#### DMCA

## RESEARCH ARTICLE Path Similarity Analysis: A Method for Quantifying Macromolecular Pathways

### Citations

255 | Replica-exchange molecular dynamics method for protein folding - Sugita, Okamoto - 1999 |

200 |
SciPy: Open source scientific tools for Python,
- Jones, Oliphant, et al.
- 2001
(Show Context)
Citation Context ...er. Using the Hausdorff and Fréchet metrics as similarity measures, we employed Ward’s method [94] in conjunction with agglomerative hierarchical clustering as implemented in the SciPy Python package =-=[95]-=-. The Ward linkage criterion specifies a minimum variance criterion that minimizes the total intra-cluster variance. In light of the focus of this paper, we restrict our study to hierarchical clusteri... |

190 |
Computing the Fréchet distance between two polygonal curves
- Alt, Godau
- 1995
(Show Context)
Citation Context ...veral distance functions that violate the triangle inequality in S1 Text. In the main part of this study, we consider two candidates for δ—the Hausdorff metric [43–45] and the discrete Fréchet metric =-=[46, 47]-=-—and illuminate situations where one might be selected in favor of the other. Given two paths as input, both metrics locate two points, one per path, corresponding to some notion of a maximal deviatio... |

117 |
Hierarchical grouping to optimize an objective function,”
- Jr
- 1963
(Show Context)
Citation Context ...s inter-cluster similarity as a function of the pairwise similarities of the objects comprising each cluster. Using the Hausdorff and Fréchet metrics as similarity measures, we employed Ward’s method =-=[94]-=- in conjunction with agglomerative hierarchical clustering as implemented in the SciPy Python package [95]. The Ward linkage criterion specifies a minimum variance criterion that minimizes the total i... |

115 |
Violin Plots: A Box PlotDensity Trace Synergism”,
- Hintze, Nelson
- 1998
(Show Context)
Citation Context ...olecular images were created with VMD [87] and the Bendix plugin [88]. Graphs were plotted with the Python libraries matplotlib [89] and seaborn [90], in particular its implementation of violin plots =-=[91]-=-. Characterizing transition paths Path similarity analysis (PSA). The Hausdorff metric, δH, and the discrete Fréchet metric, δdF, defined in Eq 2 and Eq 7, respectively, were computed as described in ... |

105 | Escaping free-energy minima - Laio, Parrinello - 2002 |

89 | A Climbing Image Nudged Elastic Band Method for Finding Saddle Points and Minimum Energy Paths. - Henkelman, Uberuaga, et al. - 2000 |

83 |
Sur quelques points du calcul fonctionnel
- Frechet
- 1906
(Show Context)
Citation Context ...veral distance functions that violate the triangle inequality in S1 Text. In the main part of this study, we consider two candidates for δ—the Hausdorff metric [43–45] and the discrete Fréchet metric =-=[46, 47]-=-—and illuminate situations where one might be selected in favor of the other. Given two paths as input, both metrics locate two points, one per path, corresponding to some notion of a maximal deviatio... |

79 | Approximate matching of polygonal shapes. - Alt, Behrends, et al. - 1995 |

75 |
Dynamic personalities of proteins
- Henzler-Wildman, Kern
(Show Context)
Citation Context ...ition and viability double-barrel z: 0 ! 4 nm Brownian + ramp 4×(2 ICs) PSA (δF), δF -δH distr/corr* (2) Methods comparison AdK closed! open various methods 3×(11 methods) PSA (δF/δH *), NCA, ζ-ρ, AA =-=(3)-=- Transition ensembles AdK closed! open DIMS, FRODA 200×(2 methods) PSA (δF *), δF-δH distr/corr* DT closed! open DIMS, FRODA 200×(2 methods) PSA (δF), δF-δH distr/corr* (4) Atomic detail from PSA AdK ... |

64 |
Investigating protein dynamics in collective coordinate space. Curr Opin Struct Biol
- Kitao, Go
- 1999
(Show Context)
Citation Context ...other common approach is principal component analysis (PCA), a tool that can be used to visualize conformational dynamics in a lower-dimensional subspace spanned by several principle components (PCs) =-=[41, 42]-=-. An important aspect of PCA is that motion along PCs can be viewed in real space, helping make complicated dynamical motions visually tractable. Using NCA, PCA or other CV approaches cannot, however,... |

63 |
Domain swapping: entangling alliances between proteins.
- MJ, Choe, et al.
- 1994
(Show Context)
Citation Context ...mation (chain A of 1AKE [83] with ligand removed). DT is believed to undergo a transition from an inactive closed conformation to an active open one, which includes a 180° rotation of a mobile domain =-=[84]-=- (Fig 2B). An open conformation was captured in a domain-swapped dimeric structure [85] and compared to the closed monomeric structure [86]. DT is divided into three domains, with the translocation (T... |

