Center for Bioinformatics, Saarland University, D-66123 Saarbrücken, Germany

Abstract

Background

Brownian Dynamics (BD) is a coarse-grained implicit-solvent simulation method that is routinely used to investigate binary protein association dynamics, but due to its efficiency in handling large simulation volumes and particle numbers it is well suited to also describe many-protein scenarios as they often occur in biological cells.

Results

Here we introduce our "brownmove" simulation package which was designed to handle many-particle problems with varying particle numbers and allows for a very flexible definition of rigid and flexible protein and polymer models. Both a Brownian and a Langevin dynamics (LD) propagation scheme can be used and hydrodynamic interactions are treated efficiently with our recently introduced TEA-HI ansatz [Geyer, Winter, JCP 130 (2009) 114905]. With simulations of constrained polymers and flexible models of spherical proteins we demonstrate that it is crucial to include hydrodynamics when multi-bead models are used in BD or LD simulations. Only then both the translational and the rotational diffusion coefficients and the timescales of the internal dynamics can be reproduced correctly. In the third example project we show how constant density boundary conditions [Geyer et al, JCP 120 (2004) 4573] can be used to set up a non-equilibrium simulation of diffusional transport across an array of fixed obstacles. Finally, we demonstrate how the agglomeration dynamics of multiple particles with attractive patches can be analysed conveniently with the help of a dynamic interaction network.

Conclusions

Combining BD and LD propagation, fast hydrodynamics, a flexible protein model, and interfaces for "open" simulation settings, our freely available "brownmove" simulation package constitutes a new platform for coarse-grained many-particle simulations of biologically relevant diffusion and transport processes.

Background

Before any reaction can occur in a biological cell between, for example, an enzyme and its substrate, or before two or more proteins can form a functional complex, the respective partners have to find each other in the crowded interior of the cell. For a full understanding of these association and dissociation processes, be it for a general picture or aimed at designing a drug that enhances or suppresses a certain reaction specifically, a lot of details need to be put together into a consistent picture, rate constants need to be determined, and effects of mutations need to be understood. Many of the details can be determined experimentally. Some of these information are microscopic detailed spatial pictures like crystal structures while others come from macroscopically measured data about turnovers or global reaction kinetics. All these parts of the puzzle can be assembled and studied conveniently by combining them into a computer model and performing simulations. In these

A Brownian dynamics simulation package therefore has to deal with many different kinds of interactions, some of which are direct physical interactions while others are a consequence of the continuum solvent ansatz. For some applications like the association of two proteins, details of their spatial forms and their charge distributions are important, whereas for other applications a model of simple spheres may already capture the important features. A general purpose BD package therefore has to be very flexible and should be easily extensible.

When we started with Brownian Dynamics simulations a few years ago

Hierarchic construction of a general particle in brownmove

**Hierarchic construction of a general particle in brownmove**. The top level "Protein" object contains one or more "Gestalt" objects which move independently and contain a shape for each interaction, which in turn contains the basic interaction entities. These are point charges for electrostatic interactions or van-der-Waals spheres for effective short range interactions. With these the forces acting on the Gestalt are calculated. The "GeomShape" object is always present and handles the conversion of the forces into a respective displacement.

While most of the interactions implemented in "brownmove" follow the usual tried strategies and approximations used in Brownian dynamics simulations of biological systems, there is only limited experience with hydrodynamic interactions (HI) in this context. The basic principles can be found in textbooks since decades, but due to the until now rather expensive numerical evaluation, simulations with HI have essentially only been performed as dedicated tests of the analytical theories while in biologically inspired projects one rather tried to get away without them.

The basic ideas of hydrodynamic interactions are simple. When a particle is moved through the solvent it drags a part of the solvent with it and thus creates a flow field that moves in the same direction as the particle. The second ingredient is Faxen's theorem which states that it takes the same force to move a particle through a solvent at rest as to keep the particle at rest in a solvent flowing around the particle with the opposite velocity. For a sphere moving with a constant velocity through a continuous solvent the flow field was calculated already by Stokes (hence the term "Stokes" friction) and can be found in most textbooks that treat fluid mechanics. A corresponding expression exists for the flow field generated by the rotation of a sphere. For translation the fluid moves in the same direction as the particle with a velocity that--to first order--decays proportional to the inverse distance. Consequently, HI are a long-range phenomenon, coupling all particles in the simulation. For accelerating particles there is, however, no closed form for the resulting flow field, because the information about any changes of the particle velocity cannot travel faster than the speed of sound, which leads to retardation effects.

In simulations with an explicit solvent the solvent molecules take care that the displacements are propagated between the particles, but in any implicit solvent method this hydrodynamic coupling has to be added back explicitly. As flow fields are associated with motion one sees that hydrodynamic interactions are not conservative forces derived from a potential energy but that they describe how the velocities of the particles influence each other. The evaluation of the actual velocity coupling is then an iterative procedure. It starts from the unperturbed, zeroth order velocities of the particles that they would have if there were no HI. From theses the resulting flow fields of the particles are evaluated and then, at the locations of the other particles, converted via Faxen's theorem into an effective force acting on them, which modifies their initial velocities. This process is iterated until convergence, which is slow due to the 1/^{-7 }can be found for example in reference

When two spherical particles come very close the above explained iteration needs to consider impractically many terms. A more efficient approach is then to expand the hydrodynamic coupling between two particles in powers of their separation, leading to the so-called lubrication corrections (see, e.g.,

From the above explanations one sees that HI add back the mechanical coupling between the particles that was lost in the implicit solvent approximation, albeit on a coarse and approximative level. Also, many proteins are not truly spherical and the correct hydrodynamics is different from the interaction of perfect spheres. The diffusional properties of non-spherical particles can be determined from a multi-bead-model

There is a number of methods that range in accuracy and effort between a fully atomistic model, which incorporates all details, and the simplified implicit solvent BD. One approach is to numerically solve the Navier-Stokes equation

Another implication which has to be considered when using the simple and efficient RPY hydrodynamics in Brownian dynamics simulations are the short timescales that are required to describe the fast protein dynamics. Then, as already mentioned above, the flow fields are not the stationary Stokes solutions in an incompressible fluid anymore. On these short time and length scales an explicitly time-dependent method should be used (see for example the discussion in

This publication, which we also use to present our "brownmove" simulation package, is organised as follows. After the above introduction, the following Results and Discussions section starts with two examples on how important hydrodynamic interactions are in coarse-grained simulations of flexible proteins. The first example investigates how the stiffness of a bead-spring polymer affects its diffusional properties, while in the second example a flexible model of a compact protein is built from a number of small beads. Both cases show that hydrodynamic interactions have to be included when both rotational and translational diffusion shall be modelled correctly. In the third example non-equilibrium diffusional transport of simplified proteins through an array of fixed obstacles is simulated. The last example then demonstrates that many-particle simulations can be conveniently analysed and understood quantitatively with the help of a dynamic interaction network. The technical details of the implementation, i.e., of the propagation algorithms, the efficient hydrodynamics, the various interactions, and the available boundary conditions are given in the Methods section after the Conclusions.

Results and Discussion

In this section we present some example scenarios to illustrate the kind of applications for which brownmove was developed. From the vast number of possible settings we chose two examples that highlight the importance of hydrodynamic interactions in coarse-grained simulations of flexible protein models and two many-particle scenarios. One describes non-equilibrium transport and the other deals with the analysis of many-particle agglomeration.

So far, two projects have been published in which the brownmove simulation package was used. These were many-particle simulations of the association of cytochrome ^{2 }patch of membrane was modelled with a planar van-der-Waals surface, the charges of the lipids were implemented via a Guy-Chapman electrostatic potential plus, for some tests, nine to 25 point charges. The rectangular simulation box was only 20 nm high, so that at the highest concentrations some 300 cytochrome

In the other project

Example 1: HI in constrained bead-spring polymers

The first example presented here is a continuation of the bead-spring-polymer simulations reported in

Two ways to implement a bead-spring polymer with confined dynamics

**Two ways to implement a bead-spring polymer with confined dynamics**. In the conventional variant shown in panel A the beads are connected by springs which are hooked up at their respective centers, and the bending angle between subsequent springs is confined harmonically by either a direct angle term or, as shown here, by additional springs between the next neighbours. "Brownmove" also allows to set up "bead train" polymers of rotating beads where the "front" of one bead is connected to the "back" of the previous bead with a short spring and the steric repulsion between the beads constricts the polymer dynamics. This is shown in panel B. For the simulations reported here the conventional setup A was used.

**Brownmove definition file for a constrained bead-spring polymer**. This (ASCII text) file defines a bead-spring polymer for a brownmove simulation with five beads connected by springs between the direct and the next neighbours. Each bead has a single van-der-Waals sphere to prevent mutual overlap between the beads, a sphere for hydrodynamic interactions, and two to four hook-up points for the connecting springs. For further details see the comments in this protein definition file.

Click here for file

**Simulation setup file for the bead-spring polymer examples**. This (ASCII text) file defines the global parameters, which polymer to use, and the boundary conditions for the bead-spring polymer example simulations. For details see the comments in the file.

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A second way to implement a polymer with constrained internal dynamics in "brownmove" is shown in Figure _{bead }of the beads was set to 1, which required a simulation timestep of Δt = 5 × 10^{-4}. The springs between the direct neighbours had a spring constant of _{N }= 1000 units and the stiffness of the angle confining next neighbour springs of length 5 was varied from _{2N }= 0.1 to 1000. The springs between the direct neighbours had a length of 2.5 such that neighbouring beads did not overlap. Overlap between all other pairs of beads was prevented by a short range repulsive potential.

