Efficient solvent-free dissipative particle dynamics for lipid bilayers.

We rigorously derived effective potentials for solvent-free DPD simulation of lipid bilayers. The derivation relies on an earlier developed hybrid particle/field method and is based on the idea that the solvent is always in local equilibrium on a coarse-grained time scale, given the instantaneous templates set by the self-assembly structure. By relating the parameters in the effective implicit-solvent potentials directly to the lipid-solvent interactions and membrane properties for the explicit solvent DPD model, we constitute an efficient and general procedure for reformulating any DPD membrane model in an implicit-solvent form. Here, we determined these membrane properties for two existing DPD models, via an analysis of membrane fluctuation spectra. Equivalent single-processor implicit- and explicit-solvent calculations show the trade-mark of implicit solvent simulation: a 20-fold reduction of the total simulation time for a system containing 92% solvent. This increased efficiency enabled us to realistically simulate the spontaneous formation of a ∼20 nm diameter vesicle on a single processor overnight. We believe that this work will contribute to an enhanced computational study of large vesicles and thus a better understanding of experimental liposome dynamics.

Stochastic quasi-Newton method: application to minimal model for proteins.

Knowledge of protein folding pathways and inherent structures is of utmost importance for our understanding of biological function, including the rational design of drugs and future treatments against protein misfolds. Computational approaches have now reached the stage where they can assess folding properties and provide data that is complementary to or even inaccessible by experimental imaging techniques. Minimal models of proteins, which make possible the simulation of protein folding dynamics by (systematic) coarse graining, have provided understanding in terms of descriptors for folding, folding kinetics, and folded states. Here we focus on the efficiency of equilibration on the coarse-grained level. In particular, we applied a new regularized stochastic quasi-Newton (S-QN) method, developed for accelerated configurational space sampling while maintaining thermodynamic consistency, to analyze the folding pathway and inherent structures of a selected protein, where regularization was introduced to improve stability. The adaptive compound mobility matrix B in S-QN, determined by a factorized secant update, gives rise to an automated scaling of all modes in the protein, in particular an acceleration of protein domain dynamics or principal modes and a slowing down of fast modes or "soft" bond constraints, similar to lincs/shake algorithms, when compared to conventional Langevin dynamics. We used and analyzed a two-step strategy. Owing to the enhanced sampling properties of S-QN and increased barrier crossing at high temperatures (in reduced units), a hierarchy of inherent protein structures is first efficiently determined by applying S-QN for a single initial structure and T=1>T(θ), where T(θ) is the collapse temperature. Second, S-QN simulations for several initial structures at very low temperature (T=0.01

Stochastic quasi-Newton molecular simulations.

We report a new and efficient factorized algorithm for the determination of the adaptive compound mobility matrix B in a stochastic quasi-Newton method (S-QN) that does not require additional potential evaluations. For one-dimensional and two-dimensional test systems, we previously showed that S-QN gives rise to efficient configurational space sampling with good thermodynamic consistency [C. D. Chau, G. J. A. Sevink, and J. G. E. M. Fraaije, J. Chem. Phys. 128, 244110 (2008)]. Potential applications of S-QN are quite ambitious, and include structure optimization, analysis of correlations and automated extraction of cooperative modes. However, the potential can only be fully exploited if the computational and memory requirements of the original algorithm are significantly reduced. In this paper, we consider a factorized mobility matrix B=JJ(T) and focus on the nontrivial fundamentals of an efficient algorithm for updating the noise multiplier J . The new algorithm requires O(n2) multiplications per time step instead of the O(n3) multiplications in the original scheme due to Choleski decomposition. In a recursive form, the update scheme circumvents matrix storage and enables limited-memory implementation, in the spirit of the well-known limited-memory Broyden-Fletcher-Goldfarb-Shanno (L-BFGS) method, allowing for a further reduction of the computational effort to O(n). We analyze in detail the performance of the factorized (FSU) and limited-memory (L-FSU) algorithms in terms of convergence and (multiscale) sampling, for an elementary but relevant system that involves multiple time and length scales. Finally, we use this analysis to formulate conditions for the simulation of the complex high-dimensional potential energy landscapes of interest.

Improved configuration space sampling: Langevin dynamics with alternative mobility.

