Identification of slow molecular order parameters for Markov model construction

“MKClconcentration(matchingtheexperimentalionicstrength). ThesystemwasthenparametrizedwiththeAMBERff99SB-ILDNforcefield [49];waterandionsweremodeledrespectivelywiththeTIP3PandJoung-Cheathamparametersets [38,37]. Thesystemthuspreparedwasfirstequilibratedfor24nsintheconstant-pressureensemble, during which it stabilized at a…”

“thetrajectory isexploredandnrepresentativepointsofthecoordinatespaceareselectedas cluster centers witha 8 clustering method. Various algorithms have been proposed in the literature, including k-means [50], k-centers[20],k-medoids[42],regularspatialclustering[83],regulartemporalclustering[83]andWard clustering[6]. The…”

“We assume that the molecular dynamics implementation is Markovian in (i.e., the time step to xt+τ is computed based on the current value of xt only), and gives rise to a unique stationary density μ(x), usually the Boltzmann density: μ(x) = Z−1e−βH (x), where H is the Hamiltonian, Z is the partition…”

“The system was then parametrized with the AMBER ff99SB-ILDN forcefield [49]; water and ions were modeled respectively with the TIP3P and Joung-Cheatham parameter sets [38, 37].”

“KID peptide analyzed below). Principal component analysis (PCA) is a frequently-used method to reducethedimensionofanorderparameterspacebyprojectingitonitslinearsubspaceofthelargestamplitude motions [4]. PCA has also been used successfully in Markov model construction [67, 65], however is suffers from the similar limit…”

“A metric often used in (2) is the normalized Euclidean metric after rigid-body translation and rotation has been removed, in short the minimal RMSD (or least RMSD) metric [39].”

“Rare conformational transitions can be explicitly traced by either single-molecule experiments [70, 88, 17, 44] or by high-throughput molecular dynamics simulations, either realized with few long trajectories [84, 48] or with many shorter trajectories [92, 11, 67, 13, 77, 12].”

“…models of molecular kinetics: Generation and validation The Journal of Chemical Physics 134, 174105 (2011); 10.1063/1.3565032 Slow dynamics in protein fluctuations revealed by time-structure based independent component analysis: The case of domain motions The Journal of Chemical Physics 134,…”

“The resulting models are often called transition networks [74, 62, 32], Diffusion maps [18, 76], Master equation models [87, 14], Markov models [64] or Markov state models [86, 16] (MSM), where “Markovianity” means that the kinetics are modeled by a memoryless jump process between states.”

“The entry represent the folded configuration of the pKID-CBP bound structure, determined through NMR [89].”

“Identification of slow molecular order parameters for Markov model construction Guillermo Pérez-Hernández, Fabian Paul, Toni Giorgino, Gianni De Fabritiis, and Frank Noé Citation: The Journal of Chemical Physics 139, 015102 (2013); View online: https://doi.org/10.1063/1.4811489 View Table of Contents: http://aip.scita…”

“…much interest (e.g. for the druggability of protein-protein interactions) to investigate whether the presence of pre-formed elements causes folded conformations to be selected from the ensemble (conformational selection) [54], or whether the binding rather occurs through induced-fit mechanics [85].”

“We have performed 7706 all-atom explicit-solvent simulations of 24 ns each using the ACEMD software [29] on the GPUGRID distributed computing platform [12], yielding a total 168 μs simulation data.”

“Like PCA, TICA [55] uses a linear transform to map the original order parameters r ( t ) to a new set of order parameters z ( t ) — the independent components (ICs).”

“All of the simulations were performed with a time step of 4 fs, enabled by the hydrogen mass repartitioning scheme [25]; long-range electrostatic forces were computed with the particle-mesh Ewald summation method.”

“The Ritz method [75] provides the optimal set of coefficients for an orthonormal basis set.”

“This is consistent with NMR experiments that have shown the N-terminal region to be approximately 50% helical in the apo-form [73].”

“Rare conformational transitions can be explicitly traced by either single-molecule experiments [70, 88, 17, 44] or by high-throughput molecular dynamics simulations, either realized with few long trajectories [84, 48] or with many shorter trajectories [92, 11, 67, 13, 77, 12].”

