Polymers and filaments in confined geometries

Imagine the filaments forming the cytoskeleton of a cell.  They are confined within the cell membrane.  Not only do these filaments influence how the cell might  behave mechanically, but how the filaments behave is coupled to the restricted space in which they grow, arrange or combine into networks.  Similarly one can think of polymers inside voids of another material, filling cracks or moving between the fillers in an elastomeric material.

For many years we have been interested in the how geometry and stiffness of polymer chains work together.  Mainly, this has been through the development of theoretical tools to describe such systems (see paper with Jerry Percus and Harry Frisch; there are others that will appear here shortly) but also recently through a string of simulations performed with Arash Azari at Stellenbosch University and the Centre for High Performance Computing in South Africa.

In a recent paper we investigated polymers of alternating stiffness all confined within a pore.  We observe how polymers become separated due to their stiffness.

Theoretical tools (using field-theoretical methods) allow careful analytical and perturbative ways to understand the role of networking, stiffness and inter- and intra-chain interactions.

Active systems

Molecular machines are responsible for a variety of non-equilibrium actions. They can make cells themselves move about, and perhaps exhibit fascinating collective motion, and the are responsible, amongst other things, for the transportation of cargo, or play a role in the tightening of the contractile ring during cells division.

A range of molecular motors move along the filamentous structures, microtubules and actin filaments, in the cell.  It is interesting and challenging to ask and understand how molecular motors when connected in different ways start modifying the properties of the networks of filaments, or how their functioning alters the mechanical and conformational, and other dynamical properties of a variety of structures.

There is a wide range of interesting introductory articles by various authors, for example, check out this one http://www.nature.com/news/the-physics-of-life-1.19105?WT.mc_id=FBK_NatureNews.

Motility assays

A system with a relatively simple set-up is an array of filaments that move on top of a flat bed of molecular motors, a so-called motility assay. When a large number of motor heads grabs, moves, detaches to a filament, they cause it to move, but also to respond to an applied force.  Under certain conditions, we found that an instability occurs [Banerjee, et al.].  Current work is related to dealing with interactions of filaments.

Above: A single filament moving on a surface with tethered motors.   Cartoon for physical content of Banerjee paper.  (Copyright Kristian Müller-Nedebock)

Stepping mechanism formalism

In the project with Janusz Meylahn we introduced novel ways to formulate the stepping action of motor heads using ideas from dynamical networking theories.  We hope to present these results very soon.

Contractile rings

Can we think about the tension in contractile rings, given the latest, fascinating experiment on filament orientation and the motors linking the rings?  This is part of the project Stanard Mebwe Pachong is investigating in her research.

Contractile rings are formed of actin filaments and molecular machines (and possible other components) to actively pinch of the cell membrane after the division of genetic material. A ring of these filaments forms and then contracts ever more tightly.

We are currently studying these systems using analytical dynamical approaches. These are complemented by some simple Langevin dynamics simulations.

 

Above: A simulations of the motions of actin filaments (some oriented clockwise and others counter-clockwise) under the action of networking and active forces. Two filaments with opposite orientations have been highlighted. (Copyright Kristian Müller-Nedebock)

Active gels

When networks of filaments are formed (which is typical of the cytoskeleton) pairs of molecular motor heads, for example, can link different strands in addition (perhaps) to other crosslinks, forming an active gel.  In past years two MSc theses on these topics have been completed in the group:  a general network formalism (using ideas from equilibrium networks in polymer physics) and a dynamical perspective.  The work was done by former members Mohau Mateyisi and Karl Möller.

Dealing with entanglements

Our recent paper (See post here; CM Rohwer, KK Müller-Nedebock “Operator Formalism for Topology-Conserving Crossing Dynamics in Planar Knot Diagrams”, Journal of Statistical Physics 159, 120-157 (2015), http://dx.doi.org/10.1007/s10955-014-1161-1) introduces a dynamical perspective to the venerable polymer physics problem of conserving the state of entanglement in polymer loops.

To understand what this means we show the example of a single strand (polymer) that forms a closed loop, but, as in the accompanying figure of the so-called trefoil knot, trefoilcannot be manipulated into a simple circular form without cutting and re-glueing the strand.  We can now ask how one can characterise the strand as it is subjected to continuous changes of its configurations due to the environment in which it finds itself. (Think of a polymer loop in a fluid.)  As it reconfigures its shape changes, but the topology of the knot must remain invariant.  The statistical physics entails computing the sum over all possible configurations maintaining the topology.  Great papers by Edwards and Frisch and Wasserman have done this elegantly.  Yet many severe and interesting mathematical challenges still face the enumeration of conformations, but also just in checking whether two knots are equivalent in a way that can be suitably implemented in statistical physics.

Our dynamical formalism is one possible approach in which one can start thinking about a dynamical route to exploring the possible permissible configurations (albeit in the reduced two-dimensional crossings in the projection of the knot only) without needing to compute and compare knot invariants.

Previously, we also published an alternative enumeration scheme to knots in which the winding number is preserved (See the related post; CM Rohwer, KK Müller-Nedebock, F-E Mpiana Mulamba “Conservation of polymer winding states: a combinatoric approach” J Phys A-Math Theor 47, 065001 (2014) DOI: http://dx.doi.org/10.1088/1751-8113/47/6/065001)