Wagner's canalization model

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Abstract

Wagner (1996, Does evolutionary plasticity evolve? Evolution 50, 1008–1023.) and Siegal and Bergman, 2002, Azevedo et al., 2006 have studied a simple model of the evolution of a network of N genes, in order to explain the observed phenomenon that systems evolve to be robust. These authors primarily considered the case N=10 and used simulations to reach their conclusions. Here we investigate this model in more detail, considering systems of different sizes with and without recombination, and with selection for convergence instead of to a specified limit. For the simpler evolutionary model lacking recombination, we analyze the system as a neutral network. This allows us to describe the equilibrium distribution networks within genotype space. Our results show that, given a sufficiently large population size, the qualitative observation that systems evolve to be robust, is itself robust, as it does not depend on the details of the model. In simple terms, robust systems have more viable offspring, so the evolution of robustness is merely selection for increased fecundity, an observation that is well known in the theory of neutral networks.

Introduction

The idea that wild-type genotypes are mutationally robust, i.e., are buffered against the effect of mutation goes back to Waddington (1942), an effect he called canalization. Waddington argued that for a well adapted trait almost all mutations with an effect are deleterious. For this reason, any modifier that reduces the effect of mutations and thus keeps the trait closer to optimum should be selected. Following the classical work of Schmalhausen (1949), Waddington (1957) and their contemporaries, research on robustness experienced a decline in the 1970s and 1980s. However, in the early 1990s as powerful molecular techniques to track and manipulate genotypes became routine, there was a renewed interest in the issue of genetic robustness.

Scharloo (1991) describes the results of a number of early experiments with Drosophila. More recently Rutherford and Lindquist (1998) and Queitsch et al. (2002) have examined the role of the heat shock protein Hsp90. All these experiments show increased phenotypic variance in populations that carry a major mutation or are exposed to environmental stress and this is interpreted as evidence for the existence of an evolutionary buffer capable of hiding and then releasing genetic variation. See Stearns (2002) and Niven (2004) for more discussion of the experiments.

The evolution of mechanisms underlying the buffering of the phenotype against genetic and environmental influences has received much theoretical attention, yet many issues remain unresolved. For discussions see Meiklejohn and Hartl (2002), Gibson and Wagner (2000), or De Visser et al. (2003). There have been a number of different approaches to modeling, see e.g., Hermisson et al. (2003), Hermisson and Wagner (2004). This paper follows the approach of Wagner (1996) and Siegal and Bergman (2002) who investigated an interacting network of N genes described by an N×N matrix where 1= on and -1= off . Masel (2004) later investigated a variant in which 1= on and 0= off.

In Section 2 we will describe the original version of Wagner's model and Masel's modification of the model. Section 3 introduces three population dynamics: one due to Wagner (1996), one to Siegal and Bergman (2002), and an intermediate model which is a hybrid of the two. Section 4 is devoted to a detailed study of the 2×2 case and a use of Markov chain theory to derive a result for the asymptotic behavior in an infinite population for a general N. In Section 5, we investigate systems with N=4,7, and 10 genes by simulation. In Section 6, we discuss the conclusions we have reached based on our analytical results and simulations.

Section snippets

The network model

Following Wagner (1996) we consider a finite population of M randomly mating individuals, each of which has an interacting network of N genes. These interactions are represented by an N×N matrix W, whose elements wij indicate the effect on gene i of the product of gene j, which may involve activation (wij>0) or repression (wij<0). Changing expression levels are modeled by the set of difference equationsSi(t+1)=σj=1NwijSj(t),where σ is the sign function; σ(x)=1 if x>0, σ(x)=-1 if x<0, and σ(0)=0

Population dynamics

For simplicity, we will describe the simulations for the 4×4 case. We consider three different evolutionary simulations. For all three, we assume the Wagner mapping for the network dynamics and for comparison in one of the scenarios we apply the 0, 1 map.

In the first simulation, which is similar to the approach of Wagner (1996), we have a fixed optimum phenotype, which without loss of generality we can suppose is Sopt=(1,1,1,1). To generate the founding population, we generate 10,000 random

Markov chain model

To gain further insight into the structure of the Wagner map, we will look at the simplest nontrivial case. In the 2×2 we can explicitly draw out what the trajectories on the {-1,1}2 state space will look like. Fig. 2 has all the possible types of systems and how many of each type occur (by permuting the positions of orbits and fixed points while still keeping the system antisymmetric). The antisymmetry is geometrically an invariance through the center of the square whose corners represent the

Path length and probability of viability for N=4,7,10

In higher dimensions it is impossible to visualize the distribution of even one row, so we instead investigate statistics associated with the mapping. We compute the mean path length at each time point for each run and then take the average over the number of runs. Path length is the number of steps that the individual takes to reach an equilibrium (fixed point), and a run simply means that a population of networks has been evolved for a fixed number of generations which is 200 for the first

Discussion

We have examined a model of the evolution of a population of genetic networks that extends earlier work of Wagner (1996) and Siegal and Bergman (2002), who investigated particular cases of the model by simulation. The aim of this investigation is to obtain a more thorough understanding of the properties of the model, in order to better understand the conclusions that can be drawn from the observed behavior. As in previous work, we find that the networks evolve to be more robust. We find these

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1

Supported by a Cornell-Sloan fellowship.

2

Partially supported by NSF grants from the probability program (0202935) and from a joint DMS/NIGMS initiative to support research in mathematical biology (0201037).

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