Reduced models of proteins and their applications

A coarse-grained approach to NMR-data-assisted modeling of protein structures

The ESCASA algorithm for analytical estimation of proton positions from coarse-grained geometry developed in our recent work has been implemented in modeling protein structures with the highly coarse-grained UNRES model of polypeptide chains (two sites per residue) and nuclear magnetic resonance (NMR) data. A penalty function with the shape of intersecting gorges was applied to treat ambiguous distance restraints, which automatically selects consistent restraints. Hamiltonian replica exchange molecular dynamics was used to carry out the conformational search. The method was tested with both unambiguous and ambiguous restraints producing good-quality models with GDT_TS from 7.4 units higher to 14.4 units lower than those obtained with the CYANA or MELD software for protein-structure determination from NMR data at the all-atom resolution. The method can thus be applied in modeling the structures of flexible proteins, for which extensive conformational search enabled by coarse-graining is more important than high modeling accuracy.

Key aspects of the past 30 years of protein design

Proteins are the workhorse of life. They are the building infrastructure of living systems; they are the most efficient molecular machines known, and their enzymatic activity is still unmatched in versatility by any artificial system. Perhaps proteins' most remarkable feature is their modularity. The large amount of information required to specify each protein's function is analogically encoded with an alphabet of just ∼20 letters. The protein folding problem is how to encode all such information in a sequence of 20 letters. In this review, we go through the last 30 years of research to summarize the state of the art and highlight some applications related to fundamental problems of protein evolution.

Evaluation of FRET X for single-molecule protein fingerprinting

Single-molecule protein identification is an unrealized concept with potentially ground-breaking applications in biological research. We propose a method called FRET X (Förster Resonance Energy Transfer via DNA eXchange) fingerprinting, in which the FRET efficiency is read out between exchangeable dyes on protein-bound DNA docking strands and accumulated FRET efficiencies constitute the fingerprint for a protein. To evaluate the feasibility of this approach, we simulated fingerprints for hundreds of proteins using a coarse-grained lattice model and experimentally demonstrated FRET X fingerprinting on model peptides. Measured fingerprints are in agreement with our simulations, corroborating the validity of our modeling approach. In a simulated complex mixture of >300 human proteins of which only cysteines, lysines, and arginines were labeled, a support vector machine was able to identify constituents with 95% accuracy. We anticipate that our FRET X fingerprinting approach will form the basis of an analysis tool for targeted proteomics.

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We propose a FRET-based single-molecule protein identification method

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Peptides are experimentally distinguishable by their fingerprints

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Our approach can classify the constituents of complex samples with 95% accuracy

Theory and Practice of Coarse-Grained Molecular Dynamics of Biologically Important Systems

Molecular dynamics with coarse-grained models is nowadays extensively used to simulate biomolecular systems at large time and size scales, compared to those accessible to all-atom molecular dynamics. In this review article, we describe the physical basis of coarse-grained molecular dynamics, the coarse-grained force fields, the equations of motion and the respective numerical integration algorithms, and selected practical applications of coarse-grained molecular dynamics. We demonstrate that the motion of coarse-grained sites is governed by the potential of mean force and the friction and stochastic forces, resulting from integrating out the secondary degrees of freedom. Consequently, Langevin dynamics is a natural means of describing the motion of a system at the coarse-grained level and the potential of mean force is the physical basis of the coarse-grained force fields. Moreover, the choice of coarse-grained variables and the fact that coarse-grained sites often do not have spherical symmetry implies a non-diagonal inertia tensor. We describe selected coarse-grained models used in molecular dynamics simulations, including the most popular MARTINI model developed by Marrink's group and the UNICORN model of biological macromolecules developed in our laboratory. We conclude by discussing examples of the application of coarse-grained molecular dynamics to study biologically important processes.

ESCASA: Analytical estimation of atomic coordinates from coarse-grained geometry for nuclear-magnetic-resonance-assisted protein structure modeling. I. Backbone and Hβ protons

A method for the estimation of coordinates of atoms in proteins from coarse-grained geometry by simple analytical formulas (ESCASA), for use in nuclear-magnetic-resonance (NMR) data-assisted coarse-grained simulations of proteins is proposed. In this paper, the formulas for the backbone H^α and amide (H^N ) protons, and the side-chain H^β protons, given the C^α -trace, have been derived and parameterized, by using the interproton distances calculated from a set of 140 high-resolution non-homologous protein structures. The mean standard deviation over all types of proton pairs in the set was 0.44 Å after fitting. Validation against a set of 41 proteins with NMR-determined structures, which were not considered in parameterization, resulted in average standard deviation from average proton-proton distances of the NMR-determined structures of 0.25 Å, compared to 0.21 Å obtained with the PULCHRA all-atom-chain reconstruction algorithm and to the 0.12 Å standard deviation of the average-structure proton-proton distance of NMR-determined ensembles. The formulas provide analytical forces and can, therefore, be used in coarse-grained molecular dynamics.

Effects of lattice constraints in coarse-grained protein models

We compare and contrast folding behavior in several coarse-grained protein models, both on- and off-lattice, in an attempt to uncover the effect of lattice constraints in these kinds of models. Using modern, extended ensemble Monte Carlo methods-Wang-Landau sampling, multicanonical sampling, replica-exchange Wang-Landau sampling, and replica-exchange multicanonical sampling, we investigate the thermodynamic and structural behavior of the protein Crambin within the context of the hydrophobic-polar, hydrophobic-"neutral"-polar (H0P), and semi-flexible H0P model frameworks. We uncover the folding process in all cases; all models undergo, at least, the two major structural transitions observed in nature-the coil-globule collapse and the folding transition. As the complexity of the model increases, these two major transitions begin to split into multi-step processes, wherein the lattice coarse-graining has a significant impact on the details of these processes. The results show that the level of structural coarse-graining is coupled to the level of interaction coarse-graining.

