Research in our group encompasses different areas of condensed matter theory, including the statistical mechanics of complex fluids, polymer physics and stochastic processes. Over the last few years, we have been especially interested in developing theoretical models of the dynamics and reactivity of biological macromolecules, with a view to understanding cellular events at the molecular level. Among the problems that we have been pursuing are the following

The ability to confine single molecules of DNA or protein to nano-sized domains, and to make measurements of their properties under these conditions, is one of the many recent technological developments that has made it possible to explore different aspects of cell biophysics in vitro. In this context, we have been trying to understand confinement effects on polymer dynamics in rigorous statistical mechanical terms, without appeal to phenomenological models. These efforts are moving along some of the following lines of research:

1. Calculations of the transport and conformational properties of continuum chains in narrow slits

   We have been able to show that on length scales of the inter-wall separation, hydrodynamic interactions between different chain segments – usually quite strong – are screened out, and that the diffusivity and relaxation times of Gaussian chains exhibit power laws in the chain length, with logarithmic corrections.

2. Calculations of flow effects on polymers in capillaries

   Although widely studied in the bulk, polymer-flow interactions have been much less studied in the presence of surfaces. Using a Rouse-Zimm approach, along with a variant of the nonlinear elastic model of chain inextensibility, we have determined the steady-state extension of a sheared polymer in a narrow cylindrical tube, and have found that the chain’s fractional extension is much smaller than its bulk value. This finding appears to reflect the screening of hydrodynamic interactions by the surface.

3. Calculations of reactivity in enclosed viscoelastic media

   Biochemical reactions inside cells are typically subject to the effects both of the cell’s boundaries and of the viscoelasticity of its contents. To probe these effects further, we have been studying how they influence one particular reaction – the diffusion-limited cyclization of long polymers. Specifically, we have determined how the cyclization time of a flexible chain scales with its length when the chain is confined to a sphere, and when its dynamics are modulated by colored Gaussian noise

4. Calculations of the stochastic binding of tethered polymers to neutral or reactive surfaces

   Communication between living cells is often mediated by the binding of proteins or other polymers to surface-attached receptors, and is typically stochastic. We have been developing methods to explore the first-passage dynamics of polymers that interact with planar surfaces, and have shown how these methods yield information about binding-time statistics that can be compared with experiment.

These various approaches to confined polymer dynamics are currently being extended in different directions.

Thermodynamics at the scale of single molecules is characterized by a marked sensitivity of its measurable properties to the effects of fluctuations. As a result, measurements of thermodynamic quantities in different samples prepared under nominally identical conditions do not always yield the same value. However, on the basis of very general principles of non-equilibrium statistical mechanics, it is believed that the distributions of these values (for selected stochastic variables) must satisfy certain mathematical constraints that are now referred to as fluctuation theorems. These theorems have been widely applied to the interpretation of thermodynamic measurements on single molecules, and research in the area of small system thermodynamics is proving to be fertile ground for generating new and interesting mathematical connections between theory and experiment. In this context, we have been using path integrals to explore the nature of the distributions that govern the fluctuations of different single-molecule systems, including the work fluctuations of an elastic dumbbell in planar elongational flow, the force fluctuations of an oscillator model of polymer stretching at constant velocity, and heat fluctuations in harmonically trapped Brownian oscillators, Brownian oscillators in magnetic fields, and coupled Brownian oscillators in dual temperature heat baths.