“The resulting models are often called transition networks [74, 62, 32], Diffusion maps [18, 76], Master equation models [87, 14], Markov models [64] or Markov state models [86, 16] (MSM), where “Markovianity” means that the kinetics are modeled by a memoryless jump process between states.”

“…and validation The Journal of Chemical Physics 134, 174105 (2011); 10.1063/1.3565032 Slow dynamics in protein fluctuations revealed by time-structure based independent component analysis: The case of domain motions The Journal of Chemical Physics 134, 065101 (2011);…”

“• The metastable states (often referred to as “free energy basins”—although we will avoid this term as it would imply the projection onto some pre-defined coordinate set), can be computed from the sign structure of the leading eigenfunctions [81, 22].”

“From this propagator, many properties can be calculated, such as transition pathways between two sets of configurations [51, 7, 67].”

“…The Journal of Chemical Physics 126, 155101 (2007); 10.1063/1.2714538 Identification of slow molecular order parameters for Markov model construction Guillermo Pérez-Hernández,1 Fabian Paul,2,a),b) Toni Giorgino,3,a) Gianni De Fabritiis,4,c) and Frank Noé1,c) 1Department of Mathematics and…”

“The TICA problem can easily be solved by subsequently solving two simple eigenvalue problems.”

“Equivalent expressions are obtained by expressing the eigenfunctions in different weighted spaced, leading to the transfer operator formulation [81], or the symmetrized propagator formulation [14].”

“Rare conformational transitions can be explicitly traced by either single-molecule experiments [70, 88, 17, 44] or by high-throughput molecular dynamics simulations, either realized with few long trajectories [84, 48] or with many shorter trajectories [92, 11, 67, 13, 77, 12].”

“Other frequently used coordinate sets are intramolecular coordinates such as dihedral angles [62] or inter-atomic distances.”

“Rare conformational transitions can be explicitly traced by either single-molecule experiments [70, 88, 17, 44] or by high-throughput molecular dynamics simulations, either realized with few long trajectories [84, 48] or with many shorter trajectories [92, 11, 67, 13, 77, 12].”

“For clustering and Markov model construction of molecular dynamics data, the packages EMMA [83], MSMbuilder [5], Wordom [82] and METAGUI [8] are currently available.”

“Conformational transitions between long-lived, or “metastable” states are essential to the function of biomolecules [27, 58, 52, 24, 78, 28, 95].”

“gle reference structure, the radius of gyration, or pre-selected distances or angles are often not good indicators of the slow eigenfunctions, and thus bear the danger of disguising the slow kinetics [45, 61, 56]. In order to avoid this, Markov model construction has focused in the last years focused on the other extr…”

“Kinetic models usually first partition the conformation space into discrete states [93, 61, 40, 33, 94, 14, 74, 56, 21, 80, 69].”

“However, there is a simple and efficient alternative to this: Problem (14) can also be solved by solving two simple eigenvalue problems using the AMUSE algorithm [47].”

“…Markovian in (i.e., the time step to xt+τ is computed based on the current value of xt only), and gives rise to a unique stationary density μ(x), usually the Boltzmann density: μ(x) = Z−1e−βH (x), where H is the Hamiltonian, Z is the partition function, and β = (kBT)−1 is the inverse temperature.”

“The resulting models are often called transition networks [74, 62, 32], Diffusion maps [18, 76], Master equation models [87, 14], Markov models [64] or Markov state models [86, 16] (MSM), where “Markovianity” means that the kinetics are modeled by a memoryless jump process between states.”

“Various algorithms have been proposed in the literature, including k-means [50], k-centers [20], k-medoids [42], regular spatial clustering[83], regular temporal clustering[83] and Ward clustering[6].”

“The TICA problem can easily be solved by subsequently solving two simple eigenvalue problems.”

“For clustering and Markov model construction of molecular dynamics data, the packages EMMA [83], MSMbuilder [5], Wordom [82] and METAGUI [8] are currently available.”