We present a new and efficient method for determining optimal configurations of a large number (N) of interacting particles. We use a coarse-grained stochastic Langevin equation in the overdamped limit to describe the dynamics of this system and replace the standard mobility by an effective space dependent inverse Hessian correlation matrix. Due to the analogy of the drift term in the Langevin equation and the update scheme in Newton's method, we expect accelerated dynamics or improved convergence in the convex part of the potential energy surface Phi. The stochastic noise term, however, is not only essential for proper thermodynamic sampling but also allows the system to access transition states in the concave parts of Phi. We employ a Broyden-Fletcher-Goldfarb-Shannon method for updating the local mobility matrix. Quantitative analysis for one and two dimensional systems shows that the new method is indeed more efficient than standard methods with constant effective friction. Due to the construction, our effective mobility adapts high values/low friction in configurations which are less optimal and low values/high friction in configurations that are more optimal.

Design of chimaeric polymersomes.

We discuss the development of hierarchical polymer particles, or variegated polymersome composites, in which at least two different components are phase separated within one polymersome chimaera. We briefly discuss the present status in experimental polymersome research, and then discuss a speculative design strategy, based on mesoscopic simulations with a dynamical variant of polymer self-consistent field theory (Mesodyn). The main conclusion is that the counter-intuitive co-assembly of demixing block copolymers is the key in controlling hierarchical structures on a mesoscopic scale. This is the classical paradox of a chimaera: the constituents live in the same scaffold, but apart. Block copolymers beyond a certain length will always split the assembly, and without further precautions, polymer based chimaerae are intrinsically unstable. To this end, we propose the application of a branched block copolymer as composite compatibilizer, glueing the separate domains together, and thereby stabilizing the chimaeric polymersome.

Structure formation in films of weakly charged block polyelectrolyte solutions.

A mean-field dynamic density functional theory is used to describe a phase diagram of concentrated solutions of weakly charged flexible block polyelectrolytes in a film. Electrostatics is taken into account by applying the local electroneutrality constraint (the Donnan membrane equilibrium approach). In the Donnan limit it is assumed that a salt added to the solution perfectly screens long-range electrostatic interactions. The phase diagram of a solution of a triblock polyelectrolyte in a film as a function of the solvent concentration and the charge of the polyelectrolyte (solvophilic) block is calculated for a given film thickness. The phase behavior of the block polyelectrolyte film arises from the interplay between surface-induced alignment and the electrostatically-driven structure formation. The observed mesoscopic structures (lamellar, perforated lamellar, cylindrical, micellar, and mixed phases) are oriented parallel to the surfaces for the considered case of morphologies unfrustrated by the film thickness. Structures with connections between parallel layers (bicontinuous, etc.) are not formed. As a result of surface-induced ordering, the region of ordered phases in a film is wider than in bulk and the phase boundary between ordered and disordered phases is more diffuse. As in the case of unconfined block polyelectrolyte solution, the solution in a film does not follow the lyotropic sequence of phases of such a block copolymer upon increase in the charge of the polyelectrolyte block. Upon changing the charge of the solvophilic copolymer block, transformations of copolymer morphology take place via change in curvature of polymeric domains. Due to confinement of a polyelectrolyte film, no swelling of solvophilic domains is observed.

Nematic-amorphous polymer interfaces in the presence of a compatibilizer.

We introduce and apply a variant of a dynamic self-consistent field simulation in two dimensions to predict the structure of interfaces between a nematic and an amorphous polymer compatibilized by a diblock copolymer. First, we investigate the effect of the nematic order on the polymer polymer interface without compatibilizer. Then we include the compatibilizer and consider two interfacial setups previously used in experiments, i.e., the bilayer setup and the trilayer setup. In the bilayer setup the diblock copolymer is mixed into the amorphous homopolymer and migrates to the interface in the course of the simulation forming a layered structure. We compare the amount of copolymer at the interface for initial concentrations of the copolymer below and above the critical micelle concentration. In the trilayer setup the initial thickness of the diblock copolymer is varied. The resulting interfacial morphology evolves in the competition between the lamellar structure induced by the interface and a micellar structure, which is intrinsic to the copolymer.

Microphase separation of weakly charged block polyelectrolyte solutions: Donnan theory for dynamic polymer morphologies.

