refs.bib

% Encoding: ISO-8859-1

@Article{Nicholls2014,
  author    = {Nicholls, A.},
  title     = {Confidence limits, error bars and method comparison in molecular modeling. {P}art 1: {T}he calculation of confidence intervals},
  journal   = {J. Comput. Aided Mol. Des.},
  year      = {2014},
  volume    = {28},
  number    = {9},
  pages     = {887--918},
  doi       = {10.1007/s10822-014-9753-z},
  owner     = {dsideriu},
  timestamp = {2018.08.27},
}

@article{Klimovich2015,
abstract = {Free energy calculations based on molecular dynamics simulations show considerable promise for applications ranging from drug discovery to prediction of physical properties and structure-function studies. But these calculations are still difficult and tedious to analyze, and best practices for analysis are not well defined or propagated. Essentially, each group analyzing these calculations needs to decide how to conduct the analysis and, usually, develop its own analysis tools. Here, we review and recommend best practices for analysis yielding reliable free energies from molecular simulations. Additionally, we provide a Python tool, alchemical-analysis.py , freely available on GitHub as part of the pymbar package (located at http://github.com/choderalab/pymbar ), that implements the analysis practices reviewed here for several reference simulation packages, which can be adapted to handle data from other packages. Both this review and the tool covers analysis of alchemical calculations generally, including free energy estimates via both thermodynamic integration and free energy perturbation-based estimators. Our Python tool also handles output from multiple types of free energy calculations, including expanded ensemble and Hamiltonian replica exchange, as well as standard fixed ensemble calculations. We also survey a range of statistical and graphical ways of assessing the quality of the data and free energy estimates, and provide prototypes of these in our tool. We hope this tool and discussion will serve as a foundation for more standardization of and agreement on best practices for analysis of free energy calculations.},
archivePrefix = {arXiv},
arxivId = {15334406},
author = {Klimovich, Pavel V. and Shirts, Michael R. and Mobley, David L.},
doi = {10.1007/s10822-015-9840-9},
eprint = {15334406},
isbn = {1573-4951 (Electronic)$\backslash$r0920-654X (Linking)},
issn = {15734951},
journal = {J. Comput. Aided Mol. Des.},
keywords = {Alchemical,Analysis tool,Binding free energy,Free energy calculation,Hydration free energy,Transfer free energy},
pmid = {25808134},
title = {{Guidelines for the analysis of free energy calculations}},
year = {2015},
volume = {29},
number = {5},
pages = {397--411}
}


@article{Kabsch1976,
abstract = {A simple procedure is derived which determines a best rotation of a given vector set into a second vector set by minimizing the weighted sum of squared deviations. The method is generalized for any given metric constraint on the transformation.},
archivePrefix = {arXiv},
arxivId = {05677394},
author = {Kabsch, W.},
doi = {10.1107/S0567739476001873},
url = {http://dx.doi.org/10.1107/s0567739476001873},
eprint = {05677394},
isbn = {0567-7394},
issn = {0567-7394},
journal = {Acta Crystallographica Section A},
number = {5},
pages = {922--923},
pmid = {96},
title = {{A solution for the best rotation to relate two sets of vectors}},
volume = {32},
year = {1976}
}

