Crystallization of asymmetric patchy models for globular proteins in solution

The ion-activated attractive patchy particle model and its application to the liquid-vapor phase transitions

Patchy particles are an intriguing subject of study and indeed a model system in the field of soft matter physics. In recent years, patchy particle models have been applied to describe a wide variety of systems, including colloidal crystals, macromolecular interactions, liquid crystals, and nanoparticle assemblies. Given the importance of the topic, rationalizing and capturing the basic features of these models is crucial to their correct application in specific systems. In this study, we extend the ion-activated attractive patchy particles model previously employed to elucidate the phase behavior of protein solutions in the presence of trivalent salts. Our extension incorporates the effect of repulsion between unoccupied and occupied binding sites, depicted as patches. Furthermore, we examine the influence of model parameters on the liquid-vapor coexistence region within the phase diagram, employing numerical methods. A deeper understanding of this model will facilitate a better comprehension of the effects observed in experiments.

Theory of density waves and organization of proteins in icosahedral virus capsids

Understanding the physical principles underlying the structural organization of the proteinaceous viral shells is of major importance to advance antiviral strategies. Here, we develop a phenomenological thermodynamic theory, which considers structures of small and middle-size icosahedral viral shells as a result of condensation of a minimum number of protein density waves on a spherical surface. Each of these irreducible critical waves has icosahedral symmetry and can be expressed as a specific series of the spherical harmonics Ylm with the same wave number l. As we demonstrate, in small viral shells self-assembled from individual proteins, the maxima of one critical density wave determine the positions of proteins, while the spatial derivatives of the second one control the protein orientations on the shell surface. In contrast to the small shells, the middle-size ones are always formed from pentamers and hexamers (referred to as capsomers). Considering all such structures deposited in the Protein Data Bank, we unexpectedly found that the positions of capsomeres in these shells correspond to the maxima of interference patterns produced by no more than two critical waves with close wave numbers. This fact allows us to explain the observed limit size of the icosahedral shells assembled from pentamers and hexamers. We also construct nonequilibrium thermodynamic potentials describing the protein crystallization and discuss the reasons behind the specific handedness of the viral shells.

Protein Association in Solution: Statistical Mechanical Modeling

Protein molecules associate in solution, often in clusters beyond pairwise, leading to liquid phase separations and high viscosities. It is often impractical to study these multi-protein systems by atomistic computer simulations, particularly in multi-component solvents. Instead, their forces and states can be studied by liquid state statistical mechanics. However, past such approaches, such as the Derjaguin-Landau-Verwey-Overbeek (DLVO) theory, were limited to modeling proteins as spheres, and contained no microscopic structure-property relations. Recently, this limitation has been partly overcome by bringing the powerful Wertheim theory of associating molecules to bear on protein association equilibria. Here, we review these developments.

The physics of empty liquids: from patchy particles to water

Empty liquids represent a wide class of materials whose constituents arrange in a random network through reversible bonds. Many key insights on the physical properties of empty liquids have originated almost independently from the study of colloidal patchy particles on one side, and a large body of theoretical and experimental research on water on the other side. Patchy particles represent a family of coarse-grained potentials that allows for a precise control of both the geometric and the energetic aspects of bonding, while water has arguably the most complex phase diagram of any pure substance, and a puzzling amorphous phase behavior. It was only recently that the exchange of ideas from both fields has made it possible to solve long-standing problems and shed new light on the behavior of empty liquids. Here we highlight the connections between patchy particles and water, focusing on the modelling principles that make an empty liquid behave like water, including the factors that control the appearance of thermodynamic and dynamic anomalies, the possibility of liquid-liquid phase transitions, and the crystallization of open crystalline structures.

The role of complementary shape in protein dimerization

Shape guides colloidal nanoparticles to form complex assemblies, but its role in defining interfaces in biomolecular complexes is less clear. In this work, we isolate the role of shape in protein complexes by studying the reversible binding processes of 46 protein dimer pairs, and investigate when entropic effects from shape complementarity alone are sufficient to predict the native protein binding interface. We employ depletants using a generic, implicit depletion model to amplify the magnitude of the entropic forces arising from lock-and-key binding and isolate the effect of shape complementarity in protein dimerization. For 13% of the complexes studied here, protein shape is sufficient to predict native complexes as equilibrium assemblies. We elucidate the results by analyzing the importance of competing binding configurations and how it affects the assembly. A machine learning classifier, with a precision of 89.14% and a recall of 77.11%, is able to identify the cases where shape alone predicts the native protein interface.

