Session X18: Biopolymers II: Simulations

Statistical Mechanics of Membrane Proteins.

The statistical mechanics of polymers is applied to the self-assembly of membrane proteins. We find a two stage process as a function of the attractive interactions between the membrane bound portions of the polymer. First individual trans-membrane helices develop connected by Gaussian loops. Secondly the alpha helices interact with each other, both entropically through the loops and enthalpically through the attractive interaction. In the phase diagram the first stage is found to be a continuous phase transition where as the aggregation of the helices is seen to be first order.   

Mechanical unfolding of proteins: reduction to a single-reaction coordinate unfolding potential, and an application of the Jarzynski Relation

Single molecule force spectroscopy (AFM, optical tweezers, etc) has revolutionized the study of many biopolymers, including DNA, RNA, and proteins. In this talk I will discuss recent work on modelling of mechanical unfolding of proteins, as often probed by AFM. I will address two issues in obtaining a coarse-grained description of protein unfolding: how to project the entire energy landscape onto an effective one dimensional unfolding potential, and how to apply the Jarzynski Relation to extract equilibrium free energies from nonequilibrium unfolding experiments.   

Boundary Element Microhydrodynamics: Stagnation of flow in protein cavities

A very precise boundary element solution of the exact Stokes flow surface integral equation has been implemented in our Fortan 90 program BEST. In our previous work (Aragon {\&} Hahn, Biophys. J. 2006, 91: 1591-1603; J. Chem. Theory and Comput. 2006, 2: 1416-1428) we obtained very precise values of the tensorial transport properties (translation, rotation, and intrinsic viscosity) for a large set of proteins with a uniform water hydration thickness of 0.11 nm. In this work, we utilize the surface stress distribution thus obtained to evaluate the flow field as a function of distance away from the hydrodynamic surface for a variety of surface features in a dimpled sphere (test case) and for the proteins myoglobin, lysozyme, and human serum albumin. We demonstrate that solvent in small to large pockets on the hydrodynamic surface moves with the protein with distances up to 2 nm for deep pockets regardless of the direction of motion of the protein. On the other hand, the fluid flow pattern on protruding portions of the hydrodynamic surface decays much more rapidly with distance from the surface. The implications of these results with respect to the amount of water of associated with the surface and the rate of transport to active enzymatic sites in stirred solutions is discussed.   

Electrostatic theory of viral self-assembly: Structure and Kinetics

Viruses self-assemble from identic capsid proteins and their genome consisting, for example, of a long single stranded (ss) RNA. For a big class of T = 3 viruses capsid proteins have long positive flexible N-terminal tails. We explore the role played by the Coulomb interaction between the N-terminal tails and negative ss RNA molecule in the kinetics of virus self-assembly. Capsid proteins stick to unassembled chain of ss RNA (antenna) and slide on it towards the self-assembly site. We show that due to such one-dimensional diffusion the virus self-assembly is more than ten times faster than the case involving only three-dimensional diffusion. In the assembled virus, the ss RNA strongly interacts with the brush of tails rooted at the inner surface of the capsid. We show that viruses are most stable when the total length of ss RNA is close to the total length of the tails. For such a structure the absolute value of total (negative) charge of ss RNA is approximately twice larger than the charge of the capsid. This conclusion agrees with available structural data.   

Electrophoresis of DNA on a disordered two-dimensional substrate

We propose a method for electrophoretic separation of DNA in which adsorbed polymers are driven over a disordered two-dimensional substrate which contains attractive sites for the polymers. Using simulations of a model for long polymer chains, we show that the mobility increases with polymer length, in contrast to gel electrophoresis techniques, and that separation can be achieved for a range of length scales. We demonstrate that the separation relies on steric interactions between polymer segments, which prevent substrate disorder sites from trapping more than one DNA segment each. Since thermal activation does not play a significant role in determining the polymer mobility, band broadening due to diffusion can be avoided in our separation method. [1] Phys. Rev. E 74, 051908 (2006).   

