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The synchrony / asynchrony dichotomy prevalent in models of genetic regulatory networks can be replaced by a parameter, s, which is the probability of a node being updated in a single time step. Here we apply the idea of such parameterized synchrony to study the dynamics of the genetic regulatory network extracted from an artificial genome model. We find that the relationship between degree of synchrony and the number of limit cycles is not linear. The number and length of limit cycles peaks at intermediate values of s. The proportion of state space explored and the length of transient trajectories also follows this pattern. The richer behavior found at intermediate values of the synchrony parameter is much more characteristic of biological systems than either full synchrony or complete asynchrony.