Self assembly of model polymers into biological random networks

The properties of biological networks, such as those found in the ocular lens capsule, are difficult to study without simplified models. Model polymers are developed, inspired by "worm-like" curve models, that are shown to spontaneously self assemble to form networks similar to those observed experimentally in biological systems. These highly simplified coarse-grained models allow the self assembly process to be studied on near-realistic time-scales. Metrics are developed (using a polygon-based framework) which are useful for describing simulated networks and can also be applied to images of real networks. These metrics are used to show the range of control that the computational polymer model has over the networks, including the polygon structure and short range order. The structure of the simulated networks are compared to previous simulation work and microscope images of real networks. The network structure is shown to be a function of the interaction strengths, cooling rates and external pressure. In addition, "pre-tangled" network structures are introduced and shown to significantly influence the subsequent network structure. The network structures obtained fit into a region of the network landscape effectively inaccessible to random (entropically-driven) networks but which are occupied by experimentally-derived configurations.

Multivalent ions and biomolecules: Attempting a comprehensive perspective

Ions are ubiquitous in nature. They play a key role for many biological processes on the molecular scale, from molecular interactions, to mechanical properties, to folding, to self-organisation and assembly, to reaction equilibria, to signalling, to energy and material transport, to recognition etc. Going beyond monovalent ions to multivalent ions, the effects of the ions are frequently not only stronger (due to the obviously higher charge), but qualitatively different. A typical example is the process of binding of multivalent ions, such as Ca2+ , to a macromolecule and the consequences of this ion binding such as compaction, collapse, potential charge inversion and precipitation of the macromolecule. Here we review these effects and phenomena induced by multivalent ions for biological (macro)molecules, from the "atomistic/molecular" local picture of (potentially specific) interactions to the more global picture of phase behaviour including, e. g., crystallisation, phase separation, oligomerisation etc. Rather than attempting an encyclopedic list of systems, we rather aim for an embracing discussion using typical case studies. We try to cover predominantly three main classes: proteins, nucleic acids, and amphiphilic molecules including interface effects. We do not cover in detail, but make some comparisons to, ion channels, colloidal systems, and synthetic polymers. While there are obvious differences in the behaviour of, and the relevance of multivalent ions for, the three main classes of systems, we also point out analogies. Our attempt of a comprehensive discussion is guided by the idea that there are not only important differences and specific phenomena with regard to the effects of multivalent ions on the main systems, but also important similarities. We hope to bridge physico-chemical mechanisms, concepts of soft matter, and biological observations and connect the different communities further.

Enhanced protein adsorption upon bulk phase separation

In all areas related to protein adsorption, from medicine to biotechnology to heterogeneous nucleation, the question about its dominant forces and control arises. In this study, we used ellipsometry and quartz-crystal microbalance with dissipation (QCM-D), as well as density-functional theory (DFT) to obtain insight into the mechanism behind a wetting transition of a protein solution. We established that using multivalent ions in a net negatively charged globular protein solution (BSA) can either cause simple adsorption on a negatively charged interface, or a (diverging) wetting layer when approaching liquid-liquid phase separation (LLPS) by changing protein concentration (cp) or temperature (T). We observed that the water to protein ratio in the wetting layer is substantially larger compared to simple adsorption. In the corresponding theoretical model, we treated the proteins as limited-valence (patchy) particles and identified a wetting transition for this complex system. This wetting is driven by a bulk instability introduced by metastable LLPS exposed to an ion-activated attractive substrate.

100th Anniversary of Macromolecular Science Viewpoint: Modeling and Simulation of Macromolecules with Hydrogen Bonds: Challenges, Successes, and Opportunities

Macromolecular materials with directional interactions such as hydrogen bonds exhibit numerous attractive features in terms of structure, thermodynamics, and dynamics. Besides enabling precise tuning of desirable geometries in the assembled state (e.g., programmable coordination numbers depending on the valency of the directional interaction), mixing in a blend/composite through stabilization via hydrogen bonds between the various components, hydrogen bonds can also impart responsiveness to external stimuli (e.g., temperature, pH). In biomacromolecules (e.g., proteins, DNA, polysaccharides), hydrogen bonds play a key role in stabilizing secondary and tertiary structures, which in turn define the function of these macromolecules. In this Viewpoint, I present the challenges, successes, and opportunities for molecular modeling and simulations to conduct fundamental and application-focused research on macromolecular materials with hydrogen bonding interactions. The past successes and limitations of atomistic simulations are discussed first, followed by highlights from recent developments in coarse-grained modeling and their use in studies of (synthetic and biologically relevant) macromolecular materials. Model development focused on polynucleotides (e.g., DNA, RNA, etc.), polypeptides, polysaccharides, and synthetic polymers at experimentally relevant conditions are highlighted. This viewpoint ends with potential future directions for macromolecular modeling and simulations with other types of directional interactions beyond hydrogen bonding.

