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].”

“Time-lagged independent component analysis (tICA) 20,21 is another common approach, which captures kinetically slow motions in a system by identifying linear combinations of features that decorrelate slowly.”

“Describing the trajectories using time-structure based independent component analysis (tICA) allows us to analyze the MD dataset in terms of its slow dynamical processes (28, 29).”

“Each protein–model ensemble comprised 32 simulations initialized from conformations selected uniformly across the two-dimensional all-atom reference TICA free-energy surface (FES). α 3D, a 73-residue three-helix bundle, showed the largest deviation from the all-atom conformational landscape among the four fast-folding…”

“The landscapes are constructed using time-lagged independent component analysis (TICA) (P´erez-Hern´andez et al. (2013)), which gives the axis of slowest dynamical motion along the trajectory.”

“VAMP and its linear cousin TICA [247, 248] incorrectly identify these outliers as important time-correlated motions, requiring intervention via careful data curation.”

“Time-lagged independent component analysis (TICA) and kernel transfer-operator methods estimate ψ j from transitions (P´erez-Hern´andez et al., 2013; Klus et al., 2020).”

“Frames are featurised generically, as sin and cos of the backbone torsions, and the slow subspace is found by time-lagged independent component analysis (TICA) [35, 36] at a lag of 2 ps.”

“Following Koopman-based approaches that identify metastable sets (Mardt et al., 2018; Pérez-Hernández et al., 2013), we construct these sets without collective variables by using the leading eigenfunctions of the Koopman generator, which capture the slow inter-basin modes.”

“These are themselves costly, but time-lagged independent component analysis (TICA) [35] provides a data-driven linear approximation from long unbiased trajectories — creating the obvious tension that TICA coordinates require ground-truth MD on the very systems for which GenMD surrogates are least…”

“PCCA+ was then used to partition the microstates into four macrostates for transition-path analysis [13–15].”

“…Kuramoto–Sivashinsky equation from t = 0 to t = 200 using Shenfun [53, 54], △ DMD or, equivalently, time-lagged independent component analysis [55, 56] are often used as a preprocessing step in order to project high-dimensional data onto low-dimensional subspaces in such a way that the slowest…”

“Such spectral kinetic scores are commonly used in MSM model selection to identify representations and discretizations that preserve the dominant slow dynamical processes and thereby improve the quality of the finite-state Markov approximation at a chosen lag time[13, 15, 16, 17].”

“Temperature and pressure in Anton simulations were controlled using the Antithetic thermostat and semi-isotropic Monte Carlo barostat within the Multigrator integration framework 37 .”

“20,21 This is in line with previous studies, which have shown that TICs often coarsely capture the slow dynamics of molecular systems, but timescale estimates and PCCA de-compositions can be improved significantly by an additional VAC step.”

“A lag time of 0.1 ns was used for dimensionality reduction [50], and trajectory data were projected onto the first two independent components (IC1 and IC2) for downstream clustering and visualization.”

“…numerous linear and non-linear techniques, such as principal component analysis (PCA), 37–40 time-lagged independent component analysis (TICA), 41,42 uniform manifold approximation and projection (UMAP), 43 and t-distributed stochastic neighbor embedding (t-SNE), 44 which project the…”

“Slow CVs are identified using time-lagged independent component analysis (tICA) 46,47 , a dimensionality reduction 5/20 method that extracts the slowest dynamical modes from trajectory data.”

“34 The  rst two TICA components were retained to represent the dominant slow conformational coordinates and were subsequently discretized by k-means clustering.”

“2(a,d)] in the leading TICA subspace [70, 71] built from sine-and cosine-encoded torsions, the directions followed a Fisher LDA sampler, and we took D = I in this feature space.”

“Specifically, We apply time-lagged independent component analysis [41] to project the trajectories into 2D space with pairwise C α coordinates.”

“Lagged operators provide another view of temporal evolution [Schmid, 2010, Williams et al., 2015, Pérez-Hernández et al., 2013, Wu and Noé, 2020, Mardt et al., 2018].”

“For each sample, 100 µ g of protein were separated in a single SDS– PAGE band, stained with Sypro Ruby, excised, and subjected to in-gel tryptic digestion as previously described (Shevchenko et al 1996, Bonzon-kulichenko et al 2011, Pérez-Hernández et al 2013).”

