Capturing protein dynamics in biological crowded environments is essential for under- standing cellular function. In this project, we have explored the dynamic properties of apoferritin in aqueous solutions under varying conditions, including different temper- ature, solvent viscosity and protein concentrations. Dynamic light scattering (DLS) was applied here at various scattering angles from 90 to 150 degrees and at temperatures 295 K and 263 K on three different samples, including one with 19.5 mg/ml apo- ferritin, 6 mg/ml NaCl and 50% glycerol in volume fraction, one with 20 mg/ml apoferritin and 6 mg/ml NaCl, and one with 196.7 mg/ml apoferritin, 6 mg/ml NaCl and 50% glycerol in volume fraction. With the intensity autocorrelation func- tions from DLS measurements, the corresponding diffusion coefficients, hydrodynamic radii and relaxation time constants for each sample under varying conditions were ex- tracted. By comparing with the previous studies, unexpectedly large hydrodynamic radii were noticed and were attributed to undissolved protein crystallites. Still, it can be indicated from our experiment that applying smaller momentum transfer, decreas- ing temperature, increasing solvent viscosity and increasing protein concentration in the solutions can slow down the diffusion dynamics of protein molecules and clusters. Especially by increasing protein concentration, the slowing down of dynamics may be due to crowding effects, as well as increased size of the crystallites. In addition, the data indicate that in all cases, larger solution viscosity can lead to slower diffusivity of proteins.

This thesis deals with several aspects of translational and reorientational dynamics of water molecules confined inside narrow carbon nanotubes. Water molecules confined in a non-polar, nanoscopic pore exhibit extremely unusual structural and dynamical properties. Adding to the list of anomalies which are already present in bulk liquid water, the confined water “chains” and “shells” springs many more surprises. The relatively weak interaction with the surrounding walls in conjuction with the strong inter-water hydrogen bonds lead o several novel structural and dynamical features, very special to this “strange” phase of water. In this thesis, we present our findings on the detailed molecular level description of translational and reorientational dynamics of this novel phase of anomalously “soft” water. Chapter 1 introduces the varied theoretical, numerical and experimental attempts to demystify the properties of bulk, interfacial and confined water. It also motivates the aspects of diffusion in low dimensional systems, which are often termed “anomalous”. In Chapter 2, we study the structure and dynamics of water molecules inside an open ended carbon nanotube placed in a bath of water molecules. The size of the nanotube allows only a single file of water molecules inside the nanotube. The water molecules inside the nanotube show solid-like ordering at room temperature, which we quantify by calculating the pair correlation function. It is shown that even for the longest observation times, the mode of diffusion of the water molecules inside the nanotube is Fickian and not sub-diffusive. We also propose a one-dimensional random walk model for the diffusion of the water molecules inside the nanotube. We find good agreement between the mean-square displacements calculated from the random walk model and from MD simulations, thereby confirming that the water molecules undergo normal-mode diffusion inside the nanotube. We attribute this behavior to strong positional correlations that cause all the water molecules inside the nanotube to move collectively as a single object. The average residence time of the water molecules inside the nanotube is shown to scale quadratically with the nanotube length. In Chapter 3, we study the diffusion of water molecules confined inside narrow (6,6) carbon nanorings. The water molecules form two oppositely polarised clusters. It is shown that the effective interaction between these two clusters is repulsive in nature. The computed mean-squared displacement (MSD) clearly shows a scaling with time, which is consistent with single file diffusion (SFD). The time up to which the water molecules undergo SFD is shown to be the lifetime of the water molecules inside these clusters. The inter-cluster repulsive interactions are electrostatic and hence long-ranged, which is in complete contrast with shorter ranged steric repulsion in other systems which exhibit SFD. In Chapter 4, we study the anisotropic orientational dynamics of water molecules confined in narrow carbon nanotubes and nanorings. We find that confinement leads to strong anisotropy in the orientational relaxation. The relaxation of the aligned dipole moments, occurring on a timescale of nanoseconds, is three order of magnitude slower than that of bulk water. In contrast, the relaxation of the vector joining the two hydrogens is ten times faster compared to bulk, with a timescale of