Polymer compaction and bridging-induced clustering of protein-inspired patchy particles

There are many proteins or protein complexes which have multiple DNA binding domains. This allows them to bind to multiple points on a DNA molecule (or chromatin fibre) at the same time. There are also many proteins which have been found to be able to compact DNAin vitro, and many others have been observed in foci or puncta when fluorescently labelled and imagedin vivo. In this work we study, using coarse-grained Langevin dynamics simulations, the compaction of polymers by simple model proteins and a phenomenon known as the 'bridging-induced attraction'. The latter is a mechanism observed in previous simulations [Brackleyet al2013Proc. Natl Acad. Sci. USA110E3605], where proteins modelled as spheres form clusters via their multivalent interactions with a polymer, even in the absence of any explicit protein-protein attractive interactions. Here we extend this concept to consider more detailed model proteins, represented as simple 'patchy particles' interacting with a semi-flexible bead-and-spring polymer. We find that both the compacting ability and the effect of the bridging-induced attraction depend on the valence of the model proteins. These effects also depend on the shape of the protein, which determines its ability to form bridges.

Obtaining Soft Matter Models of Proteins and their Phase Behavior

Globular proteins are roughly spherical biomolecules with attractive and highly directional interactions. This microscopic observation motivates describing these proteins as patchy particles: hard spheres with attractive surface patches. Mapping a biomolecule to a patchy model requires simplifying effective protein-protein interactions, which in turn provides a microscopic understanding of the protein solution behavior. The patchy model can indeed be fully analyzed, including its phase diagram. In this chapter, we detail the methodology of mapping a given protein to a patchy model and of determining the phase diagram of the latter. We also briefly describe the theory upon which the methodology is based, provide practical information, and discuss potential pitfalls. Data and scripts relevant to this work have been archived and can be accessed at https://doi.org/10.7924/r4ww7bs1p .

Using Schematic Models to Understand the Microscopic Basis for Inverted Solubility in γD-Crystallin

Inverted solubility-melting a crystal by cooling-is observed in a handful of proteins, such as carbomonoxy hemoglobin C and γD-crystallin. In human γD-crystallin, the phenomenon is associated with the mutation of the 23rd residue, a proline, to a threonine, serine, or valine. One proposed microscopic mechanism entails an increase in surface hydrophobicity upon mutagenesis. Recent crystal structures of a double mutant that includes the P23T mutation allow for a more careful investigation of this proposal. Here, we first measure the surface hydrophobicity of various mutant structures of γD-crystallin and discern no notable increase in hydrophobicity upon mutating the 23rd residue. We then investigate the solubility inversion regime with a schematic patchy particle model that includes one of three variants of temperature-dependent patch energies: two of the hydrophobic effect, and one of a more generic nature. We conclude that, while solubility inversion due to the hydrophobic effect may be possible, microscopic evidence to support it in γD-crystallin is weak. More generally, we find that solubility inversion requires a fine balance between patch strengths and their temperature-dependent component, which may explain why inverted solubility is not commonly observed in proteins. We also find that the temperature-dependent interaction has only a negligible impact on liquid-liquid phase boundaries of γD-crystallin, in line with previous experimental observations.

Temperature-Dependent Interactions Explain Normal and Inverted Solubility in a γD-Crystallin Mutant

Protein crystal production is a major bottleneck in the structural characterization of proteins. To advance beyond large-scale screening, rational strategies for protein crystallization are crucial. Understanding how chemical anisotropy (or patchiness) of the protein surface, due to the variety of amino-acid side chains in contact with solvent, contributes to protein-protein contact formation in the crystal lattice is a major obstacle to predicting and optimizing crystallization. The relative scarcity of sophisticated theoretical models that include sufficient detail to link collective behavior, captured in protein phase diagrams, and molecular-level details, determined from high-resolution structural information, is a further barrier. Here, we present two crystal structures for the P23T + R36S mutant of γD-crystallin, each with opposite solubility behavior: one melts when heated, the other when cooled. When combined with the protein phase diagram and a tailored patchy particle model, we show that a single temperature-dependent interaction is sufficient to stabilize the inverted solubility crystal. This contact, at the P23T substitution site, relates to a genetic cataract and reveals at a molecular level the origin of the lowered and retrograde solubility of the protein. Our results show that the approach employed here may present a productive strategy for the rationalization of protein crystallization.

Breakdown of the law of rectilinear diameter and related surprises in the liquid-vapor coexistence in systems of patchy particles

The phase diagram of molecular or colloidal systems depends strongly on the range and angular dependence of the interactions between the constituent particles. For instance, it is well known that the critical density of particles with "patchy" interactions shifts to lower values as the number of patches is decreased [see Bianchi et al. Phys. Rev. Lett. 97, 168301 (2006)]. Here, we present simulations that show that the phase behavior of patchy particles is even more interesting than had been appreciated. In particular, we find that, upon cooling below the critical point, the width of the liquid-vapor coexistence region of a system of particles with tetrahedrally arranged patches first increases, then decreases, and finally increases again. In other words, this system exhibits a doubly re-entrant liquid-vapor transition. As a consequence, the system exhibits a very large deviation from the law of rectilinear diameter, which assumes that the critical density can be obtained by linear extrapolation of the averages of the densities of the coexisting liquid and vapor phases. We argue that the unusual behavior of this system has the same origin as the density maximum in liquid water and is not captured by the Wertheim theory. The Wertheim theory also cannot account for our observation that the phase diagram of particles with three patches depends strongly on the geometrical distribution of the patches and on the degree to which their position on the particle surface is rigidly constrained. However, the phase diagram is less sensitive to small angular spreads in the patch locations. We argue that the phase behavior reported in this paper should be observable in experiments on patchy colloids and may be relevant for the liquid-liquid equilibrium in solutions of properly functionalized dendrimers.

