Modeling metabolic processes between molecular and systems biology.

Currently, there are two main 'schools' to handle the complexity of a living cell, the molecular biological paradigm that for each cellular function there is a key enzyme, and the complementary systems biological view that function emerges from the connectivity of the many enzymes. Here I argue that a combined molecular-systemic modeling ansatz would combine the strengths of both concepts and allow to build models that are detailed on the local level, reflect the high connectivity of cellular metabolism, and are still manageable. An overview over recent modeling advances shows that the tools and techniques are available for a hierarchic setup of even large systems form local rules which are then suited for a systemic analysis and parameterization.

Coarse-grained Brownian dynamics simulations of protein translocation through nanopores.

A crucial process in biological cells is the translocation of newly synthesized proteins across cell membranes via integral membrane protein pores termed translocons. Recent improved techniques now allow producing artificial membranes with pores of similar dimensions of a few nm as the translocon system. For the translocon system, the protein has to be unfolded, whereas the artificial pores are wide enough so that small proteins can pass through even when folded. To study how proteins permeate through such membrane pores, we used coarse-grained Brownian dynamics simulations where the proteins were modeled as single beads or bead-spring polymers for both folded and unfolded states. The pores were modeled as cylindrical holes through the membrane with various radii and lengths. Diffusion was driven by a concentration gradient created across the porous membrane. Our results for both folded and unfolded configurations show the expected reciprocal relation between the flow rate and the pore length in agreement with an analytical solution derived by Brunn et al. [Q. J. Mech. Appl. Math. 37, 311 (1984)]. Furthermore, we find that the geometric constriction by the narrow pore leads to an accumulation of proteins at the pore entrance, which in turn compensates for the reduced diffusivity of the proteins inside the pore.

Switching between crystallization and amorphous agglomeration of alkyl thiol-coated gold nanoparticles.

Crystalline and amorphous materials composed of the same atoms exhibit strikingly different properties. Likewise, the behavior of materials composed of mesoscale particles depends on the arrangement of their constituent particles. Here, we demonstrate control over particle arrangement during agglomeration. We obtain disordered and ordered agglomerates of the same alkyl thiol-coated gold nanoparticles depending on temperature and solvent. We find that ordered agglomeration occurs exclusively above the melting temperature of the ligand shells. Many-particle simulations show that the contact mechanics of the ligand shells dominate the order-disorder transition: Purely spherical particle-particle interactions yield order, whereas localized "stiction" between the ligand shells leads to disorder. This indicates that the "stickiness" and the packing of the agglomerates can be switched by the state of the ligand shells. It suggests that contact mechanics govern ordering in a wide range of nanoparticles.

Mixing normal and anomalous diffusion.

In the densely filled biological cells often subdiffusion is observed, where the average squared displacement increases slower than linear with the length of the observation interval. One reason for such subdiffusive behavior is attractive interactions between the diffusing particles that lead to temporary complex formation. Here, we show that such transient binding is not an average state of the particles but that intervals of free diffusion alternate with slower displacement when bound to neighboring particles. The observed macroscopic behavior is then the weighted average of these two contributions. Interestingly, even at very high concentrations, the unbound fraction still exhibits essentially normal diffusion.

Do we have to explicitly model the ions in brownian dynamics simulations of proteins?

Brownian dynamics (BD) is a very efficient coarse-grained simulation technique which is based on Einstein's explanation of the diffusion of colloidal particles. On these length scales well beyond the solvent granularity, a treatment of the electrostatic interactions on a Debye-Hückel (DH) level with its continuous ion densities is consistent with the implicit solvent of BD. On the other hand, since many years BD is being used as a workhorse simulation technique for the much smaller biological proteins. Here, the assumption of a continuous ion density, and therefore the validity of the DH electrostatics, becomes questionable. We therefore investigated for a few simple cases how far the efficient DH electrostatics with point charges can be used and when the ions should be included explicitly in the BD simulation. We find that for large many-protein scenarios or for binary association rates, the conventional continuum methods work well and that the ions should be included explicitly when detailed association trajectories or protein folding are investigated.

Coarse-Grained Simulations of Protein Backbone Dynamics. 1. Local Sterics Define the Dihedral Angles.

