In the thesis work we investigate four problems connected with dynamical processes in condensed medium, using different techniques of equilibrium and non-equilibrium statistical mechanics. Biology is rich in dynamical events ranging from processes involving single molecule [1] to collective phenomena [2]. In cell biology, translocation and transport processes of biological molecules constitute an important class of dynamical phenomena occurring in condensed phase. Examples include protein transport through membrane channels, gene transfer between bacteria, injection of DNA from virus head to the host cell, protein transport thorough the nuclear pores etc. We present a theoretical description of the problem of protein transport across the nuclear pore complex [3]. These nuclear pore complexes (NPCs) [4] are very selective filters that monitor the transport between the cytoplasm and the nucleoplasm. Two models have been suggested for the plug of the NPC. The first suggests that the plug is a reversible hydrogel while the other suggests that it is a polymer brush. In the thesis, we propose a model for the transport of a protein through the plug, which is treated as elastic continuum, which is general enough to cover both the models. The protein stretches the plug and creates a local deformation, which together with the protein is referred to as the bubble. The relevant coordinate describing the transport is the center of the bubble. We write down an expression for the energy of the system, which is used to analyze the motion. It shows that the bubble executes a random walk, within the gel. We find that for faster relaxation of the gel, the diffusion of the bubble is greater. Further, on adopting the same kind of free energy for the brush too, one finds that though the energy cost for the entry of the particle is small but the diffusion coefficient is much lower and hence, explanation of the rapid diffusion of the particle across the nuclear pore complex is easier within the gel model. In chemical physics, processes occurring in condensed phases like liquid or solid often involve barrier crossing. Simplest possible description of rate for such barrier crossing phenomena is given by the transition state theory [5]. One can go one step further by introducing the effect of the environment by incorporating phenomenological friction as is done in Kramer’s theory [6]. The “method of reactive flux” [7, 8] in chemical physics allows one to calculate the time dependent rate constant for a process involving large barrier by expressing the rate as an ensemble average of an infinite number of trajectories starting at the barrier top and ending on the product side at a specified later time. We compute the time dependent transmission coefficient using this method for a structureless particle surmounting a one dimensional inverted parabolic barrier. The work shows an elegant way of combining the traditional system plus reservoir model [9] and the method of reactive flux [7] and the normal mode analysis approach by Pollak [10] to calculate the time dependent transmission coefficient [11]. As expected our formula for the time dependent rate constant becomes equal to the transition state rate constant when one takes the zero time limit. Similarly Kramers rate constant is obtained by taking infinite time limit. Finally we conclude by noting that the method of analyzing the coupled Hamiltonian, introduced by Pollak is very powerful and it enables us to obtain analytical expressions for the time dependent reaction rate in case of Ohmic dissipation, even in underdamped case. The theory of first passage time [12] is one of the most important topics of research in chemical physics. As a model problem we consider a particle executing Brownian motion in full phase space with an absorbing boundary condition at a point in the position space we derive a very general expression of the survival probability and the first passage time distribution, irrespective of the statistical nature of the dynamics. Also using the prescription adopted elsewhere [13] we define a bound to the actual survival probability and an approximate first passage time distribution which are expressed in terms of the position-position, velocity-velocity and position-velocity variances. Knowledge of these variances enables one to compute the survival probability and consequently the first passage distribution function. We compute both the quantities for gaussian Markovian process and also for non-Markovian dynamics. Our analysis shows that the survival probability decays exponentially at the long time, irrespective of the nature of the dynamics with an exponent equal to the transition state rate constant [14]. Although the field of equilibrium thermodynamics and equilibrium statistical mechanics are well explored, there existed almost no theory for systems arbitrarily far from equilibrium until the advent of fluctuation theorems (FTs)[15] in mid 90�s. In general, these fluctuation theorems have provided a general prescription on energy exchanges that take place between a system and its surroundings under general nonequilibrium conditions and explain how macroscopic irreversibility appears naturally in systems that obey time reversible microscopic dynamics. Based on a Hamiltonian description we present a rigorous derivation [16] of the transient state work fluctuation theorem and the Jarzynski equality [17] for a classical harmonic oscillator linearly coupled to a harmonic heat bath, which is dragged by an external agent. Coupling with the bath makes the dynamics dissipative. Since we do not assume anything about the spectral nature of the harmonic bath the derivation is valid for a general non-Ohmic bath.

Membrane protein dynamics is of great importance for living organisms. The precise localization of proteins composing a synapse on the membrane facing a nerve terminus is essential for proper functioning of the nervous system. In muscle fibers, the nicotinic acetylcholine is densely packed under the motor nerve termini. A receptor associated protein, rapsyn, acts as a linker between the receptor and the other components of the synaptic suramolecular assembly. Advances in fluorescence microscopy have allowed to measure the behavior of a single receptor in the cell membrane. In this work single-molecule microscopy was used to track the motion of ionotropic acetylcholine (nAChR) and serotonin (5HT3R) receptors in the plasma membrane of cells. We present methods for measuring single-molecule diffusion and their analysis. Single molecule tracking has shown a high dependence of acetylcholine receptors diffusion on its associated protein rapsyn. Comparing muscle cells that either express rapsyn or are devoid of it, we found that rapsyn plays an important role on receptor immobilization. A three-fold increase of receptor mobility was observed in muscle cells devoid of rapsyn. However, in these cells, a certain fraction of immobilized receptors was also found immobile. Furthermore, nAChR were strongly confined in membrane domains of few tens of nanometers. This showed that membrane composition and membrane associated proteins influence on receptor localization. During muscle cell differentiation, the fraction of immobile nAChR diminished along with the decreasing nAChR and stable rapsyn expression levels. The importance of rapsyn in nAChR immobilization has been further confirmed by measurements in HEK 293 cells, where co-expression of rapsyn increased immobilization of the receptor. nAChR is a ligand-gated ion-channel of the Cys-loop family. In mammals, members of this receptor family share general structural and functional features. They are homo- or hetero-pentamers and form a membrane-spanning ion channel. Subunits have three major regions, an extracellular ligand binding domain, a transmembrane channel and a large intracellular loop. 5HT3R was used as a model to study the effect of this loop on receptor mobility. Single-molecule tracking experiments on receptors with progressively larger deletions in the intracellular loop did not show a dependence of the size of the loop on the diffusion coefficient of mobile receptors. However, two regions were identified to play a role in receptor mobility by changing the fractions of immobile and directed receptors. Interestingly, a prokaryotic homologue of cys-loop receptors, ELIC, devoid of a large cytoplasmic loop was found to be immobile or to show directed diffusion similar as the wild-type 5HT3R. The scaffolding protein rapsyn stabilizes nAChR clusters in a concentration dependent manner. We have measured the density and self-interactions of rapsyn using FRET microscopy. Point-mutations of rapsyn, known to provoke myopathies, destabilized rapsyn self-interactions. Rapsyn-N88K, and R91L were found at high concentration in the cytoplasm suggesting that this modification disturbs membrane association of rapsyn. A25V was found to accumulate in the endoplasmic reticulum. Fluorescent tools to measure intracellular concentration of calcium ions are of great value to study the function of neurons. Rapsyn is highly abundant at the neuromuscular junction and thus is a genuine synaptic marker. A fusion protein of rapsyn with a genetically encoded ratiometric calcium sensor has been made to probe synapse activity. This thesis has shown that the combined use of biologically relevant system and modern fluorescence microscopy techniques deliver important information on pLGIC behaviour in the cell membrane. / <p>QC 20151217</p>