Early stages of phase selection in MOF formation observed in molecular Monte Carlo simulations

Metal–organic frameworks (MOF) comprising metal nodes bridged by organic linkers show great promise because of their guest-specific gas sorption, separation, drug-delivery, and catalytic properties. The selection of metal node, organic linker, and synthesis conditions in principle offers engineered control over both structure and function. For MOFs to realise their potential and to become more than just promising materials, a degree of predictability in the synthesis and a better understanding of the self-assembly or initial growth processes is of paramount importance. Using cobalt succinate, a MOF that exhibits a variety of phases depending on synthesis temperature and ligand to metal ratio, as proof of concept, we present a molecular Monte Carlo approach that allows us to simulate the early stage of MOF assembly. We introduce a new Contact Cluster Monte Carlo (CCMC) algorithm which uses a system of overlapping “virtual sites” to represent the coordination environment of the cobalt and both metal–metal and metal–ligand associations. Our simulations capture the experimentally observed synthesis phase distinction in cobalt succinate at 348 K. To the best of our knowledge this is the first case in which the formation of different MOF phases as a function of composition is captured by unbiased molecular simulations. The CCMC algorithm is equally applicable to any system in which short-range attractive interactions are a dominant feature, including hydrogen-bonding networks, metal–ligand coordination networks, or the assembly of particles with “sticky” patches, such as colloidal systems or the formation of protein complexes.

Clusters produced in simulations of MOF assembly at metal : ligand ratios of (a) 5 : 1 and (b) 1 : 1 with identical interaction potentials.

Rate constants for proteins binding to substrates with multiple binding sites using a generalized forward flux sampling expression

To predict the response of a biochemical system, knowledge of the intrinsic and effective rate constants of proteins is crucial. The experimentally accessible effective rate constant for association can be decomposed in a diffusion-limited rate at which proteins come into contact and an intrinsic association rate at which the proteins in contact truly bind. Reversely, when dissociating, bound proteins first separate into a contact pair with an intrinsic dissociation rate, before moving away by diffusion. While microscopic expressions exist that enable the calculation of the intrinsic and effective rate constants by conducting a single rare event simulation of the protein dissociation reaction, these expressions are only valid when the substrate has just one binding site. If the substrate has multiple binding sites, a bound enzyme can, besides dissociating into the bulk, also hop to another binding site. Calculating transition rate constants between multiple states with forward flux sampling requires a generalized rate expression. We present this expression here and use it to derive explicit expressions for all intrinsic and effective rate constants involving binding to multiple states, including rebinding. We illustrate our approach by computing the intrinsic and effective association, dissociation, and hopping rate constants for a system in which a patchy particle model enzyme binds to a substrate with two binding sites. We find that these rate constants increase as a function of the rotational diffusion constant of the particles. The hopping rate constant decreases as a function of the distance between the binding sites. Finally, we find that blocking one of the binding sites enhances both association and dissociation rate constants. Our approach and results are important for understanding and modeling association reactions in enzyme-substrate systems and other patchy particle systems and open the way for large multiscale simulations of such systems.

Two time scales for self and collective diffusion near the critical point in a simple patchy model for proteins with floating bonds

Using dynamic Monte Carlo and Brownian dynamics, we investigate a floating bond model in which particles can bind through mobile bonds. The maximum number of bonds (here fixed to 4) can be tuned by appropriately choosing the repulsive, nonadditive interactions among bonds and particles. We compute the static and dynamic structure factor (intermediate scattering function) in the vicinity of the gas-liquid critical point. The static structure exhibits a weak tetrahedral network character. The intermediate scattering function shows a temporal decay deviating from a single exponential, which can be described by a double exponential decay where the two time scales differ approximately by one order of magnitude. This time scale separation is robust over a range of wave numbers. The analysis of clusters in real space indicates the formation of noncompact clusters and shows a considerable stretch in the instantaneous size distribution when approaching the critical point. The average time evolution of the largest subcluster of given initial clusters with 10 or more particles also shows a double exponential decay. The observation of two time scales in the intermediate scattering function at low packing fractions is consistent with similar findings in globular protein solutions with trivalent metal ions that act as bonds between proteins.