62 | Improved tangent estimate in the nudged elastic band method for finding minimum energy paths and saddle points. - Henkelman, Jonsson - 2000 |

61 |
A hierarchy of timescales in protein dynamics is linked to enzyme catalysis
- Henzler-Wildman, Lei, et al.
(Show Context)
Citation Context ...larity Analysis PLOS Computational Biology | DOI:10.1371/journal.pcbi.1004568 October 21, 2015 7 / 37 “AMPbd”) domain, residues 30–59, move relative to the CORE domain [73–77] around conserved hinges =-=[78]-=- (Fig 2A). The conformational change can occur in the ligand-free (apo) state as demonstrated in multiple experimental studies [78–81] and corroborated by computational analyses (reviewed by Seyler an... |

58 |
Schulten K. VMD: visual molecular dynamics
- HumphreyW
- 1996
(Show Context)
Citation Context ...yed alongside PSA for comparison.We describe how a range of conformational transition paths were generated to supply a variety of contexts in which to test PSA. Molecular images were created with VMD =-=[87]-=- and the Bendix plugin [88]. Graphs were plotted with the Python libraries matplotlib [89] and seaborn [90], in particular its implementation of violin plots [91]. Characterizing transition paths Path... |

51 | Computing discrete Fréchet distance
- Eiter, Mannila
- 1994
(Show Context)
Citation Context ...n each curve) [47] and various faster approximate solutions have been suggested [48, 49]. In this paper, however, we exclusively use the discrete Fréchet distance, δdF, with the algorithm outlined by =-=[50]-=- as it is simpler and faster to compute (in O(nm) time) than its continuous counterpart, δF. The formal definition of δdF considers two polygonal curves P and Q that are defined respectively by n andm... |

48 | Atilgan AR, Erman B. Direct evaluation of thermal fluctuations in proteins using a single-parameter harmonic potential. Folding Design - Bahar - 1997 |

42 | Accelerated molecular dynamics: a promising and efficient simulation method for biomolecules. - Hamelberg, Mongan, et al. - 2004 |

39 | RL, Demirel MC, Keskin O, Bahar I. Anisotropy of fluctuation dynamics of proteins with an elastic network model - AR, SR, et al. |

38 |
Matplotlib: A 2d graphics environment. Comput Sci Eng 9: 90–95
- JD
- 2007
(Show Context)
Citation Context ...s were generated to supply a variety of contexts in which to test PSA. Molecular images were created with VMD [87] and the Bendix plugin [88]. Graphs were plotted with the Python libraries matplotlib =-=[89]-=- and seaborn [90], in particular its implementation of violin plots [91]. Characterizing transition paths Path similarity analysis (PSA). The Hausdorff metric, δH, and the discrete Fréchet metric, δdF... |

37 | Conjugate peak refinement: an algorithm for finding reaction paths and accurate transition states in systems with many degrees of freedom. - Fischer, Karplus - 1992 |

36 | Large amplitude conformational change in proteins explored with a plastic network model: adenylate kinase - Maragakis, Karplus - 2005 |

36 | How fast-folding proteins fold.
- Lindorff-Larsen, Piana, et al.
- 2011
(Show Context)
Citation Context ...but not with Hausdorff or Fréchet metrics. Several such studies utilized native contacts-based path (dis)similarity measures [62–65]. In particular, both Graham et al. [64] and Lindorff-Larsen et al. =-=[65]-=- used dissimilarity scores to assign individual paths to folding pathways using clustering. Different methods to sample conformational transitions were compared by Huang et al. [66], who contrasted th... |

34 |
Targeted Molecular-Dynamics - a New Approach for Searching Pathways of Conformational Transitions
- Schlitter, Engels, et al.
- 1994
(Show Context)
Citation Context ...dividual paths to folding pathways using clustering. Different methods to sample conformational transitions were compared by Huang et al. [66], who contrasted the original targeted MD (TMD) algorithm =-=[13]-=- with a harmonic restraint variation of TMD (also known as “steered MD” (SMD) or “restrained TMD” (rTMD) [67])—and biased MD (BMD) approaches, and Ovchinnikov and Karplus [68], who analyzed the free e... |

29 | Movie of the structural changes during a catalytic cycle of nucleoside monophosphate kinases. - Vonrhein, Schlauderer, et al. - 1995 |

28 | Computing the Fréchet distance between simple polygons in polynomial time
- BUCHIN, BUCHIN, et al.
- 2006
(Show Context)
Citation Context ...F δH) because for convex polygonal Path Similarity Analysis PLOS Computational Biology | DOI:10.1371/journal.pcbi.1004568 October 21, 2015 5 / 37 curves the Fréchet and Hausdorff distances are equal =-=[52]-=- while for other path geometries the Fréchet distance can become arbitrarily larger than the Hausdorff distance [48]. In the case of macromolecular trajectories, the case of backtracking appears parti... |

