V}$

• Mature A₂ gametes = $Q^{\prime } = \frac{Q}{V}$

• Pre-selection A₁A₁ organisms = $d = (P^{\prime } )^{2}$

• Pre-selection A₁A₂ organisms = $h = (Q^{\prime } )^{2}$

• Pre-selection A₂A₂ organisms = $r = 2P^{\prime } Q^{\prime }$

• Post-selection A₁A₂ organisms = D = w _D d

• Post-selection A₁A₂ organisms = H = w _H h

• Post-selection A₂A₂ organisms = R = w _R r

• Average organism fitness = W = D + H+R

• Mature A₁A₂ organisms = $D^{\prime } = \frac{D}{W}$

• Mature A₁A₂ organisms = $H^{\prime } = \frac{H}{W}$

• Mature A₂A₂ organisms = $R^{\prime } = \frac{R}{W}$

• Next-generation A₁ gametes = $p^{\prime } = D^{\prime } + {\text{k}}^{\prime }$

• Next-generation A₂ gametes = $q^{\prime } = R^{\prime } + {\text{k}}H^{\prime }$

Reduced to a pair of equations, the haplo-diplont model is this:

$$\begin{gathered} \bar{p}(t + 1) = \frac{{w_{D} \left( {v_{1} p} \right)^{2} + w_{H} 2v_{1} pv_{2} qk}}{{w_{D} \left( {v_{1} p} \right)^{2} + w_{H} 2v_{1} pv_{2} qk + w_{R} \left( {v_{2} q} \right)^{2} }} \hfill \\ \bar{q}(t + 1) = \frac{{w_{H} 2v_{1} pv_{2} q(1 - k) + w_{R} \left( {v_{2} q} \right)^{2} }}{{w_{D} \left( {v_{1} p} \right)^{2} + w_{H} 2v_{1} pv_{2} q + w_{R} \left( {v_{2} q} \right)^{2} }} \hfill \\ \end{gathered} $$

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Here is a causal graph for this sort of system (Fig. 10):

Fig. 10

Causal graph of a haplo-diplont model

Note how the v ₁ and v ₂ variables, which quantify genetic effects on gametes prior to zygote formation, impact zygote frequencies. The contextualist formalism cannot quantify genetic effects on gametes prior to zygote formation by means of the α and β parameters, because the f(i)t parameters are not functions of these.

Even if we alter the f(i)t functions such that they are a function of the α and β parameters in order to accommodate gametic selection prior to zygote formation, the result is that the α and β parameters can no longer quantify differential zygote viability: poor gametic performance may demand a low value for, say, α ₂, while strong zygote performance among homozygotes for the dominant allele may demand a high value for the same quantity. Cases of antagonistic pleiotropy, in which gametic selection and zygotic selection favor rival alleles, cannot be modeled using a single set of fitness parameters: instead two sets of fitness parameters must quantify selection at each lifecycle phase, as are featured, in the form of both v’s and w’s, in the haplo-diplont model above.

Appendix 2

In their (2012) consideration of a trait-group model of altruism, an instance of a contextualist model, KGS consider a system in which groups of three particles are formed, individuals have a baseline fitness of 2, altruists contribute 2 extra offspring each to fellow group members, and incur a cost of 1 offspring for their altruism. KGS set out fixed fitness parameters for both the contextualist and the collectivist representations:

$$\begin{array}{*{20}c} {\alpha_{1} = 1} \hfill & {\alpha_{2} = 3} \hfill & {\alpha_{3} = 5} \hfill & {\beta_{0} = 2} \hfill & {\beta_{1} = 4} \hfill & {\beta_{2} = 6} \hfill \\ {\pi_{0} = 6} \hfill & {\pi_{1} = 9} \hfill & {\pi_{2} = 12} \hfill & {\phi_{1} = \frac{1}{9}} \hfill & {\phi_{2} = \frac{1}{2}} \hfill & {} \hfill \\ \end{array}$$

KGS next run their near-variant test on the trait-group model, changing things so that altruists provide a benefit of 4 to fellow group members and incur a cost of 2. They conclude from their test that both the contextualist and the collectivist parameterizations are on roughly equal footing since one must change the same number of parameters in each formalism to restore dynamical adequacy for the near-variant (Godfrey-Smith and Kerr 2012, 213).

KGS’s formal representation of the trait-group model uses fitness coefficients rather than fitness functions, such that their equations are not causally adequate. When their fitness coefficients are replaced by fitness functions that separate out the causal mechanisms into a baseline component, a benefit component, and a cost component, it becomes clear that KGS’s near variant test effectively involves changing multiple causal mechanisms at once, since they imagine altering both the benefit conferred by altruists (b) and the cost of altruism (c). Accordingly, the transformation they use to generate a near variant is not produced by an intervention. This is why KGS draw a different conclusion than the one drawn here with respect to contextualist representation of minor groups.

To see the case in detail, replace KGS’s fixed fitness coefficients for the α and β parameters in the contextualist model with the following fitness functions (with k as baseline fitness):

$$\begin{gathered} \alpha_{i} = k + b_{i} - c_{i} \hfill \\ \beta_{i} = k + b_{i} \hfill \\ k = 2 \hfill \\ b_{i} = 2i \hfill \\ c_{i} = 1 \hfill \\ \end{gathered}$$

KGS generate a near variant by means of two simultaneous interventions on the system, one setting b _i  = 4, and another setting c _i  = 2. If we contemplate running these interventions separately, then we get results that parallel the ones we got above for the D. dendriticum case: the contextualist formalism handles informally described interventions by means of changes to the values of single quantities, while the collectivist one does not do so. Indeed, KGS themselves note that “if costs alone, or benefits alone, are altered, then the contextual parametrization fares better” (2012, 214).

Appendix 3

Consider a contextualist representation of viability selection among diploids, where, if modeled using the collectivist formalism, π₀ = w_D = 4, π₁ = w_H = 5, π₂ = w_R = 6. A dynamically adequate version of the contextualist formalism for the same case, where f(i)t parameters are once again set by the Hardy–Weinberg rule, looks like this (from Eqs. 7):

$$\begin{gathered} \bar{p}(t + 1) = \frac{{\alpha_{2} p^{2} + \alpha_{1} 2pq}}{{\alpha_{2} p^{2} + \alpha_{1} 2pq + \beta_{1} 2pq + \beta_{0} q^{2} }} \hfill \\ \bar{q}(t + 1) = \frac{{\alpha_{1} 2pq + \beta_{0} q^{2} }}{{\alpha_{2} p^{2} + \alpha_{1} 2pq + \beta_{1} 2pq + \beta_{0} q^{2} }} \hfill \\ \begin{array}{*{20}c} {\alpha_{2} = 3} & {\alpha_{1} = 2.5} & {\beta_{1} = 2.5} & {\beta_{0} = 2} \\ \end{array} \hfill \\ \end{gathered} $$

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If we use the haplo-diplont model (13) from “Appendix 1” above to infer the dynamics of our diploid system with zygote fitness parameters set as above, no selection prior to zygote formation (v1 = v2), some drive k = 3/5, and p = 0.5, we infer that, after a single generation, the frequency of the A allele grows to 0.6. Reproducing this result using the contextualist model requires bumping α ₁ to 3 and reducing β ₁ to 2 (Godfrey-Smith and Kerr 2012, 212). That both parameters must be altered shows the causal inadequacy of the contextualist formalism for the diploid selection case.