“RNNs constrained by pairwise TE computed for multiple time lags could reveal coordinated dynamics across multiple timescales, similar to methods like time-lagged independent component analysis (tICA) used to identify slow conformational dynamics in proteins 56 .”

“Molecular kinetics is addressed more directly by state-specific quantities such as the committor function 13–17 or by spectral decomposition of dynamical operators 18,19 .”

“To construct an MSM, conformations are usually described by physical observables, or projected onto the directions of slowest decorrelation via TICA (Molgedey & Schuster, 1994; Schwantes & Pande, 2013; Perez-Hernandez et al., 2013).”

“Conformations were discretised into microstates by k-means clustering in the TICA space, such that conformations within the same microstate were geometrically similar and rapidly interconverting.”

“We compare PPR (seeded at the 4 state coordinates, since predicting the angular state is the primary objec-tive), standard PR, time-lagged independent component analysis (TICA) [51], PCCA+ [44], and random ordering.”

“…as k -means [20], spectral clustering [22], and density-based methods such as DBSCAN [11] and HDBSCAN [6] are often applied either directly to trajectory snapshots or to low-dimensional embeddings obtained from methods such as TICA [27], diffusion maps [7], or variational Koopman embeddings [38].”

“The resulting high-dimensional feature space was projected onto a low-dimensional kinetic subspace using time-lagged independent component analysis (TICA) [ 49 ], which isolates collective motions dominating slow conformational dynamics.”

“Time-lagged independent component analysis (TICA) [32] identifies and ranks the directions of slowest collective motion in molecular trajectories.”

“First, coarse-grained trajectories were backmapped to the atomistic level and then projected onto backbone torsional descriptors and analyzed by time-lagged independent component analysis (tICA) to identify the dominant slow conformational modes of the coarse-grained trajectories (47).”

“Other methods will generally either identify esoteric order parameters through manual selection or data driven identification (e.g., principal component analysis, PCA) that are often uninterpretable and system-specific 31 .”

“…fast-folding protein trajectories Lindorff-Larsen et al. (2011) using the Algorithm for Multiple Unknown Signals Extraction (AMUSE), which combines Principal Component Analysis (PCA) and Time-lagged Independent Component Analysis (TICA) Hyv¨arinen et al. (2001); P´erez-Hern´andez et al. (2013).”

“The resulting configurations generated by our method were then projected onto a two-dimensional space defined by time-lagged independent component analysis (TICA) for visualization [63, 64].”

“Representative examples include time-lagged independent component analysis (TICA) [6], variational approaches for Markov processes (VAMP) and their neural implementations (VAMPnets) [7], reaction-coordinate flow ideas [8], as well as Koopman/dominant-subspace learning approaches such as ISOKANN [9].”

“Dimensionality reduction was performed using a 2-step protocol: initial denoising by principal component analysis, followed by time-lagged independent component analysis (tICA) to identify the slowest collective motions [ 34 ].”

“The typical physical example involves fast oscillations inside a slowly moving envelope, where the cumulative effect of fast oscillations drives changes in the envelope [1, 2].”

“The high dimensionality of the sampled data was reduced using time-lagged Independent Component Analysis (tICA), 31 which projected the dataset onto dimensions that represented the slowest process being modeled using simulations.”

“S9A, B) were extracted from the entire set of trajectories and submitted to time-independent component analysis (tICA) 31 and subsequent MiniBatch K-643 means clustering into 500 clusters.”

“” (tICA).”

“Analysis performed on the projection to ϕ and ψ Ramachandran angles.”

“To directly compare the conformational landscapes of WT, Y220C mutant, and G245S mutant fl-p53 tetramer systems with the WT p53 DBD monomer system, the conformations explored by each system, represented by the 24 features, were used as input for time-lagged Independent Component Analysis (tICA) 30 .”

“TICA ( T ime-lagged I ndependent C omponent A nalysis) was first suggested by Molgedey and Schuster (1994) and introduced into the field of molecular dynamics and computational chemistry by Pérez-Hernández et al. (2013) and Schwantes and Pande (2013).”