Brownian dynamics simulation of restricted rotational diffusion.

The restricted rotational diffusion of an axially symmetric particle is simulated by the Brownian dynamics technique. In addition to the wobbling-in-a-cone model, several continuous potentials are considered. The particle studied is particularly simple: a sphere anchored to a point fixed in space. However, presenting the results in a convenient, reduced form, they are valid for any axially symmetric particle. From simulated rotational trajectories, we calculate (P2(cos alpha] as a function of t, where alpha is the angle between two orientations separated by time t and P2 is the second Legendre polynomial. This correlation function is closely related to time-resolved electro-optic and spectroscopic properties. Simulated results for the cone model are in excellent agreement with the quasiexact results of Lipari and Szabo (1981, J. Chem. Phys., 75:2971-2976). Thus we confirm the good performance of the simulation technique and the validity of our working conditions. Novel results are presented for continuous restricting potentials, V(theta). The (P2) results for V = 1/2K theta 2 and V = Q(1 - cos theta) are practically the same if K and Q are chosen so tht the long-time (P2) values coincide. Thus, the quadratic potential seems to be a good representation of any monotonically increasing potential. However, for an uniaxial potential such as V = Csin2 theta, the decay is appreciably faster. The (P2) decays simulated for the continuous potentials are analyzed by the monoexponential version of the cone model. We found that such an analysis produces an overestimation of the true rotational diffusion coefficient of approximately 15% only, although for uniaxial potentials the error may be larger.     

Brownian dynamics simulation of DNA condensation.

DNA condensation observed in vitro with the addition of polyvalent counterions is due to intermolecular attractive forces. We introduce a quantitative model of these forces in a Brownian dynamics simulation in addition to a standard mean-field Poisson-Boltzmann repulsion. The comparison of a theoretical value of the effective diameter calculated from the second virial coefficient in cylindrical geometry with some experimental results allows a quantitative evaluation of the one-parameter attractive potential. We show afterward that with a sufficient concentration of divalent salt (typically approximately 20 mM MgCl(2)), supercoiled DNA adopts a collapsed form where opposing segments of interwound regions present zones of lateral contact. However, under the same conditions the same plasmid without torsional stress does not collapse. The condensed molecules present coexisting open and collapsed plectonemic regions. Furthermore, simulations show that circular DNA in 50% methanol solutions with 20 mM MgCl(2) aggregates without the requirement of torsional energy. This confirms known experimental results. Finally, a simulated DNA molecule confined in a box of variable size also presents some local collapsed zones in 20 mM MgCl(2) above a critical concentration of the DNA. Conformational entropy reduction obtained either by supercoiling or by confinement seems thus to play a crucial role in all forms of condensation of DNA.     

Brownian dynamics simulation of probe diffusion in DNA: effects of probe size, charge and DNA concentration

We have used Brownian dynamics simulation to study probe diffusion in solutions of short chain DNA using our previously developed simulation algorithm. We have examined the effect of probe size, charge, and DNA concentration on the probe diffusion coefficient, with the aim of gaining insight into the diffusion of proteins in a concentrated DNA environment. In these simulations, DNA was modeled as a worm-like chain of hydrodynamically equivalent spherical frictional elements while probe particles were modeled as spheres of given charge and hydrodynamic radius. The simulations allowed for both short range Lennard-Jones interactions and long ranged electrostatic interactions between charged particles. For uncharged systems, we find that the effects of probe size and DNA concentration on the probe diffusion coefficient are consistent with excluded volume models and we interpret our results in terms of both empirical scaling laws and the predictions of scaled particle theory. For charged systems, we observe that the effects of probe size and charge are most pronounced for the smallest probes and interpret the results in terms of the probe charge density. For an ionic strength of 0.1 M we find that, below a critical probe surface charge density, the probe diffusion coefficient is largely independent of probe charge and only weakly dependent on the DNA charge. These effects are discussed in terms of the interactions between the probe and the DNA matrix and are interpreted in terms of both the underlying physics of transport in concentrated solutions and the assumptions of the simulation model.     

