Research and Projects
Fluctuating Hydrodynamics and Fluid-Structure Interactions Subject to Thermal Fluctuations
Thermal fluctuations and entropic effects play a significant role at the small physical length, time, and energy scales often relevant to phenomena in soft materials, biological systems, and microfabricated devices. Hydrodynamic theories often offer an effective kinetic description of the important roles played by solvent mediated momentum transfer that couples the motions of microstructures. To incorporate the important role of fluctuations we have developed new fluctuating hydrodynamic approaches both for theoretical investigations and computational simulations. This includes the Stochastic Immersed Boundary Method (SIB) for the efficient numerical study of dynamics of elastic structures which interact with a hydrodynamic flow subject to thermal fluctuations. We have also introduced a general framework for development of such methods referred to as the Stochastic Eulerian Lagrangian Method (SELM). We are also working on specific scientific applications with these approaches (see below research for more details).
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Dynamic Implicit Solvent Coarse-Grained Models
LAMMPS Package USER-SELM
Stochastic Immersed Boundary Methods
Stochastic Eulerian Lagrangian Methods
Many coarse-grained models have been developed that treat the solvent implicitly. These have successfully been used for many equilibrium studies in soft-condensed matter physics. However, studies involving dynamics requires incorporating the role of the solvent degrees of freedom in momentum transfer. We have developed a variety of methods to account for solvent mediated effects in such models. This includes building on our prior work on fluctuating hydrodynamics. To facilitate use on specific scientific applications, we have implemented software packages for the popular LAMMPS with our post-doc Yaohong Wang. We are investigating scientific applications in the area of lipid bilayer membranes, polymeric gels, and microfluidics. more information...
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Lipid Bilayer Membranes with Protein Inclusions
Biological lipid bilayer membranes are heterogeneous mixtures of lipid molecules and proteins. Many proteins through their geometry and specific interactions with lipids induce significant changes in the membrane material properties, which can manifest in local stiffness variations and long-range curvature. We are investigating important membrane-protein interactions and the behaviors of heterogeneous membranes important in the kinetics of many cellular processes. To capture such effects, we have developed hybrid continuum-particle descriptions that incorporate hydrodynamic coupling and thermal fluctuations. more information...
Investigations in Biology
Experimental advances are yielding a wealth of quantitative information about the mechanics and localization of processes in cell biology. This is both at the level of single molecules and at more collective levels. This presents many opportunities for theoretical approaches to play a role in the study of biological systems. In collaboration with experimentalists both on-campus at UCSB and off-campus, we are working on several projects that overlap with our other research interests. For example, we are working with the laboratory of Megan Valentine and have developed novel methods for
using passive thermal fluctuations to obtain information from assays about the mechanics of microtubule filaments of the cytoskeleton and the regulatory roles of MAPs. In previous work we have studied the kinesin motor protein and developed a coarse-grained mechanical model consistent with available optical trap experimental data. In current work, we are collaborating with the laboratory of Everett Lipman, Department of Physics, to study FRET signals which yield information about the dynamics of a single helicase motor protein as it moves along DNA.
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Osmotic Phenomena and Electrokinetics
We have developed a number of statistical mechanics approaches for the study of osmotic phenomena and electrokinetics. A central challenge is to formulate tractable mathematical models amenable to analysis and computational simulation. We have developed theory for how microscopic osmotic pressures arise and drive transport in regimes requiring effects beyond the classical the van't Hoff law. In particular, we take into account corrections arising from finite size effects, correlations, and finite-range solute interactions. We are also investigating electrokinetic phenomena in confined geometries relevant to nanofluidic and microfluidic devices.
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Numerical Methods for Stochastic Partial Differential Equations
Underlying much of our research work is the important problem of reliably and efficiently approximating numerically Stochastic Partial Differential Equations (SPDEs). SPDEs pose significant mathematical and computational challenges not present in the corresponding deterministic PDE setting. Solutions to SPDEs are often not classical requiring instead a measure on a space of generalized functions (distributions). These features pose significant challenges in the formulation of reliable numerical approximations. Also, stochastic fluctuations introduce new spatial-temporal scales and sources of stiffness. We are developing new computational and mathematical analysis approaches to cope with these issues. This includes asymptotic methods based on space and time scale separations to obtained reduced SPDE descriptions. We are also developing new computational approaches for finite difference and finite element numerical methods utilizing ideas and intuition from statistical mechanics. Our new approaches allow for general boundary conditions, complex domain geometries, and the use of adaptive spatial meshes. A key challenge these methods address is to properly treat the stochastic driving terms consistently with the introduced space-time discretization errors.
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