Vitrification and gelation in sticky spheres

Glasses and gels are the two dynamically arrested, disordered states of matter. Despite their importance, their similarities and differences remain elusive, especially at high density, where until now it has been impossible to distinguish them. We identify dynamical and structural signatures which distinguish the gel and glass transitions in a colloidal model system of hard and "sticky" spheres. It has been suggested that "spinodal" gelation is initiated by gas-liquid viscoelastic phase separation to a bicontinuous network and the resulting densification leads to vitrification of the colloid-rich phase, but whether this phase has sufficient density for arrest is unclear [M. A. Miller and D. Frenkel, Phys. Rev. Lett. 90, 135702 (2003) and P. J. Lu et al., Nature 435, 499-504 (2008)]. Moreover alternative mechanisms for arrest involving percolation have been proposed [A. P. R. Eberle et al., Phys. Rev. Lett. 106, 105704 (2011)]. Here we resolve these outstanding questions, beginning by determining the phase diagram. This, along with demonstrating that percolation plays no role in controlling the dynamics of our system, enables us to confirm spinodal decomposition as the mechanism for gelation. We are then able to show that gels can be formed even at much higher densities than previously supposed, at least to a volume fraction of ϕ = 0.59. Far from being networks, these gels apparently resemble glasses but are still clearly distinguished by the "discontinuous" nature of the transition and the resulting rapid solidification, which leads to the formation of inhomogeneous (with small voids) and far-from-equilibrium local structures. This is markedly different from the glass transition, whose continuous nature leads to the formation of homogeneous and locally equilibrated structures. We further reveal that the onset of the attractive glass transition in the form of a supercooled liquid is in fact interrupted by gelation. Our findings provide a general thermodynamic, dynamic, and structural basis upon which we can distinguish gelation from vitrification.

Self-assembly in a model colloidal mixture of dimers and spherical particles

We investigate the structure of a dilute mixture of amphiphilic dimers and spherical particles, a model relevant to the problem of encapsulating globular "guest" molecules in a dispersion. Dimers and spheres are taken to be hard particles, with an additional attraction between spheres and the smaller monomers in a dimer. Using the Monte Carlo simulation, we document the low-temperature formation of aggregates of guests (clusters) held together by dimers, whose typical size and shape depend on the guest concentration χ. For low χ (less than 10%), most guests are isolated and coated with a layer of dimers. As χ progressively increases, clusters grow in size becoming more and more elongated and polydisperse; after reaching a shallow maximum for χ≈50%, the size of clusters again reduces upon increasing χ further. In one case only (χ=50% and moderately low temperature) the mixture relaxed to a fluid of lamellae, suggesting that in this case clusters are metastable with respect to crystal-vapor separation. On heating, clusters shrink until eventually the system becomes homogeneous on all scales. On the other hand, as the mixture is made denser and denser at low temperature, clusters get increasingly larger until a percolating network is formed.

Multivalent-Ion-Activated Protein Adsorption Reflecting Bulk Reentrant Behavior

Protein adsorption at the solid-liquid interface is an important phenomenon that often can be observed as a first step in biological processes. Despite its inherent importance, still relatively little is known about the underlying microscopic mechanisms. Here, using multivalent ions, we demonstrate the control of the interactions and the corresponding adsorption of net-negatively charged proteins (bovine serum albumin) at a solid-liquid interface. This is demonstrated by ellipsometry and corroborated by neutron reflectivity and quartz-crystal microbalance experiments. We show that the reentrant condensation observed within the rich bulk phase behavior of the system featuring a nonmonotonic dependence of the second virial coefficient on salt concentration c_{s} is reflected in an intriguing way in the protein adsorption d(c_{s}) at the interface. Our findings are successfully described and understood by a model of ion-activated patchy interactions within the framework of the classical density functional theory. In addition to the general challenge of connecting bulk and interface behavior, our work has implications for, inter alia, nucleation at interfaces.

Nonclassical nucleation pathways in protein crystallization

Classical nucleation theory (CNT), which was established about 90 years ago, has been very successful in many research fields, and continues to be the most commonly used theory in describing the nucleation process. For a fluid-to-solid phase transition, CNT states that the solute molecules in a supersaturated solution reversibly form small clusters. Once the cluster size reaches a critical value, it becomes thermodynamically stable and favored for further growth. One of the most important assumptions of CNT is that the nucleation process is described by one reaction coordinate and all order parameters proceed simultaneously. Recent studies in experiments, computer simulations and theory have revealed nonclassical features in the early stage of nucleation. In particular, the decoupling of order parameters involved during a fluid-to-solid transition leads to the so-called two-step nucleation mechanism, in which a metastable intermediate phase (MIP) exists between the initial supersaturated solution and the final crystals. Depending on the exact free energy landscapes, the MIPs can be a high density liquid phase, mesoscopic clusters, or a pre-ordered state. In this review, we focus on the studies of nonclassical pathways in protein crystallization and discuss the applications of the various scenarios of two-step nucleation theory. In particular, we focus on protein solutions in the presence of multivalent salts, which serve as a model protein system to study the nucleation pathways. We wish to point out the unique features of proteins as model systems for further studies.