Mechanisms of DNA separation in entropic trap arrays: a Brownian dynamics simulation

Using Brownian dynamics simulations, we study the migration of long charged chains in an electrophoretic microchannel device consisting of an array of microscopic entropic traps with alternating deep regions and narrow constrictions. Such a device has been designed and fabricated recently by Han and Craighead [Science 288 (2000) 1026] for the separation of DNA molecules. Our simulation reproduces the experimental observation that the mobility increases with the length of the DNA. A detailed data analysis allows to identify the reasons for this behavior. Two distinct mechanisms contribute to slowing down shorter chains. One has been described earlier by Han and Craighead [Science 288 (2000) 1026]: the chains are delayed at the entrance of the constriction and escape with a rate that increases with chain length. The other, actually dominating mechanism is here reported for the first time: some chains diffuse out of their main path into the corners of the box, where they remain trapped for a long time. The probability that this happens increases with the diffusion constant, i.e., the inverse chain length.     

Adhesion of nanoparticles to vesicles: a Brownian dynamics simulation

We studied the interaction of bilayer vesicles and adhesive nanoparticles using a Brownian dynamics simulation. The nanoparticles are simple models of proteins or colloids. The adhering nanoparticle induces the morphological change of the vesicle: budding, formation of two vesicles in which only outer monolayers are connected, and fission. We also show that the nanoparticle promotes the fusion process: fusion-pore opening from a stalk intermediate, a neck-like structure that only connects outer monolayers of two vesicles. The nanoparticle bends the stalk, and induces the pore opening.     

Brownian dynamics simulation of ion flow through porin channels

Bacterial porins, which allow the passage of solutes across the outer bacterial membrane, are structurally well characterized. They therefore lend themselves to detailed studies of the determinants of ion flow through transmembraneous channels. In a comparative study, we have performed Brownian dynamics simulations to obtain statistically significant transfer efficiencies for cations and anions through matrix porin OmpF, osmoporin OmpK36, phosphoporin PhoE and two OmpF charge mutants.The simulations show that the electrostatic potential at the highly charged channel constriction serves to enhance ion permeability of either cations or anions, dependent on the type of porin. At the same time translocation of counterions is not severely impeded. At the constriction, cations and anions follow distinct trajectories, due to the segregation of basic and acidic protein residues.Simulated ion selectivity and relative conductance agree well with experimental values, and are dependent crucially on the charge constellation at the pore constriction. The experimentally observed decrease in ion selectivity and single channel conductance with increasing ionic strength is well reproduced and can be attributed to electrostatic shielding of the pore lining.     

Self-assembly of two-patch particles in solution: a Brownian dynamics simulation study

We study the self-assembly behaviour of two-patch particles with D_∞h symmetry by using Brownian dynamics simulations. The self-assembly process of two-patch particles with diverse patch coverage in two selective solvent conditions is investigated. The patchy particles in a solvent that is bad for patches but good for matrix form linear thread-like structures with low patch coverage, whereas they form 3D network structures with relatively high patch coverage on surface. For patchy particles in a solvent which is good for patches but bad for body, monolayer structures are obtained at high patch coverage, and some cluster structures emerge when surface patch coverage is low.     

A mathematical analysis for the Brownian dynamics of a DNA tether

In the single-particle tracking experiment, the internal motion of a single DNA or polymer molecule whose one end is attached to a microsphere (optical marker) and the other end is anchored to a substratum is studied (Finzi and Gelles, 1995). The stochastic Brownian dynamics of the sphere reflect the spontaneous fluctuations, thus the physical characteristics, of the DNA or polymer molecule (Qian and Elson, 1999, Qian, 2000). In this paper, two continuous models of polymer molecules, a flexible elastic string and a weakly bentable elastic rod, are analyzed. Both models are cast mathematically in terms of linear stochastic differential equations. Based on Fourier analyses, we calculate the mean square displacement (MSD) of the particle motion, the key observable in the experiment. We obtain for both models the short-time asymptotics for the MSD, as well as the long-time behavior in terms of the smallest non-zero eigenvalues. It is shown that: (i) the long-time dynamics of continuous elastic string model quantitatively agree with that of the discrete bead-spring model. (ii) The short-time MSD of both models are controlled by the tethered particle, with linear dependence on t. (iii) The two models show characteristic difference for long-time behavior: The longest relaxation time is proportional to L ² for long elastic string and to L for short elastic string, but is proportional to L ⁴ for both long and short weakly bentable rod. Received: 26 March 1998 / Revised version: 9 June 2000 / Published online: 14 September 2000     