A three-point coarse-grained model of five-water cluster with permanent dipoles and quadrupoles

As coarse-grained (CG) studies of large biomolecules increase, developments of reliable CG solvent models become particularly important. In this work, we reduce five water molecules into a three-point CG model with permanent dipole and quadrupole moments. In the CG force field, the modified Morse potential is utilized and an ideal three-water cluster is designed to derive CG-level permanent multipoles. The new CG model is parametrized on the AMOEBA polarizable force field. Various important properties of liquid water are examined to validate the new CG model. Results show that the new CG model can correctly reproduce certain important experimental properties such as density, isothermal compressibility and relative static dielectric permittivity, even better than the existing AA models. Additionally, the CPU tests reveal that the CG model can accelerate molecular dynamics simulations by a factor of 19 compared to the popular AA force field. Compared with the fix-point-charge model widely used in other CG models, the permanent-multipole-based CG model describes more rigid electrostatic attractions. This study also illustrates that the permanent multipole moments contribute a lot to the electrostatic calculations in CG simulation.

Coarse-grained (hybrid) integrative modeling of biomolecular interactions

The computational modeling field has vastly evolved over the past decades. The early developments of simplified protein systems represented a stepping stone towards establishing more efficient approaches to sample intricated conformational landscapes. Downscaling the level of resolution of biomolecules to coarser representations allows for studying protein structure, dynamics and interactions that are not accessible by classical atomistic approaches. The combination of different resolutions, namely hybrid modeling, has also been proved as an alternative when mixed levels of details are required. In this review, we provide an overview of coarse-grained/hybrid models focusing on their applicability in the modeling of biomolecular interactions. We give a detailed list of ready-to-use modeling software for studying biomolecular interactions allowing various levels of coarse-graining and provide examples of complexes determined by integrative coarse-grained/hybrid approaches in combination with experimental information.

Scale-consistent approach to the derivation of coarse-grained force fields for simulating structure, dynamics, and thermodynamics of biopolymers

In this chapter the scale-consistent approach to the derivation of coarse-grained force fields developed in our laboratory is presented, in which the effective energy function originates from the potential of mean force of the system under consideration and embeds atomistically detailed interactions in the resulting energy terms through use of Kubo's cluster-cumulant expansion, appropriate selection of the major degrees of freedom to be averaged out in the derivation of analytical approximations to the energy terms, and appropriate expression of the interaction energies at the all-atom level in these degrees of freedom. Our approach enables the developers to find correct functional forms of the effective coarse-grained energy terms, without having to import them from all-atom force fields or deriving them on a heuristic basis. In particular, the energy terms derived in such a way exhibit correct dependence on coarse-grained geometry, in particular on site orientation. Moreover, analytical formulas for the multibody (correlation) terms, which appear to be crucial for coarse-grained modeling of many of the regular structures such as, e.g., protein α-helices and β-sheets, can be derived in a systematic way. Implementation of the developed theory to the UNIfied COarse-gRaiNed (UNICORN) model of biological macromolecules, which consists of the UNRES (for proteins), NARES-2P (for nucleic acids), and SUGRES-1P (for polysaccharides) components, and is being developed in our laboratory is described. Successful applications of UNICORN to the prediction of protein structure, simulating the folding and stability of proteins and nucleic acids, and solving biological problems are discussed.

Atomic-level reconstruction of biomolecules by a rigid-fragment- and local-frame-based (RF-LF) strategy

Coarse-grained (CG) model has been a powerful tool in bridging the gap between theoretical studies and experimental phenomena in biological computing field. The reconstruction from a CG model to an atomic-detail structure is especially important in CG studies of biological systems. In this work, a rigid-fragment- and local-frame-based (RF-LF) backmapping method was proposed to achieve reverse mapping from CG models to atomic-level structures. The initial atomic-level structures were further refined to yield the final backmapping ones. With the popular Martini force field, the performance of the RF-LF method was extensively examined in the CG → AA (CG to AA) backmapping of protein/DNA/RNA systems. Besides, the RF-LF method was also extended to the backmapping of the TMFF model. Numerical results illustrate that the RF-LF backmapping method is generic and parameter-free and can provide a promising way to tackle atomic-level studies in CG models.

Recent Advances in Coarse-Grained Models for Biomolecules and Their Applications

Molecular dynamics simulations have emerged as a powerful tool to study biological systems at varied length and timescales. The conventional all-atom molecular dynamics simulations are being used by the wider scientific community in routine to capture the conformational dynamics and local motions. In addition, recent developments in coarse-grained models have opened the way to study the macromolecular complexes for time scales up to milliseconds. In this review, we have discussed the principle, applicability and recent development in coarse-grained models for biological systems. The potential of coarse-grained simulation has been reviewed through state-of-the-art examples of protein folding and structure prediction, self-assembly of complexes, membrane systems and carbohydrates fiber models. The multiscale simulation approaches have also been discussed in the context of their emerging role in unravelling hierarchical level information of biosystems. We conclude this review with the future scope of coarse-grained simulations as a constantly evolving tool to capture the dynamics of biosystems.