**Example of a bead-train polymer definition file**. This brownmove protein definition file gives an example for how a short bead-train polymer as sketched in Figure

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Figure _{CM }of the polymers scaled with their bead number _{CM }scaled as ^{-1 }independent of how flexible the polymer was. With HI and very soft springs our polymer resembles the conventional unconstrained model and, correspondingly, the scaling of _{CM }was close to the theoretical prediction of ^{-0.588 }for infinitely long polymers _{CM }decreased faster with _{2N }was increased. This effect is not very pronounced for the short polymers investigated here. Therefore, only the lowest and highest values of _{2N }= 0.1 and 1000 used here are shown in Figure _{N }and _{2N}, the polymers would start to resemble long rods for which _{CM }scales as ln(_{2N }= 1000.

Normalised center-of-mass diffusion coefficient _{CM }of bead-spring polymers vs. the number of beads,

**Normalised center-of-mass diffusion coefficient D _{CM }of bead-spring polymers vs. the number of beads, N**. Without HI,

A different behaviour was found when the rotational properties of the polymers were considered. As shown in Figure _{rot }of the orientation of the end-to-end vector was nearly insensitive to whether HI was included or not. However, now the stiffness, which determines the average length of the polymer, had a strong effect on rotation. Whereas for the short polymer with _{rot}, could not vary much, the orientational relaxation time increased by about one order of magnitude for the long _{2N }was increased from 0.1 to 1000.

Correlation time **τ**_{rot }of the orientation of the end-to-end vector vs. polymer stiffness

**Correlation time τ_{rot }of the orientation of the end-to-end vector vs. polymer stiffness**. When the spring constant of the next neighbour springs,

From these observations the following picture emerges. Translational diffusion of our polymers was only weakly affected when the more globular structure of a flexible polymer was "unfolded" by increasing the chain stiffness. However, it was crucial to include HI for the correct scaling of _{CM }with the chain length. This is even more important when proteins are simulated which can fold into a much more compact structure than these bead-spring polymers with only repulsive interactions between their beads. On the other hand, the rotational motion was very sensitive to the "folding state" of the polymer but essentially unaffected by HI. Thus, if one wants to use a bead-spring polymer as a model for a protein which can change its conformation from a folded globular to a denatured unfolded state, then HI has to be included. Otherwise, only the translational

In these bead-spring polymers the direct interactions between the beads, i.e., the springs and the van-der-Waals interactions, were very simple and thus the relative costs of including the hydrodynamic coupling were relatively high. On average, the simulations with HI took about three times as long as the corresponding simulations without HI. However, due to the O(^{2}) scaling of our TEA-HI algorithm this ratio was roughly constant for all polymer lengths and thus one can state that when a simulation can be afforded without HI, then normally the corresponding scenario with HI can now be done, too.

Example 2: Elastic proteins from small beads

The previous example shows that it is crucial to include HI when the diffusional properties of polymers or proteins are simulated with bead-spring models. The following example confirms this finding. For this we built proteins from small beads which interact via HI as pioneered by García de la Torre (see, e.g., _{bead }= 1.2 × 10^{-4 }nm^{2 }ps^{-1}. The beads were connected to their up to twelve direct neighbours by springs with a length of 5 nm. Four different particles were assembled with

Sketch of three of the elastic protein models assembled from hexagonally placed small beads

**Sketch of three of the elastic protein models assembled from hexagonally placed small beads**. The beads are held together by harmonic springs between direct neighbours which are then propagated individually. In addition to the three sizes shown here we used a fourth protein built from 39 beads. As more and more beads are used the resulting particles get closer to a spherical shape.

**Brownmove definition file for the N = 13 "elastic protein"**. For more details see the comments in this (ASCII text) file.

Click here for file

From the simulations we extracted the center-of-mass diffusion coefficient _{tr}. As shown in Figure _{tr }again scaled as ^{-1 }when HI were neglected. This is the same scaling as for the polymers which have a completely different shape. With HI included, _{tr }decreased as ^{-0.42 }with the number of beads. This is slower than in the polymer case because here the particles are much closer together on average than in the flexible, chain shaped polymers.

Scaling of the translational center-of-mass diffusion coefficient _{tr }of spherical multi-bead particles with the number of beads,

**Scaling of the translational center-of-mass diffusion coefficient D _{tr }of spherical multi-bead particles with the number of beads, N**. Without HI the usual

For comparison, the translational diffusion coefficient of a sphere scales proportional to its inverse radius _{tr }decreases with the third root of the volume. We can thus estimate that in our case where the volume is proportional to the number of beads, _{tr }should decrease as ^{-0.33}. Our results were slightly slower because both the Rotne-Prager tensor and our TEA-HI algorithm underestimate the hydrodynamic coupling at close distances

When a protein is modelled from individual beads one can adjust the diffusion coefficient _{bead }of the individual beads until a best match between an experimentally determined diffusion coefficient and the _{tr }observed in the simulation is achieved. Here we could not compare to experimental values but we ran another set of simulations without HI where _{bead }was adjusted such that the center-of-mass diffusion reproduced the values obtained with HI. The required single bead diffusion coefficients were 2.4 × 10^{-4}, 5 × 10^{-4}, 9.7 × 10^{-4}, and 1.2 × 10^{-3 }nm^{2 }ps^{-1 }for the particles with N = 4, 13, 39, and 57 beads, respectively. Consequently, in Figure _{tr }values with the faster beads coincide with the results with HI.

