Topology-based Model Research Articles

Development of a nonlinear geometric transformer model based on an incremental circuit of its magnetic core

In this paper, the investigation is focused on the topology-based geometric models of power transformers. These models take into account any type of core design, the geometrical characteristics of the magnetic core, and the nonlinear characteristics of the core material. These models differ from each other in the way in which the magnetic part is described. The methods, which have been proposed for the description of the magnetic part of a power transformer, are based on the conventional magnetic circuit of the core and present disadvantages. In this paper, a new method for the derivation of the differential equations of the magnetic core of the transformer is proposed by introducing a new auxiliary incremental circuit, more easily handled mathematically in antithesis to the conventional magnetic circuit that is used in the literature until now. Specifically, using the proposed circuit one can directly write the differential equations of the magnetic part of the transformer, the coupling between the windings of the transformer is easily calculated, the equations of the electric and the magnetic part are decoupled as well as the state equations of the electrical part can be written in standard form. The proposed method can be combined with any methodology presented in the literature on the core losses modeling without any restriction. The simulation results show forth that the proposed method of transformer modeling is suitable for transients and power quality studies. Moreover, the proposed method can be a useful tool on sensitivity analysis during the design stage of small and large power transformers as well.

Residual Structures, Conformational Fluctuations, and Electrostatic Interactions in the Synergistic Folding of Two Intrinsically Disordered Proteins

To understand the interplay of residual structures and conformational fluctuations in the interaction of intrinsically disordered proteins (IDPs), we first combined implicit solvent and replica exchange sampling to calculate atomistic disordered ensembles of the nuclear co-activator binding domain (NCBD) of transcription coactivator CBP and the activation domain of the p160 steroid receptor coactivator ACTR. The calculated ensembles are in quantitative agreement with NMR-derived residue helicity and recapitulate the experimental observation that, while free ACTR largely lacks residual secondary structures, free NCBD is a molten globule with a helical content similar to that in the folded complex. Detailed conformational analysis reveals that free NCBD has an inherent ability to substantially sample all the helix configurations that have been previously observed either unbound or in complexes. Intriguingly, further high-temperature unbinding and unfolding simulations in implicit and explicit solvents emphasize the importance of conformational fluctuations in synergistic folding of NCBD with ACTR. A balance between preformed elements and conformational fluctuations appears necessary to allow NCBD to interact with different targets and fold into alternative conformations. Together with previous topology-based modeling and existing experimental data, the current simulations strongly support an “extended conformational selection” synergistic folding mechanism that involves a key intermediate state stabilized by interaction between the C-terminal helices of NCBD and ACTR. In addition, the atomistic simulations reveal the role of long-range as well as short-range electrostatic interactions in cooperating with readily fluctuating residual structures, which might enhance the encounter rate and promote efficient folding upon encounter for facile binding and folding interactions of IDPs. Thus, the current study not only provides a consistent mechanistic understanding of the NCBD/ACTR interaction, but also helps establish a multi-scale molecular modeling framework for understanding the structure, interaction, and regulation of IDPs in general.

Synergistic folding of two intrinsically disordered proteins: searching for conformational selection

Intrinsically disordered proteins (IDPs) lack stable structures under physiological conditions but often fold into stable structures upon specific binding. These coupled binding and folding processes underlie the organization of cellular regulatory networks, and a mechanistic understanding is thus of fundamental importance. Here, we investigated the synergistic folding of two IDPs, namely, the NCBD domain of transcription coactivator CBP and the p160 steroid receptor coactivator ACTR, using a topology-based model that was carefully calibrated to balance intrinsic folding propensities and intermolecular interactions. As one of the most structured IDPs, NCBD is a plausible candidate that interacts through conformational selection-like mechanisms, where binding is mainly initiated by pre-existing folded-like conformations. Indeed, the simulations demonstrate that, even though binding and folding of both NCBD and ACTR is highly cooperative on the baseline level, the tertiary folding of NCBD is best described by the "extended conformational selection" model that involves multiple stages of selection and induced folding. The simulations further predict that the NCBD/ACTR recognition is mainly initiated by forming a mini folded core that includes the second and third helices of NCBD and ACTR. These predictions are fully consistent with independent physics-based atomistic simulations as well as a recent experimental mapping of the H/D exchange protection factors. The current work thus adds to the limited number of existing mechanistic studies of coupled binding and folding of IDPs, and provides a first direct demonstration of how conformational selection might contribute to efficient recognition of IDPs. Interestingly, even for highly structured IDPs like NCBD, the recognition is initiated by the more disordered C-terminal segment and with substantial contribution from induced folding. Together with existing studies of IDP interaction mechanisms, this argues that induced folding is likely prevalent in IDP-protein interaction, and emphasizes the importance of understanding how IDPs manage to fold efficiently upon (nonspecific) binding. Success of the current study also further supports the notion that, with careful calibration, topology-based models can be effective tools for mechanistic study of IDP interaction and regulation, especially when combined with physics-based atomistic simulations and experiments.

Conformational Flexibility and the Mechanism of Allosteric Transitions in Calmodulin

Conformational dynamics and flexibility is often essential to protein's function. Calmodulin (CaM) is a well-characterized calcium binding protein that takes functional advantage of its considerable intrinsic conformational flexibility. We investigate the allosteric open/close transition of the domains of CaM through simulations of a multiple basin, topology-based model. Our primary focus is to clarify how main chain flexibility influences the mechanism for allosteric transitions of flexible proteins. In particular, we compare the simulated transition mechanisms of the domains of CaM, which are topologically similar, but differ significantly in their conformational flexibility and dynamics. The simulated transition mechanisms are described at the residue level in terms of local structural order parameters that can be compared with predictions from a coarse grained variational model of allostery (Tripathi and Portman, J. Chem. Phys. 135 075104 (2011)).

Force-induced unfolding simulations of the human Notch1 negative regulatory region: possible roles of the heterodimerization domain in mechanosensing.

Notch receptors are core components of the Notch signaling pathway and play a central role in cell fate decisions during development as well as tissue homeostasis. Upon ligand binding, Notch is sequentially cleaved at the S2 site by an ADAM protease and at the S3 site by the γ-secretase complex. Recent X-ray structures of the negative regulatory region (NRR) of the Notch receptor reveal an auto-inhibited fold where three protective Lin12/Notch repeats (LNR) of the NRR shield the S2 cleavage site housed in the heterodimerization (HD) domain. One of the models explaining how ligand binding drives the NRR conformation from a protease-resistant state to a protease-sensitive one invokes a mechanical force exerted on the NRR upon ligand endocytosis. Here, we combined physics-based atomistic simulations and topology-based coarse-grained modeling to investigate the intrinsic and force-induced folding and unfolding mechanisms of the human Notch1 NRR. The simulations support that external force applied to the termini of the NRR disengages the LNR modules from the heterodimerization (HD) domain in a well-defined, largely sequential manner. Importantly, the mechanical force can further drive local unfolding of the HD domain in a functionally relevant fashion that would provide full proteolytic access to the S2 site prior to heterodimer disassociation. We further analyzed local structural features, intrinsic folding free energy surfaces, and correlated motions of the HD domain. The results are consistent with a model in which the HD domain possesses inherent mechanosensing characteristics that could be utilized during Notch activation. This potential role of the HD domain in ligand-dependent Notch activation may have implications for understanding normal and aberrant Notch signaling.

Topology‐based modeling of intrinsically disordered proteins: Balancing intrinsic folding and intermolecular interactions

Coupled binding and folding is frequently involved in specific recognition of so-called intrinsically disordered proteins (IDPs), a newly recognized class of proteins that rely on a lack of stable tertiary fold for function. Here, we exploit topology-based Gō-like modeling as an effective tool for the mechanism of IDP recognition within the theoretical framework of minimally frustrated energy landscape. Importantly, substantial differences exist between IDPs and globular proteins in both amino acid sequence and binding interface characteristics. We demonstrate that established Gō-like models designed for folded proteins tend to over-estimate the level of residual structures in unbound IDPs, whereas under-estimating the strength of intermolecular interactions. Such systematic biases have important consequences in the predicted mechanism of interaction. A strategy is proposed to recalibrate topology-derived models to balance intrinsic folding propensities and intermolecular interactions, based on experimental knowledge of the overall residual structure level and binding affinity. Applied to pKID/KIX, the calibrated Gō-like model predicts a dominant multistep sequential pathway for binding-induced folding of pKID that is initiated by KIX binding via the C-terminus in disordered conformations, followed by binding and folding of the rest of C-terminal helix and finally the N-terminal helix. This novel mechanism is consistent with key observations derived from a recent NMR titration and relaxation dispersion study and provides a molecular-level interpretation of kinetic rates derived from dispersion curve analysis. These case studies provide important insight into the applicability and potential pitfalls of topology-based modeling for studying IDP folding and interaction in general.

SAXS Ensemble Refinement of ESCRT-III CHMP3 Conformational Transitions

We developed and implemented an ensemble-refinement method to study dynamic biomolecular assemblies with intrinsically disordered segments. Data from small angle X-ray scattering (SAXS) experiments and from coarse-grained molecular simulations were combined by using a maximum-entropy approach. The method was applied to CHMP3 of ESCRT-III, a protein with multiple helical domains separated by flexible linkers. Based on recent SAXS data by Lata etal. (J. Mol. Biol. 378, 818, 2008), we constructed ensembles of CHMP3 at low-and high-salt concentration to characterize its closed autoinhibited state and open active state. At low salt, helix α(5) is bound to the tip of helices α(1) and α(2), in excellent agreement with a recent crystal structure. Helix α(6) remains free in solution and does not appear to be part of the autoinhibitory complex. The simulation-based ensemble refinement is general and effectively increases the resolution of SAXS beyond shape information to atomically detailed structures.

Quantitative criteria for native energetic heterogeneity influences in the prediction of protein folding kinetics

Energy landscape theory requires that the protein-folding mechanism is generally globally directed or funneled toward the native state. The collective nature of transition state ensembles further suggests that sufficient averaging of the native interactions can occur so that the knowledge of the native topology may suffice for predicting the mechanism. Nevertheless, while simple homogeneously weighted native topology-based models predict the folding mechanisms for many proteins, for other proteins knowledge of the native topology, by itself, seems not to suffice in determining the folding mechanism. Simulations of proteins with differing topologies reveal that the failure of homogeneously weighted topology-based models can, however, be completely understood within the framework of a funneled energy landscape and can be quantified by comparing the fluctuation of entropy cost for forming contacts to the expected fluctuations in contact energy. To be precise, we find the transition state ensembles of proteins with all-alpha topologies, which are more uniform in the specific entropy cost of contact formation, have transition state ensembles that are more readily perturbed by differences in energetic weights than are the transition state ensembles of proteins with significant amounts of beta-structure, where the specific entropy costs of contact formation are more widely distributed. This behavior is consistent with a random-field Ising model analogy that follows from the free energy functional approach to folding.

Characterization of Alkaline Transitions in Ferricytochrome c Using Carbon−Deuterium Infrared Probes

The alkaline-induced structural transitions of ferricytochrome c have been studied intensively as a model for how changes in metal ligation contribute to protein function and folding. Previous studies have demonstrated that multiple non-native species accumulate with increasing pH. Here, we used a combination of experiments and simulations to provide a high-resolution view of the changes associated with increasing alkaline conditions. Alkaline-induced transitions were characterized under equilibrium conditions by following changes in the IR absorptions of carbon-deuterium chromophores incorporated at Leu68, Lys72, Lys73, Lys79, and Met80. The data suggest that at least four intermediates are formed as the pH is increased prior to complete unfolding of the protein. The first alkaline transition observed appears to be driven by a single deprotonation and occurs with a midpoint of pH 8.8, but surprisingly, the intermediate formed does not appear to be one of the well-characterized lysine misligates. At higher pH, second and third deprotonations, with a combined apparent midpoint pH of 10.2, induce transitions to Lys73- or Lys79-misligated species. Interestingly, the lysine misligates appear to undergo iron reduction by the coordinated amine. A transition from the lysine misligates to another intermediate, likely a hydroxide-misligated species, is associated with a fourth deprotonation and a midpoint of pH 10.7. Finally, the protein loses tertiary structure with a fifth deprotonation that occurs with a midpoint of pH 12.7. Native topology-based models with enforced misligation are employed to help understand the structures of the observed intermediates.

Topology‐based models and NMR structures in protein folding simulations

Topology-based interaction potentials are simplified models that use the native contacts in the folded structure of a protein to define an energetically unfrustrated folding funnel. They have been widely used to analyze the folding transition and pathways of different proteins through computer simulations. Obviously, they need a reliable, experimentally determined folded structure to define the model interactions. In structures elucidated through NMR spectroscopy, a complex treatment of the raw experimental data usually provides a series of models, a set of different conformations compatible with the available experimental data. Here, we use an efficient coarse-grained simulation technique to independently consider the contact maps from every different NMR model in a protein whose structure has been resolved by the use of NMR spectroscopy. For lambda-Cro repressor, a homodimeric protein, we have analyzed its folding characteristics with a topology-based model. We have focused on the competition between the folding of the individual chains and their binding to form the final quaternary structure. From 20 different NMR models, we find a predominant three-state folding behavior, in agreement with experimental data on the folding pathway for this protein. Individual NMR models, however, show distinct characteristics, which are analyzed both at the level of the interplay between tertiary/quaternary structure formation and also regarding the thermal stability of the tertiary structure of every individual chain.

Folding of Elongated Proteins: Conventional or Anomalous?

The complexity of the mechanisms by which proteins fold has been shown by many studies to be governed by their native-state topologies. This was manifested in the ability of the native topology-based model to capture folding mechanisms and the success of folding rate predictions based on various topological measures, such as the contact order. However, while the finer details of topological complexity have been thoroughly examined and related to folding kinetics, simpler characteristics of the protein, such as its overall shape, have been largely disregarded. In this study, we investigated the folding of proteins with an unusual elongated geometry that differs substantially from the common globular structure. To study the effect of the elongation degree on the folding kinetics, we used repeat proteins, which become more elongated as they include more repeating units. Some of these have apparently anomalous experimental folding kinetics, with rates that are often less than expected on the basis of rates for globular proteins possessing similar topological complexity. Using experimental folding rates and a larger set of rates obtained from simulations, we have shown that as the protein becomes increasingly elongated, its folding kinetics becomes slower and deviates more from the rate expected on the basis of topology measures fitted for globular proteins. The observed slow kinetics is a result of a more complex pathway in which stable intermediates composed of several consecutive repeats can appear. We thus propose a novel measure, an elongation-sensitive contact order, that takes into account both the extent of elongation and the topological complexity of the protein. This new measure resolves the apparent discrimination between the folding of globular and elongated repeat proteins. Our study extends the current capabilities of folding-rate predictions by unifying the kinetics of repeat and globular proteins.

Simulations of the protein folding process using topology-based models depend on the experimental structure

Topology-based potentials (also known as Go-type models) have been widely used in the study of the protein folding problem. When a topology-based potential is applied, the structure of the native state of the protein considered has to be known in advance. This fact gives to these models a semiempirical character, and therefore the quality of the simulation results obtained for the folding transition relies, among other factors, on the accuracy of the experimental structural data employed. In this work, we use a topology-based potential to carry out folding simulations of a protein whose structure has been determined both with NMR spectroscopy and x-ray crystallography. This way, we have been able to establish to which extent the differences in the topologies of the two experimental structures, easily ignored in a standard structural analysis for this protein, affect the thermodynamic characteristics of the folding transition defined in the simulations.

A Nonoriented, Nonmanifold Boundary Representation for Geometric Applications

Topology-based geometric models are important in many fields, including CAD (computer-aided design) and computer graphics. This paper introduces a general, concise, and responsive topology-based geometric data structure, called NNB-rep (nonoriented, nonmanifold boundary representation). Its generality comes from its ability to store manifold and nonmanifold geometric models, no matter whether they are degenerate or not, as needed in the first stages of the design and modeling of engineering artifacts, in particular those with cracks and degeneracies. Unlike most boundary models, the NNB-rep does not accommodate any topologically oriented cells, hence its compactness. Furthermore, despite the classical trade-off between memory and time performance, the NNB-rep is quite responsive because it hosts an optimal incidence scheme C 9 4, which makes it adequate for a number of interactive geometric applications.

Discrete Breathers in Nonlinear Network Models of Proteins

We introduce a topology-based nonlinear network model of protein dynamics with the aim of investigating the interplay of spatial disorder and nonlinearity. We show that spontaneous localization of energy occurs generically and is a site-dependent process. Localized modes of nonlinear origin form spontaneously in the stiffest parts of the structure and display site-dependent activation energies. Our results provide a straightforward way for understanding the recently discovered link between protein local stiffness and enzymatic activity. They strongly suggest that nonlinear phenomena may play an important role in enzyme function, allowing for energy storage during the catalytic process.

Stretching of proteins in a uniform flow

Stretching of a protein by a fluid flow is compared to that in a force-clamp apparatus. The comparison is made within a simple topology-based dynamical model of a protein in which the effects of the flow are implemented using Langevin dynamics. We demonstrate that unfolding induced by a uniform flow shows a richer behavior than that in the force clamp. The dynamics of unfolding is found to depend strongly on the selection of the amino acid, usually one of the termini, which is anchored. These features offer potentially wider diagnostic tools to investigate structure of proteins compared to experiments based on the atomic force microscopy.

The Folding Energy Landscape of the Dimerization Domain of Escherichia coli Trp Repressor: A Joint Experimental and Theoretical Investigation

Enhanced structural insights into the folding energy landscape of the N-terminal dimerization domain of Escherichia coli tryptophan repressor, [2-66] 2 TR, were obtained from a combined experimental and theoretical analysis of its equilibrium folding reaction. Previous studies have shown that the three intertwined helices in [2-66] 2 TR are sufficient to drive the formation of a stable dimer for the full-length protein, [2-107] 2 TR. The monomeric and dimeric folding intermediates that appear during the folding reactions of [2-66] 2 TR have counterparts in the folding mechanism of the full-length protein. The equilibrium unfolding energy surface on which the folding and dimerization reactions occur for [2-66] 2 TR was examined with a combination of native-state hydrogen exchange analysis, pepsin digestion and matrix-assisted laser/desorption mass spectrometry performed at several concentrations of protein and denaturant. Peptides corresponding to all three helices in [2-66] 2 TR show multi-layered protection patterns consistent with the relative stabilities of the dimeric and monomeric folding intermediates. The observation of protection exceeding that offered by the dimeric intermediate in segments from all three helices implies that a segment-swapping mechanism may be operative in the monomeric intermediate. Protection greater than that expected from the global stability for a single amide hydrogen in a peptide from the C-helix possibly and another from the A-helix may reflect non-random structure, possibly a precursor for segment swapping, in the urea-denatured state. Native topology-based model simulations that correspond to a funnel energy landscape capture both the monomeric and dimeric intermediates suggested by the HX MS data and provide a rationale for the progressive acquisition of secondary structure in their conformational ensembles.

Protein folding in a force clamp

Kinetics of folding of a protein held in a force clamp are compared to an unconstrained folding. The comparison is made within a simple topology-based dynamical model of ubiquitin. We demonstrate that the experimentally observed variations in the end-to-end distance reflect microscopic events during folding. However, the folding scenarios in and out of the force clamp are distinct.

Thermal unfolding of proteins

Thermal unfolding of proteins is compared to folding and mechanical stretching in a simple topology-based dynamical model. We define the unfolding time and demonstrate its low-temperature divergence. Below a characteristic temperature, contacts break at separate time scales and unfolding proceeds approximately in a way reverse to folding. Features in these scenarios agree with experiments and atomic simulations on titin.

Mechanical unfolding of ubiquitin molecules

Mechanical stretching of ubiquitin and of its several repeats are studied through molecular-dynamics simulations. A Go-type model [H. Abe and N. Go, Biopolymers 20, 1013 (1981)] with a realistic contact map and with Lennard-Jones contact interactions is used. The model qualitatively reproduces the experimentally observed differences between force-extension patterns obtained on polyubiquitins stretched by various linkages. The terminal-to-terminal stretching of polyubiquitin results in peak forces similar to those measured for titin-based polyproteins and of a magnitude that matches measurements. Consistent with the experimental measurements, the simulated peak forces depend on the pulling speed logarithmically when thermal fluctuations are explicitly introduced. These results validate the application of topology-based models in the study of the mechanical stretching of proteins.

Mechanisms of Protein Assembly: Lessons from Minimalist Models

Many cellular functions rely on interactions among proteins and between proteins and nucleic acids. The limited success of binding predictions may suggest that the physical and chemical principles of protein binding have to be revisited to correctly capture the essence of protein recognition. In this Account, we discuss the power of reduced models to study the physics of protein assembly. Since energetic frustration is sufficiently small, native topology-based models, which correspond to perfectly unfrustrated energy landscapes, have shown that binding mechanisms are robust and governed primarily by the protein's native topology. These models impressively capture many of the binding characteristics found in experiments and highlight the fundamental role of flexibility in binding. The essential role of solvent molecules and electrostatic interactions in binding is also discussed. Despite the success of the minimally frustrated models to describe the dynamics and mechanisms of binding, the actual degree of frustration has to be explored to quantify the capacity of a protein to bind specifically to other proteins. We have found that introducing mutations can significantly reduce specificity by introducing an additional binding mode. Deciphering and quantifying the key ingredients for biological self-assembly is invaluable to reading out genomic sequences and understanding cellular interaction networks.