Dynamic Monte Carlo simulation of non-equilibrium Brownian diffusion of single-chain macromolecules

We performed dynamic Monte Carlo simulations of biased diffusion of 3D phantom single lattice polymer. We observed spontaneous deformation of polymer coil when the external driving forces exceed a critical strength. In addition, longer chains require lower critical strengths, at which their activated velocities deviate from Newtonian-fluid behaviours and merge into a master curve exhibiting shear-thinning followed with shear thickening. We attributed the cause of deformation to the random updating of monomers. The latter represents the dynamic heterogeneity along the real polymer chain, and raises a nonlinear asymmetric accumulation of local acceleration and then an internal tension between chain middle and chain end, as evidenced by our previous Brownian Dynamics simulations. Our results unravel a single-molecular-level source of nonlinear dynamics, which has been overlooked in current theoretical considerations on the basis of Rouse ideal-chain model.     

Brownian dynamics simulation of protein folding: a study of the diffusion-collision model

The dynamic aspects of protein folding are described by a series of diffusion-collision steps involving structural units (microdomains) of various sizes that combine to form the protein in its native state. A method is introduced for obtaining the rate constants for the basic diffusion-collision step by use of Brownian dynamics. The method is applied to an investigation of the folding dynamics of two α-helices connected by a flexible (random-coil) polypeptide chain. The results of this full three-dimensional treatment are compared with simplified model calculations for the diffusion-collision step. Of particular interest are the nature of the collision dynamics and the role of the intervening peptide chain.     

Brownian dynamics simulation of nucleosome formation and disruption under stretching

Using a Brownian dynamics simulation, we numerically studied the interaction of DNA with histone and proposed an octamer-rotation model to describe the process of nucleosome formation. Nucleosome disruption under stretching was also simulated. The theoretical curves of extension versus time as well as of force versus extension are consistent with previous experimental results.     

Intrachain reactions of supercoiled DNA simulated by Brownian dynamics

We considered an irreversible biochemical intrachain reaction of supercoiled DNA as a random event that occurs, with certain probability, at the instant of collision between two reactive groups bound to distant DNA sites. Using the Brownian dynamics technique, we modeled this process for a supercoiled DNA molecule of 2.5 kb length in dilute aqueous solution at an NaCl concentration of 0.1 M. We calculated the mean reaction time tau(Sigma) as a function of the intrinsic second-order rate constant k(I), the reaction radius R, and the contour separation S of the reactive groups. At the diffusion-controlled limit (k(I) --> infinity), the kinetics of reaction are determined by the mean time tau(F) of the first collision. The dependence of tau(F) on R is close to inversely proportional, implying that the main contribution to the productive collisions is made by bending of the superhelix axis. At sufficiently small k(I), the mean reaction time can be satisfactory approximated by tau(Sigma) = tau(F)(app) + 1/(k(I)c(L)), where c(L) is the local concentration of one reactive group around the other, and tau is an adjustable parameter, which we called the apparent time of the first collision. The value of tau depends on R very weakly and is approximately equal to the mean time of the first collision caused by mutual reptation of two DNA strands forming the superhelix. The quasi-one-dimensional reptation process provides the majority of productive collisions at small k(I) values.     

Solid phase DNA amplification: a Brownian dynamics study of crowding effects

Solid phase amplification (SPA), a new method to amplify DNA, is characterized by the use of surface-bound primers. This limits the amplification to two-dimensional surfaces and therefore allows the easy parallelization of DNA amplification in a single system. SPA leads to the formation of small but dense DNA brushes, called DNA colonies. For a molecule to successfully duplicate itself, it needs to bend so that its free end can find a matching primer, located on the surface. We used Brownian dynamics simulations (with a united-atom model) to model the basic kinetics of an SPA experiment. The simulations mimic the temperature cycles and the molecule duplication process found in SPA. Our results indicate that the steric interaction between molecules leads to a decreased duplication probability for molecules in the center of a colony and to an outward leaning for the molecules on the perimeter. These effects result in slower amplification (compared to solution PCR) and indicate that steric interaction alone can explain the loss of the exponential growth (characteristic of solution PCR) of the number of molecules in an SPA experiment. Furthermore, the growth of the colony as a function of the number of thermal cycles is found to be similar to the one obtained with a simple Monte Carlo simulation.     

