The purpose of this paper is to provide a comprehensive presentation and interpretation of the Ensemble Kalman Filter (EnKF) and its numerical implementation. The EnKF has a large user group, and numerous publications have discussed applications and theoretical aspects of it. This paper reviews the important results from these studies and also presents new ideas and alternative interpretations which further explain the success of the EnKF. In addition to providing the theoretical framework needed for using the EnKF, there is also a focus on the algorithmic formulation and optimal numerical implementation. A program listing is given for some of the key subroutines. The paper also touches upon specific issues such as the use of nonlinear measurements, in situ profiles of temperature and salinity, and data which are available with high frequency in time. An ensemble based optimal interpolation (EnOI) scheme is presented as a cost-effective approach which may serve as an alternative to the EnKF in some applications. A fairly extensive discussion is devoted to the use of time correlated model errors and the estimation of model bias.

The Ensemble Kalman Filter: Theoretical formulation and practical implementation

A goal in the kinetic characterization of a macromolecular system is the description of its slow relaxation processes via (i) identification of the structural changes involved in these processes and (ii) estimation of the rates or timescales at which these slow processes occur. Most of the approaches to this task, including Markov models, master-equation models, and kinetic network models, start by discretizing the high-dimensional state space and then characterize relaxation processes in terms of the eigenvectors and eigenvalues of a discrete transition matrix. The practical success of such an approach depends very much on the ability to finely discretize the slow order parameters. How can this task be achieved in a high-dimensional configuration space without relying on subjective guesses of the slow order parameters? In this paper, we use the variational principle of conformation dynamics to derive an optimal way of identifying the "slow subspace" of a large set of prior order parameters - either generic internal coordinates or a user-defined set of parameters. Using a variational formulation of conformational dynamics, it is shown that an existing method-the time-lagged independent component analysis-provides the optional solution to this problem. In addition, optimal indicators-order parameters indicating the progress of the slow transitions and thus may serve as reaction coordinates-are readily identified. We demonstrate that the slow subspace is well suited to construct accurate kinetic models of two sets of molecular dynamics simulations, the 6-residue fluorescent peptide MR121-GSGSW and the 30-residue intrinsically disordered peptide kinase inducible domain (KID). The identified optimal indicators reveal the structural changes associated with the slow processes of the molecular system under analysis.

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Identification of slow molecular order parameters for Markov model construction

Ten measures of experimental electron-density-map quality are examined and the skewness of electron density is found to be the best indicator of actual map quality. A Bayesian approach to estimating map quality is developed and used in the PHENIX AutoSol wizard to make decisions during automated structure solution.

Decision-making in structure solution using Bayesian estimates of map quality: the PHENIX autosol wizard

Markov state models (MSMs) are a powerful framework for analyzing dynamical systems, such as molecular dynamics (MD) simulations, that have gained widespread use over the past several decades. This perspective offers an overview of the MSM field to date, presented for a general audience as a timeline of key developments in the field. We sequentially address early studies that motivated the method, canonical papers that established the use of MSMs for MD analysis, and subsequent advances in software and analysis protocols. The derivation of a variational principle for MSMs in 2013 signified a turning point from expertise-driving MSM building to a systematic, objective protocol. The variational approach, combined with best practices for model selection and open-source software, enabled a wide range of MSM analysis for applications such as protein folding and allostery, ligand binding, and protein–protein association. To conclude, the current frontiers of methods development are highlighted, as well as excit...

Markov State Models: From an Art to a Science

Markov State Models (MSMs) provide an automated framework to investigate the dynamical properties of high-dimensional molecular simulations. These models can provide a human-comprehensible picture of the underlying process and have been successfully used to study protein folding, protein aggregation, protein ligand binding, and other biophysical systems. The MSM requires the construction of a discrete state-space such that two points are in the same state if they can interconvert rapidly. In the following, we suggest an improved method, which utilizes second order Independent Component Analysis (also known as time-structure based Independent Component Analysis, or tICA), to construct the state-space. We apply this method to simulations of NTL9 (provided by Lindorff-Larsen et al. Science2011, 334, 517–520) and show that the MSM is an improvement over previously built models using conventional distance metrics. Additionally, the resulting model provides insight into the role of non-native contacts by reveal...