@article{Leek2017,
abstract = {As debate rumbles on about how and how much poor statistics is to blame for poor reproducibility, Nature asked influential statisticians to recommend one change to improve science. The common theme? The problem is not our maths, but ourselves. T o use statistics well, researchers must study how scientists analyse and interpret data and then apply that information to prevent cognitive mistakes. In the past couple of decades, many fields have shifted from data sets with a dozen measurements to data sets with millions. Methods that were developed for a world with sparse and hard-to-collect informa-tion have been jury-rigged to handle bigger, more-diverse and more-complex data sets. No wonder the literature is now full of papers that use outdated statistics, misapply statistical tests and misinterpret results. The application of P values to determine whether an analysis is interesting is just one of the most visible of many shortcomings. It's not enough to blame a surfeit of data and a lack of training in analysis (J. T. Leek and R. D. Peng Proc. Natl Acad. Sci. USA 112, 1645–1646; 2015). It's also impractical to say that statistical metrics such as P values should not be used to make decisions. Sometimes a decision (editorial or funding, say) must be made, and clear guidelines are useful. The root problem is that we know very little about how people analyse and process information. An illustrative exception is graphs. Experiments show that people strug-gle to compare angles in pie charts yet breeze through comparative lengths and heights in bar charts (},
archivePrefix = {arXiv},
arxivId = {1709.07588},
author = {Leek, Jeff and McShane, Blakeley B. and Gelman, Andrew and Colquhoun, David and Nuijten, Mich{\`{e}}le B. and Goodman, Steven N.},
journal = {Nature},
doi = {10.1038/d41586-017-07522-z},
url = {http://dx.doi.org/10.1038/d41586-017-07522-z},
eprint = {1709.07588},
issn = {0028-0836},
number = {7682},
pages = {557--559},
pmid = {29189798},
title = {{Five ways to fix statistics}},
volume = {551},
year = {2017}
}

@article{Schenker1985,
author = {Schenker, Nathaniel},
doi = {10.1080/01621459.1985.10478123},
url = {http://dx.doi.org/10.1080/01621459.1985.10478123},
issn = {0162-1459},
journal = {J. Am. Stat. Assoc.},
keywords = {Bias-corrected percentile method,Nonparametric confidence intervals,Percentile method,Pivotal quantity,Resampling plans},
number = {390},
pages = {360--361},
title = {{Qualms about bootstrap confidence intervals}},
volume = {80},
year = {1985}
}

@article{Chernick2009,
author = {Chernick, Michael R. and Labudde, Robert A.},
doi = {10.1080/01966324.2009.10737767},
url = {http://dx.doi.org/10.1080/01966324.2009.10737767},
issn = {0196-6324},
journal = {Amer. J. Math. Management Sci.},
keywords = {ABC,BC,BCa,Bootstrap confidence intervals,Efron's percentile method,Iterated bootstrap,Second order bootstrap},
number = {3-4},
pages = {437--456},
title = {{Revisiting qualms about bootstrap confidence intervals}},
volume = {29},
year = {2009}
}

@article{Kolmogoroff1936,
author = {Kolmogoroff, A.},
journal = {Math. Ann.},
pages = {155--160},
title = {Zur Theorie der Markoffschen Ketten},
doi = {10.1007/bf01565412},
url = {http://dx.doi.org/10.1007/bf01565412},
volume = {112},
number = {1},
year = {1936},
issn = "0025-5831"
}

@article{Chou11,
  author={T Chou and K Mallick and R K P Zia},
  title={Non-equilibrium statistical mechanics: from a paradigmatic model to biological transport},
  journal={Rep. Prog. Phys.},
  volume={74},
  number={11},
  pages={116601},
  url={http://dx.doi.org/10.1088/0034-4885/74/11/116601},
  doi={10.1088/0034-4885/74/11/116601},
  year={2011},
  issn = {0034-4885},
  abstract={Unlike equilibrium statistical mechanics, with its well-established foundations, a similar widely accepted framework for non-equilibrium statistical mechanics (NESM) remains elusive. Here, we review some of the many recent activities on NESM, focusing on some of the fundamental issues and general aspects. Using the language of stochastic Markov processes, we emphasize general properties of the evolution of configurational probabilities, as described by master equations. Of particular interest are systems in which the dynamics violates detailed balance, since such systems serve to model a wide variety of phenomena in nature. We next review two distinct approaches for investigating such problems. One approach focuses on models sufficiently simple to allow us to find exact, analytic, non-trivial results. We provide detailed mathematical analyses of a one-dimensional continuous-time lattice gas, the totally asymmetric exclusion process. It is regarded as a paradigmatic model for NESM, much like the role the Ising model played for equilibrium statistical mechanics. It is also the starting point for the second approach, which attempts to include more realistic ingredients in order to be more applicable to systems in nature. Restricting ourselves to the area of biophysics and cellular biology, we review a number of models that are relevant for transport phenomena. Successes and limitations of these simple models are also highlighted.}
}

