Phases of Polymers and Biopolymers

In this thesis we develop coarse grained models aiming at understanding physical problems arising from phase transitions which occur at the single molecule level. The thesis will consist of two parts, grossly related to and motivated by the two subjects dealt with above. In the first half, we will focus on critical phenomena in stretching experiments, namely in DNA unzipping and polymer stretching in a bad solvent. In the second part, we will develop a model of thick polymers, with the goal of understanding the origin of the protein folds and the physics underlying the folding \u2018transition\u2019, as well as with the hope of shedding some light on some of the fundamental questions highlighted in this Introduction. In the first part of the thesis we will introduce a simple model of self-avoiding walks for DNA unzipping. In this way we can map out the phase diagram in the force vs. temperature plane. This reveals the present of an interesting cold unzipping transition. We then go on to study the dynamics of this coarse grained model. The main result which we will discuss is that the unzipping dynamics below the melting temperature obeys different scaling laws with respect to the opening above thermal denaturation, which is governed by temperature induced fluctuating bubbles. Motivated by this and by recent results from other theoretical groups, we move on to study the relation to DNA unzipping of the stretching of a homopolymer below the theta point. Though also in this case a cold unzipping is present in the phase diagram, this situation is richer from the theoretical point of view because the physics depends crucially on dimension: the underlying phase transition indeed is second order in two dimensions and first order in three. This is shown to be intimately linked to the failure of mean field in this phenomena, unlike for DNA unzipping. In particular, the globule unfolds via a series (hierarchy) of minima. In two dimensions they survive in the thermodynamic limit whereas if the dimension, d, is greater than 2, there is a crossover and for very long polymers the intermediate minima disappear. We deem it intriguing that an intermediate step in this minima hierarchy for polymers of finite length in the three-dimensional case is a regular mathematical helix, followed by a zig-zag structure. This is found to be general and almost independent of the interaction potential details. It suggests that a helix, one of the well-known protein secondary structure, is a natural choice for the ground state of a hydrophobic protein which has to withstand an effective pulling force. In the second part, we will follow the inverse route and ask for a minimal model which is able to account for the basic aspects of folding. By this, we mean a model which contains a suitable potential which has as its ground state a protein-like structure and which can account for the known thermodynamical properties of the folding transition. The existing potential which are able to do that[32] are usually constructed \u2018ad hoc\u2019 from knowledge of the native state. We stress that our procedure here is completely different and the model which we propose should be built up starting from minimal assumptions. Our main result is the following. If we throw away the usual view of a polymer as a sequence of hard spheres tethered together by a chain (see also Chapter 1) and substitute it with the notion of a flexible tube with a given thickness, then upon compaction our \u2019thick polymer\u2019 or \u2019tube\u2019 will display a rich secondary structure with protein-like helices and sheets, in sharp contrast with the degenerate and messy crumpled collapsed phase which is found with a conventional bead-and-link or bead-and-spring homopolymer model. Sheets and helices show up as the polymer gets thinner and passes from the swollen to the compact phase. In this sense the most interesting regime is a \u2018twilight\u2019 zone which consists of tubes which are at the edge of the compact phase, and we thus identify them as \u2018marginally compact strucures\u2019. Note the analogy with the result on stretching, in which the helices were in the same way the \u2018last compact\u2019 structures or the \u2018first extended\u2019 ones when the polymer is being unwinded by a force. After this property of ground states is discussed, we proceed to characterize the thermodynamics of a flexible thick polymer with attraction. The resulting phase diagram is shown to have many of the properties which are usually required from protein effective models, namely for thin polymers there is a second order collapse transition (O collapse) followed, as the temperature is lowered, by a first order transition to a semicrystalline phase where the compact phase orders forming long strands all aligned preferentially along some direction. For thicker polymers the transition to this latter phase occurs directly from the swollen phase, upon lowering T, through a first order transition resembling the folding transition of short proteins

Actomyosin contraction induces droplet motility

While cell crawling on a solid surface is relatively well understood, and relies on substrate adhesion, some cells can also swim in the bulk, through mechanisms that are still largely unclear. Here, we propose a minimal model for in-bulk self-motility of a droplet containing an isotropic and compressible contractile gel, representing a cell extract containing a disordered actomyosin network. In our model, contraction mediates a feedback loop between myosin-induced flow and advection-induced myosin accumulation, which leads to clustering and a locally enhanced flow. Interactions of the emerging clusters with the droplet membrane break flow symmetry and set the whole droplet into motion. Depending mainly on the balance between contraction and diffusion, this motion can be either straight or circular. Our simulations and analytical results provide a framework allowing to study in-bulk myosin-driven cell motility in living cells and to design synthetic motile active matter droplets

