  We present a simple theory that uses thermodynamic parameters to predict the
probability that a protein retains the wildtype structure after one or more
random amino acid substitutions. Our theory predicts that for large numbers of
substitutions the probability that a protein retains its structure will decline
exponentially with the number of substitutions, with the severity of this
decline determined by properties of the structure. Our theory also predicts
that a protein can gain extra robustness to the first few substitutions by
increasing its thermodynamic stability. We validate our theory with simulations
on lattice protein models and by showing that it quantitatively predicts
previously published experimental measurements on subtilisin and our own
measurements on variants of TEM1 beta-lactamase. Our work unifies observations
about the clustering of functional proteins in sequence space, and provides a
basis for interpreting the response of proteins to substitutions in protein
engineering applications.
