Funded Grants

Complex dynamics, phase transitions, and scaling in an evolutionary model

Mutations provide the raw material for evolution: without variability, nature would be unable to explore the "space" of evolutionary possibilities. Yet too large a mutation rate can be catastrophic, threatening the viability of lineages or even species. This has led scientists to pose the question of whether there might exist an optimal mutation rate. Drake reported similar mutation rates across a wide range of microbial genome sizes (Drake 19911; Drake et al. 1998), but later studies have not borne out the idea of a "universal" mutation rate. Instead, research suggests that mutation rates vary not only between species, but also within species. Different mutation rates can occur in different bacterial strains; those with high mutation rates can exhibit greater antibiotic resistance, and, thus, presumably, selective advantage (Denamur and Matic 2006). Mutation rates in bacteriophage T4 (Drake et al. 1969; Schapper 1998), E. coli (Sniegowski et al. 1997; Schapper 1998) and Drosophila (Nöthel 1987) can be altered by changing external conditions. It is little wonder, then, that the variability of mutation rates, and the problem of how this variability may have changed over time, is a subject of urgent debate within the evolutionary biology community (Pigliucci 2008; Pennisi 2008). Recent papers have increasingly called for the investigation of the evolution of evolvability itself -- the tantalizingly recursive possibility that the pace of evolution is itself under evolutionary control (Bell 2005; Pigliucci 2008). Surprisingly, these issues have barely begun to be addressed from the perspective of complex systems, nonlinear dynamics and statistical physics.

Several researchers have investigated the problem of optimal mutation rate from a computational perspective. Bedau and Packard (2003) developed a model in which simulated organisms ("agents") mutated by selecting different feeding strategy elements from a pool of possible behaviors; their fitness was found to be maximized for an intermediate mutation rate. Moreover, in competitions between agents with different mutation rates, those with intermediate, optimal mutation rates were the most successful. Bedau and Packard commented that "the mutation rate governs a transition between two qualitatively different phases of evolutionary dynamics", a more ordered state at low mutation rate characterized by long periods of stasis, and disordered dynamics at high mutation rates. Bedau and Packard interpreted their result as an example of adaptation to "the edge of genetic disorder", drawing a connection between their results and complex behavior at the boundary between two regimes in a phase transition.

Approaching the problem from a different perspective, Earl and Deem (2004) investigated a model of protein evolution, where fitness was determined by minimization of an energy function. Protein evolution took place on a shifting fitness landscape determined by various properties of the environment. Earl and Deem found that larger mutational shifts dominated when the landscape shifted faster, and when shifts were of larger amplitude. Like Bedau and Packard, Earl and Deem demonstrated the selectability of mutation rates, but rather than finding a universal optimum, their system evolved different mutation rates under different external conditions.

Clune et al. (2008) explored mutation rate optimization using the Avida model, in which computer programs compete as digital organisms, and their success at performing certain computational operations serves as a measure of fitness. Again, fitness was found to be maximized at an intermediate mutation rate. Like Bedau and Packard, Clune et al. allowed their digital organisms to compete against each other. On a smooth landscape, organisms with mutation rates close to the fitness-maximizing intermediate value tended to be the survivors. On a rugged landscape, however, the surviving mutation rates were significantly lower than the optimal value. Like the Earl and Deem study, this suggests that different mutation rates may be optimal under different circumstances.

Here, I propose to take a complex systems approach to investigating the role of mutations in driving evolutionary dynamics in a computational model. The work proposed here will differ from the studies cited above in two key aspects. First, I will focus on the role of maximum mutation size (here called the mutability μ) in the formation of clusters of organisms, thus linking the problem of optimal mutability to another crucial problem at the interface between complex dynamics and evolutionary biology: speciation. Secondly, I will investigate phase transition behavior as the system undergoes a transition from scattered, unclustered individuals to well-defined, structured groups, as the mutability is increased. These proposed studies will lay the necessary groundwork for my research group's long-term goal of bringing a complex systems / statistical physics approach to bear on what is perhaps the central problem of collective dynamics in evolutionary biology: the controversial problem of multiple levels of selection.

In a preliminary study (Dees and Bahar 2010), we investigated the role of mutability in an agent-based model incorporating the three fundamental criteria necessary for Darwinian evolution: variability, heritability and overpopulation (Lewontin 1970). Organisms exist in a two-dimensional morphospace, with each dimension representing an arbitrary trait. At each time step, a new generation of organisms is generated, using an assortative mating algorithm (de Cara et al. 2008). The morphospace also has a third dimension, a "fitness landscape", represented by the color scale in Figures 1b and c, and corresponding to the number of offspring a parent will produce. Depending on the implementation of the model, the landscape may either be shifted periodically or modulated by negative feedback from the local density of organisms. Coordinates of offspring are determined as a function of the parents' coordinates, modified with a random mutation selected from a uniform distribution (or, in other simulations, from a normal distribution) and limited by the maximum mutation size, the mutability parameter μ.

The model does not contain an explicit representation of genetics, and thus may be considered to lump together genetic and epigenetic effects. The choice to define the model in this way was quite deliberate; while detail is inevitably sacrificed by the absence of explicit genetics, the model has the advantage of being amenable to a rigorous comparison with mathematical models of directed percolation (see below). Clusters, the analog of species in this model, are determined by an algorithm motivated by the concept of biological species, defined as a reproductively isolated group (Mallet 1995, de Aguiar et al. 2009). Details of the clustering algorithm can be found in Dees and Bahar (2010).

In all simulations, the population quickly devolved to extinction for small μ. For large μ, the system maintained a fluctuating population. For an intermediate range of values (~μcrit), a high degree of contingency was observed in the outcomes (Figure 1). Simulations over a range of μ values show that the mean population and mean number of clusters increase sharply for an intermediate range of μ values (Figure 2); clusters also become more dense and tightly packed (see Dees and Bahar 2010). These measures may be considered as order parameters characterizing the system; the large fluctuations observed around μcrit are precisely what would be expected in a critical phase transition, as is the "contingent" behavior illustrated in Figure 1.

Another striking preliminary finding is that clustering occurs even on an entirely flat landscape, i.e., one in which every organism has identical fitness. This result is particularly intriguing in that it provides a model of speciation driven by "drift" alone, rather than by natural selection. The idea of speciation on a "neutral" landscape has been extensively debated over recent decades, most vociferously since Hubbell's generalization of neutral theory (Hubbell, 2001; Alonso et al. 2006; Leigh 2007), and is a problem of immense importance to the ecological community. A recent computational study by de Aguiar et al. (2009) demonstrated that a model incorporating assortative mating on a neutral landscape could still undergo speciation. Our result confirms that of de Aguiar, but has the advantage of being demonstrated in a model which is simpler and requires fewer assumptions. Again, this provides the advantage of accessibility to investigation from the perspective of statistical physics and complex systems theory. Evolutionary problems have occasionally been explored by physicists in the context of self-organized criticality and complex systems theory; Bedau and Packard (2003) refer to this issue in their discussion of a transition to the "edge of disorder". One of the most oft-cited examples of an attempt to place evolutionary dynamics in the context of the physics of criticality is the "sandpile" model of punctuated equilibrium developed by Sneppen and Bak, in which bursts of extinction in simulated species follow punctuated, scale-free, power-law behavior (Bak and Sneppen 1993; Sneppen et al. 1995). The present model, however, is particularly unique and powerful in that it allows the investigation of critical scaling and phase transition behavior in the context of a model of speciation -- in other words, a model for the crystallizing of one evolutionary scale (individuals) into another (species).