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Improvements in Markov State Model Construction Reveal Many Non-Native Interactions in the Folding of NTL9

Protein dynamics on a long time scale was investigated using all-atom molecular dynamics (MD) simulation and time-structure based independent component analysis (tICA). We selected the lysine-, arginine-, ornithine-binding protein (LAO) as a target protein and focused on its domain motions in the open state. A MD simulation of the LAO in explicit water was performed for 600 ns, in which slow and large-amplitude domain motions of the LAO were observed. After extracting domain motions by rigid-body domain analysis, the tICA was applied to the obtained rigid-body trajectory, yielding slow modes of the LAO's domain motions in order of decreasing time scale. The slowest mode detected by the tICA represented not a closure motion described by a largest-amplitude mode determined by the principal component analysis but a twist motion with a time scale of tens of nanoseconds. The slow dynamics of the LAO were well described by only the slowest mode and were characterized by transitions between two basins. The results show that tICA is promising for describing and analyzing slow dynamics of proteins.

Slow dynamics in protein fluctuations revealed by time-structure based independent component analysis: The case of domain motions

We recently proposed the method of time-structure based independent component analysis (tICA) to examine the slow dynamics involved in conformational fluctuations of a protein as estimated by molecular dynamics (MD) simulation [Y. Naritomi and S. Fuchigami, J. Chem. Phys. 134, 065101 (2011)]. Our previous study focused on domain motions of the protein and examined its dynamics by using rigid-body domain analysis and tICA. However, the protein changes its conformation not only through domain motions but also by various types of motions involving its backbone and side chains. Some of these motions might occur on a slow time scale: we hypothesize that if so, we could effectively detect and characterize them using tICA. In the present study, we investigated slow dynamics of the protein backbone using MD simulation and tICA. The selected target protein was lysine-, arginine-, ornithine-binding protein (LAO), which comprises two domains and undergoes large domain motions. MD simulation of LAO in explicit water was performed for 1 μs, and the obtained trajectory of C(α) atoms in the backbone was analyzed by tICA. This analysis successfully provided us with slow modes for LAO that represented either domain motions or local movements of the backbone. Further analysis elucidated the atomic details of the suggested local motions and confirmed that these motions truly occurred on the expected slow time scale.

Slow dynamics of a protein backbone in molecular dynamics simulation revealed by time-structure based independent component analysis

Classification and Annotation of the Relationship between Protein Structural Change and Ligand Binding

High-speed (HS) atomic force microscopy (AFM) is a prominent imaging technology that observes large-scale structural dynamics of biomolecules near the physiological condition, but the AFM data are limited to the surface shape of specimens. Rigid-body fitting methods were developed to obtain molecular structures that fit to an AFM image, without accounting for conformational changes. Here, we developed a method to fit flexibly a three-dimensional (3D) biomolecular structure into an AFM image. First, we describe a method to produce a pseudo-AFM image from a given 3D structure in a differentiable form. Then, using a correlation function between the experimental AFM image and the computational pseudo-AFM image, we developed a flexible fitting molecular dynamics (MD) simulation method by which we obtain protein structures that well fit to the given AFM image. We first test it with a twin experiment; using an AFM image produced from a protein structure different from its native conformation as a reference, we performed the flexible fitting MD simulations to sample conformations that fit well the reference AFM image, and the method was confirmed to work well. Then, parameter dependence in the protocol was discussed. Finally, we applied the method to a real experimental HS-AFM image for a flagellar protein FlhA, demonstrating its applicability. We also test the rigid-body fitting of a molecular structure to an AFM image. Our method will be a general tool for dynamic structure modeling based on HS-AFM images and is publicly available through the CafeMol software.

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Sotaro Fuchigami

Papers

Slow dynamics in protein fluctuations revealed by time-structure based independent component analysis: The case of domain motions

Slow dynamics of a protein backbone in molecular dynamics simulation revealed by time-structure based independent component analysis

Classification and Annotation of the Relationship between Protein Structural Change and Ligand Binding

Flexible Fitting of Biomolecular Structures to Atomic Force Microscopy Images via Biased Molecular Simulations.

CLASP2 Has Two Distinct TOG Domains That Contribute Differently to Microtubule Dynamics.