A mean-field dynamic density functional theory for the phase behavior of concentrated weakly charged block polyelectrolyte solutions is developed, using the Donnan membrane equilibrium approach to account for electrostatic interactions. In this limit all long-range electrostatic interactions are canceled and the net charge density in any region on a coarse-grained scale is zero. The phase diagram of a model triblock polyelectrolyte in solution as a function of the charge of the solvophilic block and the solvent concentration is established. Different mesoscopic structures (lamellar, bicontinuous, hexagonal, micellar, and dispersed coexisting phases) are formed depending on the copolymer charge asymmetry. It is found that upon changing the charge of the solvophilic copolymer block the polyelectrolyte solution does not follow the lyotropic sequence of phases of this polymer. Upon increase in the charge of the solvophilic blocks, changes in copolymer morphology take place by means of change in curvature of polymeric domains.

Orientations of the lamellar phase of block copolymer melts under oscillatory shear flow.

We develop a theory to describe the reorientation phenomena in the lamellar phase of block copolymer melts under reciprocating shear flow. We show that, similar to the steady shear, the oscillating flow anisotropically suppresses fluctuations and gives rise to the [parallel]--> [perpendicular] transition. The experimentally observed high-frequency reverse transition is explained in terms of interaction between the melt and the shear-cell walls.

Influence of confinement on the orientational phase transitions in the lamellar phase of a block-copolymer melt under shear flow.

In this paper, we incorporate some real-system effects into the theory of orientational phase transitions under shear flow [M. E. Cates and S. T. Milner, Phys. Rev. Lett. 62 1856 (1989) and G. H. Fredrickson, J. Rheol. 38, 1045 (1994)]. In particular, we study the influence of the shear-cell boundaries on the orientation of the lamellar phase. We predict that at low shear rates, the parallel orientation appears to be stable. We show that there is a critical value of the shear rate at which the parallel orientation loses its stability and the perpendicular one appears immediately below the spinodal. We associate this transition with a crossover from the fluctuation to the mean-field behavior. At lower temperatures, the stability of the parallel orientation is restored. We find that the region of stability of the perpendicular orientation rapidly decreases as shear rate increases. This behavior might be misinterpreted as an additional perpendicular to parallel transition recently discussed in literature.

Microhydrodynamics simulation of protein crystallization. I. Static calculations.

A computer simulation method is proposed to study the effects of hydrodynamic interactions on protein crystallization. It is a combination of Stokesian dynamics and continuum hydrodynamics and is referred to as "microhydrodynamics." The method is checked against analytical expressions for Stokes drag and diffusion coefficients for unit spheres. For a number of protein molecules the diffusion coefficients have been calculated and compared with experimental values. It is shown that the method works well for stationary calculations. Using dynamical calculations interacting protein molecules will be simulated to study the events in the early stages of protein crystallization.

Mesoscopic theories for protein crystal growth.

A computer-simulation method is proposed for studying the hydrodynamic interactions of rigid protein molecules. It is a combination of Stokes dynamics and continuum hydrodynamics. The Stokes equations of motion for the protein molecules, the creeping-flow equation for the solvent together with the no-slip boundary conditions give a complete representation of the system. The resulting three-dimensional boundary-value problem can be rewritten in a two-dimensional form (without any loss of information) considering the surfaces of the particles only. Then, by solving the equations on discrete surface elements, the so-called mobility matrix is determined in which all hydrodynamic interactions are included. Finally, after calculation of the conservative forces and the stochastic force, the new velocities of the protein molecules can be determined. The simulation method can be applied to arbitrary particle shapes. It can also handle arbitrary flow fields, and the effects of applying a flow field to the system can be studied. From analysis of the trajectories, information can be gained on the kinetics and thermodynamics in the early stages of the crystallization process.

Interfacial thermodynamics of protein adsorption and ion co-adsorption. III. Electrochemistry of bovine serum albumin adsorption on silver iodide.

An experimental analysis of charge regulation in protein adsorption is presented. The model system consists of colloidal particles of the slightly water soluble salt silver iodide as the adsorbent and the protein bovine serum albumin as the adsorbate. Protein adsorption experiments corroborate earlier findings that albumin adsorbs maximally close to the isoelectric point of the protein. The adsorption is reversible with respect to protein-protein exchange. The charge regulation is studied by novel potentiometric titrations. The Galvani potential of the adsorbent, partially covered with protein, is varied by the addition of AgNO3/KI while the pH is kept constant by means of a pH-stat. It is shown that the ion co-adsorption is a linear decreasing function of the blank surface charge density. The results are consistent with thermodynamics: for the first time a few phenomenological linkage relations between the ion co-adsorptions and chemical potentials are verified experimentally. The charge regulation is interpreted in terms of a contact layer model, which explains the ion co-adsorption by compounded ion exchange equilibria in the small layer of atomic contact between adsorbed protein and surface.