Characterization of Lipid-Protein Interactions and Lipid-Mediated Modulation of Membrane Protein Function through Molecular Simulation

The cellular membrane constitutes one of the most fundamental compartments of a living cell, where key processes such as selective transport of material and exchange of information between the cell and its environment are mediated by proteins that are closely associated with the membrane. The heterogeneity of lipid composition of biological membranes and the effect of lipid molecules on the structure, dynamics, and function of membrane proteins are now widely recognized. Characterization of these functionally important lipid-protein interactions with experimental techniques is however still prohibitively challenging. Molecular dynamics (MD) simulations offer a powerful complementary approach with sufficient temporal and spatial resolutions to gain atomic-level structural information and energetics on lipid-protein interactions. In this review, we aim to provide a broad survey of MD simulations focusing on exploring lipid-protein interactions and characterizing lipid-modulated protein structure and dynamics that have been successful in providing novel insight into the mechanism of membrane protein function.

The role of chain-stiffness in lattice protein models: A replica-exchange Wang-Landau study

Using Monte Carlo simulations, we investigate simple, physically motivated extensions to the hydrophobic-polar lattice protein model for the small (46 amino acid) protein Crambin. We use two-dimensional replica-exchange Wang-Landau sampling to study the effects of a bond angle stiffness parameter on the folding and uncover a new step in the collapse process for particular values of this stiffness parameter. A physical interpretation of the folding is developed by analysis of changes in structural quantities, and the free energy landscape is explored. For these special values of stiffness, we find non-degenerate ground states, a property that is consistent with behavior of real proteins, and we use these unique ground states to elucidate the formation of native contacts during the folding process. Through this analysis, we conclude that chain-stiffness is particularly influential in the low energy, low temperature regime of the folding process once the lattice protein has partially collapsed.

A general method for the derivation of the functional forms of the effective energy terms in coarse-grained energy functions of polymers. I. Backbone potentials of coarse-grained polypeptide chains

A general and systematic method for the derivation of the functional expressions for the effective energy terms in coarse-grained force fields of polymer chains is proposed. The method is based on the expansion of the potential of mean force of the system studied in the cluster-cumulant series and expanding the all-atom energy in the Taylor series in the squares of interatomic distances about the squares of the distances between coarse-grained centers, to obtain approximate analytical expressions for the cluster cumulants. The primary degrees of freedom to average about are the angles for collective rotation of the atoms contained in the coarse-grained interaction sites about the respective virtual-bond axes. The approach has been applied to the revision of the virtual-bond-angle, virtual-bond-torsional, and backbone-local-and-electrostatic correlation potentials for the UNited RESidue (UNRES) model of polypeptide chains, demonstrating the strong dependence of the torsional and correlation potentials on virtual-bond angles, not considered in the current UNRES. The theoretical considerations are illustrated with the potentials calculated from the ab initiopotential-energysurface of terminally blocked alanine by numerical integration and with the statistical potentials derived from known protein structures. The revised torsional potentials correctly indicate that virtual-bond angles close to 90° result in the preference for the turn and helical structures, while large virtual-bond angles result in the preference for polyproline II and extended backbone geometry. The revised correlation potentials correctly reproduce the preference for the formation of β-sheet structures for large values of virtual-bond angles and for the formation of α-helical structures for virtual-bond angles close to 90°.

On Stable States in a Topologically Driven Protein Folding Model

Theoretical models of protein folding often make simplifying assumptions that allow analysis, yielding interesting theoretical results. In this article, we study models where folding dynamics is primarily driven by local topological features in an iterative manner. We illustrate the merit of the proposed approach through its ability to simulate realistic protein folding processes even when the sequence content information is reduced to just hydrophobic and polar. We then analyze our models and show that under our simple assumptions, certain structures are inherently unstable, and that determining whether structures can be stable is an [Formula: see text]-hard problem. Interestingly, we find that when our model has only two amino acids, the problem becomes solvable in polynomial time.

Joint-based description of protein structure: its application to the geometric characterization of membrane proteins

A macroscopic description of a protein structure allows an understanding of the protein conformations in a more simplistic manner. Here, a new macroscopic approach that utilizes the joints of the protein secondary structures as a basic descriptor for the protein structure is proposed and applied to study the arrangement of secondary structures in helical membrane proteins. Two types of dihedral angle, Ω and λ, were defined based on the joint points of the transmembrane (TM) helices and loops, and employed to analyze 103 non-homologous membrane proteins with 3 to 14 TM helices. The Ω-λ plot, which is a distribution plot of the dihedral angles of the joint points, identified the allowed and disallowed regions of helical arrangement. Analyses of consecutive dihedral angle patterns indicated that there are preferred patterns in the helical alignment and extension of TM proteins, and helical extension pattern in TM proteins is varied as the size of TM proteins increases. Finally, we could identify some symmetric protein pairs in TM proteins under the joint-based coordinate and 3-dimensional coordinates. The joint-based approach is expected to help better understand and model the overall conformational features of complicated large-scale proteins, such as membrane proteins.

On the polymer physics origins of protein folding thermodynamics

A remarkable feature of the spontaneous folding of many small proteins is the striking similarity in the thermodynamics of the folding process. This process is characterized by simple two-state thermodynamics with large and compensating changes in entropy and enthalpy and a funnel-like free energy landscape with a free-energy barrier that varies linearly with temperature. One might attribute the commonality of this two-state folding behavior to features particular to these proteins (e.g., chain length, hydrophobic/hydrophilic balance, attributes of the native state) or one might suspect that this similarity in behavior has a more general polymer-physics origin. Here we show that this behavior is also typical for flexible homopolymer chains with sufficiently short range interactions. Two-state behavior arises from the presence of a low entropy ground (folded) state separated from a set of high entropy disordered (unfolded) states by a free energy barrier. This homopolymer model exhibits a funneled free energy landscape that reveals a complex underlying dynamics involving competition between folding and non-folding pathways. Despite the presence of multiple pathways, this simple physics model gives the robust result of two-state thermodynamics for both the cases of folding from a basin of expanded coil states and from a basin of compact globule states.

