Simulating rare events using a Weighted Ensemble-based string method

We introduce an extension to the Weighted Ensemble (WE) path sampling method to restrict sampling to a one dimensional path through a high dimensional phase space. Our method, which is based on the finite-temperature string method, permits efficient sampling of both equilibrium and non-equilibrium systems. Sampling obtained from the WE method guides the adaptive refinement of a Voronoi tessellation of order parameter space, whose generating points, upon convergence, coincide with the principle reaction pathway. We demonstrate the application of this method to several simple, two-dimensional models of driven Brownian motion and to the conformational change of the nitrogen regulatory protein C receiver domain using an elastic network model. The simplicity of the two-dimensional models allows us to directly compare the efficiency of the WE method to conventional brute force simulations and other path sampling algorithms, while the example of protein conformational change demonstrates how the method can be used to efficiently study transitions in the space of many collective variables

Characterization of the PCMBS-dependent modification of KCa3.1 channel gating.

Intermediate conductance, calcium-activated potassium channels are gated by the binding of intracellular Ca(2+) to calmodulin, a Ca(2+)-binding protein that is constitutively associated with the C terminus of the channel. Although previous studies indicated that the pore-lining residues along the C-terminal portion of S6 contribute to the activation mechanism, little is known about whether the nonluminal face of S6 contributes to this process. Here we demonstrate that the sulfhydral reagent, parachloromercuribenze sulfonate (PCMBS), modifies an endogenous cysteine residue predicted to have a nonluminal orientation (Cys(276)) along the sixth transmembrane segment (S6). Modification of Cys(276) manipulates the steady-state and kinetic behavior of the channel by shifting the gating equilibrium toward the open state, resulting in a left shift in apparent Ca(2+) affinity and a slowing in the deactivation process. Using a six-state gating scheme, our analysis shows that PCMBS slows the transition between the open state back to the third closed state. Interpreting this result in the context of the steady-state and kinetic data suggests that PCMBS functions to shift the gating equilibrium toward the open state by disrupting channel closing. In an attempt to understand whether the nonluminal face of S6 participates in the activation mechanism, we conducted a partial tryptophan scan of this region. Substituting a tryptophan for Leu(281) recapitulated the effect on the steady-state and kinetic behavior observed with PCMBS. Considering the predicted nonluminal orientation of Cys(276) and Leu(281), a simple physical interpretation of these results is that the nonluminal face of S6 forms a critical interaction surface mediating the transition into the closed conformation, suggesting the nonluminal C-terminal portion of S6 is allosterically coupled to the activation gate

A Continuum Method for Determining Membrane Protein Insertion Energies and the Problem of Charged Residues

Continuum electrostatic approaches have been extremely successful at describing the charged nature of soluble proteins and how they interact with binding partners. However, it is unclear whether continuum methods can be used to quantitatively understand the energetics of membrane protein insertion and stability. Recent translation experiments suggest that the energy required to insert charged peptides into membranes is much smaller than predicted by present continuum theories. Atomistic simulations have pointed to bilayer inhomogeneity and membrane deformation around buried charged groups as two critical features that are neglected in simpler models. Here, we develop a fully continuum method that circumvents both of these shortcomings by using elasticity theory to determine the shape of the deformed membrane and then subsequently uses this shape to carry out continuum electrostatics calculations. Our method does an excellent job of quantitatively matching results from detailed molecular dynamics simulations at a tiny fraction of the computational cost. We expect that this method will be ideal for studying large membrane protein complexes

A Computational Model of Protein Induced Membrane Morphology with Geodesic Curvature Driven Protein-Membrane Interface

Continuum or hybrid modeling of bilayer membrane morphological dynamics induced by embedded proteins necessitates the identification of protein-membrane interfaces and coupling of deformations of two surfaces. In this article we developed (i) a minimal total geodesic curvature model to describe these interfaces, and (ii) a numerical one-one mapping between two surface through a conformal mapping of each surface to the common middle annulus. Our work provides the first computational tractable approach for determining the interfaces between bilayer and embedded proteins. The one-one mapping allows a convenient coupling of the morphology of two surfaces. We integrated these two new developments into the energetic model of protein-membrane interactions, and developed the full set of numerical methods for the coupled system. Numerical examples are presented to demonstrate (1) the efficiency and robustness of our methods in locating the curves with minimal total geodesic curvature on highly complicated protein surfaces, (2) the usefulness of these interfaces as interior boundaries for membrane deformation, and (3) the rich morphology of bilayer surfaces for different protein-membrane interfaces

Continuum approaches to understanding ion and peptide interactions with the membrane.

Experimental and computational studies have shown that cellular membranes deform to stabilize the inclusion of transmembrane (TM) proteins harboring charge. Recent analysis suggests that membrane bending helps to expose charged and polar residues to the aqueous environment and polar head groups. We previously used elasticity theory to identify membrane distortions that minimize the insertion of charged TM peptides into the membrane. Here, we extend our work by showing that it also provides a novel, computationally efficient method for exploring the energetics of ion and small peptide penetration into membranes. First, we show that the continuum method accurately reproduces energy profiles and membrane shapes generated from molecular simulations of bare ion permeation at a fraction of the computational cost. Next, we demonstrate that the dependence of the ion insertion energy on the membrane thickness arises primarily from the elastic properties of the membrane. Moreover, the continuum model readily provides a free energy decomposition into components not easily determined from molecular dynamics. Finally, we show that the energetics of membrane deformation strongly depend on membrane patch size both for ions and peptides. This dependence is particularly strong for peptides based on simulations of a known amphipathic, membrane binding peptide from the human pathogen Toxoplasma gondii. In total, we address shortcomings and advantages that arise from using a variety of computational methods in distinct biological contexts

Inhibitor binding mode and allosteric regulation of Na+-glucose symporters.

Sodium-dependent glucose transporters (SGLTs) exploit sodium gradients to transport sugars across the plasma membrane. Due to their role in renal sugar reabsorption, SGLTs are targets for the treatment of type 2 diabetes. Current therapeutics are phlorizin derivatives that contain a sugar moiety bound to an aromatic aglycon tail. Here, we develop structural models of human SGLT1/2 in complex with inhibitors by combining computational and functional studies. Inhibitors bind with the sugar moiety in the sugar pocket and the aglycon tail in the extracellular vestibule. The binding poses corroborate mutagenesis studies and suggest a partial closure of the outer gate upon binding. The models also reveal a putative Na+ binding site in hSGLT1 whose disruption reduces the transport stoichiometry to the value observed in hSGLT2 and increases inhibition by aglycon tails. Our work demonstrates that subtype selectivity arises from Na+-regulated outer gate closure and a variable region in extracellular loop EL5

Biophysics at the coffee shop: lessons learned working with George Oster

Over the past 50 years, the use of mathematical models, derived from physical reasoning, to describe molecular and cellular systems has evolved from an art of the few to a cornerstone of biological inquiry. George Oster stood out as a pioneer of this paradigm shift from descriptive to quantitative biology not only through his numerous research accomplishments, but also through the many students and postdocs he mentored over his long career. Those of us fortunate enough to have worked with George agree that his sharp intellect, physical intuition and passion for scientific inquiry not only inspired us as scientists but also greatly influenced the way we conduct research. We would like to share a few important lessons we learned from George in honor of his memory and with the hope that they may inspire future generations of scientists.Comment: 22 pages, 3 figures, accepted in Molecular Biology of the Cel