Here, we present a coarse-grained model targeted for implicit solvent simulations of unfolded or intrinsically disordered proteins. The hierarchical model with its nonspherical building blocks allows one to reproduce the local dynamics of the backbone with simple harmonic bonds and steric collisions between a small number of atoms at the correct off-center positions on the building blocks. Here in part 1, we also describe the implementation of the global shape of the protein chain and the extended local interactions that add a first secondary structure bias, which will subsequently be augmented by additional hydrophobic interactions, hydrogen bonds, and dipole dipole couplings along the backbone. Due to its hierarchical setup, the model has a near-atomistic resolution on the local scale and the overall numerical efficiency of a coarse-grained model such that even long protein chains can be simulated efficiently.

Many-particle Brownian and Langevin Dynamics Simulations with the Brownmove package.

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.

Adhesive water networks facilitate binding of protein interfaces.

Water structure has an essential role in biological assembly. Hydrophobic dewetting has been documented as a general mechanism for the assembly of hydrophobic surfaces; however, the association mechanism of hydrophilic interfaces remains mysterious and cannot be explained by simple continuum water models that ignore the solvent structure. Here we study the association of two hydrophilic proteins using unbiased extensive molecular dynamics simulations that reproducibly recovered the native bound complex. The water in the interfacial gap forms an adhesive hydrogen-bond network between the interfaces stabilizing early intermediates before native contacts are formed. Furthermore, the interfacial gap solvent showed a reduced dielectric shielding up to distances of few nanometres during the diffusive phase. The interfacial gap solvent generates an anisotropic dielectric shielding with a strongly preferred directionality for the electrostatic interactions along the association direction.

Bridging the gap: linking molecular simulations and systemic descriptions of cellular compartments.

Metabolic processes in biological cells are commonly either characterized at the level of individual enzymes and metabolites or at the network level. Often these two paradigms are considered as mutually exclusive because concepts from neither side are suited to describe the complete range of scales. Additionally, when modeling metabolic or regulatory cellular systems, often a large fraction of the required kinetic parameters are unknown. This even applies to such simple and extensively studied systems like the photosynthetic apparatus of purple bacteria. Using the chromatophore vesicles of Rhodobacter sphaeroides as a model system, we show that a consistent kinetic model emerges when fitting the dynamics of a molecular stochastic simulation to a set of time dependent experiments even though about two thirds of the kinetic parameters in this system are not known from experiment. Those kinetic parameters that were previously known all came out in the expected range. The simulation model was built from independent protein units composed of elementary reactions processing single metabolites. This pools-and-proteins approach naturally compiles the wealth of available molecular biological data into a systemic model and can easily be extended to describe other systems by adding new protein or nucleic acid types. The automated parameter optimization, performed with an evolutionary algorithm, reveals the sensitivity of the model to the value of each parameter and the relative importances of the experiments used. Such an analysis identifies the crucial system parameters and guides the setup of new experiments that would add most knowledge for a systemic understanding of cellular compartments. The successful combination of the molecular model and the systemic parametrization presented here on the example of the simple machinery for bacterial photosynthesis shows that it is actually possible to combine molecular and systemic modeling. This framework can now straightforwardly be applied to other currently less well characterized but biologically more relevant systems.

An O(N2) approximation for hydrodynamic interactions in Brownian dynamics simulations.

In the Ermak-McCammon algorithm for Brownian dynamics, the hydrodynamic interactions (HIs) between N spherical particles are described by a 3Nx3N diffusion tensor. This tensor has to be factorized at each time step with a runtime of O(N(3)), making the calculation of the correlated random displacements the bottleneck for many-particle simulations. Here we present a faster algorithm for this step, which is based on a truncated expansion of the hydrodynamic multiparticle correlations as two-body contributions. The comparison to the exact algorithm and to the Chebyshev approximation of Fixman verifies that for bead-spring polymers this approximation yields about 95% of the hydrodynamic correlations at an improved runtime scaling of O(N(2)) and a reduced memory footprint. The approximation is independent of the actual form of the hydrodynamic tensor and can be applied to arbitrary particle configurations. This now allows to include HI into large many-particle Brownian dynamics simulations, where until now the runtime scaling of the correlated random motion was prohibitive.

Graph Measures Reveal Fine Structure of Complexes Forming in Multiparticle Simulations.

Modern simulation techniques are beginning to be used to study the dynamic assembly and disassembly of multiprotein systems. In these many-particle simulations it can be very tedious to monitor the formation of specific structures such as fully assembled protein complexes or virus capsids above a background of monomers and partial complexes. However, such analyses can be performed conveniently when the spatial configuration is mapped onto a dynamically updated interaction graph. On the example of Monte Carlo simulations of spherical particles with either isotropic or directed mutual attractions, we demonstrate that this combined strategy allows for an efficient and also detailed analysis of complex formation in many-particle systems.