The rotational diffusion coefficient _{rot }= k_{B}T/8πη^{3 }of a sphere of radius ^{-1/3 }behaviour of _{tr }we consequently find that the rotational relaxation time τ_{rot }= 1/2_{rot }scales proportional to _{tr}^{-3 }for a spherical particle. Figure _{rot }only scaled as _{bead }was modified as explained above such that for each particle size the correct _{tr }was obtained. Then, however, rotation was about five times too fast for all our test particles. This again shows that it is essential to include HI in multiple-bead models of, e.g. proteins in order to correctly describe

Rotational relaxation time of the end-to-end vector, t_{rot}, vs. the center-of-mass diffusion coefficient _{tr }for multi-bead particles of different sizes

**Rotational relaxation time of the end-to-end vector, t _{rot}, vs. the center-of-mass diffusion coefficient D_{tr }for multi-bead particles of different sizes N**. With HI the expected

However, as shown by Elcock

Another reason to use multi-bead models is that they allow to describe the internal flexibility of a protein, where often conformational changes occur upon binding or during the catalytic activity or, most prominently, when a protein folds. In these scenarios it would be a bad idea to rescale the single bead diffusion coefficients because even though the overall diffusive behaviour might be described better, the internal dynamics would take place on the too fast timescale of the individual beads. Our "spherical" particles do not undergo conformational changes but it is nevertheless interesting to investigate the dynamics of the individual beads. At very short timescales the elastic springs do not affect the small thermal fluctuations of the beads whereas for very long observation intervals the beads are effectively frozen to their positions in the particle. We thus expect that _{bead }extracted from a simulation decreases with increasing observation interval from the single bead value _{bead}(Δt) to the center-of-mass _{tr}.

For this we determined the apparent single bead diffusion coefficients _{bead }(Δt) = <^{2}(Δ^{2}(Δ_{bead }= 20 kDa. The corresponding velocity relaxation time is then τ_{rel }= 0.99 ps which is about the length of the shortest analysis timestep used here. The filled symbols in Figure _{bead}>(Δ^{-4 }nm^{2 }ps^{-1 }for very short intervals to the long-time center-of-mass value _{tr}. The behaviour of the apparent <_{bead}>(Δ_{bead }is rescaled to compensate for the omission of HI in such a multi-bead model the internal dynamics will be too fast in relation to the diffusional center-of-mass displacement. One can expect that then also smaller domains of a few connected beads move too fast in comparison to larger domains. This may affect for example the folding trajectories of proteins or the relative ordering in a sequence of conformational changes of a protein as demonstrated in _{bead}> dramatically decreased when. Δ_{rel }the apparent <_{bead}> = 2.3 × 10^{-5 }nm^{2 }ps^{-1 }was nearly one order of magnitude smaller than the specified long time value of _{bead }= 1.2 × 10^{-4 }nm^{2 }ps^{-1}. It is interesting to note that this slowing down of the short-time dynamics was essentially unaffected by HI. Apparently, in a BD or LD simulation the hydrodynamic velocity coupling requires some time to build up before it affects the relative motion of the individual beads.

Effective average diffusion coefficient <_{bead}> of the individual beads in a multi-bead model vs. the length of the observation interval Δ

**Effective average diffusion coefficient < D _{bead}> of the individual beads in a multi-bead model vs. the length of the observation interval Δt**. With BD the dynamics of the mass-less beads is determined on short time scales by the fast thermal motion of the individual beads while over longer periods the beads have to follow the center-of-mass motion of the complete particle (filled symbols). Consequently, over long intervals Δ

Comparing the runtimes for the various protein sizes we again find that the effort per timestep scales quadratically with the number of beads,

Here, however, a word of caution is required. From a theoretical point of view it is not correct to use RPY hydrodynamics which are based on stationary flow fields together with the LD propagation algorithm with its acceleration phases. Here an explicitly time dependent ansatz for HI should be used

Probably, one should use a different interpretation for

Example 3: Diffusional transport around fixed obstacles

The third example presented here relates to diffusional transport in a cell where many fixed obstacles like the cytosceleton or small vesicles obstruct the free diffusion of the soluble proteins. Our (non-equilibrium) setup is sketched in Figure _{x }= 30 nm and area _{y}_{z }= (20 nm)^{2 }was placed between two reservoirs with fixed densities ρ, of which one was set to a finite value ρ_{0 }= 4 × 10^{-4 }nm^{-3 }= 0.67 mM and the other to ρ = 0. After an equilibration phase a constant diffusion current developed which depends on the density difference ρ_{0 }and on the number and size of the fixed obstacles in the simulation volume. Here, we used one or two layers of nine obstacles in the (2D periodic) y-z-plane. The diffusing "proteins" had a radius of _{0 }= 10^{-4 }nm^{2 }ps^{-1}. They were uncharged and their van-der-Waals shapes prevented a mutual overlap. The obstacles were placed on a 3 × 3 rectangular grid either at _{o }= 7.4 nm. More details can be found in the actual setup and particle definition files given as additional files

Diffusional transport through an array of fixed obstacles

**Diffusional transport through an array of fixed obstacles**. The 2D periodic simulation box is bounded in the third dimension by two reservoirs with fixed densities ρ = ρ_{0 }and ρ = 0 between which a stationary diffusion current develops. This current depends on the number and size of the fixed spherical obstacles (grey) and on their interaction with the small mobile particles (red). For more details see text.