Shape of DNA in a box

The statistical and mechanical properties of DNA are known to change dramatically when the persistence length of DNA is comparable to the linear dimensions of the viral capsid and DNA is strongly bent. Based on modeling semiflexible DNA by worm-like chain model, we make use of the Bond Fluctuation Algorithm to study the behavior of DNA in confinement. The effective persistence length is extracted from tangent-tangent correlation function. We will present results for (a) the dependence of effective persistence length on the bare persistence length and (b) a shape transition accompanying increasing confinement.   

Polymer dynamics in a tight squeeze

Chromosomal DNA is confined to a space roughly an order of magnitude smaller than its natural radius of gyration due to the nuclear or cellular membrane. In addition to altering the observed static distributions of the chromosome, this confinement has the potential produce dynamics that differ from the Rouse/Zimm dynamics of the free chain. We propose a model for studying the dynamics of polymers under confinement that can be extended more generally to particle diffusion in a crowded environment. We find that as the size of the confining box is reduced, initially the relaxation times of the Rouse modes decrease due to the reduced phase space accessible to the polymer. However, in the strongly confined regime the relaxation times increase with decreasing box size due to jamming. We demonstrate this non-monotonic behavior using a lattice toy model as well as Monte Carlo simulations.   

$\lambda$-DNA thermal migration in a microchannel

The Ludwig-Soret effect, the migration of a species due to a temperature gradient, has been extensively studied without a complete picture of its cause. Many particle and polymer solutions have been used to study the phenomena; here we investigate the dynamics of DNA confined to a microchannel and subjected to a thermal gradient using a combination of Brownian dyanmics and the lattice Boltzmann method. We observe that the DNA molecules will migrate to colder regions of the channel, an observation also made in the experiments of Duhr, et al (Eur. Phys. J. E, \textbf{15}, (2004)). In fact, the thermal diffusion coefficient found for a range of temperature gradients and DNA molecular weights agrees nearly quantitatively with the experimental value. Furthermore, we use the simulation to understand how the interaction of the DNA with the solvent leads to thermal migration via the thermal fluctuations-fluid momentum flux coupling.   

Force-extension relation of DNA-histone complexes

In eukaryotic cells, DNA is packaged inside the nucleus in the form of chromatin, a structure whose basic repeat unit, known as the nucleosome, consists of DNA wrapped around a cylindrical complex of histone proteins. In order for the cell to function properly, these nucleosome complexes must be stable at equilibrium. At the same time, the cell must be able to gain access to the genomic information contained within the DNA, which it can achieve by exerting forces on the nucleosomes that cause the DNA to unwrap from the histones. Single molecule mechanical manipulation techniques, in which DNA/histone complexes are disrupted by an external force, can provide information not only about the equilibrium structure of these complexes, but also about the forces and displacements required to access the DNA in the nucleosome. In this talk, we derive the force-extension relation for these complexes. We allow for the DNA to unwrap from the histones in both a continuous and discontinuous fashion; that is, we allow the histones to ``pop'' off of the DNA, releasing a large amount of DNA in the process. We also include the conformational fluctuations of the unwrapped portions of the DNA.   

Monitoring the Bending Stiffness of DNA

In eukaryotic cells, the accessibility of genomic sequences provides an inherent regulation mechanism for gene expression through variations in bending stiffness encoded by the nucleic acid sequence. Cyclization of dsDNA is the prevailing method for determining DNA bending stiffness. Recent cyclization data for short dsDNA raises several fundamental questions about the soundness of the cyclization method, particularly in cases where the probability of highly bent DNA conformations is low. We herein evaluate the role of T4 DNA ligase in the cyclization reaction by inserting an environmental sensitive base analogue, 2-amino purine, to the DNA molecule. By monitoring the 2-AP fluorescence under standard cyclization conditions, it is found that in addition to trapping highly-bent cyclic DNA conformations, T4 DNA ligase enhances the apparent base pair flip out rate, thus exaggerating the measured flexibility. This result is further confirmed using fluorescence anisotropy experiments. We show that fluorescence resonance energy transfer (FRET) measurements on suitably labeled dsDNA provides an alternative approach for quantifying the bending stiffness of short fragments. DNA bending stiffness results obtained using FRET are compared with literature values.   

