A weak characterization of slow variables in stochastic dynamical systems

We present a novel characterization of slow variables for continuous Markov processes that provably preserve the slow timescales. These slow variables are known as reaction coordinates in molecular dynamical applications, where they play a key role in system analysis and coarse graining. The defining characteristics of these slow variables is that they parametrize a so-called transition manifold, a low-dimensional manifold in a certain density function space that emerges with progressive equilibration of the system's fast variables. The existence of said manifold was previously predicted for certain classes of metastable and slow-fast systems. However, in the original work, the existence of the manifold hinges on the pointwise convergence of the system's transition density functions towards it. We show in this work that a convergence in average with respect to the system's stationary measure is sufficient to yield reaction coordinates with the same key qualities. This allows one to accurately predict the timescale preservation in systems where the old theory is not applicable or would give overly pessimistic results. Moreover, the new characterization is still constructive, in that it allows for the algorithmic identification of a good slow variable. The improved characterization, the error prediction and the variable construction are demonstrated by a small metastable system

Mesoscopic model for DNA G-quadruplex unfolding

[EN] Genomes contain rare guanine-rich sequences capable of assembling into four-stranded helical structures, termed G-quadruplexes, with potential roles in gene regulation and chromosome stability. Their mechanical unfolding has only been reported to date by all-atom simulations, which cannot dissect the major physical interactions responsible for their cohesion. Here, we propose a mesoscopic model to describe both the mechanical and thermal stability of DNA G-quadruplexes, where each nucleotide of the structure, as well as each central cation located at the inner channel, is mapped onto a single bead. In this framework we are able to simulate loading rates similar to the experimental ones, which are not reachable in simulations with atomistic resolution. In this regard, we present single-molecule force-induced unfolding experiments by a high-resolution optical tweezers on a DNA telomeric sequence capable of adopting a G-quadruplex conformation. Fitting the parameters of the model to the experiments we find a correct prediction of the rupture-force kinetics and a good agreement with previous near equilibrium measurements. Since G-quadruplex unfolding dynamics is halfway in complexity between secondary nucleic acids and tertiary protein structures, our model entails a nanoscale paradigm for non-equilibrium processes in the cell.Work supported by the Spanish Ministry of Economy and Competitiveness (MINECO), grant No. FIS2014-55867, co-financed by FEDER funds. We also thank the support of the Aragon Government and Fondo Social Europeo to FENOL group. Work in J.R.A.-G. laboratory was supported by a grant from MINECO, No. MAT2015-71806-R).Bergues-Pupo, A.; Gutiérrez, I.; Arias-Gonzalez, JR.; Falo, F.; Fiasconaro, A. (2017). Mesoscopic model for DNA G-quadruplex unfolding. Scientific Reports. 7:1-13. https://doi.org/10.1038/s41598-017-10849-2S1137Arias-Gonzalez, J. R. Single-molecule portrait of DNA and RNA double helices. Integr. Biol. 6, 904 (2014).Burge, S., Parkinson, G. N., Hazel, P., Todd, A. K. & Neidle, S. Quadruplex DNA: sequence, topology and structure. Nucleic Acids Res. 34, 5402 (2006).Lam, E. Y., Beraldi, D., Tannahill, D. & Balasubramanian, S. G-quadruplex structures are stable and detectable in human genomic DNA. Nat. Commun. 4, 1796 (2013).Siddiqui-Jain, A., Grand, C. L., Bearss, D. J. & Hurley, L. H. Direct evidence for a G-quadruplex in a promoter region and its targeting with a small molecule to repress c-MYC transcription. Proc. Natl. Acad. Sci. USA 99, 11593 (2002).Endoh, T. & Sugimoto, N. Mechanical insights into ribosomal progression overcoming RNA G-quadruplex from periodical translation suppression in cells. Sci. Rep. 6, 1 (2016).Hänsel-Hertsch, R., Di Antonio, M. & Balasubramanian, S. DNA G-quadruplexes in the human genome: detection, functions and therapeutic potential. Nat. Rev. Mol. Cell Biol. 18, 279 (2017).de Messieres, M., Chang, J. C., Brawn-Cinani, B. & La Porta, A. Single-molecule study of G-quadruplex disruption using dynamic force spectroscopy. Phys. Rev. Lett. 109, 058101 (2012).Koirala, D. et al. A single-molecule platform for investigation of interactions between G-quadruplexes and small-molecule ligands. Nat. Chem. 3, 782 (2011).Long, X. et al. Mechanical unfolding of human telomere G-quadruplex DNA probed by integrated fluorescence and magnetic tweezers spectroscopy. Nucleic Acids Res. 41, 2746 (2013).Ghimire, C. et al. Direct Quantification of Loop Interaction and pi-pi Stacking for G-Quadruplex Stability at the Submolecular Level. J. Am. Chem. Soc. 136, 15544 (2014).Garavís, M. et al. Mechanical Unfolding of Long Human Telomeric RNA (TERRA). Chem. Commun. 