On the effects of PufX on the absorption properties of the light-harvesting complexes of Rhodobacter sphaeroides.

Some species of purple bacteria as, e.g., Rhodobacter sphaeroides contain the protein PufX. Concurrently, the light harvesting complexes 1 (LH1) form dimers of open rings. In mutants without PufX, the LH1s are closed rings and photosynthesis breaks down, because the ubiquinone exchange at the reaction center is blocked. However, the main purpose of the LH1 is light harvesting. We therefore investigate the effects that the PufX-induced dimerization has on the absorption properties of the core complexes. Calculations with a dipole model, which compare the photosynthetic efficiency of various configurations of monomeric and dimeric core complexes, show that the dimer can absorb photons directly into the reaction centers more efficiently, but that the performance of the more sophisticated dimeric LH1 antenna degrades faster with structural perturbations. The calculations predict an optimal orientation of the reaction centers relative to the LH1 dimer, which agrees well with the experimentally found configuration. Based on experimental observations indicating that the dimeric core complexes are indeed rather rigid, we hypothesize that in PufX(+) species the association between the LH1 and the reaction centers is enhanced. This mechanical stabilization of the core complexes would lead to the observed quinone blockage, when PufX is missing.

Molecular stochastic simulations of chromatophore vesicles from Rhodobacter sphaeroides.

A kinetic model is presented for photosynthetic processes under varying illumination based on the recently introduced steady state model of the photosynthetic chromatophore vesicles of the purple bacterium Rhodobacter sphaeroides. A stochastic simulation system is built up from independent copies of the different transmembrane proteins, each encapsulating its own set of binding sites and internal states. The proteins are then connected through pools for each of the metabolites. A number of steady state and time-dependent scenarios are presented showing that even under steady state conditions the stochastic model exhibits a different behavior than a continuous description. We find that the electronic coupling between the light harvesting complexes increases the efficiency of the core complexes which eventually allows the bacteria to bridge short illumination outages at already lower light intensities. Some new experiments are proposed by which the DeltapH dependent characteristic of the bc(1) complex or the proton buffering capacity of the vesicle could be determined.

A spatial model of the chromatophore vesicles of Rhodobacter sphaeroides and the position of the Cytochrome bc1 complex.

The photosynthetic apparatus of purple bacteria is generally considered a well-studied and understood system. However, recent atomic force microscopy images of flattened chromatophore vesicles from Rhodobacter sphaeroides restarted a debate about the stoichiometry and positions of the membrane proteins, with the interpretations of the observed images only partly being in agreement with earlier models. The most puzzling observation from the recent images is that the Cytochrome bc(1) complex, which is a central part of the photosynthetic apparatus, seems to be missing on the chromatophore vesicles, even when these were extracted from photosynthetically grown bacteria. From the available information on the geometry of the vesicle and of the proteins we reconstructed here a three-dimensional model vesicle at molecular resolution. Its central feature, also determining its diameter of approximately 45 nm, is an equatorial array of LH1 dimers, lined by a region of LH2 rings. This naturally puts the Cytochrome bc(1) complexes and the ATPase at the vesicle's poles. This spatial model may explain why the vesicle's endcaps with the bc(1) complexes are lost during the preparatory steps of the imaging process together with the ATPase and are therefore absent from the available images.

Reconstruction of a kinetic model of the chromatophore vesicles from Rhodobacter sphaeroides.

We present a molecular model of a chromatophore vesicle from Rhodobacter sphaeroides. These vesicles are ideal benchmark systems for molecular and systemic simulations, because they have been well studied, they are small, and they are naturally separated from their cellular environment. To set up a photosynthetic chain working under steady-state conditions, we compiled from the experimental literature the specific activities and geometries that have been determined for their constituents. This data then allowed defining the stoichiometries for all membrane proteins. This article contains the kinetic part of the reconstructed model, while the spatial reconstruction is presented in a companion article. By considering the transport properties of the Cytochrome c(2) and ubiquinone pools, we show that their size and oxidation states allow for an efficient buffering of the statistical fluctuations that arise from the small size of the vesicles. Stoichiometric and kinetic considerations indicate that a typical chromatophore vesicle of Rb. sphaeroides with a diameter of 45 nm should contain approximately five bc(1) monomers.