On the size of knots in ring polymers

We give two different, statistically consistent definitions of the length l of a prime knot tied into a polymer ring. In the good solvent regime the polymer is modelled by a self avoiding polygon of N steps on cubic lattice and l is the number of steps over which the knot ``spreads'' in a given configuration. An analysis of extensive Monte Carlo data in equilibrium shows that the probability distribution of l as a function of N obeys a scaling of the form p(l,N) ~ l^(-c) f(l/N^D), with c ~ 1.25 and D ~ 1. Both D and c could be independent of knot type. As a consequence, the knot is weakly localized, i.e. ~ N^t, with t=2-c ~ 0.75. For a ring with fixed knot type, weak localization implies the existence of a peculiar characteristic length l^(nu) ~ N^(t nu). In the scaling ~ N^(nu) (nu ~0.58) of the radius of gyration of the whole ring, this length determines a leading power law correction which is much stronger than that found in the case of unrestricted topology. The existence of such correction is confirmed by an analysis of extensive Monte Carlo data for the radius of gyration. The collapsed regime is studied by introducing in the model sufficiently strong attractive interactions for nearest neighbor sites visited by the self-avoiding polygon. In this regime knot length determinations can be based on the entropic competition between two knotted loops separated by a slip link. These measurements enable us to conclude that each knot is delocalized (t ~ 1).Comment: 29 pages, 14 figure

Abundance of unknots in various models of polymer loops

A veritable zoo of different knots is seen in the ensemble of looped polymer chains, whether created computationally or observed in vitro. At short loop lengths, the spectrum of knots is dominated by the trivial knot (unknot). The fractional abundance of this topological state in the ensemble of all conformations of the loop of $N$ segments follows a decaying exponential form, $\sim \exp (-N/N_0)$, where $N_0$ marks the crossover from a mostly unknotted (ie topologically simple) to a mostly knotted (ie topologically complex) ensemble. In the present work we use computational simulation to look closer into the variation of $N_0$ for a variety of polymer models. Among models examined, $N_0$ is smallest (about 240) for the model with all segments of the same length, it is somewhat larger (305) for Gaussian distributed segments, and can be very large (up to many thousands) when the segment length distribution has a fat power law tail.Comment: 13 pages, 6 color figure

Genomics and drug profiling of fatal TCF3-HLF-positive acute lymphoblastic leukemia identifies recurrent mutation patterns and therapeutic options.

TCF3-HLF-positive acute lymphoblastic leukemia (ALL) is currently incurable. Using an integrated approach, we uncovered distinct mutation, gene expression and drug response profiles in TCF3-HLF-positive and treatment-responsive TCF3-PBX1-positive ALL. We identified recurrent intragenic deletions of PAX5 or VPREB1 in constellation with the fusion of TCF3 and HLF. Moreover somatic mutations in the non-translocated allele of TCF3 and a reduction of PAX5 gene dosage in TCF3-HLF ALL suggest cooperation within a restricted genetic context. The enrichment for stem cell and myeloid features in the TCF3-HLF signature may reflect reprogramming by TCF3-HLF of a lymphoid-committed cell of origin toward a hybrid, drug-resistant hematopoietic state. Drug response profiling of matched patient-derived xenografts revealed a distinct profile for TCF3-HLF ALL with resistance to conventional chemotherapeutics but sensitivity to glucocorticoids, anthracyclines and agents in clinical development. Striking on-target sensitivity was achieved with the BCL2-specific inhibitor venetoclax (ABT-199). This integrated approach thus provides alternative treatment options for this deadly disease

Effect of Knotting on the Shape of Polymers

Momentary configurations of long polymers at thermal equilibrium usually deviate from spherical symmetry and can be better described, on average, by a prolate ellipsoid. The asphericity and nature of asphericity (or prolateness) that describe these momentary ellipsoidal shapes of a polymer are determined by specific expressions involving the three principal moments of inertia calculated for configurations of the polymer. Earlier theoretical studies and numerical simulations have established that as the length of the polymer increases, the average shape for the statistical ensemble of random configurations asymptotically approaches a characteristic universal shape that depends on the solvent quality. It has been established, however, that these universal shapes differ for linear, circular, and branched chains. We investigate here the effect of knotting on the shape of cyclic polymers modeled as random isosegmental polygons. We observe that random polygons forming different knot types reach asymptotic shapes that are distinct from the ensemble average shape. For the same chain length, more complex knots are, on average, more spherical than less complex knots