Phase behavior and bulk structural properties of a microphase former with anisotropic competing interactions: A density functional theory study

Using classical density functional theory, we investigate systems exhibiting interactions where a short-range anisotropic attractive force competes with a long-range spherically symmetric repulsive force. The former is modelled within Wertheim's first-order perturbation theory for patchy particles, and the repulsive part is assumed to be a Yukawa potential which is taken into account via a mean-field approximation. From previous studies of systems with spherically symmetric competing interactions, it is well known that such systems can exhibit stable bulk cluster phases (microphase separation) provided that the repulsion is sufficiently weak compared to the attraction. For the present model system, we find rich phase diagrams including both reentrant clustering and liquid-gas binodals. In particular, the model predicts inhomogeneous bulk phases at extremely low packing fractions, which cannot be observed in systems with isotropic competing interactions.

What macromolecular crystallogenesis tells us - what is needed in the future

Crystallogenesis is a longstanding topic that has transformed into a discipline that is mainly focused on the preparation of crystals for practising crystallo-graphers. Although the idiosyncratic features of proteins have to be taken into account, the crystallization of proteins is governed by the same physics as the crystallization of inorganic materials. At present, a diversified panel of crystallization methods adapted to proteins has been validated, and although only a few methods are in current practice, the success rate of crystallization has increased constantly, leading to the determination of ∼105 X-ray structures. These structures reveal a huge repertoire of protein folds, but they only cover a restricted part of macromolecular diversity across the tree of life. In the future, crystals representative of missing structures or that will better document the structural dynamics and functional steps underlying biological processes need to be grown. For the pertinent choice of biologically relevant targets, computer-guided analysis of structural databases is needed. From another perspective, crystallization is a self-assembly process that can occur in the bulk of crowded fluids, with crystals being supramolecular assemblies. Life also uses self-assembly and supramolecular processes leading to transient, or less often stable, complexes. An integrated view of supramolecularity implies that proteins crystallizing either in vitro or in vivo or participating in cellular processes share common attributes, notably determinants and antideterminants that favour or disfavour their correct or incorrect associations. As a result, under in vivo conditions proteins show a balance between features that favour or disfavour association. If this balance is broken, disorders/diseases occur. Understanding crystallization under in vivo conditions is a challenge for the future. In this quest, the analysis of packing contacts and contacts within oligomers will be crucial in order to decipher the rules governing protein self-assembly and will guide the engineering of novel biomaterials. In a wider perspective, understanding such contacts will open the route towards supramolecular biology and generalized crystallogenesis.

Criticality of colloids with three distinct interaction patches: As simple as A,B,C?

We systematically study the phase behavior of a simple model of associating fluids which consists of hard spherical particles with three short-ranged attractive sites on their surfaces (sticky spots or patches), of types A,B, and C, that can form bonds with energy ε_{ij} (i,j=A,B,C). We consider realizations of the model with one, two, or three nonzero ε_{ij}. Using Wertheim's first order perturbation theory of association, we establish the minimum requirements on the bond energies for the model to exhibit a liquid-vapor critical point, and investigate the nature of criticality in each case. As a preliminary, we rigorously show that, within this theory, particles with M identical sites do not condense if M<3, a result that was previously conjectured, but never proved.

Limiting the valence: advancements and new perspectives on patchy colloids, soft functionalized nanoparticles and biomolecules

Artistic representation of limited valance units consisting of a soft core (in blue) and a small number of flexible bonding patches (in orange).

Modeling phase transitions in mixtures of β-γ lens crystallins

We analyze the experimentally determined phase diagram of a γD-βB1 crystallin mixture. Proteins are described as dumbbells decorated with attractive sites to allow inter-particle interaction. We use thermodynamic perturbation theory to calculate the free energy of such mixtures and, by applying equilibrium conditions, also the compositions and concentrations of the co-existing phases. Initially we fit the Tcloudversus packing fraction η measurements for a pure (x2 = 0) γD solution in 0.1 M phosphate buffer at pH = 7.0. Another piece of experimental data, used to fix the model parameters, is the isotherm x2vs. η at T = 268.5 K, at the same pH and salt content. We use the conventional Lorentz-Berthelot mixing rules to describe cross interactions. This enables us to determine: (i) model parameters for pure βB1 crystallin protein and to calculate; (ii) complete equilibrium surface (Tcloud-x2-η) for the crystallin mixtures. (iii) We present the results for several isotherms, including the tie-lines, as also the temperature-packing fraction curves. Good agreement with the available experimental data is obtained. An interesting result of these calculations is evidence of the coexistence of three phases. This domain appears for the region of temperatures just out of the experimental range studied so far. The input parameters, leading good description of experimental data, revealed a large difference between the numbers of the attractive sites for γD and βB1 proteins. This interesting result may be related to the fact that γD has a more than nine times smaller quadrupole moment than its partner in the mixture.

