Protein Collapse is Encoded in the Folded State Architecture
Folded states of single domain globular proteins, the workhorses in cells, are compact with high packing density. It is known that the radius of gyration, Rg, of both the folded and unfolded (created by adding denaturants) states increase as N{nu} where N is the number of amino acids in the protein. The values of the celebrated Flory exponent{nu} are, respectively, [Formula], and {approx} 0.6 in the folded and unfolded states, which coincide with those found in homopolymers in poor and good solvents, respectively. However, the extent of compaction of the unfolded state of a protein under low denaturant concentration, conditions favoring the formation of the folded state, is unknown. This problem which goes to the heart of how proteins fold, with implications for the evolution of foldable sequences, is unsolved. We develop a theory based on polymer physics concepts that uses the contact map of proteins as input to quantitatively assess collapsibility of proteins. The model, which includes only two-body excluded volume interactions and attractive interactions reflecting the contact map, has only expanded and compact states. Surprisingly, we find that although protein collapsibility is universal, the propensity to be compact depends on the protein architecture. Application of the theory to over two thousand proteins shows that the extent of collapsibility depends not only on N but also on the contact map reflecting the native fold structure. A major prediction of the theory is that {beta}-sheet proteins are far more collapsible than structures dominated by -helices. The theory and the accompanying simulations, validating the theoretical predictions, fully resolve the apparent controversy between conclusions reached using different experimental probes assessing the extent of compaction of a couple proteins. As a by product, we show that the theory correctly predicts the scaling of the collapse temperature of homopolymers as a function of the number of monomers. By calculating the criterion for collapsibility as a function of protein length we provide quantitative insights into the reasons why single domain proteins are small and the physical reasons for the origin of multi-domain proteins. We also show that non-coding RNA molecules, whose collapsibility is similar to proteins with {beta}-sheet structures, must undergo collapse prior to folding, adding support to \"Compactness Selection Hypothesis\" proposed in the context of RNA compaction.