@article{Schappals2017,
abstract = {Thermodynamic properties are often modeled by classical force fields which describe the interactions on the atomistic scale. Molecular simulations are used for retrieving thermodynamic data from such models, and many simulation techniques and computer codes are available for that purpose. In the present round robin study, the following fundamental question is addressed: Will different user groups working with different simulation codes obtain coinciding results within the statistical uncertainty of their data? A set of 24 simple simulation tasks is defined and solved by five user groups working with eight molecular simulation codes: DL{\_}POLY, GROMACS, IMC, LAMMPS, ms2, NAMD, Tinker, and TOWHEE. Each task consists of the definition of (1) a pure fluid that is described by a force field and (2) the conditions under which that property is to be determined. The fluids are four simple alkanes: ethane, propane, n-butane, and iso-butane. All force fields consider internal degrees of freedom: OPLS, TraPPE, and a modified OPLS version with bond stretching vibrations. Density and potential energy are determined as a function of temperature and pressure on a grid which is specified such that all states are liquid. The user groups worked independently and reported their results to a central instance. The full set of results was disclosed to all user groups only at the end of the study. During the study, the central instance gave only qualitative feedback. The results reveal the challenges of carrying out molecular simulations. Several iterations were needed to eliminate gross errors. For most simulation tasks, the remaining deviations between the results of the different groups are acceptable from a practical standpoint, but they are often outside of the statistical errors of the individual simulation data. However, there are also cases where the deviations are unacceptable. This study highlights similarities between computer experiments and laboratory experiments, which are both subject not only to statistical error but also to systematic error.},
author = {Schappals, Michael and Mecklenfeld, Andreas and Kr{\"{o}}ger, Leif and Botan, Vitalie and K{\"{o}}ster, Andreas and Stephan, Simon and Garc{\'{i}}a, Edder J. and Rutkai, Gabor and Raabe, Gabriele and Klein, Peter and Leonhard, Kai and Glass, Colin W. and Lenhard, Johannes and Vrabec, Jadran and Hasse, Hans},
doi = {10.1021/acs.jctc.7b00489},
url = {http://dx.doi.org/10.1021/acs.jctc.7b00489},
issn = {1549-9618},
journal = {J. Chem. Theory Comput.},
number = {9},
pages = {4270--4280},
title = {{Round Robin Study: Molecular Simulation of Thermodynamic Properties from Models with Internal Degrees of Freedom}},
volume = {13},
year = {2017}
}

@ARTICLE{PatroneUQreview,
   author = {{Patrone}, P.~N. and {Dienstfrey}, A.},
    title = "{Uncertainty Quantification for Molecular Dynamics}",
  journal = {ArXiv e-prints},
archivePrefix = "arXiv",
   eprint = {1801.02483},
 primaryClass = "physics.comp-ph",
 keywords = {Physics - Computational Physics},
     year = 2018,
    month = jan,
   url = {https://arxiv.org/abs/1801.02483},
  adsnote = {Provided by the SAO/NASA Astrophysics Data System}
}

@Inbook{PatroneAIAA,
author={Patrone, Paul
and Kearsley, Anthony
and Dienstfrey, Andrew},
title={2018 AIAA Non-Deterministic Approaches Conference},
chapter={The role of data analysis in uncertainty quantification: {C}ase studies for materials modeling},
series={AIAA SciTech Forum},
year={2018},
month={Jan},
day={07},
publisher={American Institute of Aeronautics and Astronautics},
doi={10.2514/6.2018-0927},
Url={http://dx.doi.org/10.2514/6.2018-0927}
}

@BOOK{Leimkuhler,
  Author = {Leimkuhler, Ben and Matthews, Charles},
  title     = "Molecular Dynamics with Deterministic and Stochastic Numerical Methods",
  isbn      = "978-3-319-16374-1",
  year      = 2015,
  publisher = "Springer International Publishing",
  address   = "Switzerland"
}