Multi-Conformation Monte Carlo: A Method for Introducing Flexibility in Efficient Simulations of Many-Protein Systems

We present a novel multi-conformation Monte Carlo simulation method that enables the modeling of protein-protein interactions and aggregation in crowded protein solutions. This approach is relevant to a molecular-scale description of realistic biological environments, including the cytoplasm and the extracellular matrix, which are characterized by high concentrations of biomolecular solutes (e.g., 300-400 mg/mL for proteins and nucleic acids in the cytoplasm of Escherichia coli). Simulation of such environments necessitates the inclusion of a large number of protein molecules. Therefore, computationally inexpensive methods, such as rigid-body Brownian dynamics (BD) or Monte Carlo simulations, can be particularly useful. However, as we demonstrate herein, the rigid-body representation typically employed in simulations of many-protein systems gives rise to certain artifacts in protein-protein interactions. Our approach allows us to incorporate molecular flexibility in Monte Carlo simulations at low computational cost, thereby eliminating ambiguities arising from structure selection in rigid-body simulations. We benchmark and validate the methodology using simulations of hen egg white lysozyme in solution, a well-studied system for which extensive experimental data, including osmotic second virial coefficients, small-angle scattering structure factors, and multiple structures determined by X-ray and neutron crystallography and solution NMR, as well as rigid-body BD simulation results, are available for comparison.

Tuning protein-protein interactions using cosolvents: specific effects of ionic and non-ionic additives on protein phase behavior

Cosolvents are routinely used to modulate the (thermal) stability of proteins and, hence, their interactions with proteins have been studied intensely. However, less is known about their specific effects on protein-protein interactions, which we characterize in terms of the protein phase behavior. We analyze the phase behavior of lysozyme solutions in the presence of sodium chloride (NaCl), guanidine hydrochloride (GuHCl), glycerol, and dimethyl sulfoxide (DMSO). We experimentally determined the crystallization boundary (XB) and, in combination with data on the cloud-point temperatures (CPTs), the crystallization gap. In agreement with other studies, our data indicate that the additives might affect the protein phase behavior through electrostatic screening and additive-specific contributions. At high salt concentrations, where electrostatic interactions are screened, both the CPT and the XB are found to be linear functions of the additive concentration. Their slopes quantify the additive-specific changes of the phase behavior and thus of the protein-protein interactions. While the specific effect of NaCl is to induce attractions between proteins, DMSO, glycerol and GuHCl (with increasing strength) weaken attractions and/or induce repulsions. Except for DMSO, changes of the CPT are stronger than those of the XB. Furthermore, the crystallization gap widens in the case of GuHCl and glycerol and narrows in the case of NaCl. We relate these changes to colloidal interaction models, namely square-well and patchy interactions.

A versatile model for soft patchy particles with various patch arrangements

We propose a simple and general mesoscale soft patchy particle model, which can felicitously describe the deformable and surface-anisotropic characteristics of soft patchy particles. This model can be used in dynamics simulations to investigate the aggregation behavior and mechanism of various types of soft patchy particles with tunable number, size, direction, and geometrical arrangement of the patches. To improve the computational efficiency of this mesoscale model in dynamics simulations, we give the simulation algorithm that fits the compute unified device architecture (CUDA) framework of NVIDIA graphics processing units (GPUs). The validation of the model and the performance of the simulations using GPUs are demonstrated by simulating several benchmark systems of soft patchy particles with 1 to 4 patches in a regular geometrical arrangement. Because of its simplicity and computational efficiency, the soft patchy particle model will provide a powerful tool to investigate the aggregation behavior of soft patchy particles, such as patchy micelles, patchy microgels, and patchy dendrimers, over larger spatial and temporal scales.

Equivalence of Brownian dynamics and dynamic Monte Carlo simulations in multicomponent colloidal suspensions

We propose a simple but powerful theoretical framework to quantitatively compare Brownian dynamics (BD) and dynamic Monte Carlo (DMC) simulations of multicomponent colloidal suspensions. By extending our previous study focusing on monodisperse systems of rodlike colloids, here we generalize the formalism described there to multicomponent colloidal mixtures and validate it by investigating the dynamics in isotropic and liquid crystalline phases containing spherical and rodlike particles. In order to investigate the dynamics of multicomponent colloidal systems by DMC simulations, it is key to determine the elementary time step of each species and establish a unique timescale. This is crucial to consistently study the dynamics of colloidal particles with different geometry. By analyzing the mean-square displacement, the orientation autocorrelation functions, and the self part of the van Hove correlation functions, we show that DMC simulation is a very convenient and reliable technique to describe the stochastic dynamics of any multicomponent colloidal system. Our theoretical formalism can be easily extended to any colloidal system containing size and/or shape polydisperse particles.

