Coarse-grained models for proteins

Coarse-grained models for proteins and biomolecular aggregates have recently enjoyed renewed interest. Coarse-grained representations combined with enhanced computer power currently allow the simulation of systems of biologically relevant size (submicrometric) and timescale (microsecond or millisecond). Although these techniques still cannot be considered as predictive as all-atom simulations, noticeable advances have recently been achieved, mainly concerning the use of more rigorous parameterization techniques and novel algorithms for sampling configurational space. Moreover, the simulation size scales and timescales coincide with those that can be reached with the most advanced spectroscopic techniques, making it possible to directly compare simulation and experiment.     

Coarse-grained strategy for modeling protein stability in concentrated solutions

We present a coarse-grained approach for modeling the thermodynamic stability of single-domain globular proteins in concentrated aqueous solutions. Our treatment derives effective protein-protein interactions from basic structural and energetic characteristics of the native and denatured states. These characteristics, along with the intrinsic (i.e., infinite dilution) thermodynamics of folding, are calculated from elementary sequence information using a heteropolymer collapse theory. We integrate this information into Reactive Canonical Monte Carlo simulations to investigate the connections between protein sequence hydrophobicity, protein-protein interactions, protein concentration, and the thermodynamic stability of the native state. The model predicts that sequence hydrophobicity can affect how protein concentration impacts native-state stability in solution. In particular, low hydrophobicity proteins are primarily stabilized by increases in protein concentration, whereas high hydrophobicity proteins exhibit richer nonmonotonic behavior. These trends appear qualitatively consistent with the available experimental data. Although factors such as pH, salt concentration, and protein charge are also important for protein stability, our analysis suggests that some of the nontrivial experimental trends may be driven by a competition between destabilizing hydrophobic protein-protein attractions and entropic crowding effects.     

Stochastic models for aggregation processes

Three models are presented, which describe the aggregation of objects into groups and the distributions of groups sizes and group numbers within habitats. The processes regarded are pure accumulation processes which involve only formation and invasion of groups. Invasion represents the special case of fusion when only single objects - and not groups - join a group of certain size. The basic model is derived by a single parameter, the formation probability q, which represents the probability of an object to form a new group. A novel, discrete and finite distribution that results for the group sizes is deduced from this aggregation process and it is shown that it converges to a geometric distribution if the number of objects tends to infinity. Two extensions of this model, which both converge to the Waring distribution, are added: the model can be extended either with a beta distributed formation probability or with the assumption that the invasion probability depends on the group size. Relationships between the limiting distributions involved are discussed.     

Coarse-grained models for simulations of multiprotein complexes: application to ubiquitin binding

We develop coarse-grained models and effective energy functions for simulating thermodynamic and structural properties of multiprotein complexes with relatively low binding affinity (K_d > 1 μM) and apply them to binding of Vps27 to membrane-tethered ubiquitin. Folded protein domains are represented as rigid bodies. The interactions between the domains are treated at the residue level with amino-acid-dependent pair potentials and Debye-Hückel-type electrostatic interactions. Flexible linker peptides connecting rigid protein domains are represented as amino acid beads on a polymer with appropriate stretching, bending, and torsion-angle potentials. In simulations of membrane-attached protein complexes, interactions between amino acids and the membrane are described by residue-dependent short-range potentials and long-range electrostatics. We parameterize the energy functions by fitting the osmotic second virial coefficient of lysozyme and the binding affinity of the ubiquitin-CUE complex. For validation, extensive replica-exchange Monte Carlo simulations are performed of various protein complexes. Binding affinities for these complexes are in good agreement with the experimental data. The simulated structures are clustered on the basis of distance matrices between two proteins and ranked according to cluster population. In ∼ 70% of the complexes, the distance root-mean-square is less than 5 Å from the experimental structures. In ∼ 90% of the complexes, the binding interfaces on both proteins are predicted correctly, and in all other cases at least one interface is correct. Transient and nonspecifically bound structures are also observed. With the validated model, we simulate the interaction between the Vps27 multiprotein complex and a membrane-tethered ubiquitin. Ubiquitin is found to bind preferentially to the two UIM domains of Vps27, but transient interactions between ubiquitin and the VHS and FYVE domains are observed as well. These specific and nonspecific interactions are found to be positively cooperative, resulting in a substantial enhancement of the overall binding affinity beyond the ∼ 300 μM of the specific domains. We also find that the interactions between ubiquitin and Vps27 are highly dynamic, with conformational rearrangements enabling binding of Vps27 to diverse targets as part of the multivesicular-body protein-sorting pathway.     

