Improved Measures for the Shape of a Disordered Polymer To Test a Mean-Field Theory of Collapse

Changes in volume play a significant but often under-emphasized role in governing the stability of proteins in the cell.1!6 When proteins fold from extended unfolded state to a compact or folded native state, there is a concomitant decrease in the characteristic volume occupied by the protein as probed by osmolyte particles, and thus an increase in the volume available to osmolytes or crowding particles. In such processes that involve changes in polymer density concomitant with collapse, the effective size of the polymer is of interest. Accurate measures of the size of a self-avoiding walk, which can approximate configurations in the unfolded ensemble of a protein, have a long history.7!9 Early on, it was recognized by Kuhn using probability arguments for ideal chains, that ideal chains had inherent anisotropy. The configurations of polymer chains are better represented by generalizing quantities such as the end to end distance or radius of gyration, which are amenable to analytic formulation11!13 to tensor quantities such as the radius of gyration tensor S (defined in 5 below), first studied by !Solc and Stockmayer for on-lattice ideal chains. Invariant polynomials constructed from the components of the radius of gyration tensor S allowed the anisotropy of polymer configurations to be treated analytically by field theoretical methods such as ε = 4 ! d expansions or 1/d expansions. One such anisotropy parameter is the asphericityΔ (0eΔe 1), defined through ratios of the trace (tr) of the radius of gyration tensor: