Thermodynamics of Gō-type models for protein folding

Computational Modeling of DNA 3D Structures: From Dynamics and Mechanics to Folding

DNA carries the genetic information required for the synthesis of RNA and proteins and plays an important role in many processes of biological development. Understanding the three-dimensional (3D) structures and dynamics of DNA is crucial for understanding their biological functions and guiding the development of novel materials. In this review, we discuss the recent advancements in computer methods for studying DNA 3D structures. This includes molecular dynamics simulations to analyze DNA dynamics, flexibility, and ion binding. We also explore various coarse-grained models used for DNA structure prediction or folding, along with fragment assembly methods for constructing DNA 3D structures. Furthermore, we also discuss the advantages and disadvantages of these methods and highlight their differences.

Behavior of Proteins under Pressure from Experimental Pressure-Dependent Structures

Structure-based models are coarse-grained representations of the interactions responsible for the protein folding process. In their simplest form, they use only the native contact map of a given protein to predict the main features of its folding process by computer simulation. Given their limitations, these models are frequently complemented with sequence-dependent contributions or additional information. Specifically, to analyze the effect of pressure on the folding/unfolding transition, special forms of these interaction potentials are employed, which may a priori determine the outcome of the simulations. In this work, we have tried to keep the original simplicity of structure-based models. Therefore, we have used folded structures that have been experimentally determined at different pressures to define native contact maps and thus interactions dependent on pressure. Despite the apparently tiny structural differences induced by pressure, our simulation results provide different thermodynamic and kinetic behaviors, which roughly correspond to experimental observations (when there is a possible comparison) of two proteins used as benchmarks, hen egg-white lysozyme and dihydrofolate reductase. Therefore, this work shows the feasibility of using experimental native structures at different pressures to analyze the global effects of this physical property on the protein folding process.

Computational methods in the study of self-entangled proteins: a critical appraisal

The existence of self-entangled proteins, the native structure of which features a complex topology, unveils puzzling, and thus fascinating, aspects of protein biology and evolution. The discovery that a polypeptide chain can encode the capability to self-entangle in an efficient and reproducible way during folding, has raised many questions, regarding the possible function of these knots, their conservation along evolution, and their role in the folding paradigm. Understanding the function and origin of these entanglements would lead to deep implications in protein science, and this has stimulated the scientific community to investigate self-entangled proteins for decades by now. In this endeavour, advanced experimental techniques are more and more supported by computational approaches, that can provide theoretical guidelines for the interpretation of experimental results, and for the effective design of new experiments. In this review we provide an introduction to the computational study of self-entangled proteins, focusing in particular on the methodological developments related to this research field. A comprehensive collection of techniques is gathered, ranging from knot theory algorithms, that allow detection and classification of protein topology, to Monte Carlo or molecular dynamics strategies, that constitute crucial instruments for investigating thermodynamics and kinetics of this class of proteins.

Hydrophobic confinement modulates thermal stability and assists knotting in the folding of tangled proteins

There is growing support for the idea that the in vivo folding process of knotted proteins is assisted by chaperonins, but the mechanism of chaperonin assisted folding remains elusive. Here, we conduct extensive Monte Carlo simulations of lattice and off-lattice models to explore the effects of confinement and hydrophobic intermolecular interactions with the chaperonin cage in the folding and knotting processes. We find that moderate to high protein-cavity interactions (which are likely to be established in the beginning of the chaperonin working cycle) cause an energetic destabilization of the protein that overcomes the entropic stabilization driven by excluded volume, and leads to a decrease of the melting temperature relative to bulk conditions. Moreover, mild-to-moderate hydrophobic interactions with the cavity (which would be established later in the cycle) lead to a significant enhancement of knotting probability in relation to bulk conditions while simultaneously moderating the effect of steric confinement in the enhancement of thermal stability. Our results thus indicate that the chaperonin may be able to assist knotting without simultaneously thermally stabilizing potential misfolded states to a point that would hamper productive folding thus compromising its functional role.

Design of a structure-based model for protein folding from flexible conformations

We introduce a coarse-grained, structure-based model for protein folding that considers the flexibility of the native state in the definition of the model interactions.

