Kinetics of Loop Formation in Polymer Chains

We investigate the kinetics of loop formation in flexible ideal polymer chains (Rouse model), and polymers in good and poor solvents. We show for the Rouse model, using a modification of the theory of Szabo, Schulten, and Schulten, that the time scale for cyclization is $\tau_c\sim \tau_0 N^2$ (where $\tau_0$ is a microscopic time scale and $N$ is the number of monomers), provided the coupling between the relaxation dynamics of the end-to-end vector and the looping dynamics is taken into account. The resulting analytic expression fits the simulation results accurately when $a$, the capture radius for contact formation, exceeds $b$, the average distance between two connected beads. Simulations also show that, when $a < b$, $\tau_c\sim N^{\alpha_\tau}$, where $1.5<{\alpha_\tau}\le 2$ in the range $7<N<200$ used in the simulations. By using a diffusion coefficient that is dependent on the length scales $a$ and $b$ (with $a<b$), which captures the two-stage mechanism by which looping occurs when $a < b$, we obtain an analytic expression for $\tau_c$ that fits the simulation results well. The kinetics of contact formation between the ends of the chain are profoundly affected when interactions between monomers are taken into account. Remarkably, for $N < 100$ the values of $\tau_c$ decrease by more than two orders of magnitude when the solvent quality changes from good to poor. Fits of the simulation data for $\tau_c$ to a power law in $N$ ($\tau_c\sim N^{\alpha_\tau}$) show that $\alpha_\tau$ varies from about 2.4 in a good solvent to about 1.0 in poor solvents. Loop formation in poor solvents, in which the polymer adopts dense, compact globular conformations, occurs by a reptation-like mechanism of the ends of the chain.Comment: 30 pages, 9 figures. Revised version includes a new figure (8) and minor changes to the tex

KINETICS OF POLYMER CYCLIZATION REACTION AND NOVEL COVALENT DNA CROSS-LINKING ASSAYS

In this dissertation I first do an extensive review of polymer cyclization kinetics. Different theories of polymer cyclization kinetics, their assumptions and their predictions are presented along with the predictions of computer simulations. In addition, the experimental results for synthetic and biological polymers are summarized

Theoretical Study of Pulled Polymer Loops as a Model for Fission Yeast Chromosome

In this thesis, we study the physics of the pulled polymer loops motivated by a biological problem of chromosome alignment during meiosis in fission yeast. During prophase I of meiotic fission yeast, the chromosomes form a loop structure by binding their telomeres to the Spindle Pole Body (SPB). SPB nucleates the growth of microtubules in the cytoplasm. Molecular motors attached to the cell membrane can exert the force on the microtubules and thus pull the whole nucleus. The nucleus performs oscillatory motion from one to the other end of the elongated zygote cell. Experimental evidence suggests that these oscillations facilitate homologous chromosome alignment which is required for the gene recombination. Our goal is to understand the physical mechanism of this alignment. We thus propose a model of pulled polymer loops to represent the chromosomal motion during oscillations. Using a freely-jointed bead-rod model for the pulled polymer loop, we solve the equilibrium statistics of the polymer configurations both in 1D and 3D. In 1D, we find a peculiar mapping of the bead-rod system to a system of particles on a lattice. Utilizing the wealth of tools of the particle system, we solve exactly the 1D stationary measure and map it back to the polymer system. To address the looping geometry, the Brownian Bridge technique is employed. The mean and variance of beads position along the loop are discussed in detail both in 1D and 3D. We then can calculate the three-dimensional statistics of the distance between corresponding beads from a pair of loops in order to discuss the pairing problem of homologous chromosomes. The steady-state shape of a three-dimensional pulled polymer loop is quantified using the descriptors based on the gyration tensor. Beyond the steady state statistics, the relaxation dynamics of the pinned polymer loop in a constant external force field is discussed. In 1D we show the mapping of polymer dynamics to the well-known Asymmetric Simple Exclusion Process (ASEP) model. Our pinned polymer loop is mapped to a half-filled ASEP with reflecting boundaries. We solve the ASEP model exactly by using the generalized Bethe ansatz method. Thus with the help of the ASEP theory, the relaxation time of the polymer problem can be calculated analytically. To test our theoretical predictions, extensive simulations are performed. We find that our theory of relaxation time fit very well to the relaxation time of a 3D polymer in the direction of the external force field. Finally, we discuss the relevance of our findings to the problem of chromosome alignment in fission yeast