@article{Rizzi2,
   author = "Rizzi, F. and Jones, R. E. and Debusschere, B. J. and Knio, O. M.",
   title = "Uncertainty quantification in MD simulations of concentration driven ionic flow through a silica nanopore. {II.} {U}ncertain potential parameters",
   journal = "J. Chem. Phys.",
   year = "2013",
   volume = "138",
   number = "19", 
   eid = 194105,
   pages = "194105",
   doi = "10.1063/1.4804669",
   url = "http://dx.doi.org/10.1063/1.4804669",
   issn = "0021-9606"
}

@article{Rizzi3,
author = {F. Rizzi and H. N. Najm and B. J. Debusschere and K. Sargsyan and M. Salloum and H. Adalsteinsson and O. M. Knio},
title = {Uncertainty Quantification in MD Simulations. {Part I: F}orward Propagation},
journal = {Multiscale Model. Simul.},
volume = {10},
number = {4},
pages = {1428--1459},
year = {2012},
doi = {10.1137/110853169},
url = {http://dx.doi.org/10.1137/110853169},
issn = {1540-3459},
eprint = { 
        http://dx.doi.org/10.1137/110853169   
}
}

@article{Rizzi4,
author = {F. Rizzi and H. N. Najm and B. J. Debusschere and K. Sargsyan and M. Salloum and H. Adalsteinsson and O. M. Knio},
title = {Uncertainty Quantification in MD Simulations. {Part II: B}ayesian Inference of Force-Field Parameters},
journal = {Multiscale Model. Simul.},
volume = {10},
number = {4},
pages = {1460--1492},
year = {2012},
doi = {10.1137/110853170},
eprint = { 
        http://dx.doi.org/10.1137/110853170
},
url = { http://dx.doi.org/10.1137/110853170},
issn = "1540-3459"
}

@article{Grossfield2009,
abstract = {Growing computing capacity and algorithmic advances have facilitated the study of increasingly large biomolecular systems at longer timescales. However, with these larger, more complex systems come questions about the quality of sampling and statistical convergence. What size systems can be sampled fully? If a system is not fully sampled, can certain "fast variables" be considered well-converged? How can one determine the statistical significance of observed results? The present review describes statistical tools and the underlying physical ideas necessary to address these questions. Basic definitions and ready-to-use analyses are provided, along with explicit recommendations. Such statistical analyses are of paramount importance in establishing the reliability of simulation data in any given study.},
annote = {PMCID: PMC2865156},
author = {Grossfield, Alan and Zuckerman, Daniel M},
doi = {10.1016/s1574-1400(09)00502-7},
institution = {University of Rochester Medical Center, Department of Biochemistry and Biophysics, Box 712, Rochester, N.Y., 14642, USA, 585-276-4193.},
journal = {Annu. Rep. Comput. Chem.},
pages = {23--48},
pmid = {20454547},
title = {{Quantifying uncertainty and sampling quality in biomolecular simulations.}},
Url = {http://dx.doi.org/10.1016/s1574-1400(09)00502-7},
volume = {5},
number = {},
year = {2009},
issn = "1574-1400"
}