Steric confinement and enhanced local flexibility assist knotting in simple models of protein folding

The chaperonin complex GroEL-GroES is able to accelerate the folding process of knotted proteins considerably. However, the folding mechanism inside the chaperonin cage is elusive. Here we use a combination of lattice and off-lattice Monte Carlo simulations of simple Gō models to study the effect of physical confinement and local flexibility on the folding process of protein model systems embedding a trefoil knot in their native structure. This study predicts that steric confinement plays a specific role in the folding of knotted proteins by increasing the knotting probability for very high degrees of confinement. This effect is observed for protein MJ0366 even above the melting temperature for confinement sizes compatible with the size of the GroEL/GroES chaperonin cage. An enhanced local flexibility produces the same qualitative effects on the folding process. In particular, we observe that knotting probability increases up to 40% in the transition state of protein MJ0366 when flexibility is enhanced. This is underlined by a structural change in the transition state, which becomes devoid of helical content. No relation between the knotting mechanism and flexibility was found in the context of the off-lattice model adopted in this work.

Knotted proteins: A tangled tale of Structural Biology

Knotted proteins have their native structures arranged in the form of an open knot. In the last ten years researchers have been making significant efforts to reveal their folding mechanism and understand which functional advantage(s) knots convey to their carriers. Molecular simulations have been playing a fundamental role in this endeavor, and early computational predictions about the knotting mechanism have just been confirmed in wet lab experiments. Here we review a collection of simulation results that allow outlining the current status of the field of knotted proteins, and discuss directions for future research.

A multiscale approach to simulating the conformational properties of unbound multi-C₂H₂ zinc finger proteins

The conformational properties of unbound multi-Cys2 His2 (mC2H2) zinc finger proteins, in which zinc finger domains are connected by flexible linkers, are studied by a multiscale approach. Three methods on different length scales are utilized. First, atomic detail molecular dynamics simulations of one zinc finger and its adjacent flexible linker confirmed that the zinc finger is more rigid than the flexible linker. Second, the end-to-end distance distributions of mC2H2 zinc finger proteins are computed using an efficient atomistic pivoting algorithm, which only takes excluded volume interactions into consideration. The end-to-end distance distribution gradually changes its profile, from left-tailed to right-tailed, as the number of zinc fingers increases. This is explained by using a worm-like chain model. For proteins of a few zinc fingers, an effective bending constraint favors an extended conformation. Only for proteins containing more than nine zinc fingers, is a somewhat compacted conformation preferred. Third, a mesoscale model is modified to study both the local and the global conformational properties of multi-C2H2 zinc finger proteins. Simulations of the CCCTC-binding factor (CTCF), an important mC2H2 zinc finger protein for genome spatial organization, are presented.

How determinant is N-terminal to C-terminal coupling for protein folding?

The existence of native interactions between the protein termini is a major determinant of the free energy barrier in a two-state folding transition being therefore a critical modulator of protein folding cooperativity.

Using VIPT-jump to distinguish between different folding mechanisms: application to BBL and a Trpzip

Protein folding involves a large number of sequential molecular steps or conformational substates. Thus, experimental characterization of the underlying folding energy landscape for any given protein is difficult. Herein, we present a new method that can be used to determine the major characteristics of the folding energy landscape in question, e.g., to distinguish between activated and barrierless downhill folding scenarios. This method is based on the idea that the conformational relaxation kinetics of different folding mechanisms at a given final condition will show different dependences on the initial condition. We show, using both simulation and experiment, that it is possible to differentiate between disparate kinetic folding models by comparing temperature jump (T-jump) relaxation traces obtained with a fixed final temperature and varied initial temperatures, which effectively varies the initial potential (VIP) of the system of interest. We apply this method (hereafter refer to as VIPT-jump) to two model systems, tryptophan zipper (Trpzip)-2c and BBL, and our results show that BBL exhibits characteristics of barrierless downhill folding, whereas Trpzip-2c folding encounters a free energy barrier. In addition, using the T-jump data of BBL we are able to provide, via Langevin dynamics simulations, a realistic estimate of its conformational diffusion coefficient.

