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.

A multiscale model of mechanotransduction by the ankyrin chains of the NOMPC channel.

Our senses of touch and hearing are dependent on the conversion of external mechanical forces into electrical impulses by the opening of mechanosensitive channels in sensory cells. This remarkable feat involves the conversion of a macroscopic mechanical displacement into a subnanoscopic conformational change within the ion channel. The mechanosensitive channel NOMPC, responsible for hearing and touch in flies, is a homotetramer composed of four pore-forming transmembrane domains and four helical chains of 29 ankyrin repeats that extend 150 Å into the cytoplasm. Previous work has shown that the ankyrin chains behave as biological springs under extension and that tethering them to microtubules could be involved in the transmission of external forces to the NOMPC gate. Here we combine normal mode analysis (NMA), full-atom molecular dynamics simulations, and continuum mechanics to characterize the material properties of the chains under extreme compression and extension. NMA reveals that the lowest-frequency modes of motion correspond to fourfold symmetric compression/extension along the channel, and the lowest-frequency symmetric mode for the isolated channel domain involves rotations of the TRP domain, a putative gating element. Finite element modeling reveals that the ankyrin chains behave as a soft spring with a linear, effective spring constantof 22 pN/nm for deflections ≤15 Å. Force-balance analysis shows that the entire channel undergoes rigid body rotation during compression, and more importantly, each chain exerts a positive twisting moment on its respective linker helices and TRP domain. This torque is a model-independent consequence of the bundle geometry and would cause a clockwise rotation of the TRP domain when viewed from the cytoplasm. Force transmission to the channel for compressions >15 Å depends on the nature of helix-helix contact. Our work reveals that compression of the ankyrin chains imparts a rotational torque on the TRP domain, which potentially results in channel opening.

New Continuum Approaches for Determining Protein-Induced Membrane Deformations.

The influence of the membrane on transmembrane proteins is central to a number of biological phenomena, notably the gating of stretch activated ion channels. Conversely, membrane proteins can influence the bilayer, leading to the stabilization of particular membrane shapes, topological changes that occur during vesicle fission and fusion, and shape-dependent protein aggregation. Continuum elastic models of the membrane have been widely used to study protein-membrane interactions. These mathematical approaches produce physically interpretable membrane shapes, energy estimates for the cost of deformation, and a snapshot of the equilibrium configuration. Moreover, elastic models are much less computationally demanding than fully atomistic and coarse-grained simulation methodologies; however, it has been argued that continuum models cannot reproduce the distortions observed in fully atomistic molecular dynamics simulations. We suggest that this failure can be overcome by using chemically and geometrically accurate representations of the protein. Here, we present a fast and reliable hybrid continuum-atomistic model that couples the protein to the membrane. We show that the model is in excellent agreement with fully atomistic simulations of the ion channel gramicidin embedded in a POPC membrane. Our continuum calculations not only reproduce the membrane distortions produced by the channel but also accurately determine the channel's orientation. Finally, we use our method to investigate the role of membrane bending around the charged voltage sensors of the transient receptor potential cation channel TRPV1. We find that membrane deformation significantly stabilizes the energy of insertion of TRPV1 by exposing charged residues on the S4 segment to solution.

Continuum descriptions of membranes and their interaction with proteins: Towards chemically accurate models.

Biological membranes deform in response to resident proteins leading to a coupling between membrane shape and protein localization. Additionally, the membrane influences the function of membrane proteins. Here we review contributions to this field from continuum elastic membrane models focusing on the class of models that couple the protein to the membrane. While it has been argued that continuum models cannot reproduce the distortions observed in fully-atomistic molecular dynamics simulations, we suggest that this failure can be overcome by using chemically accurate representations of the protein. We outline our recent advances along these lines with our hybrid continuum-atomistic model, and we show the model is in excellent agreement with fully-atomistic simulations of the nhTMEM16 lipid scramblase. We believe that the speed and accuracy of continuum-atomistic methodologies will make it possible to simulate large scale, slow biological processes, such as membrane morphological changes, that are currently beyond the scope of other computational approaches. This article is part of a Special Issue entitled: Membrane Proteins edited by J.C. Gumbart and Sergei Noskov.

Equilibrium and kinetics of DNA overstretching modeled with a quartic energy landscape.

It is well known that the dsDNA molecule undergoes a phase transition from B-DNA into an overstretched state at high forces. For some time, the structure of the overstretched state remained unknown and highly debated, but recent advances in experimental techniques have presented evidence of more than one possible phase (or even a mixed phase) depending on ionic conditions, temperature, and basepair sequence. Here, we present a theoretical model to study the overstretching transition with the possibility that the overstretched state is a mixture of two phases: a structure with portions of inner strand separation (melted or M-DNA), and an extended phase that retains the basepair structure (S-DNA). We model the double-stranded DNA as a chain composed of n segments of length l, where the transition is studied by means of a Landau quartic potential with statistical fluctuations. The length l is a measure of cooperativity of the transition and is key to characterizing the overstretched phase. By analyzing the different values of l corresponding to a wide spectrum of experiments, we find that for a range of temperatures and ionic conditions, the overstretched form is likely to be a mix of M-DNA and S-DNA. For a transition close to a pure S-DNA state, where the change in extension is close to 1.7 times the original B-DNA length, we find l ? 25 basepairs regardless of temperature and ionic concentration. Our model is fully analytical, yet it accurately reproduces the force-extension curves, as well as the transient kinetic behavior, seen in DNA overstretching experiments.

Competition between supercoils and toroids in single molecule DNA condensation.

The condensation of free DNA into toroidal structures in the presence of multivalent ions and polypeptides is well known. Recent single molecule experiments have shown that condensation into toroids occurs even when the DNA molecule is subjected to tensile forces. Here we show that the combined tension and torsion of DNA in the presence of condensing agents dramatically modifies this picture by introducing supercoiled DNA as a competing structure in addition to toroids. We combine a fluctuating elastic rod model of DNA with phenomenological models for DNA interaction in the presence of condensing agents to compute the minimum energy configuration for given tension and end-rotations. We show that for each tension there is a critical number of end-rotations above which the supercoiled solution is preferred and below which toroids are the preferred state. Our results closely match recent extension rotation experiments on DNA in the presence of spermine and other condensing agents. Motivated by this, we construct a phase diagram for the preferred DNA states as a function of tension and applied end-rotations and identify a region where new experiments or simulations are needed to determine the preferred state.

The dependence of DNA supercoiling on solution electrostatics.

We develop an elastic-isotropic rod model for twisted DNA in the plectonemic regime. We account for DNA elasticity, electrostatic interactions and entropic effects due to thermal fluctuations. We apply our model to single-molecule experiments on a DNA molecule attached to a substrate at one end, while subjected to a tensile force and twisted by a given number of turns at the other end. The free energy of the DNA molecule is minimized subject to the imposed end rotations. We compute values of the torsional stress, radius, helical angle and key features of the rotation-extension curves. We also include in our model the end loop energetic contributions and obtain estimates for the jumps in the external torque and extension of the DNA molecule seen in experiments. We find that, while the general trends seen in experiments are captured simply by rod mechanics, the details can be accounted for only with the proper choice of electrostatic and entropic interactions. We perform calculations with different ionic concentrations and show that our model yields excellent fits to mechanical data from a large number of experiments. Our methods also allow us to consider scenarios where we have multiple plectonemes or a series of loops forming in the DNA instead of plectonemes. For a given choice of electrostatic and entropic interactions, we find there is a range of forces in which the two regimes can coexist due to thermal motion.