“Conformational transitions between long-lived, or “metastable” states are essential to the function of biomolecules [27, 58, 52, 24, 78, 28, 95].”

“of only ˝ = 2 ns [67]. Stock and co-workers have exploredthepossibilityofusingdihedralanglesasaninputtoPCA(dPCA).Asangularcoordinates, theycannotbeaveragedinthesamewaylikenonperiodiccoordinates. Ref. [3]thussuggeststouse the sine and cosine of backbone dihedral angles as input coordinates. It is found for hepta-alanin…”

“These rare transitions are ubiquitously found in biomolecular processes, including folding [34, 44], complex conformational rearrangements between native protein substates [26, 63], and ligand binding [68].”

“Kinetic models usually first partition the conformation space into discrete states [93, 61, 40, 33, 94, 14, 74, 56, 21, 80, 69].”

“pf.edu *frank.noe@fu-berlin.de 1 arXiv:1302.6614v1 [physics.chem-ph] 26 Feb 2013 1 Introduction Conformationaltransitionsbetweenlong-lived,or“metastable” statesareessentialtothefunctionof biomolecules[27,58,52,24,78,28,95]. Theseraretransitionsareubiquitouslyfoundinbiomolecularprocesses,includingfolding[34,44], comple…”

“Conformational transitions between long-lived, or “metastable” states are essential to the function of biomolecules [27, 58, 52, 24, 78, 28, 95].”

“Rare conformational transitions can be explicitly traced by either single-molecule experiments [70, 88, 17, 44] or by high-throughput molecular dynamics simulations, either realized with few long trajectories [84, 48] or with many shorter trajectories [92, 11, 67, 13, 77, 12].”

“…Published by the American Institute of Physics Articles you may be interested in Markov models of molecular kinetics: Generation and validation The Journal of Chemical Physics 134, 174105 (2011); 10.1063/1.3565032 Slow dynamics in protein fluctuations revealed by…”

“These rare transitions are ubiquitously found in biomolecular processes, including folding [34, 44], complex conformational rearrangements between native protein substates [26, 63], and ligand binding [68].”

“The resulting models are often called transition networks [74, 62, 32], Diffusion maps [18, 76], Master equation models [87, 14], Markov models [64] or Markov state models [86, 16] (MSM), where “Markovianity” means that the kinetics are modeled by a memoryless jump process between states.”

“From this propagator, many properties can be calculated, such as transition pathways between two sets of configurations [51, 7, 67].”

“Identification of slow molecular order parameters for Markov model construction Guillermo Pérez-Hernández, Fabian Paul, Toni Giorgino, Gianni De Fabritiis, and Frank Noé Citation: The Journal of Chemical Physics 139, 015102 (2013); View online: https://doi.org/10.1063/1.4811489 View Table of Contents: http://aip.scita…”

“n many examples, especially protein folding (see [9, 72] and references therein). Recent applications include folding of MR121-GSGS-W peptide[72], folding of FiP35 WW domain, GTT, NTL9, and protein G [6] and discovery of cryptic allosteric sites in -lactamase, interleukin-2, and RNase H [10]. However, minimal RMSD ten…”

“inatespaceareselectedas cluster centers witha 8 clustering method. Various algorithms have been proposed in the literature, including k-means [50], k-centers[20],k-medoids[42],regularspatialclustering[83],regulartemporalclustering[83]andWard clustering[6]. The k-meansalgorithmrequiresthecoordinatespacetobeavectorspace…”

“The resulting models are often called transition networks [74, 62, 32], Diffusion maps [18, 76], Master equation models [87, 14], Markov models [64] or Markov state models [86, 16] (MSM), where “Markovianity” means that the kinetics are modeled by a memoryless jump process between states.”

“The resulting models are often called transition networks [74, 62, 32], Diffusion maps [18, 76], Master equation models [87, 14], Markov models [64] or Markov state models [86, 16] (MSM), where “Markovianity” means that the kinetics are modeled by a memoryless jump process between states.”

“• From the largest m Eigenvalues and their associated Eigenfunctions, a rank-m propagator can be assembled that can describe the dynamics slower than timescale t m [46].”