Molecular motors robustly drive active gels to a critically connected state

Living systems often exhibit internal driving: active, molecular processes drive nonequilibrium phenomena such as metabolism or migration. Active gels constitute a fascinating class of internally driven matter, where molecular motors exert localized stresses inside polymer networks. There is evidence that network crosslinking is required to allow motors to induce macroscopic contraction. Yet a quantitative understanding of how network connectivity enables contraction is lacking. Here we show experimentally that myosin motors contract crosslinked actin polymer networks to clusters with a scale-free size distribution. This critical behavior occurs over an unexpectedly broad range of crosslink concentrations. To understand this robustness, we develop a quantitative model of contractile networks that takes into account network restructuring: motors reduce connectivity by forcing crosslinks to unbind. Paradoxically, to coordinate global contractions, motor activity should be low. Otherwise, motors drive initially well-connected networks to a critical state where ruptures form across the entire network.Comment: Main text: 21 pages, 5 figures. Supplementary Information: 13 pages, 8 figure

Statistical mechanics models in protein association problems

Doctor of PhilosophyDepartment of PhysicsJeremy D. SchmitProtein-Protein interactions can lead to disordered states such as precipitates or gels, or to ordered states such as crystals or microtubules. In order to study the different natures of protein-protein interactions we have developed statistical mechanics models in order to interpret the varied behavior of different protein systems. The main point will be to develop theoretical models that infer the time a length scales that characterize the dynamics of the systems analyzed. This approach seek to facilitate a connection to simulations and experiments, where a high resolution analysis in length and time is possible, since the theories can provide insights about the relevant time and length scales, and also about issues that can appear when studying these systems. The first system studied is monoclonal antibodies in solution. Antibody solutions deviate from the dynamical and rheological response expected for globular proteins, especially as volume fraction is increased. Experimental evidence shows that antibodies can reversibly bind to each other via F[subscript]ab and F[subscript]c domains, and form larger structures (clusters) of several antibodies. Here we present a microscopic equilibrium model to account for the distribution of cluster sizes. Antibody clusters are modeled as polymers that can grow via reversible bonds either between two F[subscript]ab domains or between a F[subscript]ab and a F[subscript]c. We propose that the dynamical and rheological behavior is determined by molecular entanglements of the clusters. This entanglement does not occur at low concentrations where antibody-antibody binding contributes to the viscosity by increasing the effective size of the particles. The model explains the observed shear-thinning behavior of antibody solutions. The second system is protein condensates inside living cells. Biomolecule condensates appear throughout the cell serving a wide variety of functions, but it is not clear how functional properties show in the concentrated network inside the condensate droplets. Here we model disordered proteins as linear polymers formed by "stickers" evenly spaced by "spacers". The spacing between stickers gives rise to different network toplogies inside the condensate droplet, determining distinguishing properties such us density and client binding. The third system is protein-protein binding in a salt solutions. Biomolecular simulations are typically performed in an aqueous environment where the number of ions remains fixed for the duration of the simulation, generally with a number of salt pairs intended to match the macroscopic salt concentration. In contrast, real biomolecules experience local ion environments where the salt concentration is dynamic and may differ from bulk. We develop a statistical mechanics model to account for fluctuations of ions concentrations, and study how it affects the free energy of protein-protein binding

