Interdisciplinary Centre for Mathematical and Computational Modeling, University of Warsaw, Żwirki i Wigury 93, 02-089 Warsaw, Poland

Abstract

Biochemical reactions in living systems occur in complex, heterogeneous media with total concentrations of macromolecules in the range of 50 - 400

Introduction

Intracellular organelles are packed with small solutes, macromolecules, membranes and skeletal proteins. Such crowding is characteristic not only of the cell's interior but also for extracellular tissues

More and more experimental methods have become available to investigate macromolecules under in vivo conditions. These include nuclear magnetic resonance

Theories of diffusion for colloidal soft matter are well established. Within their framework, the behavior of colloidal systems (both dilute and concentrated), consisting of mesoscopically large colloidal particles dispersed in low-molecular-weight solvent, can be simulated, reproduced, and more importantly predicted

Simulation methods used to study the in vivo diffusion of macromolecules should take into account their biological localization, sizes and shapes, and positions in three dimensional space. Intermolecular interactions (both specific and non-specific, repulsive and attractive) should be also included

Diffusion in dilute and crowded solutions

In dilute solutions, diffusional properties of molecules treated as rigid bodies are determined by their sizes, shapes, temperature, and solvent viscosity. The complete information required to describe translational and rotational motions of Brownian particles in dilute solutions is contained in their diffusion tensors. Single-particle diffusion tensor,

The diffusion tensors of molecules can be obtained from rigid-body hydrodynamic calculations ^{tt}^{rr }^{trans}

with the ^{trans }

In biological setup, solutes interact with their environments and other molecules which affects their Brownian motion, thus single-particle diffusion tensors are no longer the unique determinants of diffusion. Nevertheless, the relation between the

Particle-based Brownian dynamics algorithms

Brownian dynamics is a stochastic simulation approach with continuum space and time. In BD the molecules exhibit noise as they are propagated according to the Langevin equation

Brownian dynamics of rigid, arbitrarily shaped objects

The propagation scheme for a number of rigid bodies, described either at atomic level or with coarse-grained models, can be written as

where

Here,

A simplified version of the above algorithm

Brownian dynamics with hydrodynamic interactions

When hydrodynamic interactions (HI) are considered in BD simulations, the diffusion tensor of the entire system (6N×6N, with N being the number of spherical molecules or the number of spherical pseudo-atoms in case of more elaborate coarse-grained molecular models

When the Rotne-Prager-Yamakwa

Intermolecular interactions

The influence of environments on the diffusion of macromolecules manifests through intermolecular forces acting on individual particles. The types and functional forms of interactions included in BD simulations vary depending on the level of detail used to describe the studied systems. Different types of intermolecular interactions that can be modeled in BD simulations are briefly described below.

Electrostatics

When diffusing molecules are described in a reduced way with spherical subunits (e. g., as in case of model spherical proteins _{i}_{j}_{i}_{j}

Where ϵ_{o }_{ij }

A more sophisticated approach to treat electrostatic interactions in BD simulations of rigid molecules with atomic detail

Lennard-Jones potentials

Non-specific interactions between molecules can be modeled with the Lennard-Jones type functions that are commonly used in molecular dynamics simulations to compute the van der Waals interactions between atoms; at large intermolecular separations molecules attract each other, while at small separations the interactions are repulsive and molecules are impenetrable. The Lennard-Jones functional form and parameters for BD simulations with coarse-grained spherical models

In BD simulations employing all-atom models, the forces acting between the surface atoms of molecules can be evaluated using standard (6/12) Lennard-Jones potentials:

However, the well depth ϵ_{LJ }

Hydrophobic effects

Nonpolar (hydrophobic) interactions can be included in atomistic BD simulations by assuming their proportionality to the amount of the solvent accessible surface area (SASA) that gets buried when molecules come close to each other. However, computing of SASA can be slow and to make SASA-based approaches efficient, approximations are needed. Examples of how to incorporate the hydrophobic effects based on SASA in BD simulations can be found in the work of Elcock and McCammon

While SASA-based models are commonly used to evaluate nonpolar interactions, it has been shown that including the solvent accessible volume (SAV) terms may be essential to accurately model the nonpolar solvation, especially at atomic-length scales

It was also proposed that hydrophobic interactions can be accounted for by modifying the van der Waals interactions and assigning more favorable interaction energies to nonpolar surface atoms than to polar ones

Hydrodynamic interactions

In BD simulations water molecules are not treated explicitly but the influence of solvent on the diffusion of suspended molecules can be included by proper treatment of hydrodynamic interactions. However, even though their importance is widely recognized

In the Ermak-McCammon algortihm ^{3 }(with N being the number of particles), while evaluating direct intermolecular interactions, that are assumed pairwise, scales as N^{2}. It is possible to lower the cost of evaluating HI using the Chebyshev approximation proposed by Fixman ^{2.5}). Recently, Geyer and Winter proposed a faster algorithm that scales as N^{2 }and is based on a truncated expansion of the hydrodynamic multi-particle correlations as two-body contributions

An additional computational cost arises when HI are evaluated in finite simulation boxes containing molecules. This cost is due to the fact that for evaluating diffusion tensors Ewald summation has to be used

The treatment of hydrodynamics within the framework of the Oseen, Rotne-Prager or Rotne-Prager-Yamakwa tensors is appropriate for very dilute systems capturing only the far-field, two-body hydrodynamic effects. However, in crowded biological systems, the many-body HI and lubrication forces (resulting from the thin layer of solvent that separates the nearly touching particles) may significantly influence diffusion. One crude way to include the many-body HI in BD simulations is to use mean-field approaches

The most advanced approach to treating HI, used recently in BD simulations of a biological system ^{1.25}log(N).