49, 6397 (2013).Fonseca Guerra, C., Zijlstra, H., Paragi, G. & Bickelhaupt, F. M. Telomere structure and stability: covalency in hydrogen bonds, not resonance assistance, causes cooperativity in guanine quartets. Chemistry-A European Journal 17, 12612 (2011).Yurenko, Y. P., Novotn, J., Sklen, V. & Marek, R. Exploring non-covalent interactions in guanine-and xanthine-based model DNA quadruplex structures: a comprehensive quantum chemical approach. Phys. Chem. Chem. Phys. 16, 2072 (2014).Poudel, L. et al. Implication of the solvent effect, metal ions and topology in the electronic structure and hydrogen bonding of human telomeric G-quadruplex DNA. Phys. Chem. Chem. Phys. 18, 21573 (2016).Li, M. H., Luo, Q., Xue, X. G. & Li, Z. S. Toward a full structural characterization of G-quadruplex DNA in aqueous solution: Molecular dynamics simulations of four G-quadruplex molecules. J. Mol. Struct-Theochem. 952, 96 (2010).Islam, B. et al. Conformational dynamics of the human propeller telomeric DNA quadruplex on a microsecond time scale. Nucleic Acids Res. 41, 2723 (2013).Stadlbauer, P., Krepl, M., Cheatham, T. E., Koca, J. & Sponer, J. Structural dynamics of possible late-stage intermediates in folding of quadruplex DNA studied by molecular simulations. Nucleic Acids Res. 41, 7128 (2013).Li, H., Cao, E. & Gisler, T. Force-induced unfolding of human telomeric G-quadruplex: a steered molecular dynamics simulation study. Biochem. Bioph. Res. Co. 379, 70 (2009).Yang, C., Jang, S. & Pak, Y. Multiple stepwise pattern for potential of mean force in unfolding the thrombin binding aptamer in complex with Sr2+. J. Chem. Phys. 135, 225104 (2011).Bergues-Pupo, A. E., Arias-Gonzalez, J. R., Morón, M. C., Fiasconaro, A. & Falo, F. Role of the central cations in the mechanical unfolding of DNA and RNA G-quadruplexes. Nucleic Acids Res. 43, 7638 (2015).Linak, M. C., Tourdot, R. & Dorfman, K. D. Moving beyond Watson-Crick models of coarse grained DNA dynamics. J. Chem Phys. 135, 205102 (2011).Rebi, M., Mocci, F., Laaksonen, A. & Ulin, J. Multiscale simulations of human telomeric G-quadruplex DNA. J. Phys. Chem. B 119, 105 (2014).Stadlbauer, P. et al. Coarse-Grained Simulations Complemented by Atomistic Molecular Dynamics Provide New Insights into Folding and Unfolding of Human Telomeric G-Quadruplexes. J. Chem. Theory Comput. 12, 6077 (2016).Parkinson, G. N., Lee, M. P. & Neidle, S. Crystal structure of parallel quadruplexes from human telomeric DNA. Nature 417, 876 (2002).Bhattacharya, D., Arachchilageand, G. M. & Basu, S. Metal Cations in G-Quadruplex Folding and Stability. Frontiers in Chemistry 4, 38 (2016).de Lorenzo, S., Ribezzi-Crivellari, M., Arias-Gonzalez, J. R., Smith, S. B. & Ritort, F. A Temperature-Jump Optical Trap for Single-Molecule Manipulation. Biophys. J. 108, 2854 (2015).Smith, S. B., Cui, Y. & Bustamante, C. Optical-trap force transducer that operates by direct measurement of light momentum. Methods Enzymol. 361, 134 (2003).Mergny, J. L., Phan, A. T. & Lacroix, L. Following G-quartet formation by UV-spectroscopy. FEBS letters 435, 74 (1998).Torrie, G. M. & Valleau, J. P. Nonphysical sampling distributions in Monte Carlo free-energy estimation: umbrella sampling. J. Comput. Phys. 23, 187 (1977).Kumar, S., Bouzida, D., Swendsen, R. H., Kollman, P. A. & Rosenberg, J. M. The weighted histogram analysis method for free-energy calculations on biomolecules I. The method. J. Comput. Chem. 13, 1011 (1992).Evans, E. & Ritchie, K. Dynamic strength of molecular adhesion bonds. Biophys. J. 72, 1541 (1997).Dudko, O. K., Hummer, G. & Szabo, A. Intrinsic rates and activation free energies from single-molecule pulling experiments. Phys. Rev. Lett. 96, 108101 (2006).Friddle, R. W., Noy, A. & De Yoreo, J. J. Interpreting the widespread nonlinear force spectra of intermolecular bonds. Proc. Natl. Acad. Sci. 109, 13573 (2012)

In Silico Elucidation of the Recognition Dynamics of Ubiquitin

Elucidation of the mechanism of biomacromolecular recognition events has been a topic of intense interest over the past century. The inherent dynamic nature of both protein and ligand molecules along with the continuous reshaping of the energy landscape during the binding process renders it difficult to characterize this process at atomic detail. Here, we investigate the recognition dynamics of ubiquitin via microsecond all-atom molecular dynamics simulation providing both thermodynamic and kinetic information. The high-level of consistency found with respect to experimental NMR data lends support to the accuracy of the in silico representation of the conformational substates and their interconversions of free ubiquitin. Using an energy-based reweighting approach, the statistical distribution of conformational states of ubiquitin is monitored as a function of the distance between ubiquitin and its binding partner Hrs-UIM. It is found that extensive and dense sampling of conformational space afforded by the µs MD trajectory is essential for the elucidation of the binding mechanism as is Boltzmann sampling, overcoming inherent limitations of sparsely sampled empirical ensembles. The results reveal a population redistribution mechanism that takes effect when the ligand is at intermediate range of 1–2 nm from ubiquitin. This mechanism, which may be depicted as a superposition of the conformational selection and induced fit mechanisms, also applies to other binding partners of ubiquitin, such as the GGA3 GAT domain

On Conduction in a Bacterial Sodium Channel

Voltage-gated Na+-channels are transmembrane proteins that are responsible for the fast depolarizing phase of the action potential in nerve and muscular cells. Selective permeability of Na+ over Ca2+ or K+ ions is essential for the biological function of Na+-channels. After the emergence of the first high-resolution structure of a Na+-channel, an anionic coordination site was proposed to confer Na+ selectivity through partial dehydration of Na+ via its direct interaction with conserved glutamate side chains. By combining molecular dynamics simulations and free-energy calculations, a low-energy permeation pathway for Na+ ion translocation through the selectivity filter of the recently determined crystal structure of a prokaryotic sodium channel from Arcobacter butzleri is characterised. The picture that emerges is that of a pore preferentially occupied by two ions, which can switch between different configurations by crossing low free-energy barriers. In contrast to K+-channels, the movements of the ions appear to be weakly coupled in Na+-channels. When the free-energy maps for Na+ and K+ ions are compared, a selective site is characterised in the narrowest region of the filter, where a hydrated Na+ ion, and not a hydrated K+ ion, is energetically stable

QM/MM MD and Free Energy Simulations of G9a-Like Protein (GLP) and Its Mutants: Understanding the Factors that Determine the Product Specificity

Certain lysine residues on histone tails could be methylated by protein lysine methyltransferases (PKMTs) using S-adenosyl-L-methionine (AdoMet) as the methyl donor. Since the methylation states of the target lysines play a fundamental role in the regulation of chromatin structure and gene expression, it is important to study the property of PKMTs that allows a specific number of methyl groups (one, two or three) to be added (termed as product specificity). It has been shown that the product specificity of PKMTs may be controlled in part by the existence of specific residues at the active site. One of the best examples is a Phe/Tyr switch found in many PKMTs. Here quantum mechanical/molecular mechanical (QM/MM) molecular dynamics (MD) and free energy simulations are performed on wild type G9a-like protein (GLP) and its F1209Y and Y1124F mutants for understanding the energetic origin of the product specificity and the reasons for the change of product specificity as a result of single-residue mutations at the Phe/Tyr switch as well as other positions. The free energy barriers of the methyl transfer processes calculated from our simulations are consistent with experimental data, supporting the suggestion that the relative free energy barriers may determine, at least in part, the product specificity of PKMTs. The changes of the free energy barriers as a result of the mutations are also discussed based on the structural information obtained from the simulations. The results suggest that the space and active-site interactions around the ε-amino group of the target lysine available for methyl addition appear to among the key structural factors in controlling the product specificity and activity of PKMTs

A practical guide to the simultaneous determination of protein structure and dynamics using metainference

Accurate protein structural ensembles can be determined with metainference, a Bayesian inference method that integrates experimental information with prior knowledge of the system and deals with all sources of uncertainty and errors as well as with system heterogeneity. Furthermore, metainference can be implemented using the metadynamics approach, which enables the computational study of complex biological systems requiring extensive conformational sampling. In this chapter, we provide a step-by-step guide to perform and analyse metadynamic metainference simulations using the ISDB module of the open-source PLUMED library, as well as a series of practical tips to avoid common mistakes. Specifically, we will guide the reader in the process of learning how to model the structural ensemble of a small disordered peptide by combining state-of-the-art molecular mechanics force fields with nuclear magnetic resonance data, including chemical shifts, scalar couplings and residual dipolar couplings.Comment: 49 pages, 9 figure

Characterizing Structural Transitions Using Localized Free Energy Landscape Analysis

Structural changes in molecules are frequently observed during biological processes like replication, transcription and translation. These structural changes can usually be traced to specific distortions in the backbones of the macromolecules involved. Quantitative energetic characterization of such distortions can greatly advance the atomic-level understanding of the dynamic character of these biological processes.Molecular dynamics simulations combined with a variation of the Weighted Histogram Analysis Method for potential of mean force determination are applied to characterize localized structural changes for the test case of cytosine (underlined) base flipping in a GTCAGCGCATGG DNA duplex. Free energy landscapes for backbone torsion and sugar pucker degrees of freedom in the DNA are used to understand their behavior in response to the base flipping perturbation. By simplifying the base flipping structural change into a two-state model, a free energy difference of upto 14 kcal/mol can be attributed to the flipped state relative to the stacked Watson-Crick base paired state. This two-state classification allows precise evaluation of the effect of base flipping on local backbone degrees of freedom.The calculated free energy landscapes of individual backbone and sugar degrees of freedom expectedly show the greatest change in the vicinity of the flipping base itself, but specific delocalized effects can be discerned upto four nucleotide positions away in both 5' and 3' directions. Free energy landscape analysis thus provides a quantitative method to pinpoint the determinants of structural change on the atomic scale and also delineate the extent of propagation of the perturbation along the molecule. In addition to nucleic acids, this methodology is anticipated to be useful for studying conformational changes in all macromolecules, including carbohydrates, lipids, and proteins

Analyzing and Biasing Simulations with PLUMED

This chapter discusses how the PLUMED plugin for molecular dynamics can be used to analyze and bias molecular dynamics trajectories. The chapter begins by introducing the notion of a collective variable and by then explaining how the free energy can be computed as a function of one or more collective variables. A number of practical issues mostly around periodic boundary conditions that arise when these types of calculations are performed using PLUMED are then discussed. Later parts of the chapter discuss how PLUMED can be used to perform enhanced sampling simulations that introduce simulation biases or multiple replicas of the system and Monte Carlo exchanges between these replicas. This section is then followed by a discussion on how free-energy surfaces and associated error bars can be extracted from such simulations by using weighted histogram and block averaging techniques

Direct observation of topoisomerase IA gate dynamics

Type IA topoisomerases cleave single-stranded DNA and relieve negative supercoils in discrete steps corresponding to the passage of the intact DNA strand through the cleaved strand. Although type IA topoisomerases are assumed to accomplish this strand passage via a protein-mediated DNA gate, opening of this gate has never been observed. We developed a single-molecule assay to directly measure gate opening of the Escherichia coli type IA topoisomerases I and III. We found that after cleavage of single-stranded DNA, the protein gate opens by as much as 6.6 nm and can close against forces in excess of 16 pN. Key differences in the cleavage, ligation, and gate dynamics of these two enzymes provide insights into their different cellular functions. The single-molecule results are broadly consistent with conformational changes obtained from molecular dynamics simulations. These results allowed us to develop a mechanistic model of interactions between type IA topoisomerases and single-stranded DNA

Allosteric effects in cyclophilin mutants may be explained by changes in nano-microsecond time scale motions

The relationship between molecular motion and catalysis in enzymes is debated. Here, simulations of cyclophilin A and three catalytically-impaired mutants reveal a nanosecond-scale interconversion between active and inactive conformations, orders of magnitude faster than previously suggested