22 |
Re®ned structure of dimeric diphtheria toxin at 2.0 AÊ resolution
- MJ, Choe, et al.
- 1994
(Show Context)
Citation Context ...on from an inactive closed conformation to an active open one, which includes a 180° rotation of a mobile domain [84] (Fig 2B). An open conformation was captured in a domain-swapped dimeric structure =-=[85]-=- and compared to the closed monomeric structure [86]. DT is divided into three domains, with the translocation (T) domain, residues 179–379, being responsible for the majority of the opening and unrol... |

21 | Approximating the Frechet distance for realistic curves in near linear time.
- Driemel, Har-Peled, et al.
- 2010
(Show Context)
Citation Context ...ist to solve this difficult problem in O(nm log nm) time for polygonal curves (where n andm are the number of vertices in each curve) [47] and various faster approximate solutions have been suggested =-=[48, 49]-=-. In this paper, however, we exclusively use the discrete Fréchet distance, δdF, with the algorithm outlined by [50] as it is simpler and faster to compute (in O(nm) time) than its continuous counterp... |

18 | Domain closure in adenylate kinase joints on either side of two helices close like neighboring fingers - Gerstein, Schulz, et al. - 1993 |

18 |
Adenylate kinase motions during catalysis: an energetic counterweight balancing substrate binding. Structure
- CW, GJ, et al.
- 1996
(Show Context)
Citation Context ...e is a particularly suitable model system for studying general conformational transitions [12]. We produced transition paths between an open conformation of AdK [represented by chain A of PDB id 4AKE =-=[77]-=- from the Protein Data Bank [82] (PDB)], and a closed conformation (chain A of 1AKE [83] with ligand removed). DT is believed to undergo a transition from an inactive closed conformation to an active ... |

18 | Re®ned structure of monomeric diphtheria toxin at 2.3 AÊ resolution
- MJ, Eisenberg
- 1994
(Show Context)
Citation Context ... open one, which includes a 180° rotation of a mobile domain [84] (Fig 2B). An open conformation was captured in a domain-swapped dimeric structure [85] and compared to the closed monomeric structure =-=[86]-=-. DT is divided into three domains, with the translocation (T) domain, residues 179–379, being responsible for the majority of the opening and unrolling conformational motion about the receptor-bindin... |

17 | The Frechet distance revisited and extended.
- Har-Peled, Raichel
- 2011
(Show Context)
Citation Context ...ist to solve this difficult problem in O(nm log nm) time for polygonal curves (where n andm are the number of vertices in each curve) [47] and various faster approximate solutions have been suggested =-=[48, 49]-=-. In this paper, however, we exclusively use the discrete Fréchet distance, δdF, with the algorithm outlined by [50] as it is simpler and faster to compute (in O(nm) time) than its continuous counterp... |

15 | Improved sampling methods for molecular simulation - Lei, Duan - 2007 |

15 | Finding transition pathways using the string method with swarms of trajectories - Pan, Sezer, et al. - 2008 |

13 | Illuminating the mechanistic roles of enzyme conformational dynamics. - Hanson - 2007 |

12 |
Biomolecular simulation: a computational microscope for molecular biology.
- RO, RM, et al.
- 2012
(Show Context)
Citation Context ...ale and beyond. Equilibrium molecular dynamics (MD) is arguably one of the most robust approaches to simulating macromolecular dynamics, in large part due to the availability of full atomistic detail =-=[4]-=-. It has has been the workhorse tool for studying the protein structure–function connection [5]. However, conformational transitions are rare events: crossing events in the transition region of phase ... |

12 | Native Contacts Determine Protein Folding Mechanisms in Atomistic Simulations - Best, Hummer, et al. - 2013 |

12 | Protein structure-structure alignment with discrete Fréchet distance
- Jiang, Xu, et al.
(Show Context)
Citation Context ...g handwritten documents [54], and comparing trajectories of moving objects in geographic information systems [55]. Both metrics have also found applications in biology for protein structure alignment =-=[56, 57]-=- and protein homology analysis [58, 59]. To our knowledge, the Hausdorff and Fréchet metrics have not been widely used as general tools to quantify macromolecular pathways. However, recently two studi... |

11 | Fréchet distance with speed limits
- Maheshwari, Sack, et al.
- 2011
(Show Context)
Citation Context ... and we stress that this study does not purport to exhaust all applications of PSA, nor represent an optimized application. Other path metrics—e.g., Fréchet with speed limits, direction-based Fréchet =-=[69]-=-, or Fréchet with shortcuts for the analysis of noisy data [70]—may offer advantages in carrying out various analyses. The Hausdorff distance can be generalized as well to measure, for instance, dista... |

