### Brownian and Langevin dynamics simulations

BD and Langevin dynamics (LD) simulation methods may be applied for the purpose of studying the motion and the interactions of biological macromolecules in solvent. The macromolecules are modelled as particles diffusing in a solvent that is modelled as a continuum that exerts frictional and random, stochastic forces on the particles. Common to these methods is the possibility of accessing phenomena whose time-scale is much greater than that usually achievable in atomistic molecular dynamics simulations. A range of methodological developments were presented at the BDBDB2 meeting and these have been implemented in a number of new software packages, as well as in existing packages. The approaches differ in the strategies used to minimize computational cost and, at the same time, retain sufficient detail and accuracy for studying the process of interest. The strategies used to extend the space and time-scale of the simulations can be divided into three categories: physical approximation, smart algorithms and parallelization. The different software packages take advantage of one or more of these strategies.

Many physical approximations have been described to reduce the complexity of the system simulated. When the internal degrees of freedom are not fundamental for describing the process studied, the macromolecules can be considered as rigid bodies. This approximation, which greatly reduces the complexity, allows the atomic details of the macromolecules to be retained. Atomically detailed rigid-body BD simulations have been implemented, for example, in Macrodox [1], UHBD [2] and SDA [3]. Paolo Mereghetti (HITS, Heidelberg, Germany) described extensions of the latter to the simulation of solutions of many protein molecules (with SDAMM) [4]. Adrian Elcock (University of Iowa, USA) described the ground-breaking application of this type of model to simulate a crowded cytoplasm-like environment made up of about fifty different types of macromolecules that occur in *Escherichia coli* [5, 6].

One can further reduce the level of detail by keeping the rigid body representation and coarse-graining the atomistic details. For example, the representation of a molecule by a simple sphere with an excluded volume interaction or a sphere with a reactive patch interacting with a Coulomb potential, has been employed for analysing diffusional association processes [7].

In many cases, such as macromolecular folding processes or binding by induced fit or conformational selection, the rigid body approximation breaks down and a strategy that explicitly treats internal flexibility is required. A coarse grained representation is frequently used. Typically, groups of atoms are represented as beads interacting via a set of interactions that have been parameterized using more accurate methods or experimental information. Coarse grained models are implemented in the BD simulation codes: UHBD [2], BD_BOX (developed in Joanna Trylska's group at the University of Warsaw, Poland; unpublished), BrownDye [8], BrownMove [9] and Simulflex [10].

Smart algorithms are important for achieving computational efficiency. Gary Huber (University of California, San Diego, USA), for example, described several algorithms implemented in BrownDye [8], including an adaptive timestep procedure, charge lumping and a collision detection algorithm.

Parallelization and making use of state-of-the art hardware is equally important. In the BD_BOX software, Maciej Dlugosz (University of Warsaw, Poland) has made extensive use of GPU programming and parallel programming with the Message Passing Interface (MPI) and the shared memory openMP approaches. BD_BOX is intended to be an engine that allows the simulation of very large biomolecular systems treated as coarse grained polymers in implicit solvent.

In BD simulations, the solvent is treated implicitly, that is, the solvent granularity is neglected. In some cases, particular attention should be paid to the treatment of solvent-solute interactions. For example, Daria Kokh (Heidelberg Institute for Theoretical Studies, Germany) showed that, to properly describe the adsorption of proteins to metal surfaces with a continuum model using BD simulations, specific properties of the hydration shell on metal surfaces should be accounted for by including additional, semi-empirically parameterized terms in the protein-surface forces [11, 12]. Often, hydrodynamic interactions (HI) are neglected. The question of the importance of HI, and how they can be treated in BD simulations, came up several times during the meeting and it will be discussed in the following section.

### The importance of the solvent: hydrodynamic interactions (HI)

Understanding the effects of HI on the diffusion and association of macromolecules in complex environments (e.g. within the cell) is non-trivial since the importance of HI strongly depends on the properties of the system itself (e.g. the dimension of the particles, covalent linking of particles, net charge, solute concentration, boundaries, etc.). Adrian Elcock found that HI are fundamental for reproducing experimental diffusional properties (translational and rotational diffusion) of flexible interconnected polymer chains by performing simulations of folding proteins with and without HI [13]. Moreover, Elcock demonstrated the importance of HI for the reaction rates computed using BD [6, 14]. In particular, he showed that the absence of HI tends to contribute to the overestimation of the reaction rates [14].

