  A fundamental question for evolutionary biology is why rates of evolution
vary dramatically between proteins. Perhaps surprisingly, it is controversial
how much a protein's functional importance affects its rate of evolution. In
most studies, functional importance has been measured on the coarse scale of
protein knock-outs, while evolutionary rate has been measured on the fine scale
of amino acid substitutions. Here we introduce dynamical influence, a finer
measure of protein functional importance. To measure dynamical influence, we
first use detailed biochemical models of particular reaction networks to
measure the influence of each reaction rate constant on network dynamics. We
then define the dynamical influence of a protein to be the average influence of
the rate constants for all reactions it is involved in.
  Using models of a dozen biochemical systems and sequence data from
vertebrates, we show that dynamical influence and evolutionary rate are
negatively correlated; proteins with greater dynamical influence evolve more
slowly. We also show that proteins with greater dynamical influence are not
more likely to be essential. This suggests that there are many cellular
reactions whose presence is essential for life, but whose quantitative rate is
relatively unimportant to fitness. We also provide evidence that the effect of
dynamical influence on evolutionary rate is independent of protein expression
level, expression specificity, gene compactness, and reaction degree. Dynamical
influence offers a finer view of functional importance, and our results suggest
that focusing on essentiality may have previously led to an underestimation of
the role function plays in protein evolution.