11 | Jaywalking your dog: computing the Fréchet distance with shortcuts
- Driemel, Har-Peled
(Show Context)
Citation Context ...applications of PSA, nor represent an optimized application. Other path metrics—e.g., Fréchet with speed limits, direction-based Fréchet [69], or Fréchet with shortcuts for the analysis of noisy data =-=[70]-=-—may offer advantages in carrying out various analyses. The Hausdorff distance can be generalized as well to measure, for instance, distances between surfaces (instead of 1D curves) [71]. The multitud... |

11 |
Structure of the complex between adenylate kinase from Escherichia coli and the inhibitor Ap5A refined at 1.9 A resolution. A model for a catalytic transition state
- CW, GE
- 1992
(Show Context)
Citation Context ...s [12]. We produced transition paths between an open conformation of AdK [represented by chain A of PDB id 4AKE [77] from the Protein Data Bank [82] (PDB)], and a closed conformation (chain A of 1AKE =-=[83]-=- with ligand removed). DT is believed to undergo a transition from an inactive closed conformation to an active open one, which includes a 180° rotation of a mobile domain [84] (Fig 2B). An open confo... |

10 | Computer simulations of protein folding by targeted molecular dynamics. Proteins-Structure Function and Genetics
- Ferrara, Apostolakis, et al.
- 2000
(Show Context)
Citation Context ...ere compared by Huang et al. [66], who contrasted the original targeted MD (TMD) algorithm [13] with a harmonic restraint variation of TMD (also known as “steered MD” (SMD) or “restrained TMD” (rTMD) =-=[67]-=-)—and biased MD (BMD) approaches, and Ovchinnikov and Karplus [68], who analyzed the free energy profiles along the transition tubes surrounding the paths produced by several TMD variants. The use of ... |

8 | On searching in, sampling of, and dynamically moving through conformational space of biomolecular systems: A review - Christen, Gunsteren |

8 | de Groot BL: Geometry-based sampling of conformational transitions in proteins. Structure - Seeliger, Haas - 1993 |

6 | Hyperdynamics: Accelerated molecular dynamics of infrequent events - AF - 1997 |

6 |
Delarue M. MinActionPath: maximum likelihood trajectory for largescale structural transitions in a coarse-grained locally harmonic energy landscape
- Franklin, Koehl, et al.
(Show Context)
Citation Context ...t – – – Cα-only GOdMD[101] bonds: inf. sq-well – NBF: Go-like + ENM-MetaD ANMP[102] double-well ANM – Emix = min{Ui, Uf} iENM[103] double-well ANM – Emix = F(Ui, Uf) (arbitrary), collision penalty MAP=-=[31]-=- two ANMs, OM dynamics overdamped Langevin‡ minimum OM action! 2 ODEs+BCs! path MENM-SD/SP[104] double-well ANM – Emix = β −1ln{exp[−β(Ui + i)] + exp(−β(Uf + f)]} All MD-based methods use atomic res... |

6 | de Angulo R, Guieysse D, Remaud-Simeon M, Tran V: A path planning approach for computing large-amplitude motions of flexible molecules - Cortes, Simeon |

6 |
Oono Y, Schulten K. Principal component analysis and long time protein dynamics
- MA, Wriggers
(Show Context)
Citation Context ...other common approach is principal component analysis (PCA), a tool that can be used to visualize conformational dynamics in a lower-dimensional subspace spanned by several principle components (PCs) =-=[41, 42]-=-. An important aspect of PCA is that motion along PCs can be viewed in real space, helping make complicated dynamical motions visually tractable. Using NCA, PCA or other CV approaches cannot, however,... |

6 | Bounding the Fréchet distance by the Hausdorff distance. In
- Alt, Knauer
- 2001
(Show Context)
Citation Context ...Fréchet distance as simply the Fréchet metric (distance) with symbol δF for brevity. The Fréchet distance is bounded from below by the Hausdorff distance for any given pair of piecewise-linear curves =-=[51]-=- (δF δH) because for convex polygonal Path Similarity Analysis PLOS Computational Biology | DOI:10.1371/journal.pcbi.1004568 October 21, 2015 5 / 37 curves the Fréchet and Hausdorff distances are equ... |

6 | Exploring the conformational transitions of biomolecular systems using a simple two-state anisotropic network model
- Das, Gur, et al.
(Show Context)
Citation Context ...c contacts* overlap/angle/H-bond constraints Morph[72] CHARMM/XPLOR† – energy minimization of intermediate snapshots LinInt – – – Cα-only GOdMD[101] bonds: inf. sq-well – NBF: Go-like + ENM-MetaD ANMP=-=[102]-=- double-well ANM – Emix = min{Ui, Uf} iENM[103] double-well ANM – Emix = F(Ui, Uf) (arbitrary), collision penalty MAP[31] two ANMs, OM dynamics overdamped Langevin‡ minimum OM action! 2 ODEs+BCs! path... |