**Simulation setup file for the diffusion-between-obstacles examples**. Brownmove simulation setup file for a 2D periodic box with two oppositely placed constant density interfaces and an array of obstacles as sketched in Figure

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**Brownmove definition file for a simulation box with two constant density interfaces and a central array of fixed spherical obstacles (see Figure **

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**Brownmove particle definition file for a minimal spherical, uncharged, van-der-Waals particle used in the diffusion-between-obstacles example**.

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In our "brownmove" simulations the obstacles were implemented as a "Wall" object with a "Gestalt" that consisted of nine (18) van-der-Waals spheres at fixed positions. As these fixed structures do not move anyway, no mutual interactions between them are evaluated, so that the number of pairwise forces _{int }that has to be determined at each timestep is given by

where _{p }is the number of mobile particles and _{o }is the number of fixed obstacles. This means that for larger systems the runtime still scales with _{p }+ _{o})^{2}) if the obstacles were implemented as mobile particles that are confined to their locations by, e.g., harmonic constraints. One caveat though is that in "brownmove" hydrodynamic interactions between moving proteins and fixed obstacles are not implemented yet. Consequently, in this project no HI was used. With non-interacting particles and no obstacles in the simulation volume the resulting diffusion current _{D }= _{0 }ρ_{0 }/_{x }would be _{D }= 6.7 × 10^{-10 }ps^{-1 }nm^{-2}, i.e., particles would arrive at the ρ = 0 boundary with a rate of _{D }= _{y}_{z}_{D }= 1.1 × 10^{-6 }ps^{-1}. With our finite sized particles we obtained _{D }= 1.2 × 10^{-6 }ps^{-1 }from a control simulation without obstacles. All simulations were run for 10 million timesteps of Δ_{L}(_{L}(_{D }through the simulation volume with the respective array of obstacles and the intersect with the x-axis corresponds to the time _{D }that the particles needed to cover the distance _{x }around the obstacles. The results are given in table _{p }of mobile particles in each of the simulations. The investigated configurations were a single or a double layer of obstacles with radii _{o }= 5 to 7.3 nm and either purely repulsive short range interactions between all particles or with an additional attractive term between the obstacles and the diffusing proteins.

Diffusive particle transport through an array of fixed obstacles.

**Setup**

_{D }[μs^{-1}]

_{D }[μs]

no obstacles (control)

20

1.2

8

single layer, repulsive, _{o }= 5 nm

17

0.69

8

double layer, repulsive, _{0 }= 5 nm

15

0.55

12

double layer, repulsive, _{0 }= 6 nm

11

0.16

10

single layer, attractive, _{o }= 5 nm

22

1.2

5

double layer, attractive, _{0 }= 5 nm

25

1.2

5.5

double layer, attractive, _{0 }= 6 nm

25

1.0

6

double layer, attractive, _{0 }= 7.3 nm

27

0.72

7

Number of particles in the simulation volume, _{p}, rate of particles that crossed the simulation volume, _{D}, and the time _{D }that the particles needed to diffuse through the simulation box for the different setups that were simulated.

The results in table _{p}, and the diffusion rate _{D }decreased as expected with each additional layer of obstacles and also with the obstacle radius _{0}. It also took the particles slightly longer to travel through the simulation volume when the second layer was added because now the shorter direct path was blocked and the particles had to diffuse around the obstacles. When the obstacle radius was increased beyond _{0 }≈ 6.5 nm the diffusion current was blocked within the scope of our simulation. Then also _{D }become very large and no particles reached the ρ = 0 side within the given simulation duration of 100 μs. Intrestingly when a short range attractive interaction was added between the proteins and the obstacles, diffusion was much less hindered. Even two layers of obstacles with _{0 }= 5 nm did not reduce the diffusion rate _{D}. Due to the attraction the proteins stayed closer to the obstacles which reduced the effective "bulk" density. Then, more particles were inserted and _{p }increased. When the proteins are attracted to the obstacles they temporarily slide along their surfaces and are thus effectively funnelled through the gaps between the obstacles. Consequently, it took them less time to pass the obstacle "barrier" and _{D }decreased considerably. To shut off the particle transport the now attractive obstacles had to be made so large that the proteins did not fit through the pores anymore. This occurred at _{o }≥ 7.4 nm. The efficiency of the surface-induced funneling was so high that even at _{0 }= 7.3 nm, where the pores between the obstacles were only slightly larger than the proteins, the diffusion rate was about as high as with a single layer of much smaller repulsive obstacles.