The shadow map: a general contact definition for capturing the dynamics of biomolecular folding and function

Structure-based models (SBMs) are simplified models of the biomolecular dynamics that arise from funneled energy landscapes. We recently introduced an all-atom SBM that explicitly represents the atomic geometry of a biomolecule. While this initial study showed the robustness of the all-atom SBM Hamiltonian to changes in many of the energetic parameters, an important aspect, which has not been explored previously, is the definition of native interactions. In this study, we propose a general definition for generating atomically grained contact maps called "Shadow". The Shadow algorithm initially considers all atoms within a cutoff distance and then, controlled by a screening parameter, discards the occluded contacts. We show that this choice of contact map is not only well behaved for protein folding, since it produces consistently cooperative folding behavior in SBMs but also desirable for exploring the dynamics of macromolecular assemblies since, it distributes energy similarly between RNAs and proteins despite their disparate internal packing. All-atom structure-based models employing Shadow contact maps provide a general framework for exploring the geometrical features of biomolecules, especially the connections between folding and function.

Improvement of structure-based potentials for protein folding by native and nonnative hydrogen bonds

Pure Gō models (where every native interaction equally stabilizes the folded state) have widely proved their convenience in the computational investigation of protein folding. However, a chemistry-based description of the real interactions also provides a desirable tune in the analysis of the folding process, and thus some hybrid Gō potentials that combine both aspects have been proposed. Among all the noncovalent interactions that contribute to protein folding, hydrogen bonds are the only ones with a partial covalent character. This feature makes them directional and, thus, more difficult to model as part of the coarse-grained descriptions that are typically employed in Gō models. Thanks to a simplified but rigorous representation of backbone hydrogen bonds that we have recently proposed, we present in this article a combined potential (Gō + backbone hydrogen bond) to study the thermodynamics of protein folding in the frame of very simple simulation models. We show that the explicit inclusion of hydrogen bonds leads to a systematic improvement in the description of protein folding. We discuss a representative set of examples (from two-state folders to downhill proteins, with different types of native structures) that reveal a relevant agreement with experimental data.

Roles of non-native hydrogen-bonding interaction in helix-coil transition of a single polypeptide as revealed by comparison between Gō-like and non-Gō models

To explore the role of non-native interactions in the helix-coil transition, a detailed comparison between a Gō-like model and a non-Gō model has been performed via lattice Monte Carlo simulations. Only native hydrogen bonding interactions occur in the Gō-like model, and the non-native ones with sequence interval more than 4 is also included into the non-Gō model. Some significant differences between the results from those two models have been found. The non-native hydrogen bonds were found most populated at temperature around the helix-coil transition. The rearrangement of non-native hydrogen bonds into native ones in the formation of alpha-helix leads to the increase of susceptibility of chain conformation, and even two peaks of susceptibility of radius of gyration versus temperature exist in the case of non-Gō model for a non-short peptide, while just a single peak exists in the case of Gō model for a single polypeptide chain with various chain lengths. The non-native hydrogen bonds have complicated the temperature-dependence of Zimm-Bragg nucleation constant. The increase of relative probability of non-native hydrogen bonding for long polypeptide chains leads to non-monotonous chain length effect on the transition temperature.

The nucleation mechanism of protein folding: a survey of computer simulation studies

The nucleation mechanism of protein folding, originally proposed by Baldwin in the early 1970s, was firstly observed by Shakhnovich and co-workers two decades later in the context of Monte Carlo simulations of a simple lattice model. At about the same time the extensive use of φ-value analysis provided the first experimental evidence that the folding of Chymotrypsin-inhibitor 2, a small single-domain protein, which folds with two-state kinetics, is also driven by a nucleation mechanism. Since then, the nucleation mechanism is generally considered the most common form of folding mechanism amongst two-state proteins. However, recent experimental data has put forward the idea that this may not necessarily be so, since the accuracy of the experimentally determined φ values, which are used to identify the critical (i.e. nucleating) residues, is typically poor. Here, we provide a survey of in silico results on the nucleation mechanism, ranging from simple lattice Monte Carlo to more sophisticated off-lattice molecular dynamics simulations, and discuss them in light of experimental data.

