Coalescence Model for Crumpled Globules Formed in Polymer Collapse

Abstract

The rapid collapse of a polymer, due to external forces or changes in solvent, yields a long-lived "crumpled globule." The conjectured fractal structure shaped by hierarchical collapse dynamics has proved difficult to establish, even with large simulations. To unravel this puzzle, we study a coarse-grained model of in-falling spherical blobs that coalesce upon contact. Distances between pairs of monomers are assigned upon their initial coalescence, and do not "equilibrate" subsequently. Surprisingly, the model reproduces quantitatively the dependence of distance on segment length, suggesting that the slow approach to scaling is related to the wide distribution of blob sizes. DOI: 10.1103/PhysRevLett.115.088303 PACS numbers: 82.35.-x, 68.43.Jk, 82.35.Pq The rapid collapse of a polymer into a dense globule is a long-standing problem [1][2][3][4][5][6][7][8][9][10][11][12]. Such a collapse may be triggered by changes in solvent quality, causing the polymer to reduce its solvent-exposed surface area by forming a dense globule. A polymer may also be condensed by active forces, as in the rearrangement of DNA by proteins in the cell nucleus [2]. The rapid collapse does not allow sufficient time for formation of topological entanglements, which abound in an equilibrated compact globule [1][2][3][4]6,13]. It is suggested [1] that during collapse segments of the polymer initially condense to spherelike "blobs," which coarsen upon contact to form larger blobs. At any given time during the process the state is then assumed to be characterized by a single length scale [1,[7][8][9], e.g., the typical size of the blobs or the width of the tube connecting the blobs [see, e.g., Fig. 1(a) [14]]. A central assumption is that when two blobs coalesce they remain more or less segregated within the newly formed structure. This is due to the slow relaxation processes within blobs, and due to topological constraints that prevent polymer segments forming the blobs from freely mixing-unlike a melt of independent polymer segments with open ends [1,[15][16][17][18][19][20]. The final configuration is thus predicted to be a constant density, self-similar, hierarchical structure, known as the "crumpled globule" or "fractal globule" [1-6,11]. The end-to-end distance r m of segments of length m in the resulting globule is predicted to scale as r m ∼ m 1=d in d space dimensions (throughout the Letter d ¼ 3). This is in contrast to the equilibrium state reached at much later times [10], where r m ∼ m 1=2 for small m, saturating at the globule size r max ¼ N 1=d , where N is the length of the polymer [21].These predictions have been tested in several simulations of polymer compaction [2,[4][5][6]11], which generally confirm that the rapidly collapsed state is not entangled, and is indeed different from the equilibrium globule. However, they do not agree upon its fractal nature. In particular, the expected scaling r m ∼ m 1=d has not been conclusively confirmed, even with the largest size simulations (recently extended to polymers of up to 250 ...

Kinetic mechanisms of crumpled globule formation

“…After α L reaches its minimum, it increases due to mixing in the already collapsed "blobs" up to the maximal value α L = 1/2, corresponding to the equilibrium globule state. Theoretical studies and computer simulation results show [11][12][13]16], that chain segments, which form distinct blobs, remain segregated for a relatively long time and can not freely mix immediately after coagulation of blobs due to topological constraints. This happens because chain can not cross itself, and there are no open chain ends (except the two terminal ends), as in the polymer solution of many chains.…”

“…Blobs grow in size continuously, merging together, and the chain collapses globally. However, size distribution of blobs during collapse under active compression can be very nontrivial [12]. It is also unclear how chain structure changes in time inside blobs during collapse, and theoretical works do not describe this evolution.…”

“…It is also unclear how chain structure changes in time inside blobs during collapse, and theoretical works do not describe this evolution. It was suggested, that the partially mixed crumpled globule is stabilized on a large scale by topological constraints [11][12][13] after global collapse of the chain. Regulation of stability of the crumpled globule steady-state is also an important unsolved problem [10,11].…”

Concepts of polymer statistical topology

“…Studying the relaxation properties of the elastic network of contacts in a crumpled globule, we showed that the dynamic properties of hierarchically folded polymer chains in globular phase are similar to those of natural molecular machines (like myosin, for example). We discuss the potential ways of implementations of such artificial molecular machine in computer and real experiments, paying attention to the conditions necessary for stabilization of crumples under the fractal globule formation in the polymer chain collapse [64] (see also the work on coalescence of crumples in polymer collapse [96]). …”

FISH-ing for captured contacts: towards reconciling FISH and 3C

“…For most such polymer systems, at distances much larger than the persistence length, mean spatial distance, R, monotonically increases, and contact frequency, P c , monotonically decreases as a function of the number of intervening monomers, s. Given a pair of loci, the number of monomers between them, s, fully specifies their contact probability and spatial distance. For example, in a random walk, the distance between monomers, R(s) is proportional to s 1/2 , while the contact probability is inversely proportional to the volume occupied by a section of the polymer, P c (s) ~ R(s) -3 = s -3/2 Slightly different, yet still directly proportional, relationships can be seen in other polymer ensembles, including the fractal globule and unknotted globules (Bunin and Kardar, 2015;Imakaev et al, 2015a) (to be discussed further in future work). Here, we see this typical, monotonic behavior, in the region of our simulated polymer that lies far from the dynamic loop (Supplemental Fig 2).…”