In this simplified scenario both the mobile "proteins" and the obstacles were perfect spheres without any surface roughness and without hydrodynamic interactions which would slow down the protein diffusion close to the large obstacles

Example 4: Particle agglomeration networks

In the last example we present simulations of the agglomeration of simple particles with a binding patch and how such many-particle simulations can be analysed conveniently with the help of a dynamic interaction network. This idea was previously introduced in reference _{tr }= 1.2 × 10^{-4 }nm^{2 }ps^{-1 }and _{rot }= 2.26 × 10^{-5 }ps^{-1}. With the mass of a protein of that size of

Simulation of particle agglomeration and analysis via a dynamic network

**Simulation of particle agglomeration and analysis via a dynamic network**. Sketch of the particles composed of two mutually displaced van-der-Waals spheres (A). As indicated in B, the particles can bind to each other with their red sides. The spatial snapshots are mapped onto an interaction network by using a distance criterion. The resulting dynamic network is then used to analyse the simulation (C).

The simulation was performed with 27 particles in a cubic simulation box of 30 nm length with 3D periodic boundary conditions for 20 μs. Every 10 ns the positions and orientations of the particles were saved to disk. The particle definition and the brownmove simulation setup are given as additional files

**Brownmove particle definition file for the agglomeration-with-network-analysis example simulation**. As sketched in Figure

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**Brownmove simulation setup file for the agglomeration-with-network-analysis example simulation**.

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To demonstrate the convenience of such a network analysis we first present in Figure

Two snapshots of a simulation of particle agglomeration

**Two snapshots of a simulation of particle agglomeration**. At

Using the dynamic network it is now actually quite easy to visualise the clustering dynamics quantitatively. Figure

Network analysis of a particle agglomeration simulation

**Network analysis of a particle agglomeration simulation**. Panel A shows which cluster sizes occurred during the simulation. The two arrows denote the time points of the two snapshots shown in figure 11. The other two panels give the average and the maximal degree, <

Panel B of Figure

A third view onto the simulation is presented in panel C which gives the number of clusters (connected components) of sizes larger than a single particle, #CC(

In general we find that the more local measures such as the degree of a node give less information about the overall state of the simulation than the more global ones like the cluster size distribution. Other helpful measures which were not presented here include the clustering coefficient which quantifies how well or how regular the neighbourhood of a given particle is connected, or the distribution of shortest paths which gives a measure about how densely packed the clusters are.

Conclusions

In this publication we gave four simple examples for simulation projects that can be performed "out of the box" with our brownmove simulation package. The main features presented here are the fast hydrodynamics algorithm, the easy and flexible setup of mobile proteins and of fixed structures in the simulation volume, and how a dynamic network can be used to conveniently analyse a many particle simulation quantitatively and to visualise the fast changes of the time dependent spatial properties.

In the first example we investigated how the stiffness of a bead-spring polymer which was controlled by additional springs between next neighbours, affects the translational and rotational diffusion coefficients. This model system can be compared to a denatured protein in unfolded states ranging from a molten globule like structure of the completely flexible polymer up to a rather rod-like structure when the 1-3-springs are made stiffer. In experiments one finds that a more stretched configuration generally diffuses slower. When hydrodynamic interactions (HI) are omitted--as was often done in BD simulations of protein--the long-time translational diffusion coefficient was completely insensitive to the "folding state" of the polymer. Only when HI were included we observed that the more stretched stiffer polymers diffused slower than the more compact flexible versions. Interestingly, the rotational diffusion, characterised by the relaxation time τ_{rot }of the end-to-end vector, was only very weakly affected by HI. While the (constrained) bead-spring polymer can be interpreted as an unfolded protein, the flexible particles of our second example represent folded proteins. They were built from different numbers of small beads placed on a hexagonal close packing lattice and connected by springs to their direct neighbours. Again we examined translational and rotational diffusion and compared our results to the predictions for a sphere with an equivalent radius. In the simulations with HI the predicted scaling both of the center-of-mass diffusion coefficient _{tr }and of the orientational relaxation time τ_{rot }with the number of beads was observed, whereas without HI we could only reproduce either _{tr }or τ_{rot }by rescaling the diffusion coefficients of the individual beads, but not both at the same time. Generally, without HI either rotation was too fast or translation too slow. From these two examples one finds that HI must be included in simulations of flexible protein models when translation, rotation, and internal dynamics are investigated. With HI, the translational diffusion coefficient of the individual beads is then the only parameter that needs to be adjusted. The rotational properties and also the relative timing of internal motions then come "for free". This was already briefly discussed recently by Frembgen-Kesner and Elcock ^{2}) scaling. For these two examples with their very simple beads the inclusion of HI only slowed down the simulations by a factor of about three. When the individual sub units are more complex, i.e., when additionally point charges are included or more than a single van-der-Waals sphere are used to model non-spherical blocks, then the relative costs of HI can even decrease to less than ten percent of the total simulation time. This had been the case for our recently published simulation of a small peptide