Coarse-grained strategy for modeling protein stability in concentrated solutions. III: directional protein interactions

We extend our coarse-grained modeling strategy described in parts I and II of this investigation to account for nonuniform spatial distributions of hydrophobic residues on the solvent-exposed surfaces of native proteins. Within this framework, we explore how patchy surfaces can influence the solvent-mediated protein-protein interactions, and the unfolding and self-assembly behaviors of proteins in solution. In particular, we compare the equilibrium unfolding and self-assembly trends for three model proteins that share the same overall sequence hydrophobicity, but exhibit folded configurations with different solvent-exposed native-state surface morphologies. Our model provides new insights into how directional interactions can affect native-state protein stability in solution. We find that strongly-directional attractions between native molecules with patchy surfaces can help stabilize the folded conformation through the formation of self-assembled clusters. In contrast, native proteins with more uniform surfaces are destabilized by protein-protein attractions involving the denatured state. Finally, we discuss how the simulation results provide insights into the experimental solution behaviors of several proteins that display directional interactions in their native states.     

Cellular strategies for controlling protein aggregation

The aggregation of misfolded proteins is associated with the perturbation of cellular function, ageing and various human disorders. Mounting evidence suggests that protein aggregation is often part of the cellular response to an imbalanced protein homeostasis rather than an unspecific and uncontrolled dead-end pathway. It is a regulated process in cells from bacteria to humans, leading to the deposition of aggregates at specific sites. The sequestration of misfolded proteins in such a way is protective for cell function as it allows for their efficient solubilization and refolding or degradation by components of the protein quality-control network. The organized aggregation of misfolded proteins might also allow their asymmetric distribution to daughter cells during cell division.     

Exploring protein aggregation and self-propagation using lattice models: phase diagram and kinetics

Many seemingly unrelated neurodegenerative disorders, such as amyloid and prion diseases, are associated with propagating fibrils whose structures are dramatically different from the native states of the corresponding monomers. This observation, along with the experimental demonstration that any protein can aggregate to form either fibrils or amorphous structures (inclusion bodies) under appropriate external conditions, suggest that there must be general principles that govern aggregation mechanisms. To probe generic aspects of prion-like behavior we use the model of Harrison, Chan, Prusiner, and Cohen. In this model, aggregation of a structure, that is conformationally distinct from the native state of the monomer, occurs by three parallel routes. Kinetic partitioning, which leads to parallel assembly pathways, occurs early in the aggregation process. In all pathways transient unfolding precedes polymerization and self-propagation. Chain polymerization is consistent with templated assembly, with the dimer being the minimal nucleus. The kinetic effciency of R(n-1) + G --> R(n) (R is the aggregation prone state and G is either U, the unfolded state, or N, the native state of the monomer) is increased when polymerization occurs in the presence of a "seed" (a dimer). These results support the seeded nucleated-polymerization model of fibril formation in amyloid peptides. To probe generic aspects of aggregation in two-state proteins, we use lattice models with side chains. The phase diagram in the (T,C) plane (T is the temperature and C is the polypeptide concentration) reveals a bewildering array of "phases" or structures. Explicit computations for dimers show that there are at least six phases including ordered structures and amorphous aggregates. In the ordered region of the phase diagram there are three distinct structures. We find ordered dimers (OD) in which each monomer is in the folded state and the interaction between the monomers occurs via a well-defined interface. In the domain-swapped structures a certain fraction of intrachain contacts are replaced by interchain contacts. In the parallel dimers the interface is stabilized by favorable intermolecular hydrophobic interactions. The kinetics of folding to OD shows that aggregation proceeds directly from U in a dynamically cooperative manner without populating partially structured intermediates. These results support the experimental observation that ordered aggregation in the two-state folders U1A and CI2 takes place from U. The contrasting aggregation processes in the two models suggest that there are several distinct mechanisms for polymerization that depend not only on the polypeptide sequence but also on external conditions (such as C, T, pH, and salt concentration).     

Computational models for the prediction of polypeptide aggregation propensity

In amyloid fibrils, beta-strand conformations of polypeptide chains, or segments thereof, are perpendicular to the fibril axis, but knowledge of their three dimensional structure at atomic level of detail is scarce. Two types of computational approaches have been developed recently for investigating the aggregation propensity of peptides and proteins and identifying the segments most prone to form fibrils (hot spots). The physicochemical properties of the natural amino acids (e.g. beta-propensity, hydrophobicity, aromatic content and charge) have been used to derive phenomenological models able to predict changes in aggregation rate upon mutation, as well as absolute rates and hot spots. Applications of these models to entire proteomes have provided evidence that intrinsically disordered proteins are less amyloidogenic than globular proteins. In the second type of approach, amyloidogenic polypeptides have been decomposed into overlapping segments, and atomistic simulations of three or more copies of each segment have been performed to obtain insights into aggregation propensity and structural details of the ordered aggregates (e.g. turn regions).     

Coarse-grained free energy functions for studying protein conformational changes: a double-well network model

In this work, a double-well network model (DWNM) is presented for generating a coarse-grained free energy function that can be used to study the transition between reference conformational states of a protein molecule. Compared to earlier work that uses a single, multidimensional double-well potential to connect two conformational states, the DWNM uses a set of interconnected double-well potentials for this purpose. The DWNM free energy function has multiple intermediate states and saddle points, and is hence a “rough” free energy landscape. In this implementation of the DWNM, the free energy function is reduced to an elastic-network model representation near the two reference states. The effects of free energy function roughness on the reaction pathways of protein conformational change is demonstrated by applying the DWNM to the conformational changes of two protein systems: the coil-to-helix transition of the DB-loop in G-actin and the open-to-closed transition of adenylate kinase. In both systems, the rough free energy function of the DWNM leads to the identification of distinct minimum free energy paths connecting two conformational states. These results indicate that while the elastic-network model captures the low-frequency vibrational motions of a protein, the roughness in the free energy function introduced by the DWNM can be used to characterize the transition mechanism between protein conformations.     

Coarse-grained strategy for modeling protein stability in concentrated solutions. II: phase behavior

We use highly efficient transition-matrix Monte Carlo simulations to determine equilibrium unfolding curves and fluid phase boundaries for solutions of coarse-grained globular proteins. The model we analyze derives the intrinsic stability of the native state and protein-protein interactions from basic information about protein sequence using heteropolymer collapse theory. It predicts that solutions of low hydrophobicity proteins generally exhibit a single liquid phase near their midpoint temperatures for unfolding, while solutions of proteins with high sequence hydrophobicity display the type of temperature-inverted, liquid-liquid transition associated with aggregation processes of proteins and other amphiphilic molecules. The phase transition occurring in solutions of the most hydrophobic protein we study extends below the unfolding curve, creating an immiscibility gap between a dilute, mostly native phase and a concentrated, mostly denatured phase. The results are qualitatively consistent with the solution behavior of hemoglobin (HbA) and its sickle variant (HbS), and they suggest that a liquid-liquid transition resulting in significant protein denaturation should generally be expected on the phase diagram of high-hydrophobicity protein solutions. The concentration fluctuations associated with this transition could be a driving force for the nonnative aggregation that can occur below the midpoint temperature.     