For dilute solutions (< 0.1 volume fraction) of interacting unbound proteins, the effect of HI on diffusional properties is less crucial. Paolo Mereghetti showed how, in dilute regimes, the concentration dependent diffusion coefficient of lysozyme and BPTI solutions can be reproduced without explicitly including HI. These results agree with those obtained previously in similar simulations by McGuffee and Elcock [5]. Further, if one is only interested in equilibrium thermodynamic properties, HI do not play any role and can be neglected. Elcock showed how BD simulations without HI of a model of *E. coli* cytoplasm successfully describe the relative thermodynamic stabilities of proteins measured in *E. coli*.

Implementing HI in simulations is challenging as the canonical approach requires the factorization of a 3N × 3N diffusion tensor, which is an O(N^{3}) problem (N is the number of particles). Efficient procedures to reduce the computational time were discussed by Jose Garcia de la Torre (University of Murcia, Spain) [15]; and Thiamer Geyer (Saarland University, Germany) described a new approximate method for computing the hydrodynamic coupling of the random displacements which scales as N^{2} and is valid for HI that are not too strong [9, 16]. This approach is advantageous for simulations since it reduces the cost of computing HI to the same order as the computation of the direct forces.

In a dense environment, the correct reproduction of dynamic properties can be expected to depend on accurate modelling of HI. Indeed, beside the far-field part of HI, usually modelled with the Rotne-Prager [17] Yamakawa [18] tensor, near-field many-body interactions, so called lubrication forces [19], become important. As shown by Gerhard Naegele (Institute of Solid State Research, Forschungszentrum Jülich, Germany), neglecting the near-field part leads to unphysical behaviour, such as negative sedimentation coefficients, or inaccurate estimates of diffusional properties [20, 21]. To take care of both far-field as well as near-field HI, accelerated Stokesian dynamics, developed by Banchio and Brady [20], can be used. Recently, Ando and Skolnick performed Stokesian dynamics simulations of macromolecular motions in models of *E. coli* cytoplasm [21] and found the accurate treatment of HI important for reproducing measured protein diffusion coefficients.

### Continuum and hybrid methods

BD treats the primary solute species explicitly, and the solvent implicitly. That is, BD is based on a Langevin type formulation of time-dependent statistical mechanics [22]. As has been noted, this represents a coarse-graining of molecular dynamics type treatments, in which both the solute and solvent particles are commonly treated explicitly. An even greater degree of coarse-graining yields fully continuum level treatments of all diffusing solute and solvent species, corresponding to a Fokker-Planck type formulation of time-dependent statistical mechanics. The simplest example is the treatment of diffusing solutes in terms of the Smoluchowski diffusion equation, i.e. as a time-varying or steady-state concentration or distribution function that depends on spatial coordinates [23].

The continuum level treatments of diffusion have both advantages and disadvantages relative to BD treatments. Continuum level treatments offer computational efficiencies when very large numbers of simple (ideally, non-interacting, point-like) solutes are involved. Indeed, such descriptions are often amenable to analytical study. One familiar result is the Smoluchowski second-order rate constant for solute reaction with a perfectly absorbing, spherical target [24]. More complicated model systems can sometimes be dealt with by numerical solution of the relevant partial differential equations - the Smoluchowski equation or, for charged solutes, the Poisson-Nernst-Planck equation [25]. The numerical methods available include finite difference and finite element methods, among others [26]. Such methods have been applied, for example, to account for realistic anatomical structure in simulations of the diffusion of neurotransmitters in synapses [27] and of calcium ions in cardiac myocytes [28].

On the other hand, BD treatments offer advantages when the solute molecules are substantially non-spherical, are flexible, or have anisotropic interactions. The number of coordinates required to describe such systems in continuum terms grows rapidly as these factors are added in. Also, in the case of low concentrations of solute particles in critical regions, the Brownian treatments account for stochastic effects in the most natural way.

An appealing prospect for future work is the development of hybrid models, in which continuum type treatments can be used in some parts of space and BD treatments in other parts of space (e.g., in the region of an enzyme's active site, if stochastic effects may be important). An early effort in this direction has been described by the Helms group [29]. Another type of hybrid model that has proven very insightful utilizes a Fourier decomposition of continuous lipid bilayers plus Brownian timesteps to describe dynamical processes in biological membranes [30].