Prediction of Structural and Functional Aspects of Protein

To predict the structure of protein from a primary amino acid sequence is computationally difficult. An investigation of the methods and algorithms used to predict protein structure and a thorough knowledge of the function and structure of proteins are critical for the advancement of biology and the life sciences as well as the development of better drugs, higher-yield crops, and even synthetic bio-fuels. To that end, this chapter sheds light on the methods used for protein structure prediction. This chapter covers the applications of modeled protein structures and unravels the relationship between pure sequence information and three-dimensional structure, which continues to be one of the greatest challenges in molecular biology. With this resource, it presents an all-encompassing examination of the problems, methods, tools, servers, databases, and applications of protein structure prediction, giving unique insight into the future applications of the modeled protein structures. In this chapter, current protein structure prediction methods are reviewed for a milieu on structure prediction, the prediction of structural fundamentals, tertiary structure prediction, and functional imminent. The basic ideas and advances of these directions are discussed in detail.

Performance of protein-structure predictions with the physics-based UNRES force field in CASP11

Participating as the Cornell-Gdansk group, we have used our physics-based coarse-grained UNited RESidue (UNRES) force field to predict protein structure in the 11th Community Wide Experiment on the Critical Assessment of Techniques for Protein Structure Prediction (CASP11). Our methodology involved extensive multiplexed replica exchange simulations of the target proteins with a recently improved UNRES force field to provide better reproductions of the local structures of polypeptide chains. All simulations were started from fully extended polypeptide chains, and no external information was included in the simulation process except for weak restraints on secondary structure to enable us to finish each prediction within the allowed 3-week time window. Because of simplified UNRES representation of polypeptide chains, use of enhanced sampling methods, code optimization and parallelization and sufficient computational resources, we were able to treat, for the first time, all 55 human prediction targets with sizes from 44 to 595 amino acid residues, the average size being 251 residues. Complete structures of six single-domain proteins were predicted accurately, with the highest accuracy being attained for the T0769, for which the CαRMSD was 3.8 Å for 97 residues of the experimental structure. Correct structures were also predicted for 13 domains of multi-domain proteins with accuracy comparable to that of the best template-based modeling methods. With further improvements of the UNRES force field that are now underway, our physics-based coarse-grained approach to protein-structure prediction will eventually reach global prediction capacity and, consequently, reliability in simulating protein structure and dynamics that are important in biochemical processes.

AVAILABILITY AND IMPLEMENTATION

Freely available on the web at http://www.unres.pl/ CONTACT: has5@cornell.edu.

Insights into Cotranslational Nascent Protein Behavior from Computer Simulations

Regulation of protein stability and function in vivo begins during protein synthesis, when the ribosome translates a messenger RNA into a nascent polypeptide. Cotranslational processes involving a nascent protein include folding, binding to other macromolecules, enzymatic modification, and secretion through membranes. Experiments have shown that the rate at which the ribosome adds amino acids to the elongating nascent chain influences the efficiency of these processes, with alterations to these rates possibly contributing to diseases, including some types of cancer. In this review, we discuss recent insights into cotranslational processes gained from molecular simulations, how different computational approaches have been combined to understand cotranslational processes at multiple scales, and the new scenarios illuminated by these simulations. We conclude by suggesting interesting questions that computational approaches in this research area can address over the next few years.

Tabulation as a high-resolution alternative to coarse-graining protein interactions: Initial application to virus capsid subunits

Traditional coarse-graining based on a reduced number of interaction sites often entails a significant sacrifice of chemical accuracy. As an alternative, we present a method for simulating large systems composed of interacting macromolecules using an energy tabulation strategy previously devised for small rigid molecules or molecular fragments [S. Lettieri and D. M. Zuckerman, J. Comput. Chem. 33, 268-275 (2012); J. Spiriti and D. M. Zuckerman, J. Chem. Theory Comput. 10, 5161-5177 (2014)]. We treat proteins as rigid and construct distance and orientation-dependent tables of the interaction energy between them. Arbitrarily detailed interactions may be incorporated into the tables, but as a proof-of-principle, we tabulate a simple α-carbon Gō-like model for interactions between dimeric subunits of the hepatitis B viral capsid. This model is significantly more structurally realistic than previous models used in capsid assembly studies. We are able to increase the speed of Monte Carlo simulations by a factor of up to 6700 compared to simulations without tables, with only minimal further loss in accuracy. To obtain further enhancement of sampling, we combine tabulation with the weighted ensemble (WE) method, in which multiple parallel simulations are occasionally replicated or pruned in order to sample targeted regions of a reaction coordinate space. In the initial study reported here, WE is able to yield pathways of the final ∼25% of the assembly process.