Monitoring protein aggregation and toxicity in Alzheimer's disease mouse models using in vivo imaging

Aggregation of amyloid beta peptide into senile plaques and hyperphosphorylated tau protein into neurofibrillary tangles in the brain are the pathological hallmarks of Alzheimer's disease. Despite over a century of research into these lesions, the exact relationship between pathology and neurotoxicity has yet to be fully elucidated. In order to study the formation of plaques and tangles and their effects on the brain, we have applied multiphoton in vivo imaging of transgenic mouse models of Alzheimer's disease. This technique allows longitudinal imaging of pathological aggregation of proteins and the subsequent changes in surrounding neuropil neurodegeneration and recovery after therapeutic interventions.     

Local models of cell aggregation kinetics

Mathematical models of the phenomena of the sorting out of embryonic cells are derived which predict multiple clustering of like cell types into many small islands. The experiments of Trinkaus and Lentz, and other essentially two-dimensional cell sorting results, provide examples. Two continuous models are introduced, both based on the Smoluchowski theory of Brownian motion. A mass-interaction model predicts local clumping of the internally segregating cell type within a continuum of the other type. A diffusion model simulates the suspension-aggregate experiments of Roth and Weston. A discrete model of sorting out, the exchange model, is re-derived in a clearer manner than before, and certain recent criticisms are responded to. Results in recent two-dimensional cell sorting experiments are discussed and a stability criterion for patterns of multiple clusters is introduced. The site frequency model is re-examined and criticized in terms of its application to the Roth-Weston experiments.     

Coarse-grained lattice model simulations of sequence-structure fitness of a ribosome-inactivating protein

Many realistic protein-engineering design problems extend beyond the computational limits of what is considered practical when applying all-atom molecular-dynamics simulation methods. Lattice models provide computationally robust alternatives, yet most are regarded as too simplistic to accurately capture the details of complex designs. We revisit a coarse-grained lattice simulation model and demonstrate that a multiresolution modeling approach of reconstructing all-atom structures from lattice chains is of sufficient accuracy to resolve the comparability of sequence-structure modifications of the ricin A-chain (RTA) protein fold. For a modeled structure, the unfolding-folding transition temperature was calculated from the heat capacity using either the potential energy from the lattice model or the all-atom CHARMM19 force-field plus a generalized Born solvent approximation. We found, that despite the low-resolution modeling of conformational states, the potential energy functions were capable of detecting the relative change in the thermodynamic transition temperature that distinguishes between a protein design and the native RTA fold in excellent accord with reported experimental studies of thermal denaturation. A discussion is provided of different sequences fitted to the RTA fold and a possible unfolding model.     

Non-native protein aggregation kinetics

Experimental kinetics of non-native protein aggregation are of practical importance in that they help dictate viable processing, formulation, and storage conditions for biotechnology products, and appear to play a role in determining the onset of a number of diseases. Fundamentally, aggregation kinetics provide insights into the identity of key intermediates in the process, and quantitative tests of available models of aggregation. Although aggregation kinetics often display seemingly disparate behaviors across different proteins and sample conditions, this review illustrates how many of these can be understood within a general framework that treats aggregation as a multi-stage process, and how most available kinetic models of aggregation can be grouped hierarchically in terms of which stage(s) they include. This provides an aid for workers seeking a mechanistic interpretation of in vitro aggregation kinetics, for discriminating among competing models, and in designing experiments to assess in vitro protein stability. Limitations and the utility of purely kinetic approaches to studying aggregation, clarifications of common misperceptions regarding experimental aggregation kinetics, and some outstanding challenges in the field are briefly discussed.     

Role for the alpha-helix in aberrant protein aggregation

Is the alpha-helix structure capable of triggering the formation of aberrant protein aggregates? To answer this question, we investigate the in vitro aggregation of tau protein in the presence of the helix-inducing agent TFE. Tau is a natively unfolded protein that binds to microtubules and forms aggregates in Alzheimer's disease. We find that full-length tau has residual alpha-helix structure, which is further enhanced by three mutations involved in genetic neurological disorders. TFE concentrations matching an alpha-helical content of 40% in full-length tau and the triple mutant induce the formation of aggregates that are morphologically and structurally heterogeneous. A simple dilution experiment reveals that heterogeneity results from the competition between alpha-helical fibrillar aggregates and more classical amyloid-like aggregates. The alpha-helical aggregates are more resilient to dilution and have the spectroscopic features of alpha-helical coiled coils. We propose a general mechanism by which intrinsically stable alpha-helices can associate into aggregates with only coarse coiled-coil symmetry. In tau, high intrinsic alpha-helix stability and coarse coiled-coil symmetry could be byproducts of its biological function.     

