A Generic Neutral Model for Measuring Excess Evolutionary Activity of Genotypes

We introduce and study a simple generic model of neutral evolution of genotypes, designed to provide a feasible and general method for quantifying excess evolutionary activity|the extent to which evolutionary activity is the product of adaptive evolution. We compare the behavior of the generic neutral model against two other models: Packard's agent-based model of the evolution of sensory-motor functionality and a neutral \shadow" of Packard's model. Diversity and evolutionary activity of these three models across the mutation rate spectrum illustrate the feasibility and general applicability of the generic neutral model, con rm the appropriateness of using neutral models to quantify the extent of the continual adaptive success of genotypes, and reveal power-law dependences of evolutionary activity on mutation rate. 1 The Need for Neutral Models Although it is commonly accepted that adaptive evolution produces much of the structure and functionality in complex systems [6, 4], it is often di cult to distinguish adaptive change from other evolutionary phenomena such as random genetic drift [3]. Some even question whether adaptations can be objectively identi ed at all [3]. The ultimate goal of this paper is to facilitate the investigation of universal laws of adaptive evolution. Toward this end, this paper aims to develop statistics for objectively identifying and quantifying adaptive evolutionary activity, especially statistics feasible and general enough to apply to a broad enough range of natural and arti cial evolutionary systems. This paper illustrates evolutionary activity statistics in the context of a simple arti cial model of sensory-motor evolution|Packard's Bugs model| and we apply the method to a broadly applicable level of analysis|whole genotypes. In this setting we do nd simple law-like regularities involving adaptive evolution. Part of what makes this especially interesting is that evolutionary activity statistics apply to myriad evolutionary systems at myriad levels of analysis, so we can investigate whether the same regulatities hold in evolving systems in general. We use the approach of Bedau and Packard [1, 2] to identify the extent to which a system's evolutionary dynamics depend on adaptation rather than other evolutionary forces like chance and necessity. That is, we screen o the e ect of non-adaptive evolutionary forces by comparing the evolutionary dynamics observed in target evolutionary systems with those observed in analogous evolutionary systems in which adaptive evolution cannot happen. We term these non-adaptive evolutionary data lters \neutral models" of evolution. Filtering observed data with a neutral model yields a measure of excess evolutionary activity|that activity due to adaptation. In e ect, neutral models are null hypotheses against which the action of adaptive evolution stands out in relief. One method for making neutral models is to craft a system that \shadows" the target evolutionary system in all relevant respects except that a shadow genotype's presence or concentration or longevity cannot be due to the genotype's adaptive signi cance [2]. Since such \neutral shadows" are speci cally tailored to the target system of interest, they create sharp no-adaptation null hypotheses. But because they are tailor-made for speci c target systems, studying new target systems requires constructing new shadow models, and it is vexing to compare models that shadow di erent target systems. An obvious way to address these problems is to create a simple generic neutral model|one neutral model that reasonably approximates a host of di erent shadow models. The immediate goal of the present paper is to de ne and study such a generic neutral model. There are a series of steps involved in proving the value of this generic model, such as comparing it with many di erent shadow models, discerning how its behavior depends on crucial model parameters, exploring how adaptation alters its behavior, and connecting it with related theoretical and empirical work. The present paper takes the rst step in this process by comparing the generic neutral model with a simple evolutionary system and its neutral shadow. 2 Evolutionary Activity Statistics Evolutionary activity statistics are computed from data obtained by observing an evolving system. We view an evolving system as a population of components participating in a cycle of birth, life and death, with each component largely determined by inherited traits. Birth and mutation introduce innovations into the population. Adaptive innovations persist in the population because of their bene cial e ects for component survival or reproduction, and non-adaptive innovations either disappear or persist passively. The idea behind evolutionary activity is to identify innovations that persist and continue to be signi cant. Counters are attached to components for bookkeeping purposes, to update each component's current activity as the component persists. If the components are passed along during reproduction, the corresponding counters are inherited with the components, maintaining an increasing count for an entire lineage. Previous work has studied components on the level of individual alleles [1] as well as genotypes [2] and taxonomic families [2]. For simplicity, here we restrict our attention to entire genotypes. To measure activity contributions we attach a counter to each component of the system, ai(t), where i labels the component and t labels time. A component's activity increases over time as follows, ai(t) = P k t i(k), where i(k) is the activity increment for component i at time k. Various activity incrementation functions i(t) can be used, depending on the nature of the components and the purposes at hand. Since in the present context more adaptive genotypes tend to persist longer, it's natural to measure a component's contribution to the system's evolutionary activity simply by its age. So we choose an activity incrementation function that increases a component's activity counter by one unit for each time step that it exists: i(t) = 1 if component i exists at t 0 otherwise : (1) Though there are ways to re ne this simple counting method [1, 2], this version facilitates direct comparison with many other systems. Now, we can de ne various statistics based on the components in a system and their activity counters. There are various ways to quantify diversity and evolutionary activity (e.g., [1, 2]). Here we choose statistics that make it easy to compare diversity and evolutionary activity across a wide variety of evolving systems. Perhaps the simplest statistic is the system's diversity, D(t), which is the number of components present at time t, D(t) = #fi : ai(t) > 0g ; (2) where #f g denotes set cardinality. A measure of the continual adaptive success of the components in the system at a given time is provided by the total cumulative evolutionary activity, A(t), which simply sums the evolutionary activity of all the components at a given time: