Simulation of protein association: Kinetic pathways towards crystal contacts.

We conducted molecular dynamics simulations combined with distance-based umbrella sampling and forward flux sampling to investigate the early stages of protein crystallization. Formation of contacts with long-range interactions and/or an exposed position on the protein surface was kinetically preferred over more stable hydrophobic contacts with a shorter attractive range, while the thermodynamic stability of the protein crystal was provided by hydrophobic interactions. Contacts with a large interaction area showed complex dissociation pathways that were not detected by distance-based umbrella sampling. Instead, forward flux sampling simulations of contact dissociation identified long-range attractive interactions.

Role of geometrical shape in like-charge attraction of DNA.

While the phenomenon of like-charge attraction of DNA is clearly observed experimentally and in simulations, mean-field theories fail to predict it. Kornyshev et al. argued that like-charge attraction is due to DNA's helical geometry and hydration forces. Strong-coupling (SC) theory shows that attraction of like-charged rods is possible through ion correlations alone at large coupling parameters, usually by multivalent counterions. However for SC theory to be applicable, counterion-counterion correlations perpendicular to the DNA strands need to be sufficiently small, which is not a priori the case for DNA even with trivalent counterions. We study a system containing infinitely long DNA strands and trivalent counterions by computer simulations employing varying degrees of coarse-graining. Our results show that there is always attraction between the strands, but its magnitude is indeed highly dependent on the specific shape of the strand. While discreteness of the charge distribution has little influence on the attractive forces, the role of the helical charge distribution is considerable: charged rods maintain a finite distance in equilibrium, while helices collapse to close contact with a phase shift of π, in full agreement with SC predictions. The SC limit is applicable because counterions strongly bind to the charged sites of the helices, so that helix-counterion interactions dominate over counterion-counterion interactions. Thus DNA's helical geometry is not crucial for like-charge DNA attraction, but strongly enhances it, and electrostatic interactions in the strong-coupling limit are sufficient to explain this attraction.

Two-stage crystallization of charged colloids under low supersaturation conditions.

We report simulations on the homogeneous liquid-fcc nucleation of charged colloids for both low and high contact energy values. As a precursor for crystal formation, we observe increased local order at the position where the crystal will form, but no correlations with the local density. Thus, the nucleation is driven by order fluctuations rather than density fluctuations. Our results also show that the transition involves two stages in both cases, first a transition of liquid → bcc, followed by a bcc → hcp/fcc transition. Both transitions have to overcome free energy barriers, so that a spherical bcc-like cluster is formed first, in which the final fcc structure is nucleated mainly at the surface of the crystallite. This means that the second stage bcc-fcc phase transition is a heterogeneous nucleation in the partially grown solid phase, even though we start from a homogeneous bulk liquid. The height of the bcc → hcp/fcc free energy barrier strongly depends on the contact energies of the colloids. For low contact energy this barrier is low, so that the bcc → hcp/fcc transition occurs spontaneously. For the higher contact energy, the second barrier is too high to be crossed spontaneously by the colloidal system. However, it was possible to ratchet the system over the second barrier and to transform the bcc nuclei into the stable hcp/fcc phase. The transitions are dominated by the first liquid-bcc transition and can be described by classical nucleation theory using an effective surface tension.

Kinetic dielectric decrement revisited: phenomenology of finite ion concentrations.

With the help of a recently developed non-equilibrium approach, we investigate the ionic strength dependence of the Hubbard-Onsager dielectric decrement. We compute the depolarization of water molecules caused by the motion of ions in sodium chloride solutions from the dilute regime (0.035 M) up close to the saturation concentration (4.24 M), and find that the kinetic decrement displays a strong non-monotonic behavior, in contrast to the prediction of available models. We introduce a phenomenological modification of the Hubbard-Onsager continuum theory, which takes into account the screening due to the ionic cloud at the mean-field level and, which is able to describe the kinetic decrement at high concentrations including the presence of a pronounced minimum.

Induction of entropic segregation: the first step is the hardest.

In confinement, overlapping polymers experience entropic segregating forces that tend to demix them. This plays a role during cell replication, where it facilitates the segregation of daughter chromosomes. It has been argued that these forces are strong enough to explain chromosome segregation in elongated bacteria such as E. coli without the need for additional active mechanisms [S. Jun and B. Mulder, Proc. Natl. Acad. Sci. U. S. A., 2006, 103, 12388]. However, entropic segregation can only set in after the initial symmetry has been broken. We demonstrate that the timescale for this induction phase is exponentially growing in the chain length, while the actual segregation time scales only quadratically in the chain length. Thus the induction quickly becomes the dominating, slow process, and makes entropic segregation much less efficient than previously thought. The slow induction might also explain the long delay in chromosome segregation observed in experiments on E. coli.

Hydrodynamic interactions slow down crystallization of soft colloids.

Colloidal suspensions are often argued to be an ideal model for studying phase transitions such as crystallization, as they have the advantage of tunable interactions and experimentally tractable time and length scales. Because crystallization is assumed to be unaffected by details of particle transport other than the bulk diffusion coefficient, findings are frequently argued to be transferable to pure melts without solvent. In this article, we present molecular dynamics simulations of crystallization in a suspension of colloids with Yukawa interactions which challenge this assumption. In order to investigate the role of hydrodynamic interactions mediated by the solvent, we model the solvent both implicitly and explicitly, using Langevin dynamics and the fluctuating lattice Boltzmann method, respectively. Our simulations show a significant reduction of the crystal growth velocity due to hydrodynamic interactions even at moderate hydrodynamic coupling. This slowdown is accompanied by a reduction of the width of the layering region in front of the growing crystal. Thus the dynamics of a colloidal suspension differ strongly from that of a melt, making it less useful as a model for solvent-free melts than previously thought.

Nudged-elastic band used to find reaction coordinates based on the free energy.

Transition paths characterize chemical reaction mechanisms. In this paper, we present a new method to find mean reaction paths based on the free energy. A nudged elastic band (NEB) is optimized using gradients and Hessians of the free energy, which are obtained from umbrella integration. The transition state can be refined by a Newton-Raphson search starting from the highest point of the NEB path. All optimizations are done using Cartesian coordinates. Independent molecular dynamics (MD) runs are performed at each image used to discretize the path. This makes the method intrinsically parallel. In contrast to other free energy methods, the algorithm does not become more expensive when including more degrees of freedom in the active space. The method is applied to the alanine-dipeptide as a test case and compared to pathways that have been derived from metadynamics and forward flux sampling.

Automatic, optimized interface placement in forward flux sampling simulations.

Forward flux sampling (FFS) provides a convenient and efficient way to simulate rare events in equilibrium or non-equilibrium systems. FFS ratchets the system from an initial state to a final state via a series of interfaces in phase space. The efficiency of FFS depends sensitively on the positions of the interfaces. We present two alternative methods for placing interfaces automatically and adaptively in their optimal locations, on-the-fly as an FFS simulation progresses, without prior knowledge or user intervention. These methods allow the FFS simulation to advance efficiently through bottlenecks in phase space by placing more interfaces where the probability of advancement is lower. The methods are demonstrated both for a single-particle test problem and for the crystallization of Yukawa particles. By removing the need for manual interface placement, our methods both facilitate the setting up of FFS simulations and improve their performance, especially for rare events which involve complex trajectories through phase space, with many bottlenecks.

Comparison of scalable fast methods for long-range interactions.

Based on a parallel scalable library for Coulomb interactions in particle systems, a comparison between the fast multipole method (FMM), multigrid-based methods, fast Fourier transform (FFT)-based methods, and a Maxwell solver is provided for the case of three-dimensional periodic boundary conditions. These methods are directly compared with respect to complexity, scalability, performance, and accuracy. To ensure comparable conditions for all methods and to cover typical applications, we tested all methods on the same set of computers using identical benchmark systems. Our findings suggest that, depending on system size and desired accuracy, the FMM- and FFT-based methods are most efficient in performance and stability.

How close to two dimensions does a Lennard-Jones system need to be to produce a hexatic phase?

We report on a computer simulation study of a Lennard-Jones liquid confined in a narrow slit pore with tunable attractive walls. In order to investigate how freezing in this system occurs, we perform an analysis using different order parameters. Although some of the parameters indicate that the system goes through a hexatic phase, other parameters do not. This shows that to be certain whether a system of a finite particle number has a hexatic phase, one needs to study not only a large system, but also several order parameters to check all necessary properties. We find that the Binder cumulant is the most reliable one to prove the existence of a hexatic phase. We observe an intermediate hexatic phase only in a monolayer of particles confined such that the fluctuations in the positions perpendicular to the walls are less than 0.15 particle diameters, i.e., if the system is practically perfectly 2D.

Internal dynamics of supercoiled DNA molecules.

The intramolecular diffusive motion within supercoiled DNA molecules is of central importance for a wide array of gene regulation processes. It has recently been shown, using fluorescence correlation spectroscopy, that plasmid DNA exhibits unexpected acceleration of its internal diffusive motion upon supercoiling to intermediate density. Here, we present an independent study that shows a similar acceleration for fully supercoiled plasmid DNA. We have developed a method that allows fluorescent labeling of a 200-bp region, as well as efficient supercoiling by Escherichia coli gyrase. Compared to plain circular or linear DNA, the submicrosecond motion within the supercoiled molecules appears faster by up to an order of magnitude. The mean-square displacement as a function of time reveals an additional intermediate regime with a lowered scaling exponent compared to that of circular DNA. Although this unexpected behavior is not fully understood, it could be explained by conformational constraints of the DNA strand within the supercoiled topology in combination with an increased apparent persistence length.

Electrostatic layer correction with image charges: a linear scaling method to treat slab 2D+h systems with dielectric interfaces.

A fast algorithm for dealing with electrostatic interactions in partially periodic systems that are confined along the nonperiodic direction by two planar dielectric interfaces is presented. The method is a generalization of the electrostatic layer correction (ELC) method of Arnold et al. [J. Chem. Phys. 117, 2496 (2002)], and employs an exact relation between the 2D+h system and a three-dimensional (3D) periodic system. The terms connecting the two systems can be evaluated linearly in the number of charges. Thus, the method shows overall the scaling of the underlying method employed to handle the Coulombic 3D case. Moreover, our algorithm can accurately handle multiple polarization image charges due to the dielectric interfaces as well as all the periodic images due to the periodic boundary conditions and has full control over the errors depending on the underlying method used for the 3D periodic case.

Unexpected relaxation dynamics of a self-avoiding polymer in cylindrical confinement.

We report extensive simulations of the relaxation dynamics of a self-avoiding polymer confined inside a cylindrical pore. In particular, we concentrate on examining how confinement influences the scaling behavior of the global relaxation time of the chain, tau, with the chain length N and pore diameter D. An earlier scaling analysis based on the de Gennes blob picture led to tau approximately N(2)D(13). Our numerical effort that combines molecular dynamics and Monte Carlo simulations, however, consistently produces different tau results for N up to 2000. We argue that the previous scaling prediction is only asymptotically valid in the limit N"D(53)"1, which is currently inaccessible to computer simulations and, more interestingly, is also difficult to reach in experiments. Our results are thus relevant for the interpretation of recent experiments with DNA in nano- and microchannels.

Time scale of entropic segregation of flexible polymers in confinement: implications for chromosome segregation in filamentous bacteria.

We report molecular dynamics simulations of the segregation of two overlapping chains in cylindrical confinement. We find that the entropic repulsion between chains can be sufficiently strong to cause segregation on a time scale that is short compared to the one for diffusion. This result implies that entropic driving forces are sufficiently strong to cause rapid bacterial chromosome segregation.

ICMMM2D: an accurate method to include planar dielectric interfaces via image charge summation.

An extension of the MMM2D method, called ICMMM2D, is presented to deal with the electrostatic interactions in partially periodic systems that are confined along one direction by two planar dielectric interfaces. The method handles all electrostatic interactions that are induced by the presence of dielectric interfaces by an image charge method. Our method accurately treats repeated image charges under the planar dielectric interfaces as well as the periodic images that are due to the periodic boundary conditions along the other two directions. The scaling of the method with the number of charges, N, is still N5/3, and the overhead involved approximately doubles the CPU time compared to the original MMM2D method. The errors are fully under control and the error bounds can be preset up to computer accuracy.

Confined space and effective interactions of multiple self-avoiding chains.

We study the link between three seeming-disparate cases of self-avoiding polymers: strongly overlapping multiple chains in dilute solution, chains under spherical confinement, and the onset of semidilute solutions. Our main result is that the free energy for overlapping n chains is independent of chain length and scales as n9/4, slowly crossing over to n3, as n increases. For strongly confined polymers inside a spherical cavity, we show that rearranging the chains does not cost an additional free energy. Our results imply that, during cell cycle, global reorganization of eukaryotic chromosomes in a large cell nucleus could be readily achieved.

MMM1D: a method for calculating electrostatic interactions in one-dimensional periodic geometries.

We present a new method to accurately calculate the electrostatic energy and forces on charges in a system with periodic boundary conditions in one of three spatial dimensions. We transform the Coulomb sum via a convergence factor into a series of fast decaying functions similar to the Lekner method. Rigorous error bounds for the energies and the forces are derived and numerically verified. The method has a computational complexity of O(N(2)), but is faster and easier to use than previously reported methods.