The third example demonstrates how constant density boundary conditions

In the last example we simulated a scenario where particles with a "sticky" patch formed temporary agglomerates. Usually such simulations with their fast and frequent association and dissociation events are tedious to analyse. Following a recent project

The examples presented here are rather templates than actual projects. However, they not only emphasise the importance of HI but they also demonstrate that our freely available "brownmove" simulation package is very flexible and allows to easily investigate a number of scenarios for which a specialised software had to be written before. This especially applies to many-particle simulations with different kinds of particles in "open" systems where, e.g., non-equilibrium transport or reactions, or the association of particles from a large bulk phase onto a surface are considered.

Methods

Particle setup and interactions

As mentioned above, the particles in a "brownmove" simulation are set up hierarchically from various "objects" (see Figure

To model a particle, first a "Protein" object is defined. When this Protein object describes a rigid particle like a folded protein or a colloidal particle then it contains a single "Gestalt" object, which in turn contains the "Shape" objects that are responsible for the interactions. To implement a bead-spring polymer or a flexible protein, the "Protein" object contains multiple "Gestalt" objects and the definitions of the springs between these Gestalten. With this local definition of the connecting springs a single template molecule can be set up from which then multiple copies are drawn and inserted into the simulation during runtime. A "Protein" object thus defines an entity which is inserted or removed from a simulation as a whole.

The next level in the model hierarchy are the independently moving "Gestalt" objects. They keep track of their position and orientation and contain the "Shape" objects. The "GeomShape" is always present. It handles the generation of the random forces and then converts the total force accumulated during the current timestep into the corresponding displacement. Here the Langevin and the Brownian Dynamics propagation schemes are implemented. The other Shape objects model the physical interactions with the other particles, that is, during the force evaluation every pair of "Gestalt" objects compares whether they have Shapes of the same type. Then these two Shapes calculate their contribution to the mutual interaction.

Currently, five types of interactions are implemented. These are shielded Coulombic interactions between sets of point charges in the "EstatShape", effective short range van-der-Waals type interactions in the "vdWShape", bonds from harmonic and quartic terms in the "BondShape", external forces via the "ExternalShape", and hydrodynamic interactions via the "HIShape" objects.

Electrostatic interactions are implemented in a Debye-Hückel model via point charges which interact via a screened Coulomb potential Φ_{ik}_{i}_{k}

Here, ε is the relative dielectric constant of water and ε_{0 }the vacuum dielectric constant. The shielding from ions in the solvent is captured in the inverse Debye length κ= 1/_{D}. In most cases, the point charges are embedded in the proteins where there are no counter ions to shield the interaction. This is accounted for by _{ik}_{i}_{k}
_{i}_{k}

In brownmove the short range interactions between the surfaces of the proteins or colloidal particles are modelled phenomenologically. For this, an arbitrary number of so-called van-der-Waals spheres can be defined in the "vdWShape" objects, which then interact via a Lennard-Jones-type potential depending on the closest distance _{12 }between their surfaces.

For each van-der-Waals sphere a position within the "vdWShape", a radius, and a "colour" index are specified. For each pair of colours different interaction parameters can be specified to model, e.g., short-range hydrophobic attractions or purely repulsive hydrophilic interaction patches. For numerical stability, the diverging Lennard-Jones interaction can be linearized when the spheres overlap more than a specified distance.

Bonds between "Gestalt" objects of the same "Protein" can be hooked up at arbitrary positions on the "BondShapes". This means that bonds are not restricted to the centers of a "Gestalt" but that they can be attached eccentrically. Currently, harmonic and quartic terms are implemented for the bond potentials.

Similar "hooks" for externally specified forces are provided by the "ExternalShape" objects. Currently implemented are a harmonic potential which allows to confine the position of a "Gestalt" to a certain position, a constant vectorial force which can be used to model, e.g., gravitational forces or a constant fluid velocity, and a shear field.

Langevin and Brownian propagation schemes

To derive the implicit solvent propagation scheme for Brownian and Langevin Dynamics simulation, we start from Newton's equations of motion for the full system of the large proteins or colloidal particles and the many small solvent molecules. In this equation the forces _{i}_{i}

Here, _{r }= -γ_{i}

Here the indices _{ik}) which has 3_{i }which lead to the same displacements. With the relation between the friction coefficient γ_{i }and the self diffusion coefficient _{ii }= k_{B}T/γ_{i }we get

Then the many-particle Newton equation (4) reduces to a Langevin equation with a friction term and the effective forces _{i }_{ik}_{i}

Assuming that the force _{i }remains constant during a short time interval _{0}. For convenience we drop the coordinate index