Liquid-liquid phase separation in biology

Cells organize many of their biochemical reactions in non-membrane compartments. Recent evidence has shown that many of these compartments are liquids that form by phase separation from the cytoplasm. Here we discuss the basic physical concepts necessary to understand the consequences of liquid-like states for biological functions.

Monte Carlo simulation of kinetically slowed down phase separation

Supercooled colloidal or molecular systems at low densities are known to form liquid, crystalline or glassy drops, which may remain isolated for a long time before they aggregate. This paper analyses the properties of this large time window, and how it can be tackled by computer simulation. We use single-particle and virtual move Monte Carlo simulations of short-range attractive spheres which are undercooled to the temperature region, where the spinodal intersects the attractive glass line. We study two different systems and we report the following kinetic behavior. A low-density system is shown to exhibit universal linear growth regimes under single-particle Monte Carlo correlating the growth rate to the local structure. These regimes are suppressed under collective motion, where droplets aggregate into a single large disordered domain. It is shown that the aggregation can be avoided and linear regimes recovered, if long-range repulsion is added to the short-range attraction. The results provide an insight into the behavior of the virtual move algorithm generating cluster moves according to the local forcefields. We show that different choices of maximum Monte Carlo displacement affect the dynamical trajectories but lead to the same kinetically slowed down or arrested states.

Ion-activated attractive patches as a mechanism for controlled protein interactions

The understanding of protein interactions to control phase and nucleation behavior of protein solutions is an important challenge for soft matter, biological and medical research. Here, we present ion bridges of multivalent cations between proteins as an ion-activated mechanism for patchy interaction that is directly supported by experimental findings in protein crystals. A deep understanding of experimentally observed phenomena in protein solutions--including charge reversal, reentrant condensation, metastable liquid-liquid phase separation, cluster formation and different pathways of crystallization--is gained by an analytic model that directly displays parameter dependencies and physical connections. The direct connection between experiment and theory provides a conceptual framework for future experimental, computational and theoretical research on topics such as rational design of phase behavior and crystallization pathways on the basis of the statistical physics of patchy particles.

Statistical analysis of crystallization database links protein physico-chemical features with crystallization mechanisms

X-ray crystallography is the predominant method for obtaining atomic-scale information about biological macromolecules. Despite the success of the technique, obtaining well diffracting crystals still critically limits going from protein to structure. In practice, the crystallization process proceeds through knowledge-informed empiricism. Better physico-chemical understanding remains elusive because of the large number of variables involved, hence little guidance is available to systematically identify solution conditions that promote crystallization. To help determine relationships between macromolecular properties and their crystallization propensity, we have trained statistical models on samples for 182 proteins supplied by the Northeast Structural Genomics consortium. Gaussian processes, which capture trends beyond the reach of linear statistical models, distinguish between two main physico-chemical mechanisms driving crystallization. One is characterized by low levels of side chain entropy and has been extensively reported in the literature. The other identifies specific electrostatic interactions not previously described in the crystallization context. Because evidence for two distinct mechanisms can be gleaned both from crystal contacts and from solution conditions leading to successful crystallization, the model offers future avenues for optimizing crystallization screens based on partial structural information. The availability of crystallization data coupled with structural outcomes analyzed through state-of-the-art statistical models may thus guide macromolecular crystallization toward a more rational basis.

Spontaneous chiral resolution in two-dimensional systems of patchy particles

Short ranged potentials and their anisotropy produce spontaneous chiral resolution in a two dimensional model of patchy particles introduced in this paper. This model could represent an equimolar binary mixture (racemic mixture) of two kinds of chiral molecules (enantiomers) adsorbed to a bi-dimensional domain where only lateral short ranged interactions are present. Most racemic mixtures undergo chiral resolution due to their spatial anisotropy, the combined effect of long range forces and the thermodynamic conditions. The patchy particles are modeled as a hard disk and four different bonding sites located to produce chirality. Phase behavior and structural properties are analysed using Discontinuous Molecular Dynamics in the canonical ensemble. When the four patchy particles are separated by the angles {60°, 120°, 60°, 120°}, spontaneous chiral resolution is produced, given by the formation of homochiral clusters, if started from the corresponding racemic mixture. Gel behavior is also obtained in all the systems for low temperatures and low densities.

Characterizing protein crystal contacts and their role in crystallization: rubredoxin as a case study

The fields of structural biology and soft matter have independently sought out fundamental principles to rationalize protein crystallization. Yet the conceptual differences and the limited overlap between the two disciplines have thus far prevented a comprehensive understanding of the phenomenon to emerge. We conduct a computational study of proteins from the rubredoxin family that bridges the two fields. Using atomistic simulations, we characterize the protein crystal contacts, and accordingly parameterize patchy particle models. Comparing the phase diagrams of these schematic models with experimental results enables us to critically examine the assumptions behind the two approaches. The study also reveals features of protein–protein interactions that can be leveraged to crystallize proteins more generally.