Efficient conformational space exploration in ab initio protein folding simulation

Ab initio protein folding simulation largely depends on knowledge-based energy functions that are derived from known protein structures using statistical methods. These knowledge-based energy functions provide us with a good approximation of real protein energetics. However, these energy functions are not very informative for search algorithms and fail to distinguish the types of amino acid interactions that contribute largely to the energy function from those that do not. As a result, search algorithms frequently get trapped into the local minima. On the other hand, the hydrophobic-polar (HP) model considers hydrophobic interactions only. The simplified nature of HP energy function makes it limited only to a low-resolution model. In this paper, we present a strategy to derive a non-uniform scaled version of the real 20×20 pairwise energy function. The non-uniform scaling helps tackle the difficulty faced by a real energy function, whereas the integration of 20×20 pairwise information overcomes the limitations faced by the HP energy function. Here, we have applied a derived energy function with a genetic algorithm on discrete lattices. On a standard set of benchmark protein sequences, our approach significantly outperforms the state-of-the-art methods for similar models. Our approach has been able to explore regions of the conformational space which all the previous methods have failed to explore. Effectiveness of the derived energy function is presented by showing qualitative differences and similarities of the sampled structures to the native structures. Number of objective function evaluation in a single run of the algorithm is used as a comparison metric to demonstrate efficiency.

Tunable Coarse Graining for Monte Carlo Simulations of Proteins via Smoothed Energy Tables: Direct and Exchange Simulations

Many commonly used coarse-grained models for proteins are based on simplified interaction sites and consequently may suffer from significant limitations, such as the inability to properly model protein secondary structure without the addition of restraints. Recent work on a benzene fluid (Lettieri S.; Zuckerman D. M.J. Comput. Chem.2012, 33, 268-275) suggested an alternative strategy of tabulating and smoothing fully atomistic orientation-dependent interactions among rigid molecules or fragments. Here we report our initial efforts to apply this approach to the polar and covalent interactions intrinsic to polypeptides. We divide proteins into nearly rigid fragments, construct distance and orientation-dependent tables of the atomistic interaction energies between those fragments, and apply potential energy smoothing techniques to those tables. The amount of smoothing can be adjusted to give coarse-grained models that range from the underlying atomistic force field all the way to a bead-like coarse-grained model. For a moderate amount of smoothing, the method is able to preserve about 70-90% of the α-helical structure while providing a factor of 3-10 improvement in sampling per unit computation time (depending on how sampling is measured). For a greater amount of smoothing, multiple folding-unfolding transitions of the peptide were observed, along with a factor of 10-100 improvement in sampling per unit computation time, although the time spent in the unfolded state was increased compared with less smoothed simulations. For a β hairpin, secondary structure is also preserved, albeit for a narrower range of the smoothing parameter and, consequently, for a more modest improvement in sampling. We have also applied the new method in a "resolution exchange" setting, in which each replica runs a Monte Carlo simulation with a different degree of smoothing. We obtain exchange rates that compare favorably to our previous efforts at resolution exchange (Lyman E.; Zuckerman D. M.J. Chem. Theory Comput.2006, 2, 656-666).

Effect of single-site mutations on hydrophobic-polar lattice proteins

We developed a heuristic method for determining the ground-state degeneracy of hydrophobic-polar (HP) lattice proteins, based on Wang-Landau and multicanonical sampling. It is applied during comprehensive studies of single-site mutations in specific HP proteins with different sequences. The effects in which we are interested include structural changes in ground states, changes of ground-state energy, degeneracy, and thermodynamic properties of the system. With respect to mutations, both extremely sensitive and insensitive positions in the HP sequence have been found. That is, ground-state energies and degeneracies, as well as other thermodynamic and structural quantities, may be either largely unaffected or may change significantly due to mutation.

A unified coarse-grained model of biological macromolecules based on mean-field multipole-multipole interactions

A unified coarse-grained model of three major classes of biological molecules—proteins, nucleic acids, and polysaccharides—has been developed. It is based on the observations that the repeated units of biopolymers (peptide groups, nucleic acid bases, sugar rings) are highly polar and their charge distributions can be represented crudely as point multipoles. The model is an extension of the united residue (UNRES) coarse-grained model of proteins developed previously in our laboratory. The respective force fields are defined as the potentials of mean force of biomacromolecules immersed in water, where all degrees of freedom not considered in the model have been averaged out. Reducing the representation to one center per polar interaction site leads to the representation of average site–site interactions as mean-field dipole–dipole interactions. Further expansion of the potentials of mean force of biopolymer chains into Kubo’s cluster-cumulant series leads to the appearance of mean-field dipole–dipole interactions, averaged in the context of local interactions within a biopolymer unit. These mean-field interactions account for the formation of regular structures encountered in biomacromolecules, e.g., α-helices and β-sheets in proteins, double helices in nucleic acids, and helicoidally packed structures in polysaccharides, which enables us to use a greatly reduced number of interacting sites without sacrificing the ability to reproduce the correct architecture. This reduction results in an extension of the simulation timescale by more than four orders of magnitude compared to the all-atom representation. Examples of the performance of the model are presented.

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How good are simplified models for protein structure prediction?

Protein structure prediction (PSP) has been one of the most challenging problems in computational biology for several decades. The challenge is largely due to the complexity of the all-atomic details and the unknown nature of the energy function. Researchers have therefore used simplified energy models that consider interaction potentials only between the amino acid monomers in contact on discrete lattices. The restricted nature of the lattices and the energy models poses a twofold concern regarding the assessment of the models. Can a native or a very close structure be obtained when structures are mapped to lattices? Can the contact based energy models on discrete lattices guide the search towards the native structures? In this paper, we use the protein chain lattice fitting (PCLF) problem to address the first concern; we developed a constraint-based local search algorithm for the PCLF problem for cubic and face-centered cubic lattices and found very close lattice fits for the native structures. For the second concern, we use a number of techniques to sample the conformation space and find correlations between energy functions and root mean square deviation (RMSD) distance of the lattice-based structures with the native structures. Our analysis reveals weakness of several contact based energy models used that are popular in PSP.

Coarse-grained models of the proteins backbone conformational dynamics

Coarse-grained models are more and more frequently used in the studies of the proteins structural and dynamic properties, since the reduced number of degrees of freedom allows to enhance the conformational space exploration. This chapter attempts to provide an overview of the various coarse-grained models that were applied to study the functional conformational changes of the polypeptides main chain around their native state. It will more specifically discuss the methods used to represent the protein backbone flexibility and to account for the physico-chemical interactions that stabilize the secondary structure elements.

Revised Backbone-Virtual-Bond-Angle Potentials to Treat the l- and d-Amino Acid Residues in the Coarse-Grained United Residue (UNRES) Force Field

Continuing our effort to introduce d-amino-acid residues in the united residue (UNRES) force field developed in our laboratory, in this work the Cα ··· Cα ··· Cα backbone-virtual-bond-valence-angle (θ) potentials for systems containing d-amino-acid residues have been developed. The potentials were determined by integrating the combined energy surfaces of all possible triplets of terminally blocked glycine, alanine, and proline obtained with ab initio molecular quantum mechanics at the MP2/6-31G(d,p) level to calculate the corresponding potentials of mean force (PMFs). Subsequently, analytical expressions were fitted to the PMFs to give the virtual-bond-valence potentials to be used in UNRES. Alanine represented all types of amino-acid residues except glycine and proline. The blocking groups were either the N-acetyl and N',N'-dimethyl or N-acetyl and pyrrolidyl group, depending on whether the residue next in sequence was an alanine-type or a proline residue. A total of 126 potentials (63 symmetry-unrelated potentials for each set of terminally blocking groups) were determined. Together with the torsional, double-torsional, and side-chain-rotamer potentials for polypeptide chains containing d-amino-acid residues determined in our earlier work (Sieradzan et al. J. Chem. Theory Comput., 2012, 8, 4746), the new virtual-bond-angle (θ) potentials now constitute the complete set of physics-based potentials with which to run coarse-grained simulations of systems containing d-amino-acid residues. The ability of the extended UNRES force field to reproduce thermodynamics of polypeptide systems with d-amino-acid residues was tested by comparing the experimentally measured and the calculated free energies of helix formation of model KLALKLALxxLKLALKLA peptides, where x denotes any d- or l- amino-acid residue. The obtained results demonstrate that the UNRES force field with the new potentials reproduce the changes of free energies of helix formation upon d-substitution but overestimate the free energies of helix formation. To test the ability of UNRES with the new potentials to reproduce the structures of polypeptides with d-amino-acid residues, an ab initio replica-exchange folding simulation of thurincin H from Bacillus thuringiensis, which has d-amino-acid residues in the sequence, was carried out. UNRES was able to locate the native α-helical hairpin structure as the dominant structure even though no native sulfide-carbon bonds were present in the simulation.

Design of a rotamer library for coarse-grained models in protein-folding simulations

Rotamer libraries usually contain geometric information to trace an amino acid side chain, atom by atom, onto a protein backbone. These libraries have been widely used in protein design, structure refinement and prediction, homology modeling, and X-ray and NMR structure validation. However, they usually present too much information and are not always fully compatible with the coarse-grained models of the protein geometry that are frequently used to tackle the protein-folding problem through molecular simulation. In this work, we introduce a new backbone-dependent rotamer library for side chains compatible with low-resolution models in polypeptide chains. We have dispensed with an atomic description of proteins, representing each amino acid side chain by its geometric center (or centroid). The resulting rotamers have been estimated from a statistical analysis of a large structural database consisting of high-resolution X-ray protein structures. As additional information, each rotamer includes the frequency with which it has been found during the statistical analysis. More importantly, the library has been designed with a careful control to ensure that the vast majority of side chains in protein structures (at least 95% of residues) are properly represented. We have tested our library using an independent set of proteins, and our results support a good correlation between the reconstructed centroids from our rotamer library and those in the experimental structures. This new library can serve to improve the definition of side chain centroids in coarse-grained models, avoiding at the same time an excessive additional complexity in a geometric model for the polypeptide chain.

Evolutionary Algorithm in the Optimization of a Coarse-Grained Force Field

Simulations using residue-scale coarse-grained models of biomolecules are less computationally demanding than simulations employing full-atomistic force fields. However, the coarse-grained models are often difficult and tedious to parametrize for certain applications. Therefore, a systematic and objective method to help develop or adapt the coarse-grained models is needed. We present an automatic method that implements an evolutionary algorithm to find a set of optimal force field parameters for a one-bead coarse-grained model. In addition to an optimized force field, parameter correlations and significance of the potential energy terms can be determined. The method is applied to two classes of problems: the dynamics of an RNA helix and the RNA structure prediction.

Construction of an intermediate-resolution lattice model and re-examination of the helix-coil transition: a dynamic Monte Carlo simulation

In protein modeling, spatial resolution and computational efficiency are always incompatible. As a compromise, an intermediate-resolution lattice model has been constructed in the present work. Each residue is decomposed into four basic units, i.e. the α-carbon group, the carboxyl group, the imino group, and the side-chain group, and each basic coarse-grained unit is represented by a minimum cubic box with eight lattice sites. The spacing of the lattice is about 0.56 Å, holding the highest spatial resolution for the present lattice protein models. As the first report of this new model, the helix-coil transition of a polyalanine chain was examined via dynamic Monte Carlo simulation. The period of formed α-helix was about 3.68 residues, close to that of a natural α-helix. The resultant backbone motion was found to be in the realistic regions of the conformational space in the Ramachandran plot. Helix propagation constant and nucleation constant were further determined through the dynamic hydrogen bonding process and torsional angle variation, and the results were used to make comparison between classical Zimm-Bragg theory and Lifson-Roig theory based on the Qian-Schellman relationship. The simulation results confirmed that our lattice model can reproduce the helix-coil transition of polypeptide and construct a moderately fine α-helix conformation without significantly weakening the priority in efficiency for a lattice model.

Designing a polycationic probe for simultaneous enrichment and detection of microRNAs in a nanopore

The nanopore sensor can detect cancer-derived nucleic acid biomarkers such as microRNAs (miRNAs), providing a noninvasive tool potentially useful in medical diagnostics. However, the nanopore-based detection of these biomarkers remains confounded by the presence of numerous other nucleic acid species found in biofluid extracts. Their nonspecific interactions with the nanopore inevitably contaminate the target signals, reducing the detection accuracy. Here we report a novel method that utilizes a polycationic peptide-PNA probe as the carrier for selective miRNA detection in the nucleic acid mixture. The cationic probe hybridized with microRNA forms a dipole complex, which can be captured by the pore using a voltage polarity that is opposite the polarity used to capture negatively charged nucleic acids. As a result, nontarget species are driven away from the pore opening, and the target miRNA can be detected accurately without interference. In addition, we demonstrate that the PNA probe enables accurate discrimination of miRNAs with single-nucleotide difference. This highly sensitive and selective nanodielectrophoresis approach can be applied to the detection of clinically relevant nucleic acid fragments in complex samples.

On the characterization and software implementation of general protein lattice models

models of proteins have been widely used as a practical means to computationally investigate general properties of the system. In lattice models any sterically feasible conformation is represented as a self-avoiding walk on a lattice, and residue types are limited in number. So far, only two- or three-dimensional lattices have been used. The inspection of the neighborhood of alpha carbons in the core of real proteins reveals that also lattices with higher coordination numbers, possibly in higher dimensional spaces, can be adopted. In this paper, a new general parametric lattice model for simplified protein conformations is proposed and investigated. It is shown how the supporting software can be consistently designed to let algorithms that operate on protein structures be implemented in a lattice-agnostic way. The necessary theoretical foundations are developed and organically presented, pinpointing the role of the concept of main directions in lattice-agnostic model handling. Subsequently, the model features across dimensions and lattice types are explored in tests performed on benchmark protein sequences, using a Python implementation. Simulations give insights on the use of square and triangular lattices in a range of dimensions. The trend of potential minimum for sequences of different lengths, varying the lattice dimension, is uncovered. Moreover, an extensive quantitative characterization of the usage of the so-called "move types" is reported for the first time. The proposed general framework for the development of lattice models is simple yet complete, and an object-oriented architecture can be proficiently employed for the supporting software, by designing ad-hoc classes. The proposed framework represents a new general viewpoint that potentially subsumes a number of solutions previously studied. The adoption of the described model pushes to look at protein structure issues from a more general and essential perspective, making computational investigations over simplified models more straightforward as well.

Consistent View of Protein Fluctuations from All-Atom Molecular Dynamics and Coarse-Grained Dynamics with Knowledge-Based Force-Field

It is widely recognized that atomistic Molecular Dynamics (MD), a classical simulation method, captures the essential physics of protein dynamics. That idea is supported by a theoretical study showing that various MD force-fields provide a consensus picture of protein fluctuations in aqueous solution [Rueda, M. et al. Proc. Natl. Acad. Sci. U.S.A. 2007, 104, 796-801]. However, atomistic MD cannot be applied to most biologically relevant processes due to its limitation to relatively short time scales. Much longer time scales can be accessed by properly designed coarse-grained models. We demonstrate that the aforementioned consensus view of protein dynamics from short (nanosecond) time scale MD simulations is fairly consistent with the dynamics of the coarse-grained protein model - the CABS model. The CABS model employs stochastic dynamics (a Monte Carlo method) and a knowledge-based force-field, which is not biased toward the native structure of a simulated protein. Since CABS-based dynamics allows for the simulation of entire folding (or multiple folding events) in a single run, integration of the CABS approach with all-atom MD promises a convenient (and computationally feasible) means for the long-time multiscale molecular modeling of protein systems with atomistic resolution.

A generic force field for protein coarse-grained molecular dynamics simulation

Coarse-grained (CG) force fields have become promising tools for studies of protein behavior, but the balance of speed and accuracy is still a challenge in the research of protein coarse graining methodology. In this work, 20 CG beads have been designed based on the structures of amino acid residues, with which an amino acid can be represented by one or two beads, and a CG solvent model with five water molecules was adopted to ensure the consistence with the protein CG beads. The internal interactions in protein were classified according to the types of the interacting CG beads, and adequate potential functions were chosen and systematically parameterized to fit the energy distributions. The proposed CG force field has been tested on eight proteins, and each protein was simulated for 1000 ns. Even without any extra structure knowledge of the simulated proteins, the Cα root mean square deviations (RMSDs) with respect to their experimental structures are close to those of relatively short time all atom molecular dynamics simulations. However, our coarse grained force field will require further refinement to improve agreement with and persistence of native-like structures. In addition, the root mean square fluctuations (RMSFs) relative to the average structures derived from the simulations show that the conformational fluctuations of the proteins can be sampled.