Arc-repressor dimerization on DNA: folding rate enhancement by colocalization

Multimeric proteins are ubiquitous in many cellular processes that require high levels of regulation. Eukaryotic gene expression is often regulated by a mechanism of combinatorial control that involves the binding of dimeric transcription factors to DNA together with the coordinated activity of additional proteins. In this study, we investigated the dimerization of the Arc-repressor on DNA with the aim of achieving microscopic insight into the possible advantages of interacting with DNA as a complex rather than as a monomeric single-domain protein. We used a computational coarse-grained model in which the protein dynamics was governed by native interactions and protein-DNA interactions were dictated by electrostatic forces. Inspired by previous experimental work that showed an enhanced refolding rate for the Arc-repressor in the presence of DNA and other polyanions, we focused on the mechanism and kinetics of the assembly of Arc monomers in the presence of single- (ssDNA) and double-stranded DNA (dsDNA) molecules in a low-salt concentration environment. The electrostatic interactions that attract the protein to the dsDNA were shown to be fundamental in colocalizing the unfolded Arc chains and in accelerating refolding. Arc monomers bind the dsDNA efficiently and nonspecifically, and search for each other via one-dimensional diffusion. The fastest folding of Arc is observed for DNA of 30 bp. Longer DNA is significantly less efficient in accelerating the Arc refolding rate, since the two subunits search distinct regions of the one-dimensional DNA and are therefore much less colocalized. The probability that the two unfolded chains will meet on 200 bp DNA is similar to that in the bulk. The colocalization of Arc subunits on ssDNA results in much faster folding compared to that obtained on dsDNA of the same length. Differences in the rate of Arc refolding, cooperativity, and the structure of its transition state ensemble introduced by ssDNA and dsDNA molecules demonstrate the important role of colocalization in biological self-assembly processes.

Insights from coarse-grained Gō models for protein folding and dynamics

Exploring the landscape of large scale conformational changes such as protein folding at atomistic detail poses a considerable computational challenge. Coarse-grained representations of the peptide chain have therefore been developed and over the last decade have proved extremely valuable. These include topology-based Gō models, which constitute a smooth and funnel-like approximation to the folding landscape. We review the many variations of the Gō model that have been employed to yield insight into folding mechanisms. Their success has been interpreted as a consequence of the dominant role of the native topology in folding. The role of local contact density in determining protein dynamics is also discussed and is used to explain the ability of Gō-like models to capture sequence effects in folding and elucidate conformational transitions.

Folding pathway of the b1 domain of protein G explored by multiscale modeling

The understanding of the folding mechanisms of single-domain proteins is an essential step in the understanding of protein folding in general. Recently, we developed a mesoscopic CA-CB side-chain protein model, which was successfully applied in protein structure prediction, studies of protein thermodynamics, and modeling of protein complexes. In this research, this model is employed in a detailed characterization of the folding process of a simple globular protein, the B1 domain of IgG-binding protein G (GB1). There is a vast body of experimental facts and theoretical findings for this protein. Performing unbiased, ab initio simulations, we demonstrated that the GB1 folding proceeds via the formation of an extended folding nucleus, followed by slow structure fine-tuning. Remarkably, a subset of native interactions drives the folding from the very beginning. The emerging comprehensive picture of GB1 folding perfectly matches and extends the previous experimental and theoretical studies.

Influence of the chain stiffness on the thermodynamics of a Gō-type model for protein folding

The relative importance of local and long range interactions in the characteristics of the protein folding process has long been a matter of controversy. Computer simulations based on Gō-type models have been widely used to study this topic, but without much agreement on which type of interactions is more relevant for the foldability of a protein. In this work, the authors also employ a topology-based potential and simulation model to analyze the influence of local and long range interactions on the thermodynamics of the folding transition. The former are mainly used to control the degree of flexibility (or stiffness) of the chain, mostly appreciable in the unfolded (noncompact) state. Our results show the different effects that local and nonlocal interactions have on the entropy and the energy of the system. This implies that a balance between both types of interactions is required, so that a free energy barrier exists between the native and the denatured states. The variations in the contribution of both types of interactions have also a direct effect on the stability of the chain conformations, including the possible appearance of thermodynamic folding intermediates.

Downhill versus two-state protein folding in a statistical mechanical model

The authors address the problem of downhill protein folding in the framework of a simple statistical mechanical model, which allows an exact solution for the equilibrium and a semianalytical treatment of the kinetics. Focusing on protein 1BBL, a candidate for downhill folding behavior, and comparing it to the WW domain of protein PIN1, a two-state folder of comparable size, the authors show that there are qualitative differences in both the equilibrium and kinetic properties of the two molecules. However, the barrierless scenario which would be expected if 1BBL were a true downhill folder is observed only at low enough temperature.