These two equations can now directly be used to propagate the particles in the implicit-solvent approximation. This Langevin Dynamics (LD) propagation scheme assumes that the solvent can be substituted by time averaged random kicks and Stokesian friction and that Δ_{rel }=

where the velocity follows the force instantaneously and the displacement due to the external forces increases linearly with the timestep Δ

While in practical applications both for LD and BD there is the usual upper limit for the integration timestep where the numerical accuracy deteriorates, there is also a conceptual lower limit for Δ_{rel}. For BD simulations with a single particle type one usually finds an integration timestep which is large enough to be conceptually usable and also short enough for a numerically stable propagation, but this may not work any more when proteins of different sizes are considered. Then, a timestep which is stable enough for the faster smaller particles may be unphysically short for the slower larger ones. For more details on this problem see reference

In brownmove both algorithms are implemented and can even be used within the same simulation. When a mass is defined for a given particle then the LD propagation scheme is used by the "GeomShape" object, otherwise the BD algorithm. Even though the LD equations of motion (8) and (9) look more complicated than the simple BD equations (10), the additional numerical costs are negligible, because most of the terms are constants and the effort for the actual propagation step scales linearly with the number of particles, while the evaluation of the pairwise forces scales quadratically. We therefore advocate to always use the LD propagation scheme for which only the easily controllable numerical accuracy puts a constraint on the timestep.

Fast Hydrodynamics

Hydrodynamic interactions, which describe the coupling of the particle velocities via the displaced solvent, are modelled in "brownmove" via the Rotne-Prager-Yamakawa (RPY) tensor extended to handle rotation and particles of different radii

The effective hydrodynamically corrected external forces ^{eff }acting on particle

Applying the hydrodynamic coupling to the random forces is not that straightforward due to "temperature conservation"--the self diffusion of the particles which is a measure for their temperature is, for lower concentrations, not affected by the hydrodynamic interactions

The normalisation factors _{i }and the weights _{ik }can be determined approximately and the resulting truncated expansion approximation hydrodynamics (TEA-HI) recovers at least 90% of the correlations at a runtime scaling which increases only quadratically with the particle number _{ii }= 1 the normalisation factors of equation (12) can be determined from

and the quadratic equation

where ε = <_{ik }/_{ii}> is the average of the normalised off-diagonal entries of the diffusion matrix. Apart from the faster evaluation, this approximate form of the HI has the advantage that the sums in equations (12), (13), and (14) can be evaluated simultaneously with very low memory requirements from the temporarily set up two-body submatrices of the diffusion tensor. With this even large many-particle simulations fit into the fast level-3 cache of current CPUs.

In "brownmove" the above TEA-HI algorithm is implemented in the "HiShape" objects in which currently a single hydrodynamic sphere can be defined. How the hydrodynamic coupling, which is based on an instantaneous flow field, can be combined with the LD algorithm is explained in detail in reference

Coming back to the problem of bead overlap, in the original algorithm of Ermak and McCammon the results degrade with increasing overlap because the RPY tensor does not cover this regime whereas with Fixman's Chebychev approximation additionally the convergence becomes slower due to the diverging range of the eigenvectors when the spheres start to overlap

Boundary conditions

In "brownmove" various boundary condition can be used. The most simple scenario is an infinite simulation volume which would be used, e.g., to verify the long time diffusional behaviour of a flexible protein assembled from multiple sub-units. To confine the particles, "brownmove" allows to specify combinations of simple reflecting walls, one, two, or three dimensional periodic setups, and walls with a "Gestalt". By defining a "Gestalt" for a wall, not only planar van-der-Waals surfaces can be defined but also static structures like membrane proteins built from van-der-Waals spheres and point charges. A "Wall" object can thus also be used to model a rigid network of microtubili or non-mobile vesicles around which the proteins have to find their way. When periodic boundary conditions are specified, "brownmove" uses image particles during the evaluation of the interactions.

A special type of boundary condition is implemented with the use of particle acceptors and injectors, which allow to define constant density reservoirs and implement reactions at a membrane. The idea of a constant density interface is the following

Data analysis

For maximal flexibility the output of a brownmove simulation is not directly saved to disk, which would often result in unnecessarily large output files. The particle positions are rather piped into a command that is specified in the setup file. In the most simple case this command dumps the particle positions to a file. If, e.g., from a simulation of a bead-spring polymer only the center of mass and the vector from the first to the last bead is required at each output interval, the output command would extract that information on the fly and only save the processed output to disk. The output commands can be simple scripts, full-fledged analysis tools, or even multiple analysis programs chained together via pipes.

Availability

The brownmove simulation package which was presented here is freely available for academic use. The latest version can be downloaded together with documentation and some examples at

Authors' contributions

TG wrote the simulation and analysis software, conceived, performed, and analyzed the simulations, and wrote the manuscript.

Acknowledgements

The author thanks Jörg Niggemann and Christian Gorba for fruitful discussions on the design of the brownmove package and for help in the initial steps of the C++ implementation. Uwe Winter implemented the first versions of the Langevin propagation and of the fast HI approximation.