6 |
Predicting order of conformational changes during protein conformational transitions using an interpolated elastic network model. Proteins: Struct. Funct
- Tekpinar, Zheng
- 2010
(Show Context)
Citation Context ...rph[72] CHARMM/XPLOR† – energy minimization of intermediate snapshots LinInt – – – Cα-only GOdMD[101] bonds: inf. sq-well – NBF: Go-like + ENM-MetaD ANMP[102] double-well ANM – Emix = min{Ui, Uf} iENM=-=[103]-=- double-well ANM – Emix = F(Ui, Uf) (arbitrary), collision penalty MAP[31] two ANMs, OM dynamics overdamped Langevin‡ minimum OM action! 2 ODEs+BCs! path MENM-SD/SP[104] double-well ANM – Emix = β −1l... |

5 | Enhanced sampling in molecular dynamics using metadynamics, replica-exchange and temperature acceleration - Abrams, Bussi |

5 | Vanden-Eijnden E, Ciccotti G. String method in collective variables: Minimum free energy paths and isocommittor surfaces - Maragliano, Fischer |

5 | Computing the Hausdorff distance between curved objects - Alt, Scharf |

5 | Protein local structure alignment under the discrete Fréchet distance
- Zhu
- 2007
(Show Context)
Citation Context ...g handwritten documents [54], and comparing trajectories of moving objects in geographic information systems [55]. Both metrics have also found applications in biology for protein structure alignment =-=[56, 57]-=- and protein homology analysis [58, 59]. To our knowledge, the Hausdorff and Fréchet metrics have not been widely used as general tools to quantify macromolecular pathways. However, recently two studi... |

5 |
Madej T. Analysis of protein homology by assessing the (dis)similarity in protein loop regions
- AR
(Show Context)
Citation Context ...ring trajectories of moving objects in geographic information systems [55]. Both metrics have also found applications in biology for protein structure alignment [56, 57] and protein homology analysis =-=[58, 59]-=-. To our knowledge, the Hausdorff and Fréchet metrics have not been widely used as general tools to quantify macromolecular pathways. However, recently two studies employed Fréchet distances to assess... |

5 |
Comparison of three perturbation molecular dynamics methods for modeling conformational transitions
- Huang, Ozkirimli, et al.
- 2009
(Show Context)
Citation Context ...orff-Larsen et al. [65] used dissimilarity scores to assign individual paths to folding pathways using clustering. Different methods to sample conformational transitions were compared by Huang et al. =-=[66]-=-, who contrasted the original targeted MD (TMD) algorithm [13] with a harmonic restraint variation of TMD (also known as “steered MD” (SMD) or “restrained TMD” (rTMD) [67])—and biased MD (BMD) approac... |

5 | Induced-fit movements in adenylate kinases - GE, CW, et al. - 1990 |

5 | Activation energy of catalysisrelated domain motion in E. coli adenylate kinase - Shapiro, Meirovitch - 2006 |

4 |
Perahia D, Ghélis C. Conformational dynamics and enzyme activity. Biochimie
- JM
- 1998
(Show Context)
Citation Context ... a0 < a1 and Q : [b0, b1]! R3N, b0 < b1 that are parameterized with a real parameter, the continuous Fréchet distance corresponds to finding two specific continuous and monotonous parameterizations α:=-=[0, 1]-=-! [a0, a1] and β:[0, 1]! [b0, b1] (the “schedules” of the man and the dog along their paths) so that the largest point distance d for a given set of parameterizations is minimized [47], dFðP;QÞ mina... |

4 | Vaart A, Karplus M: Minimum free energy pathways and free energy profiles for conformational transitions based on atomistic molecular dynamics simulations - der |

4 |
Speranskiy K, Thorpe MF. Generating stereochemically acceptable protein pathways
- DW
(Show Context)
Citation Context ...ng (R) domain, residues 380–535, and the catalytic (C) domain, residues 1–178. The conformational transition of a DT monomer was simulated previously and considered challenging for simulation methods =-=[39, 72]-=-. We simulated transition pathways of DT between a closed and open conformation based on chain A from the monomeric structure (PDB id: 1MDT [86]) and chain A from the domain-swapped dimeric structure ... |

4 |
Structural similarity of loops in protein families: toward the understanding of protein evolution
- AR, Madej
(Show Context)
Citation Context ...ring trajectories of moving objects in geographic information systems [55]. Both metrics have also found applications in biology for protein structure alignment [56, 57] and protein homology analysis =-=[58, 59]-=-. To our knowledge, the Hausdorff and Fréchet metrics have not been widely used as general tools to quantify macromolecular pathways. However, recently two studies employed Fréchet distances to assess... |

4 | Analysis and elimination of a bias in targeted molecular dynamics simulations of conformational transitions: application to calmodulin - Ovchinnikov, Karplus - 2012 |

4 | Domain closure in adenylate kinase. Biochemistry - MA, EV, et al. - 1996 |

4 | NMR identification of transient complexes critical to adenylate kinase catalysis - Adén, Wolf-Watz |

4 |
Denning EJ, Woolf TB, Beckstein O. MDAnalysis: A Toolkit for the Analysis of Molecular Dynamics Simulations
- Michaud-Agrawal
(Show Context)
Citation Context ...ctively, were computed as described in the Introduction. Further details on the numerical implementation are provided in S2 Text. Both metrics are implemented as part of the MDAnalysis Python package =-=[92]-=- in the module MDAnalysis. Fig 2. Macromolecular transitions studied with PSA. (A) The closed$ open transition for adenylate kinase (AdK) involves the hinge-like motion of the LID (green) and NMP (cya... |

3 |
A theoretical view of protein dynamics
- Orozco
- 2014
(Show Context)
Citation Context ...es to simulating macromolecular dynamics, in large part due to the availability of full atomistic detail [4]. It has has been the workhorse tool for studying the protein structure–function connection =-=[5]-=-. However, conformational transitions are rare events: crossing events in the transition region of phase space take place much faster than the (waiting) time scales of metastable equilibria, often by ... |

3 |
Equilibrium sampling in biomolecular simulations. Annu Rev Biophys
- DM
(Show Context)
Citation Context ...ing in trajectory-based (i.e., dynamics-based) methods [13– 21] is accelerated by reducing the computational cost per time step and/or by minimizing the total number of time steps needed for sampling =-=[10]-=-. Non-dynamical approaches can be roughly divided into the class of minimum (free) energy path (MEP/MFEP) methods [22–30], including elastic network model (ENM) approaches [31–35], and prior-informati... |

3 |
Path corrected functionals of stochastic trajectories: towards relative free energy and reaction coordinate calculations. Chem Phys Lett
- TB
- 1998
(Show Context)
Citation Context ...und in S5 Text. Two of the tested methods are based on MD combined with perturbation techniques (perturbation MD) designed to drive transitions between initial and final states. Two MDmethods —DIMSMD =-=[15, 105]-=- (implemented in CHARMM c36b2 [113]) and TMD (implemented in NAMD 2.10 [114])—used the all-atom CHARMM22/CMAP force field [115, 116] with Langevin dynamics and implicit solvents (ACE [117] in CHARMM, ... |

3 | de Groot BL: The atomistic mechanism of conformational transition in adenylate kinase: a TEE-REX molecular dynamics study - MB |

3 | Dellago C, Geissler PL. Transition path sampling: Throwing ropes over rough mountain passes - PG, Chandler |

3 |
Comparing images using the Hausdorff distance.
- DP, GA, et al.
- 1993
(Show Context)
Citation Context ...d as dHðP;QÞ maxfdhðP j QÞ; dhðQ j PÞg; ð2Þ where dhðP j QÞ max p2P min q2Q dðp; qÞ ð3Þ is the directed Hausdorff distance from P toQ, and d is a distance metric onR3N (measuring point distances) =-=[43]-=-; the vertical bar (PjQ) emphasizes that δh(PjQ) is not commutative. The function δh(PjQ) selects the point p 2 P, among all points in P, with the most distant nearest neighbor q 2 Q (as measured by... |

3 |
Bhattacharyya C. Fréchet Distance Based Approach for Searching Online Handwritten Documents
- Sriraghavendra, Karthik
(Show Context)
Citation Context ...hes. The Hausdorff metric has found applications in image comparison [43], while the orientation-dependent Fréchet metric has been used for handwriting recognition and searching handwritten documents =-=[54]-=-, and comparing trajectories of moving objects in geographic information systems [55]. Both metrics have also found applications in biology for protein structure alignment [56, 57] and protein homolog... |

3 |
Denning EJ, Perilla JR, Woolf TB: Zipping and unzipping of adenylate kinase: atomistic insights into the ensemble of open-closed transitions
- Beckstein
(Show Context)
Citation Context ...licitly illustrate the uses and limitations of heuristic collective variables, and to make a connection with previous work, we examine the AdK closed$ open transition (Fig 2A) in 2D angle-angle space =-=[99]-=-. The NMP-CORE angle θNMP is formed by the geometric centers of residues 115–125 (CORE-LID), 90–100 (CORE), and 35–55 (NMP) of E. coli AdK. Likewise, θLID is defined as the angle between residues 179–... |

3 |
Hospital A, Gelpi JL, Orozco M. Finding conformational transition pathways from discrete molecular dynamics simulations
- Sfriso, Emperador, et al.
(Show Context)
Citation Context ...Resa Name Force field/potentialb Solvent energeticsc Mixing function/other energeticsd all-atom DIMS[105] CHARMM22/CMAP ACS/ACE2 IS T = 300 K rTMD[67] CHARMM22/CMAP Generalized Born IS T = 300 K MDdMD=-=[100]-=- bonds/angles: inf. sq-well Lazaridis-Karplus IS NBF: simple vdW/electrostatic, T = 300 K FRODA[39] stereochemical constraints hydrophobic contacts* overlap/angle/H-bond constraints Morph[72] CHARMM/X... |