Single polymer dynamics in semi-dilute solutions: linear and ring polymers

Synthetic and biological polymers are ubiquitous in nature and modern technologies. Traditional characterization methods of polymeric materials rely on bulk level measurements that can provide useful information on material properties. However, these methods generally cannot access underlying molecular information, such as polymer conformation, distributions in molecular behavior, and the role of intermolecular interactions in non-equilibrium flows. Over the past two decades, single molecule techniques have been established to investigate molecular-level dynamics, thereby allowing direct access to polymer chain relaxation mechanisms and polymer non-linear response under a variety of flows. Despite recent progress in the field of single polymer dynamics, however, the vast majority of single molecule studies has focused on dilute solutions of linear polymers. In this thesis, we effectively extend single molecule imaging to increasingly complex polymeric systems of increasing polymer concentration and more complex chain architectures. In this way, we aim to address several fundamental questions, including how do polymer concentration and chain architecture affect dynamics at the single chain level? We address these questions using a combination of single molecule experiments and Brownian dynamics simulations. In one project, we performed a series of single molecule experiments by systematically increasing polymer concentration to the semi-dilute untangled regime. Based on these results, we obtained a scaling relation for longest polymer relaxation time as a function of concentration, and these results are compared to blob scaling theories. We further studied single polymer dynamics upon a step-strain deformation in planar extensional flow, including both transient and steady state polymer extension. Experimental data are compared to results from large-scale Brownian dynamics simulations that include intra- and intermolecular hydrodynamic interactions and excluded volume interactions, work performed in collaboration with the Prakash group at Monash University. In this way, we obtain parameter-free predictions of polymer dynamics in non-dilute flows using the method successive fine-graining. Remarkably, our results show a close comparison between experiments and simulation, which provides a solid understanding of polymer dynamics in the semi-dilute concentration regime, both near equilibrium under strong flow. In the second project, we studied the impact of circular polymer or ring polymer topology on single chain dynamics in extensional flows. Single molecule experiments revealed that ring polymers stretch differently compared to linear polymers in extensional flows in the context of the coil-stretch transition. Interestingly, we found that the ring structure exhibits a strong hydrodynamic coupling between the two strands of a stretched ring, which leads to a "slow-down" of the coil-stretch transition and a looping effect of rings under strong extensional flow. Moving beyond our work on single chain dynamics in dilute and semi-dilute solutions, we further sought to identify how molecular-scale interactions are translated into collective non-Newtonian fluid properties. In particular, we developed a new technique to directly measure normal stresses or extensional viscosity in microfluidic devices by coupling the Stokes trap with particle tracking. Here, we study the phenomenon of flow-induced particle migration to measure polymer-induced solution stresses and extensional viscosity in semi-dilute solutions of DNA and synthetic polymers. We combined the automated hydrodynamic trap, which is a home-built microfluidic hydro- dynamic trap, and a piezo-nano positioning stage to directly observe particle migration in polymer solution undergoing planar extensional flow. Experimental data was analyzed in the context of a second-order fluid model in order to determine normal stress. Finally, extensional viscosity was deduced from particle migration experiments, and these results showed favorable comparison to extensional viscosity measurements determined with the optically-detected elastocapillary self-thinning dripping-onto-substrate (ODES-DOS) extensional rheometer. Overall, this thesis aims to provide a fundamental molecular picture of polymer dynamics in the semi-dilute concentration regime and for different polymer architectures. Combining single molecule fluorescence microscopy, Brownian dynamics simulation, 3D particle tracking, and continuum-level constitutive equations, we are able to provide an informative physical picture of polymers in non-Newtonian semi-dilute polymer solutions. From a broad view, this work provides a starting point to relate macroscopic stress response to a microscopic or molecular-level interactions, thereby providing a new perspective to understand non-linear polymer properties

The physics of active polymers and filaments

Active matter agents consume internal energy or extract energy from the environment for locomotion and force generation. Already rather generic models, such as ensembles of active Brownian particles, exhibit phenomena, which are absent at equilibrium, in particular motility-induced phase separation and collective motion. Further intriguing nonequilibrium effects emerge in assemblies of bound active agents as in linear polymers or filaments. The interplay of activity and conformational degrees of freedom gives rise to novel structural and dynamical features of individual polymers as well as in interacting ensembles. Such out-of-equilibrium polymers are an integral part of living matter, ranging from biological cells with filaments propelled by motor proteins in the cytoskeleton, and RNA/DNA in the transcription process, to long swarming bacteria and worms such as Proteus mirabilis and Caenorhabditis elegans, respectively. Even artificial active polymers have been synthesized. The emergent properties of active polymers or filaments depend on the coupling of the active process to their conformational degrees of freedom, aspects which are addressed in this article. The theoretical models for tangentially and isotropically self-propelled or active-bath driven polymers are presented, both in presence and absence of hydrodynamic interactions. The consequences for their conformational and dynamical properties are examined, emphasizing the strong influence of the coupling between activity and hydrodynamic interactions. Particular features of emerging phenomena, induced by steric and hydrodynamic interactions, are highlighted. Various important, yet theoretically unexplored, aspects are featured and future challenges are discussed.Comment: The following article has been accepted by The Journal of Chemical Physics. After it is published, it will be found at https://aip.scitation.org/journal/jc

How important are fluctuations in the treatment of internal friction in polymers?

The Rouse model with internal friction (RIF), a widely used theoretical framework to interpret the effects of internal friction on conformational transitions in biomolecules, is shown to be an approximate treatment that is based on preaveraging internal friction. By comparison with Brownian dynamics simulations of an exact coarse-grained model that incorporates fluctuations in internal friction, the accuracy of the preaveraged model predictions is examined both at and away from equilibrium. While the two models predict intrachain autocorrelations that approach each other for long enough chain segments, they differ in their predictions for shorter segments. Furthermore, the two models differ qualitatively in their predictions for the chain extension and viscosity in shear flow, which is taken to represent a prototypical out-of-equilibrium condition.Comment: 10 pages, 5 figures, additional supplemental materia