Unfortunately, modeling of HI in BD simulations is currently limited to systems composed of coarse-grained particles (either spherical molecules or composed of a small number of spherical units - pseudo-atoms). While such models perform well for colloids, a question yet unanswered is whether they are sufficient for crowded, heterogeneous biological systems.

Applications to diffusion in crowded environments

Typically, the setup of BD simulations aimed at investigating diffusion in crowded solutions is the following. A number of molecules (usually 10^{2 }- 10^{3}), described either at the atomistic level or with coarse-grained models, is contained inside a primary simulation box. Periodic boundaries are applied to reduce the edge effects and the influence of the finite size of the system. BD trajectories of all molecules are generated with the algorithm of choice. The BD simulation time scale should be long enough so the relative displacements in the system be at least comparable with the dimensions of molecules; the simulation time scales range from microseconds to milliseconds, while typical time steps are about a few picoseconds. When one considers the number of simulated molecules, the need to describe them in as detailed way as possible, and the complexity of interactions calculated in a single simulation step, it becomes clear that BD simulations of crowded systems require a lot of computational resources.

Below we present a few recent (except for one) examples of BD simulations of macromolecular diffusion in crowded media. We describe the studies that best illustrate the potential and shortcomings of the BD technique applied to in vivo-like systems and the ones that introduce a significant element of novelty. Some of these computational studies relate to experiments - experimental data were either used for parameterization or for direct comparison with simulations.

Beginnings

One of the first studies that used the BD technique to investigate macromolecular diffusion in congested media was that of Dwyer and Bloomfield

Simulations of the diffusion of BSA in the DNA solution conducted at low ionic strength of 0.01 M gave worse agreement with experiments which the authors attributed to the lack of HI and limitations of the applied electrostatic model. The work of Dwyer and Bloomfield is significant because it shows that even very simple modeling approaches can accurately predict the behavior of rather complex systems.

We note that the models similar to the one of Dwyer and Bloomfield are still used, for example, in computational studies of facilitated diffusion of proteins on DNA. The description of facilitated diffusion phenomenon and models of non-specific protein-DNA interactions can be found in references

Protein-protein association in crowded environments

Most biological reactions require some sort of diffusion-mediated encounter. Here we describe BD applications that directly investigate the influence of crowded environments on diffusional encounter and association of molecules.

Sun and Weinstein

One of the conclusions of the work of Sun and Weinstein

Similar observations were made by Kim and Yethiraj

In general, the above simulation results are in agreement with purely theoretical predictions

Modeling bacterial cytoplasm

In the recent work of Elcock's

Intermolecular interactions included electrostatics, modeled using the effective charges approach of Gabadoulline and Wade

While adjusting the short-range attractive interactions led to reproducing the in vivo GFP diffusion coefficient, it has been later pointed out

The approach to model diffusion in the E. coli cytoplasm taken by Ando and Skolnick _{α }protein atoms and the P, C4', N1, and N9 atoms of nucleic acids. The authors dealt with the anisotropy of macromolecular shapes using the rigid body BD algorithm

Based on simulations of molecular-shaped and equivalent sphere systems, the authors showed that the effects of macromolecular shapes on long-time translational diffusion are rather small in the studied concentration range of 250 - 350

Both studies described above

Concluding remarks

This short review focuses on the current status of the particle-based Brownian dynamics technique and its applicability to model diffusive phenomena in biological environments. We described a few BD studies of three-dimensional macromolecular diffusion in aqueous cellular compartments under molecular crowding conditions. However, there are various other BD applications, not presented here, that consider diffusive phenomena in cellular architectures, such as the cytoskeleton

We presented different modeling approaches, from simple excluded-volume models to most sophisticated atomic level descriptions of the in vivo-like systems. The complexity of intra and extracellular environments, their dynamics, and the number of different interactions that influence the diffusive behavior of macromolecules, make computational studies of crowding extremely difficult. While it is desirable to construct models that treat biological systems at atomic resolutions and include sophisticated interactions

Obviously, current BD algorithms need development. These include accurate and computationally efficient models of direct, both specific and non-specific, intermolecular interactions as well as HI models that up to date have been largely neglected in the simulations of biological systems.

BD simulations should be (at least ideally) able to reproduce and predict the in vivo dynamics. However, the computational cost of sophisticated BD simulations may be too high, even with the growth of computing power and available modern technologies. Thus, the long-term goal should be to systematically develop efficient algorithms interfacing different modeling approaches - from atomistic to coarse-grained models of macromolecules, mesoscopic models of biological environments, and appropriate boundary conditions for different cell compartments. New approaches and models should be verified experimentally to establish their quality and predictive power.

Competing interests

The authors declare that they have no competing interests.

Authors' contributions

MD studied literature and designed manuscript content. MD and JT wrote the paper.

Acknowledgements

The authors acknowledge support from University of Warsaw (BST 2010, G31-4), Polish Ministry of Science and Higher Education (N N301 245236, N N519 646640) and Foundation for Polish Science (Focus program and TEAM/2009-3/8 project co-financed by European Regional Development Fund operated within Innovative Economy Operational Programme).