“Conformational transitions between long-lived, or “metastable” states are essential to the function of biomolecules [27, 58, 52, 24, 78, 28, 95].”

“…Chemical Physics 126, 155101 (2007); 10.1063/1.2714538 Identification of slow molecular order parameters for Markov model construction Guillermo Pérez-Hernández,1 Fabian Paul,2,a),b) Toni Giorgino,3,a) Gianni De Fabritiis,4,c) and Frank Noé1,c) 1Department of Mathematics and Computer Science,…”

“MR121-GSGSW is a well-studied fluorescent peptide that has been extensively characterized by experiments [59], simulations [19] and also Markov models [60, 65].”

“PCA has also been used successfully in Markov model construction [67, 65], however is suffers from the similar limitations like minimal RMSD, as there is no general guarantee that large-amplitude motions are associated with slow transitions.”

“The resulting models are often called transition networks [74, 62, 32], Diffusion maps [18, 76], Master equation models [87, 14], Markov models [64] or Markov state models [86, 16] (MSM), where “Markovianity” means that the kinetics are modeled by a memoryless jump process between states.”

“Rare conformational transitions can be explicitly traced by either single-molecule experiments [70, 88, 17, 44] or by high-throughput molecular dynamics simulations, either realized with few long trajectories [84, 48] or with many shorter trajectories [92, 11, 67, 13, 77, 12].”

“These rare transitions are ubiquitously found in biomolecular processes, including folding [34, 44], complex conformational rearrangements between native protein substates [26, 63], and ligand binding [68].”

“For clustering and Markov model construction of molecular dynamics data, the packages EMMA [83], MSMbuilder [5], Wordom [82] and METAGUI [8] are currently available.”

“Identification of slow molecular order parameters for Markov model construction Guillermo Pérez-Hernández, Fabian Paul, Toni Giorgino, Gianni De Fabritiis, and Frank Noé Citation: The Journal of Chemical Physics 139, 015102 (2013); View online: https://doi.org/10.1063/1.4811489 View Table of Contents: http://aip.scita…”

“Kinetic models usually first partition the conformation space into discrete states [93, 61, 40, 33, 94, 14, 74, 56, 21, 80, 69].”

“Rare conformational transitions can be explicitly traced by either single-molecule experiments [70, 88, 17, 44] or by high-throughput molecular dynamics simulations, either realized with few long trajectories [84, 48] or with many shorter trajectories [92, 11, 67, 13, 77, 12].”

“[30, 13], the binding of a small ligand was analyzed by using the three-dimensional positions of the ligand with respect to the protein.”

“Conformational transitions between long-lived, or “metastable” states are essential to the function of biomolecules [27, 58, 52, 24, 78, 28, 95].”

“The approximation error of all of the above quantities can be cast in terms of the approximation error of the eigenvalues and eigenfunctions [79, 23, 72, 71].”

“…correlation functions (e.g. fluorescence correlation, intermediate scattering function in dynamic neutron or X-ray scattering) can be computed as a sum of single-exponential relaxations with timescales computed from λ i and amplitudes from the φ i and the experimental observable [65, 43, 15].”

“The TICA problem can easily be solved by subsequently solving two simple eigenvalue problems.”

“MR121-GSGSW is a well-studied fluorescent peptide that has been extensively characterized by experiments [59], simulations [19] and also Markov models [60, 65].”

“Mitsutake et al [53] used relaxation mode analysis, a related technique, to analyze the dynamics of Met-enkephalin.”

“Kinetic models usually first partition the conformation space into discrete states [93, 61, 40, 33, 94, 14, 74, 56, 21, 80, 69].”

“In subsequent work [35], the approach was extended to the “dihedral PCA by parts” method.”

“MR121-GSGSW is a well-studied fluorescent peptide that has been extensively characterized by experiments [59], simulations [19] and also Markov models [60, 65].”

“To promote simultaneous sampling of high-and low-probability configurations, we included an RC loss using two slow collective coordinates that have been used earlier in the analysis of this BPTI simulation (supplementary materials) ( 29 , 30 ).”

“[21 ].”