Approximate scaling laws for MRSD and non-crossing distance per residue

<p><b>, across proteins and for domains within a single protein</b>. (A) MRSD (blue circles) as a function of chain length for our protein dataset. The slope of the best fit line on the log-log plot gives the power law scaling: . Non-crossing distance per residue (red circles) <i>vs. </i> shows much larger scatter across native topologies, but follows an approximate scaling law which is superextensive, indicating an increasing importance of chain non-crossing per residue as system size is increased. At system sizes larger than , even minimal motion is dominated by entanglement. (B) Same quantities as in panel (A), but for the domains in proteins 2A5E and 2HA8. The scaling laws are different than in panel (A), and show stronger chain-length dependence. For 2HA8, domains 1, 2 and 1-2 together (the full protein) are considered; for 2A5E domains 1,2,3,4, 1-2, 2-3, 3-4, 1-2-3, 2-3-4, and 1-2-3-4 (the full protein) are considered. Based on these scaling laws found by building up proteins from subdomains, at system sizes larger than , minimal pathways become entanglement-dominated. (C) Schematic renderings of the domains, color-coded in 2HA8 (left) and 2A5E (right).