Topological patterns in two-dimensional gel electrophoresis of DNA knots

Gel electrophoresis is a powerful experimental method to probe the topology of DNA and other biopolymers. While there is a large body of experimental work which allows us to accurately separate different topoisomers of a molecule, a full theoretical understanding of these experiments has not yet been achieved. Here we show that the mobility of DNA knots depends crucially and subtly on the physical properties of the gel, and in particular on the presence of dangling ends. The topological interactions between these and DNA molecules can be described in terms of an “entanglement number”, and yield a non-monotonic mobility at moderate fields. Consequently, in two-dimensional electrophoresis, gel bands display a characteristic arc pattern; this turns into a straight line when the density of dangling ends vanishes. We also provide a novel framework to accurately predict the shape of such arcs as a function of molecule length and topological complexity, which may be used to inform future experiments

Simplifying Topological Entanglements by Entropic Competition of Slip-Links

Topological entanglements are abundant, and often detrimental, in polymeric systems in biology and materials science. Here we theoretically investigate the topological simplification of knots by diffusing slip-links (SLs), which may represent biological or synthetic molecules, such as proteins on the genome or cyclodextrines in slide-ring gels. We find that SLs entropically compete with knots and can localise them, greatly facilitating their downstream simplification by transient strand-crossing. We further show that the efficiency of knot localisation strongly depends on the topology of the SL network and, informed by our findings, discuss potential strategies to control the topology of biological and synthetic materials

Competition between local erasure and long-range spreading of a single biochemical mark leads to epigenetic bistability

The mechanism through which cells determine their fate is intimately related to the spreading of certain biochemical (so-called epigenetic) marks along their genome. The mechanisms behind mark spreading and maintenance are not yet fully understood, and current models often assume a long-range infection-like process for the dynamics of marks, due to the polymeric nature of the chromatin fibre which allows looping between distant sites. While these existing models typically consider antagonising marks, here we propose a qualitatively different scenario which analyses the spreading of a single mark. We define a 1D stochastic model in which mark spreading/infection occurs as a long-range process whereas mark erasure/recovery is a local process, with an enhanced rate at boundaries of infected domains. In the limiting case where our model exhibits absorbing states, we find a first-order-like transition separating the marked/infected phase from the unmarked/recovered phase. This suggests that our model, in this limit, belongs to the long-range compact directed percolation universality class. The abrupt nature of the transition is retained in a more biophysically realistic situation when a basal infection/recovery rate is introduced (thereby removing absorbing states). Close to the transition there is a range of bistability where both the marked/infected and unmarked/recovered states are metastable and long lived, which provides a possible avenue for controlling fate decisions in cells. Increasing the basal infection/recovery rate, we find a second transition between a coherent (marked or unmarked) phase, and a mixed, or random, one.Comment: 11 pages, 7 figures, 2 appendice

Motility-induced phase separation and coarsening in active matter

Active systems, or active matter, are self-driven systems which live, or function, far from equilibrium - a paradigmatic example which we focus on here is provided by a suspension of self-motile particles. Active systems are far from equilibrium because their microscopic constituents constantly consume energy from the environment in order to do work, for instance to propel themselves. The nonequilibrium nature of active matter leads to a variety of non-trivial intriguing phenomena. An important one which has recently been the subject of intense interest among biological and soft matter physicists is that of the so-called "motility-induced phase separation", whereby self-propelled particles accumulate into clusters in the absence of any explicit attractive interactions between them. Here we review the physics of motility-induced phase separation, and discuss this phenomenon within the framework of the classic physics of phase separation and coarsening. We also discuss theories for bacterial colonies where coarsening may be arrested. Most of this work will focus on the case of run-and-tumble and active Brownian particles in the absence of solvent-mediated hydrodynamic interactions - we will briefly discuss at the end their role, which is not currently fully understood in this context.Comment: Contribution to the special issue "Coarsening dynamics", Comptes Rendus de Physique, see https://sites.google.com/site/ppoliti/crp-special-issu