The propagation of acoustic waves along a DNA molecule in viscous environment

Low-frequency vibrations of the structural elements of the DNA molecule in a viscous solution were investigated. It was shown that shear, twist and extension waves can propagate along the axis of the molecule. The mutual influence of acoustic waves on each other was analyzed. Formulas for determining the sound velocity and the attenuation coefficients for these types of waves are obtained.     

Brownian dynamics simulation of the lateral distribution of charged membrane components

Brownian dynamics simulations were performed to study the contribution of electric interactions between charged membrane components to their lateral distribution in a two-dimensional viscous liquid (bilayer lipid membrane). The electrostatic interaction potential was derived from an analytical solution of the linearized Poisson-Boltzmann equation for point charges in an electrolyte solution — membrane — electrolyte solution system. Equilibrium as well as dynamic quantities were investigated. The lateral organization of membrane particles, modelled by mobile cylinders in a homogeneous membrane separating two electrolyte solutions was described by spatial distribution functions, diffusion coefficients and cluster statistics. Disorder, local order and crystal-like arrangements were observed as a function of the particle charge, the closest possible distances between the charges and the particle density. The simulations revealed that the system is very sensitive to the position of the charges with respect to the electrolyte solution — membrane interface. Electrostatic interactions of charges placed directly on the membrane surface were almost negligible, whereas deeper charges demonstrated pronounced interaction. Biologically relevant parameters corresponded at most to local and transient ordering. It was found that lateral electric forces can give rise to a preferred formation of clusters with an even number of constituents provided that the closest possible charge-charge distances are small. It is concluded that lateral electrostatic interactions can account for local particle aggregations, but their impact on the global arrangement and movement of membrane components is limited. Correspondence to: D. Walther     

Bimolecular reaction simulation using Weighted Ensemble Brownian dynamics and the University of Houston Brownian Dynamics program

We discuss here the implementation of the Weighted Ensemble Brownian (WEB) dynamics algorithm of Huber and Kim in the University of Houston Brownian Dynamics (UHBD) suite of programs and its application to bimolecular association problems. WEB dynamics is a biased Brownian dynamics (BD) algorithm that is more efficient than the standard Northrup-Allison-McCammon (NAM) method in cases where reaction events are infrequent because of intervening free energy barriers. Test cases reported here include the Smoluchowski rate for association of spheres, the association of the enzyme copper-zinc superoxide dismutase with superoxide anion, and the binding of the superpotent sweetener N-(p-cyanophenyl)-N'-(diphenylmethyl)-guanidinium acetic acid to a monoclonal antibody fragment, NC6.8. Our results show that the WEB dynamics algorithm is a superior simulation method for enzyme-substrate reaction encounters with large free energy barriers.     

Brownian motion of the wormlike chain and segmental diffusion of DNA

The dynamics of the wormlike chain model for a polymer in solution is investigated in the case of free torsional and no longitudinal variations. A Langevin equation is derived and solved for circularly closed chains, neglecting hydrodynamic interactions. The local diffusion behavior of particular segments is described, and it is found that the mean-square displacements are proportional to t^3/4 at short times. Also, the equilibrium correlation functions for the closed chain are derived from the dynamic model in both the discrete and wormlike cases.     

Brownian dynamics simulation of protein solutions: structural and dynamical properties

The study of solutions of biomacromolecules provides an important basis for understanding the behavior of many fundamental cellular processes, such as protein folding, self-assembly, biochemical reactions, and signal transduction. Here, we describe a Brownian dynamics simulation procedure and its validation for the study of the dynamic and structural properties of protein solutions. In the model used, the proteins are treated as atomically detailed rigid bodies moving in a continuum solvent. The protein-protein interaction forces are described by the sum of electrostatic interaction, electrostatic desolvation, nonpolar desolvation, and soft-core repulsion terms. The linearized Poisson-Boltzmann equation is solved to compute electrostatic terms. Simulations of homogeneous solutions of three different proteins with varying concentrations, pH, and ionic strength were performed. The results were compared to experimental data and theoretical values in terms of long-time self-diffusion coefficients, second virial coefficients, and structure factors. The results agree with the experimental trends and, in many cases, experimental values are reproduced quantitatively. There are no parameters specific to certain protein types in the interaction model, and hence the model should be applicable to the simulation of the behavior of mixtures of macromolecules in cell-like crowded environments.