about 150 femtoseconds. The slow dipolar relaxation is mediated by the hopping of orientational defects, which are nucleated by the water molecules outside the tube, across the linear water chain. In Chapter 5, we study the reorientational dynamics of water molecules confined inside narrow carbon nanotubes immersed in a bath of water. Our simulations show that the confined water molecules exhibit bistability in their reorientational relaxation, which proceeds by angular jumps between the two stable states. The energy barrier between these two states is about 2kBT. The effect of non-Markovian jumps shows up in the ratio of the timescales o the first and second order reorientational correlation functions, which exceeds the value of the ratio in the diffusive limit. The analytical solution of a proposed model is also presented, which qualitatively explains this “unusual” relaxation. These results will have important implications in understanding proton conduction in water-filled ion channels. In Chapter 6, we report the thermodynamic aspects of the translational and re-orientational dynamics of the strongly confined water molecules. Considering the energetics it is surprising that the water molecules spontaneously fill up the nanotube. Thus the thermodynamics of entry of water molecules in the hydrophobic cavity of nanotube. This is generally attributed to the rotational entropy gain by the water molecules on entering the tube, a fact which has not been demonstrated quantitatively so far. We show that the gain in rotational component of the entropy compensates the loss of energy of the water molecules upon entering the nanotube. In Chapter 7, we conclude by summarising the work done in the previous chapters and discuss the future course of actions. We would like to extend the studies on the diffusion of water inside finite nanotubes in the presence of bathwater outside, to nanotube lengths, where it is possible to observe the cross-over from an initial “single file” to and eventual, centre of mass dominated, “normal” diffusion. The mean field estimate of the length of the nanotube required so that one observes a crossover from the initial “single file” to “normal” diffusion at 100 ps is about 700 ˚A. Simulation of such a system would possibly provide an unambiguous answer to the question, whether it is possible to observe SFD in finite carbon nanotubes, filled with water. Regarding the reorientational dynamics, we would like to extend our understanding of the reorientational relaxation of water chains to more more complicated structures. Depending on the diameter of the confining nanotube water molecules form polygons of ice. In the present situation each water molecule can be in only two possible states of orientation. Hence, it would be interesting to predict the reorientational dynamics for other ice structures, where each water molecule can be more “orientational states”. In Chapter 8, we report a work which is unrelated to the rest of this thesis. The work has been done in collaboration with Prof. T. V. Ramakrishnan and Prof. Vijay B. Shenoy. We report a novel method for the calculation of elastic constants of a solid in the frame work of Ramakrishnan-Youssouf density functional theory. The structural aspect of the liquid to solid transition and how it affects the elastic constants of the solids is brought out very clearly. The calculation is analytical and we obtain explicit expressions for the elastic constants. The description of the solid is in terms of the structure factor, S(G), of the coexisting liquid. The elastic constants are expressed as a function of equilibrium parameters, such as c(0), relatedto the compressibility of the liquid. Another important quantity on which the elastic constants depend is the curvature, c"(|G|), of c(|q|)curve at its peak (q= G). These quantities are known experimentally for many systems, and can also be calculated accurately. The shear modulus depends only on c"(|G|), while the bulk modulus has contributions from both c"(|G|) and c(0). The obtained elastic constants do not satisfy the Cauchy relations, in that C12 is not equal to C44. Calculations have been performed for two-dimensional square and triangular lattices as well as bcc and fcc lattices in three dimensions. It is seen that in order to get good agreement between the theoretical and the experimental results of the elastic constants, three body correlations have to be introduced in the calculations for the bcc and the fcc lattices. For the last, in which two shells of reciprocal lattice vectors are appropriate, we point out the modifications needed for choosing the lattice parameter in the unstrained freezing problem. We obtain a new, first principles, quasiuniversal relation for elastic constants, scaled by the melting temperature, that is experimentally satisfied. It is similar to the famous Verlet criterion that S(|G|) = 2.9 at freezing and is free of some of the unphysical aspects of previous work.

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