The behavior of biophysical systems often is quite different when investigated on different length scales and a complete description of such a system typically requires different approaches for these different length scales. At atomic and mesoscopic scales, a considerable number of degrees of freedom are involved. The associated free energy profiles are relatively rugged with energy barriersor kinetic bottlenecks preventing efficient sampling when doing numerical simulations. To alleviate this problem, a variety of sampling strategies have been developed.

In this thesis, I will start with a general background of multi-scale modeling of biophysical systems, and review the different sampling strategies that are being used when doing simulations. Next, I will talk about three of my projects. The first two involve protein binding problems at both the microscopic and mesoscopic levels. The third one is thediffusion-driven phase separation problem inside a cross-linked network of semiflexible polymers. I will use these three projects to illustrate how one can apply multi-scale descriptions and sampling techniques to biophysical systems in and out of equilibrium.

This dissertation aims at improving the current understanding of the physics of mobile impurities in highly correlated liquid-like phases of matter. Impurity problems pose challenging and intricate questions in different realms of many-body physics. For instance, the problem of ''solvation'' of charged solutes in polar solvents, has been the subject of longstanding debates among chemical physicists. The significant role of quantum fluctuations of the solvent, as well as the break down of linear response theory, render the ordinary treatments intractable. Inspired by this complicated problem, we first attempt to understand the role of non-specific quantum fluctuations in the solvation process. To this end, we calculate the dynamic structure factor of a model polar liquid, using the classical Molecular Dynamics (MD) simulations. We verify the failure of linear response approximation in the vicinity of a hydrated electron, by comparing the outcomes of MD simulations with the predictions of linear response theory. This nonlinear behavior is associated with the pronounced peaks of the structure factor, which reflect the strong fluctuations of the local modes. A cavity picture is constructed based on heuristic arguments, which suggests that the electron, along with the surrounding polarization cloud, behave like a frozen sphere, for which the linear response theory is broken inside and valid outside. The inverse radius of the spherical region serves as a UV momentum cutoff for the linear response approximation to be applicable.

The problem of mobile impurities in polar liquids can be also addressed in the framework of the ''polaron'' problem. Polaron is a quasiparticle that typically acquires an extended state at weak couplings, and crossovers to a self-trapped state at strong couplings. Using the analytical fits to the numerically obtained charge-charge structure factor, a phenomenological approach is proposed within the Leggett's influence functional formalism, which derives the effective Euclidean action from the classical equation of motion. We calculate the effective mass of the polaron in the model polar liquid at zero and finite temperatures. The self-trapping transition of this polaron turns out to be discontinuous in certain regions of the phase diagram.

In order to systematically investigate the role of quantum fluctuations on the polaron properties, we adopt a quantum field theory which supports nearly-critical local modes: the quantum Landau-Brazovskii (QLB) model, which exhibits fluctuation-induced first order transition (weak crystallization). In the vicinity of the phase transition, the quantum fluctuations are strongly correlated; one can in principle tune the strength of these fluctuations, by adjusting the parameters close to or away from the transition point. Furthermore, sufficiently close to the transition, the theory accommodates ``soliton'' solutions, signaling the nonlinear response of the system. Therefore, the model seems to be a promising candidate for studying the effects of strong quantum fluctuations and also failure of linear response theory, in the polaron problem. We observe that at zero temperature, and away from the Brazovskii transition where the linear response approximation is valid, the localization transition of the polaron is discontinuous. Upon enhancing fluctuations|of either thermal or quantum nature|the gap of the effective mass closes at distinct second-order critical points. Sufficiently close to the Brazovskii transition where the nonlinear contributions of the field are significantly large, a new state appears in addition to extended and self-trapped polarons: an impurity-induced soliton. We interpret this as the break-down of linear response, reminiscent of what we observe in a polar liquid. Quantum LB model has been proposed to be realizable in ultracold Bose gases in cavities. We thus discuss the experimental feasibility, and propose a setup which is believed to exhibit the aforementioned polaronic and solitonic states.