3 |
Hospital A, Emperador A, Orozco M. Exploration of conformational transition pathways from coarse-grained simulations. Bioinformatics. 2013; 29: 1980–1986. doi: 10.1093/bioinformatics/btt324 PMID: 23740746
- Sfriso
(Show Context)
Citation Context ...T = 300 K FRODA[39] stereochemical constraints hydrophobic contacts* overlap/angle/H-bond constraints Morph[72] CHARMM/XPLOR† – energy minimization of intermediate snapshots LinInt – – – Cα-only GOdMD=-=[101]-=- bonds: inf. sq-well – NBF: Go-like + ENM-MetaD ANMP[102] double-well ANM – Emix = min{Ui, Uf} iENM[103] double-well ANM – Emix = F(Ui, Uf) (arbitrary), collision penalty MAP[31] two ANMs, OM dynamics... |

2 |
der Vaart A, YangW. Protein structural transitions and their functional role
- Karplus, YQ, et al.
(Show Context)
Citation Context ...ique pairs of points (i.e., number of links) satisfying the following conditions: (1) The first/last pairs correspond to the first/last points of the respective paths (a1 = b1 = 1, aL = n and bL =m); =-=(2)-=- at least one point on a path (for a pair of points, one per path) must be advanced to its successive point, i.e., (ai+1 = ai and bi+1 = bi + 1) or (ai+1 = ai + 1 and bi+1 = bi) or (ai+1 = ai + 1 and ... |

2 |
Enzymatic transition states and dynamic motion in barrier crossing
- SD, VL
(Show Context)
Citation Context ...ium simulations thus disproportionately sample metastable states instead of transition events—the so-called sampling problem—greatly limiting their ability to generate conformational transition paths =-=[6]-=-. Enhanced path-sampling methods and other computational approaches have been developed to mitigate the sampling problem inherent to macromolecular transition events, permitting the observation of phy... |

2 | Free Energies from Adaptive Reaction Coordinate Forces (FEARCF): an application to ring puckering. Mol Phys. 2009 20 Apr; 107(8–12):1243–1250. doi: 10.1080 - CB, KJ |

2 | Amplitude Elastic Motions in Proteins from a Single-Parameter, Atomic Analysis - Large |

2 | Enosh A, Schueler-Furman O, Halperin D. Rapid sampling of molecular motions with prior information constraints - Raveh |

2 | The limited role of nonnative contacts in the folding pathways of a lattice protein - BC, JP, et al. |

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The morph server: a standardized system for analyzing and visualizing macromolecular motions in a database framework
- KrebsWG
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Citation Context ...o examined closed! open DT transitions (Fig 2B), which serves as a more challenging example due to the difficultly of capturing the putative unfolding and refolding required for conformational change =-=[72]-=-. AdK is divided into three domains: the ATP-binding (or “LID”) domain, residues 122–159 in the mesophilic Escherichia coli sequence (AKeco), and the AMP-binding (or “NMP” or Path Similarity Analysis ... |

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Farztdinov G, Gutin AM, Karplus M. Protein folding bottlenecks: A lattice Monte Carlo simulation
- Shakhnovich
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Citation Context ... a residue pair whose Cα atoms are separated by a distance smaller than 8 Å. A native contact is a contact present in a reference structure. Given a transition path, the fraction of native contacts Q =-=[96]-=- is the fraction of contacts in a native structure that are present in a transition structure. We then define Q1 and Q2, for any intermediate conformer in a transition, as the fractions of native cont... |

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GN Jr, Kavraki LE. Understanding protein flexibility through dimensionality reduction
- ML, Phillips
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Citation Context ...e useful for visualizing and identifying functional protein motions, selecting the collective variables that span the projected space and adequately describe a conformational transition is nontrivial =-=[97, 98]-=-. Choosing heuristic coordinates for a given system often requires strong physical intuition, something that may be absent when studying new or complicated transitions. In general, the determination o... |

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Analyzing large-scale structural change in proteins: comparison of principal component projection and Sammon mapping. Proteins. 2006 1 Jul; 64(1):210–218. doi
- Mesentean, Fischer, et al.
(Show Context)
Citation Context ...e useful for visualizing and identifying functional protein motions, selecting the collective variables that span the projected space and adequately describe a conformational transition is nontrivial =-=[97, 98]-=-. Choosing heuristic coordinates for a given system often requires strong physical intuition, something that may be absent when studying new or complicated transitions. In general, the determination o... |

2 |
Protein conformational transitions explored by mixed elastic network models
- ZhengW, Hummer
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Citation Context ...l ANM – Emix = min{Ui, Uf} iENM[103] double-well ANM – Emix = F(Ui, Uf) (arbitrary), collision penalty MAP[31] two ANMs, OM dynamics overdamped Langevin‡ minimum OM action! 2 ODEs+BCs! path MENM-SD/SP=-=[104]-=- double-well ANM – Emix = β −1ln{exp[−β(Ui + i)] + exp(−β(Uf + f)]} All MD-based methods use atomic resolution; Morph and LinInt are the only other methods with greater than Cα resolution. Except fo... |