Producing high-accuracy lattice models from protein atomic coordinates including side chains

Lattice models are a common abstraction used in the study of protein structure, folding, and refinement. They are advantageous because the discretisation of space can make extensive protein evaluations computationally feasible. Various approaches to the protein chain lattice fitting problem have been suggested but only a single backbone-only tool is available currently. We introduce LatFit, a new tool to produce high-accuracy lattice protein models. It generates both backbone-only and backbone-side-chain models in any user defined lattice. LatFit implements a new distance RMSD-optimisation fitting procedure in addition to the known coordinate RMSD method. We tested LatFit's accuracy and speed using a large nonredundant set of high resolution proteins (SCOP database) on three commonly used lattices: 3D cubic, face-centred cubic, and knight's walk. Fitting speed compared favourably to other methods and both backbone-only and backbone-side-chain models show low deviation from the original data (~1.5 Å RMSD in the FCC lattice). To our knowledge this represents the first comprehensive study of lattice quality for on-lattice protein models including side chains while LatFit is the only available tool for such models.

Genomics-aided structure prediction

We introduce a theoretical framework that exploits the ever-increasing genomic sequence information for protein structure prediction. Structure-based models are modified to incorporate constraints by a large number of non-local contacts estimated from direct coupling analysis (DCA) of co-evolving genomic sequences. A simple hybrid method, called DCA-fold, integrating DCA contacts with an accurate knowledge of local information (e.g., the local secondary structure) is sufficient to fold proteins in the range of 1-3 Å resolution.

Comparative modeling and protein-like features of hydrophobic-polar models on a two-dimensional lattice

Lattice models of proteins have been extensively used to study protein thermodynamics, folding dynamics, and evolution. Our study considers two different hydrophobic-polar (HP) models on the 2D square lattice: the purely HP model and a model where a compactness-favoring term is added. We exhaustively enumerate all the possible structures in our models and perform the study of their corresponding folds, HP arrangements in space and shapes. The two models considered differ greatly in their numbers of structures, folds, arrangements, and shapes. Despite their differences, both lattice models have distinctive protein-like features: (1) Shapes are compact in both models, especially when a compactness-favoring energy term is added. (2) The residue composition is independent of the chain length and is very close to 50% hydrophobic in both models, as we observe in real proteins. (3) Comparative modeling works well in both models, particularly in the more compact one. The fact that our models show protein-like features suggests that lattice models incorporate the fundamental physical principles of proteins. Our study supports the use of lattice models to study questions about proteins that require exactness and extensive calculations, such as protein design and evolution, which are often too complex and computationally demanding to be addressed with more detailed models.

Exploring the energy landscapes of protein folding simulations with Bayesian computation

Nested sampling is a Bayesian sampling technique developed to explore probability distributions localized in an exponentially small area of the parameter space. The algorithm provides both posterior samples and an estimate of the evidence (marginal likelihood) of the model. The nested sampling algorithm also provides an efficient way to calculate free energies and the expectation value of thermodynamic observables at any temperature, through a simple post processing of the output. Previous applications of the algorithm have yielded large efficiency gains over other sampling techniques, including parallel tempering. In this article, we describe a parallel implementation of the nested sampling algorithm and its application to the problem of protein folding in a Gō-like force field of empirical potentials that were designed to stabilize secondary structure elements in room-temperature simulations. We demonstrate the method by conducting folding simulations on a number of small proteins that are commonly used for testing protein-folding procedures. A topological analysis of the posterior samples is performed to produce energy landscape charts, which give a high-level description of the potential energy surface for the protein folding simulations. These charts provide qualitative insights into both the folding process and the nature of the model and force field used.

Kinetic network study of the diversity and temperature dependence of Trp-Cage folding pathways: combining transition path theory with stochastic simulations

We present a new approach to study a multitude of folding pathways and different folding mechanisms for the 20-residue mini-protein Trp-Cage using the combined power of replica exchange molecular dynamics (REMD) simulations for conformational sampling, transition path theory (TPT) for constructing folding pathways, and stochastic simulations for sampling the pathways in a high dimensional structure space. REMD simulations of Trp-Cage with 16 replicas at temperatures between 270 and 566 K are carried out with an all-atom force field (OPLSAA) and an implicit solvent model (AGBNP). The conformations sampled from all temperatures are collected. They form a discretized state space that can be used to model the folding process. The equilibrium population for each state at a target temperature can be calculated using the weighted-histogram-analysis method (WHAM). By connecting states with similar structures and creating edges satisfying detailed balance conditions, we construct a kinetic network that preserves the equilibrium population distribution of the state space. After defining the folded and unfolded macrostates, committor probabilities (P(fold)) are calculated by solving a set of linear equations for each node in the network and pathways are extracted together with their fluxes using the TPT algorithm. By clustering the pathways into folding "tubes", a more physically meaningful picture of the diversity of folding routes emerges. Stochastic simulations are carried out on the network, and a procedure is developed to project sampled trajectories onto the folding tubes. The fluxes through the folding tubes calculated from the stochastic trajectories are in good agreement with the corresponding values obtained from the TPT analysis. The temperature dependence of the ensemble of Trp-Cage folding pathways is investigated. Above the folding temperature, a large number of diverse folding pathways with comparable fluxes flood the energy landscape. At low temperature, however, the folding transition is dominated by only a few localized pathways.