We eventually generalize the polaron formalism to the case of impurities that couple quadratically to a nearly-critical field; hence called the ''quadratic polaron''. The Hertz-Millis field theory and its generalization to the case of magnetic transition in helimagnets, is taken as a toy model. The phase diagram of the bare model contains both second-order and fluctuation-induced first-order quantum phase transitions. We propose a semi-classical scenario in which the impurity and the field couple quadratically. The polaron properties in the vicinity of these transitions are calculated in different dimensions. We observe that the quadratic coupling in three dimensions, even in the absence of the critical modes with finite wavelength, leads to a jump-like localization of the polaron. In lower dimensions, the transition behavior remains qualitatively similar to those in the case of linear coupling, namely the critical modes must have a finite wavelength to localize the particle.

The systems encountered in biophysics typically straddle the line between solidand fluid, and are thus subject to large deformations even at modest stress val- ues. This, along with the presence of material properties such as visco-elasticity and strain hardening, necessitate the use of non-linear elasticity when describ- ing the physics of these systems. In this dissertation, we explore the use of non-linear elasticity theory in describing soft matter physics, with a focus on the specific example of cavitation in polymer gels and similar systems. We first look at cavitation in the context of equilibrium mechanics. In sim- ple materials that obey neo-Hookean elasticity, we show that compressibility effects strongly enhance cavitation. On the other hand, cavitation phenom- ena in gels of flexible polymers in a binary solvent that phase separates are surprisingly similar to those of incompressible materials. We find that, as a function of the interfacial energy between the two solvent components, there is a sharp transition between cavitation and classical nucleation-and-growth. Fi- nally, biopolymer gels are characterized by strain hardening, and even very low levels of strain hardening are shown to suppress cavitation in polymer gel that obey Flory-Huggins theory in the absence of strain hardening. Next, we explore the dynamics of cavitation in non-linear incompressible mate- rials. We find that, while purely elastic systems can be described entirely within either a Lagrangian or Eulerian frame of reference, and viscous fluids with no elasticity can be described entirely within an Eulerian frame, visco-elastic ma- terials such a Maxwell materials cannot be fully described without making use both frames, translating between the two using a known deformation mapping.

RNA-protein binding is important in many biological processes. These processes depend on the structural properties of the RNA, DNA and proteins involved. In this thesis we have investigated the nature of these interactions at the scale of coarse grained interactions, in atomistic detail and with analytical continuum theories. By exploring the dynamics of these biopolymers at these different scales, we hope to shed light on the important phenomena and characteristics that govern the biological processes involved.

First we show that the structural flexibility of these polymers affects he binding energy measured in MD simulations and can help understand the different packaging scenarios in in-vitro viral reconstitution. Then using analytical models we show how flexibility can aid it non-specific binding given the strength of interaction compared to thermal energy between the binding of polymers.

We finish this thesis by looking at coarse grained brownian dynamics simulations for the bonds in a chain to represent the non-linear viscoelastic behavior of proteins deforming under force. We see both equilibrium mechanical induced melting of the chain and a dynamical phase transition in the response of the chain under a varying drive amplitude and frequency of and AC applied force out of equilibrium. Using the model to understand the dynamics of a deformable chain under force, we explore how this model can provide insight into the binding of deformable structure forming chains that mimic biopolymers like RNA, DNA and proteins.

The statistical physics of complex polymers with branches and circuits is the topic of this dissertation. An important motivation are large, single-stranded (ss) RNA molecules. Such molecules form complex ``secondary" and ``tertiary" structures that can be represented as branched polymers with circuits. Such structures are in part directly determined by the nucleotide sequence and in part subject to thermal fluctuations. The polymer physics literature on molecules in this class has mostly focused on randomly branched polymers without circuits while there has been minimal research on polymers with specific structures and on polymers that contain circuits. The dissertation is composed of three parts: Part I studies branched polymers with thermally fluctuating structure confined to a potential well as a simple model for the encapsidation of viral RNA. Excluded volume interactions were ignored. In Part II, I apply Flory theory to the study of the encapsidation of viral ss RNA molecules with specific branched structures, but without circuits, in the presence of excluded volume interaction. In Part III, I expand on Part II and consider complex polymers with specific structure including both branching and circuits. I introduce a method based on the mathematics of Laplacian matrices that allows me to calculate density profiles for such molecules, which was not possible within Flory theory.