1 | Chng CP. Coarse-grained models reveal functional dynamics–I. Elastic network models– theories, comparisons and perspectives - LW |

1 | Coarse-grained models reveal functional dynamics–II. Molecular dynamics simulation at the coarse-grained level–theories and biological applications - CP, LW |

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Beckstein O. Sampling large conformational transitions: adenylate kinase as a testing ground
- SL
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Citation Context ...icability to highly fluctuating paths in high dimensions. AdK’s closed/open transition (Fig 2A) is a standard test case that captures general, essential features of conformational changes in proteins =-=[12]-=-. Alongside AdK in our analysis of transition ensembles, we also examined closed! open DT transitions (Fig 2B), which serves as a more challenging example due to the difficultly of capturing the putat... |

1 | Finite temperature string method for the study of rare events - EW, Vanden-Eijnden |

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Similarity measures for protein ensembles. PLoS One
- Lindorff-Larsen, Ferkinghoff-Borg
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Citation Context ...wo protein conformers by quantifying the percentage of shared contacts. Another promising approach may be to integrate information-based metrics used for measuring the similarity of protein ensembles =-=[53]-=-. In this paper, we exclusively used the best-fit rmsd as the point metric due to its simplicity and widespread use, helping to connect with familiar intuitions and avoid obfuscating the examination o... |

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Fajer M, Meng Y, Gumbart JC, et al. Generalized Scalable Multiple Copy Algorithms for Molecular Dynamics
- JiangW, Huang
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Citation Context ...n widely used as general tools to quantify macromolecular pathways. However, recently two studies employed Fréchet distances to assess convergence of transition paths to an optimal path. Jiang et al. =-=[60]-=- used the same discrete Fréchet metric as used in this study to assess the convergence of a swarms-ofFig 1. Comparison of Hausdorff and Fréchet distance. Two paths P (green) andQ (cyan) begin at state... |

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CB. Unrestrained computation of free energy along a path
- BM, Huang, et al.
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Citation Context ...F > δH. doi:10.1371/journal.pcbi.1004568.g001 Path Similarity Analysis PLOS Computational Biology | DOI:10.1371/journal.pcbi.1004568 October 21, 2015 6 / 37 trajectories string method. Dickson et al. =-=[61]-=- employed a variation of the discrete Fréchet distance where the coupling distance was defined as the average distance between all pairs in a coupling (instead of the maximum distance as in Eq 6); thi... |

1 | Wolynes PG. Analysis of single molecule folding studies with replica correlation functions - Lenz, SS |

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Force-induced change in protein unfolding mechanism: discrete or continuous switch
- TGW, RB
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Citation Context ...ve been compared quantitatively but not with Hausdorff or Fréchet metrics. Several such studies utilized native contacts-based path (dis)similarity measures [62–65]. In particular, both Graham et al. =-=[64]-=- and Lindorff-Larsen et al. [65] used dissimilarity scores to assign individual paths to folding pathways using clustering. Different methods to sample conformational transitions were compared by Huan... |

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Precise Hausdorff distance computation between polygonal meshes. Comput Aided Geom Des. 2010 Nov; 27(8):580–591. doi: 10.1016/j.cagd.2010.04.004
- BartoňM, Elber, et al.
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Citation Context ...of noisy data [70]—may offer advantages in carrying out various analyses. The Hausdorff distance can be generalized as well to measure, for instance, distances between surfaces (instead of 1D curves) =-=[71]-=-. The multitudinous permutations that can be selected among the various path metrics, structural similarity metrics, clustering algorithms, etc. make PSA a flexible tool for trajectory analysis. Model... |

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Intuitive helix geometry analysis and abstraction
- Bendix
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Citation Context ...rison.We describe how a range of conformational transition paths were generated to supply a variety of contexts in which to test PSA. Molecular images were created with VMD [87] and the Bendix plugin =-=[88]-=-. Graphs were plotted with the Python libraries matplotlib [89] and seaborn [90], in particular its implementation of violin plots [91]. Characterizing transition paths Path similarity analysis (PSA).... |

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et al. seaborn: v0.5.0
- WaskomM, Hobson, et al.
- 2014
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Citation Context ...to supply a variety of contexts in which to test PSA. Molecular images were created with VMD [87] and the Bendix plugin [88]. Graphs were plotted with the Python libraries matplotlib [89] and seaborn =-=[90]-=-, in particular its implementation of violin plots [91]. Characterizing transition paths Path similarity analysis (PSA). The Hausdorff metric, δH, and the discrete Fréchet metric, δdF, defined in Eq 2... |