Interfacial thermodynamics of protein adsorption, ion co-adsorption and ion binding in solution. I. Phenomenological linkage relations for ion exchange in lysozyme chromatography and titration in solution.

In this paper we discuss the thermodynamics of ion binding in solution, protein adsorption and ion co-adsorption. The emphasis is on charge regulation effects. To this end, we introduce phenomenological linkage relations from which the ion binding can be calculated from the electrolyte dependency of proton titration curves and the co-adsorption from the electrolyte dependency of protein adsorption isotherms. The linkage relations are derived from classical interfacial thermodynamics, and thus offer an alternative approach as compared to the mass balance equations which are currently used in biotechnology, and Record et al.'s 1978 analysis of Wyman's Binding Polynomial for protein interactions. The co-adsorption theory is an extension of our previous analysis of ion binding in solution, which we include here for comparison of the ion co-adsorption with the ion binding in solution. The theory is applied to the chromatography of lysozyme on the strong cation exchanger 'mono S' and to the proton titration of lysozyme in solution. In the accompanying Part 2 of this paper the results are interpreted with a simple model.

Interfacial thermodynamics of protein adsorption, ion co-adsorption and ion binding in solution. II. Model interpretation of ion exchange in lysozyme chromatography.

In this paper we present a model for the ion exchange effects in protein adsorption. The model is applied to chromatography of lysozyme on strong cation exchanger 'mono S'. The experimental and general thermodynamic aspects have been discussed in Part 1, the preceding paper. The main modelling assumptions are (i) the charge regulation is confined to the small layer of contact between adsorbed protein and exchanger surface, (ii) the contact layer as a whole is electroneutral and (iii) the number of protein acid/base groups and exchanger surface acid groups which participate in the ion exchange is proportional to the area of the contact layer. The model is fitted to the experimental data by adjustment of only two or three parameters. The experimental co-adsorption numbers are very well reproduced. A few conspicuous features emerge: (i) the number of protein acid/base groups and exchanger surface acid groups in the contact layer varies with the medium conditions, such that the number is higher when the interaction between protein and exchanger surface is stronger. (ii) There is indirect evidence for structural alterations in the upper layers of the exchanger surface: the adsorbed protein is probably partly 'buried' in the surface.

Thermodynamics of ion binding by proteins. Phenomenological linkage relations for binding of electrolyte and interpretation by double layer theory.

In this paper we report a new set of thermodynamic linkage relations for the binding of electrolyte by proteins. The relations are derived for protein solutions in membrane equilibrium with a reference solution, allowing a phenomenological definition of ion binding. This is an extension of Wyman's linkage theory. The theory is applied to the electrolyte dependence of proton titration curves for bovine serum albumin in KCl solution (C. Tanford, S. A. Swanson and W. S. Shore, J. Am. Chem. Soc. 77 (1955) 6414). The curves are re-analysed in terms of Esin-Markov coefficients. In addition, we discuss the interpretation of the phenomenological K+ and Cl- binding numbers in terms of a two-state binding model, in which part of the ions are thought to adsorb on specific sites at the protein surface and/or part in the diffuse layer. It is shown that the electrolyte binds largely in the diffuse layer, especially when the protein surface charge is high.

Orientation of adsorbed cytochrome c as a function of the electrical potential of the interface studied by total internal reflection fluorescence.

A method for determination of the orientation of adsorbed structure-stable proteins using Total Internal Reflection Fluorescence is outlined. The theory has been elaborated for orientation studies on adsorbed free base cytochrome c, of which the prophyrin can be used as an intrinsic fluorescent label. The ratio of fluorescence intensities at two polarization modes of the incident light (the transverse magnetic and the transverse electric polarization mode, respectively) gives a relation between the orientation angles of the porphyrin relative to the interface. As an illustration of the theory, experimental results on the adsorption of cytochrome c at an optically transparent SnO2 film electrode are presented. It is concluded that the orientation of the molecules can only be affected by the interfacial potential during the process of adsorption, but, once adsorbed, the orientation cannot be changed anymore by variation of the potential.