Predictive models for protein crystallization

Crystallization of proteins is a nontrivial task, and despite the substantial efforts in robotic automation, crystallization screening is still largely based on trial-and-error sampling of a limited subset of suitable reagents and experimental parameters. Funding of high throughput crystallography pilot projects through the NIH Protein Structure Initiative provides the opportunity to collect crystallization data in a comprehensive and statistically valid form. Data mining and machine learning algorithms thus have the potential to deliver predictive models for protein crystallization. However, the underlying complex physical reality of crystallization, combined with a generally ill-defined and sparsely populated sampling space, and inconsistent scoring and annotation make the development of predictive models non-trivial. We discuss the conceptual problems, and review strengths and limitations of current approaches towards crystallization prediction, emphasizing the importance of comprehensive and valid sampling protocols. In view of limited overlap in techniques and sampling parameters between the publicly funded high throughput crystallography initiatives, exchange of information and standardization should be encouraged, aiming to effectively integrate data mining and machine learning efforts into a comprehensive predictive framework for protein crystallization. Similar experimental design and knowledge discovery strategies should be applied to valid analysis and prediction of protein expression, solubilization, and purification, as well as crystal handling and cryo-protection.     

Representation of Coarse-grained Potentials for Polymer Simulations

In order to be able to simulate long time and large space scale properties of polymer melts one has to resort to coarse grained models, for example by subdividing all polymers into parts and restricting attention to the center of mass positions and velocities of these parts. The dynamics of these variables is governed by Langevin equations in which the free energy obtained by integrating the remaining variables provides the potential of the conservative forces. In general this leads to many particle interactions on the coarse-grained level. Methods suggested in the literature to represent these many particle interactions by effective two body interactions are reviewed and a new method, based on the Gibbs-Bogoliubov inequality, is proposed. The reason why none of these methods is able to reproduce the pressure of the underlying atomistic model is discussed.     

RBC aggregation: laboratory data and models

The reversible aggregation of red blood cells (RBC) into linear and three-dimensional structures continues to be of basic science and clinical interest: RBC aggregation affects low shear blood viscosity and microvascular flow dynamics, and can be markedly enhanced in several clinical states. Until fairly recently, most research efforts were focused on relations between suspending medium composition (i.e., protein levels, polymer type and concentration) and aggregate formation. However, there is now an increasing amount of experimental evidence indicating that RBC cellular properties can markedly affect aggregation, with the term "RBC aggregability" coined to describe the cell's intrinsic tendency to aggregate. Variations of aggregability can be large, with some changes of aggregation substantially greater than those resulting from pathologic states. The present review provides a brief overview of this topic, and includes such areas as donor-to-donor variations, polymer-plasma correlations, effects of RBC age, effects of enzymatic treatment, and current developments related to the mechanisms involved in RBC aggregation.     

MCMC for hidden Markov models incorporating aggregation of states and filtering

This paper is concerned with the statistical analysis of single ion channel records. Single channels are modelled by using hidden Markov models and a combination of Bayesian statistics and Markov chain Monte Carlo methods. The techniques presented here provide a straightforward generalization to those in Rosales et al. (2001, Biophys. J., 80, 1088–1103), allowing to consider constraints imposed by a gating mechanism such as the aggregation of states into classes. This paper also presents an extension that allows to consider correlated background noise and filtered data, extending the scope of the analysis toward real experimental conditions. The methods described here are based on a solid probabilistic basis and are less computationally intensive than alternative Bayesian treatments or frequentist approaches that consider correlated data.