A Generic Neutral Model for Quantitative Comparison of Genotypic Evolutionary Activity

We use a new general-purpose model of neutral evolution of genotypes to make quantitative comparisons of diversity and adaptive evolutionary activity as a function of mutation rate among two versions of Packard's Bugs model and their neutral shadows. Comparing diversity and evolutionary activity of all these models across the mutation rate spectrum shows that the generic neutral model may have broad applicability in discovering quantitative laws involving adaptive evolutionary activity in di erent evolving systems. 1 The Need for a Generic Neutral Model Adaptive evolution is thought to produce much of the order and functionality evident in complex systems [9, 7, 5], but it is often di cult to distinguish adaptive change from other evolutionary phenomena such as random genetic drift and architectural necessity [6, 10], and some even question whether adaptations can be objectively identi ed at all [6]. Recent progress on identifying adaptive evolutionary phenomena includes Bedau and Packard's statistical methods for measuring adaptive evolutionary activity. Here, we apply these methods to the problem of determining how adaptive evolutionary activity depends on mutation rate. Our ultimate aim is to develop methods for objectively identifying and measuring adaptive evolutionary activity in all evolutionary systems, both natural and arti cial, so that we can seek universal laws of adaptive evolutionary activity. Here, we test such a method, applied at the level of whole genotypes, in the context of two simple models of sensory-motor evolution. But the same method can be applied at other levels of analysis in other evolutionary systems. The ultimate signi cance of this work comes from the possibility of quantitatively comparing evolutionary adaptations across all evolving systems. The centerpiece of our method is Bedau and Packard's evolutionary activity statistics. (We also measure system diversity, D, which is simply the number of di erent genotypes present in a system at a given time.) Detailed de nitions and motivations for evolutionary activity statistics are readily available elsewhere [2, 1, 3, 4, 13]. Evolutionary activity statistics aim to identify evolutionary innovations (here, new genotypes) that persist and continue to play a signi cant role in a system because of their adaptive value. These statistics fall into two broad classes: those re ecting evolutionary activity's extent and those re ecting its intensity. Here, we attend only to the extent of evolutionary activity, measuring it with mean cumulative evolutionary activity (sometimes simply called \activity"), A, which in the present context is operationalized as the mean age of the genotypes present in a system at a given time. So, in this context, the higher a system's mean activity, the higher the mean age of the system's genotypes, which means the greater the continual adaptive success of those genotypes. Intuitively, the extent of evolutionary activity concerns how much adaptive structure is present in a system; one might refer to this as the continual adaptive success of the system's components. By contrast, the intensity of evolutionary activity re ects the rate at which new adaptive structure is being created. The extent and intensity of adaptive evolutionary activity are independent. For example, if a population of highly adaptive genotypes persist inde nitely without changing and no new genotypes invade the system, then the extent of evolutionary activity is positive and perhaps grow over time, but the intensity of evolutionary activity falls to nil. To ensure that evolutionary activity statistics re ect the adaptive success of the genotypes and not non-adaptive evolutionary forces like chance and necessity, one must use non-adaptive evolutionary systems called \neutral models" as null hypotheses. That is, one must screen o the e ects of non-adaptive evolutionary forces like chance by comparing the evolutionary dynamics observed in target evolutionary systems with those observed in analogous neutral models. Such neutral analogues have heretofore been constructed by crafting systems that \shadow" the target 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 [3, 4]. Since neutral shadows are tailored to target systems, they sharply show the target systems' deviation from the no-adaptation null hypothesis. But neutral shadows have signi cant drawbacks, too, for studying a new target system involves constructing and studying a new neutral shadow, and it is vexing to make meaningful quantitative comparisons among di erent tailor-made neutral shadows. The obvious way to solve these problems is to create a generic neutral model|one neutral model that can approximate many di erent neutral shadows. The immediate goal of this paper is to de ne such a generic neutral model and test its usefulness for quantifying evolutionary activity across di erent systems. We pursue this goal by comparing the generic neutral model with two simple evolutionary systems and their neutral shadows.