Validity of the bead-spring model for describing the linear viscoelastic properties of single-strand DNA under strongly denaturing conditions

Using a normal mode analysis, we predict the infinite dilution linear viscoelastic properties of single strand-DNA molecules and compare the results to the linear viscoelastic data of Shusterman et al. (Shusterman, Alon, Gavrinyov, and Krichevsky, 2004) obtained by monitoring the diffusion of a fluorescently labeled terminus of the molecule. To compute the overall best global fit, we constrain the hydrodynamic interaction parameter, $h*$, equilibrium rms spring extension, $b$, and the number of Kuhn steps per spring, $N_{K,S}$, to be equal for the strands compared. The fits using the bead-spring model for all but 23100 base ss-DNA strands match the experimental data at long times with significant deviations at intermediate and short times. However, parameters fitted separately to all individual strand lengths predict results well. The best-fits to data for 2400 and 6700 base pairs yield $N_{K,S} \quad \sim $ 12 and $h*$ = 0.12. These values are similar to those found for conventional polymers such as polystyrene which have been successfully modeled with $N_{K,S} \quad \sim $ 7 and $h*$ = 0.15, indicating ss-DNA and polystyrene exhibit analogous hydrodynamic behavior.   

Dynamics of particles with key-lock interactions

We present a theoretical discussion of particles which interact through the reversible formation of multiple key-lock bridges. Two potential experimental realizations include DNA- grafted particles which interact with a two-dimensional DNA substrate, and particles grafted with antibodies interacting with a protein substrate. We argue there is a percolation transition characterized by the average number of bridges realized between a particle and the substrate. The transition separates a regime in which particles are localized from a diffusive regime where they explore the substrate surface through mutiple breaking and reconnecting of bridges. This diffusion behavior is dispersive, characterized by $\langle r^ {2}(t) \rangle \sim t^{\alpha}$ with $\alpha<1$. The distribution of departure times in a multi-particle system is calculated in two different models which account for the particle dynamics above and below the percolation threshold.   

Molecular Dynamics Simulation of semi-flexible filament assembly

An MD simulation has been developed to study semi-flexible filament bundling and network formation by cross-linkers in solution. We aim to model and understand the ordered mesoscale structures observed experimentally by F-actin, in which different network configurations occur for different concentrations of cross-linking protein. [1]. We use a bead-rod model for the semi-flexible filament and linkers, which can be easily adapted to different cross-linking proteins. Electrostatic interactions were shown to be the main mechanism for the aggregation process, Coulombic forces between excess charges on the proteins dominate at the long range inducing the assembly, while at short range the Van der Waals interaction and specific binding potentials of the proteins have been taken into account. We discuss the effects of screening the Coulomb interaction not only for the linker conc. at which a phase transition occurs but also on bundling and network configuration above the transition point. These results are in good agreement with experimental observations of actin filament bundling and assembly. [1] \textbf{L.S. Hirst} et al J. CHEM. PHYS. 123, 104902~(2005)   

Ion condensation near patterned surfaces

Using the exactly solvable model we have studied ion condensation near patterned surfaces. Competition between electrostatic energy of interaction with the surface and the energy of ionic crystal of correlated condensed ions result in reach phase behavior. Our results show that the structure of ionic crystal of condensed ions essentially depends on the surface charged density and the size of the pattern. Our results can contribute to understanding of different closed phenomena, like surface pattern recognition, separation and synthesis of molecules near patterned surfaces.