Stable knotted phases in semiflexible polymers
Statistical Physics, Nonlinear Dynamics, Complex Systems, Soft and Fluid Matter, Biological Physics
Final Report Abstract
In this project, we performed Monte Carlo computer simulations to study the occurrence of knots in linear polymer chains. For flexible polymers it is well known that knots of various types form with a certain probability. Recently, we found in the phase diagram of semiflexible polymers modeled by a bead-stick model well-defined regions, where knots of a specific type exist not only by chance but are thermodynamically stable. This means that almost every conformation is characterized by the same knot type. By combining the multicanoncial algorithm with the replica-exchange Monte Carlo simulation method, here we extended our previous results to longer polymers consisting of up to 42 monomers which exhibit a much richer phase structure. An important technical aspect is our careful calibration of the used Monte Carlo move set. By this means we now arrive at a more detailed understanding of the properties of these “knotted” phases, which are good examples for topologically stable phases. For example, we clarify the stability criterion and elucidate why there is a clear phase coexistence at the first-order like transition into the knotted phases, although no latent heat (in the total energy) is observable. To this end we determined two-dimensional free-energy landscapes in the ELJ − Ebend plane, where ELJ and Ebend are the Lennard-Jones and bending energies, respectively. From preliminary simulations, we suspected that knotted conformations are suppressed in the case where the bond length rb is identical to the distance rmin of the minimum of the Lennard-Jones potential acting among non-adjacent monomers. For a better support of this conjecture, we systematically varied in this project the bond length in both, beadstick and bead-spring models, and in a comparative study employing a two-dimensional variant of the replica-exchange method investigated its influence on the occurrence of stable knotted phases. As a main qualitative result we find that the region of knotted phases is wider in the bead-spring model than in the bead-stick model. We could also clarify the puzzle why in a related, but rather implicitly defined bead-spring model no stable knot phases have been reported in the literature. Our simulation results clearly show that the distinguishing parameter is indeed the ratio rb /rmin as suspected. And by (semi-) analytical considerations we present theoretical arguments why for rb /rmin ≈ 1 bent structures are more likely than knots. We also conducted extensive computer simulations for elucidating how an adsorbing substrate influences the knotting propensity of semiflexible polymers. Within a coarsegrained bead-spring model we find that already for moderate substrate attraction (compared to the Lennard-Jones attraction among the monomers) the likelihood to find stable knotted phases is rather low. The simple qualitative reason is the observation that due to the substrate attraction combined with the bending stiffness, a semiflexible polymer surprisingly rapidly forms two-dimensional patterns. Besides that we achieved a deeper understanding of the generic aspects of adsorption of semiflexible polymers in general. Our results of these studies for coarse-grained models also serve as a preparation for more demanding simulations with a focus on knot formation of chemically realistic (biological) polymers in bulk and interacting with a substrate. In the past few years quite impressive progress has been made in preparation and detection of single polymers adsorbed on surfaces, so that it is now possible to experimentally detect the conformations of small polymers with only about 20 monomers.
Publications
- Interplay of adsorption and semiflexibility: Structural behavior of grafted polymers under poor solvent conditions, Macromolecules 50, 4054–4063 (2017)
K.S. Austin, J. Zierenberg, and W. Janke
(See online at https://doi.org/10.1021/acs.macromol.6b02738) - Efficiencies of joint non-local update moves in Monte Carlo simulations of coarse-grained polymers, Comput. Phys. Commun. 224, 222–229 (2018)
K.S. Austin, M. Marenz, and W. Janke
(See online at https://doi.org/10.1016/j.cpc.2017.10.014) - Generalized ensemble computer simulations of macromolecules, invited Ising Lecture Notes 2016, in: Order, Disorder and Criticality: Advanced Problems of Phase Transition Theory, Vol. 5, ed. Y. Holovatch (World Scientific, Singapore, 2018), pp. 173–225
W. Janke
(See online at https://doi.org/10.1142/9789813232105_0004)