@Article{Grossfield-2015,
  author      = {Leioatts, Nicholas and Romo, Tod D. and Danial, Shairy Azmy and Grossfield, Alan},
  title       = {Retinal Conformation Changes Rhodopsin's Dynamic Ensemble.},
  journal     = {Biophys. J.},
  year        = {2015},
  volume      = {109},
  number      = {3},
  pages       = {608--617},
  abstract    = {G protein-coupled receptors are vital membrane proteins that allosterically transduce biomolecular signals across the cell membrane. However, the process by which ligand binding induces protein conformation changes is not well understood biophysically. Rhodopsin, the mammalian dim-light receptor, is a unique test case for understanding these processes because of its switch-like activity; the ligand, retinal, is bound throughout the activation cycle, switching from inverse agonist to agonist after absorbing a photon. By contrast, the ligand-free opsin is outside the activation cycle and may behave differently. We find that retinal influences rhodopsin dynamics using an ensemble of all-atom molecular dynamics simulations that in aggregate contain 100 ?s of sampling. Active retinal destabilizes the inactive state of the receptor, whereas the active ensemble was more structurally homogenous. By contrast, simulations of an active-like receptor without retinal present were much more heterogeneous than those containing retinal. These results suggest allosteric processes are more complicated than a ligand inducing protein conformational changes or simply capturing a shifted ensemble as outlined in classic models of allostery.},
  doi         = {10.1016/j.bpj.2015.06.046},
  institution = {Biophysics, University of Rochester Medical Center, Rochester, New York. Electronic address: alan_grossfield@urmc.rochester.edu.},
  language    = {eng},
  medline-pst = {ppublish},
  owner       = {alan},
  pii         = {S0006-3495(15)00655-4},
  pmid        = {26244742},
  timestamp   = {2015.10.29},
  Url         = {http://dx.doi.org/10.1016/j.bpj.2015.06.046},
  issn = "0006-3495"
}

@Article{Romo2011,
  author   = {Romo, Tod D. and Grossfield, Alan},
  title    = {{Block covariance overlap method and convergence in molecular dynamics simulation}},
  journal  = {J. Chem. Theory Comput.},
  year     = {2011},
  volume   = {7},
  number   = {8},
  pages    = {2464--2472},
  issn     = {1549-9618},
  abstract = {Molecular dynamics (MD) is a powerful tool for understanding the fluctuations of biomolecular systems. It is, however, subject to statistical errors in its sampling of the underlying distribution of states. One must understand these errors in order to draw meaningful conclusions from the simulation. This is becoming ever more critical as MD simulations of even larger systems are attempted. We present here a new method for determining the extent of convergence that relies on measures of the fluctuation space sampled by the simulation without any a priori knowledge of states or partitioning of the configuration space. This method reveals long correlation times, even for simple systems, and suggests caution when interpreting macromolecular simulations. We also compare this method with previous efforts to characterize the sampling of MD simulation.},
  doi      = {10.1021/ct2002754},
  url      = {http://dx.doi.org/10.1021/ct2002754},
  pmid     = {26606620},
}

@Article{Zhang2010,
  author  = {Zhang, Xin and Bhatt, Divesh and Zuckerman, D M},
  title   = {{Automated sampling assessment for molecular simulations using the effective sample size}},
  journal = {J. Chem. Theory Comput.},
  year    = {2010},
  volume  = {6},
  number = {10},
  pages   = {3048--3057},
  annote  = {PMCID: PMC3017371},
  doi = {10.1021/ct1002384},
  url = {http://dx.doi.org/10.1021/ct1002384},
  issn = "1549-9618"
}

@article{Nemec2017,
abstract = {Molecular dynamics (MD) simulation is a natural method for the study of flexible molecules but at the same time is limited by the large size of the conformational space of these molecules. We ask by how much the MD sampling quality for flexible molecules can be improved by two means: the use of diverse sets of trajectories starting from different initial conformations to detect deviations between samples and sampling with enhanced methods such as accelerated MD (aMD) or scaled MD (sMD) that distort the energy landscape in controlled ways. To this end, we test the effects of these approaches on MD simulations of two flexible biomolecules in aqueous solution, Met-Enkephalin (5 amino acids) and HIV-1 gp120 V3 (a cycle of 35 amino acids). We assess the convergence of the sampling quantitatively with known, extensive measures of cluster number Nc and cluster distribution entropy Sc and with two new quantities, conformational overlap Oconf and density overlap Odens, both conveniently ranging from 0 to 1. These ...},
author = {Nemec, Mike and Hoffmann, Daniel},
doi = {10.1021/acs.jctc.6b00823},
url = {http://dx.doi.org/10.1021/acs.jctc.6b00823},
isbn = {1549-9618},
issn = {1549-9618},
journal = {J. Chem. Theory Comput.},
number = {2},
pages = {400--414},
pmid = {28085284},
title = {{Quantitative Assessment of Molecular Dynamics Sampling for Flexible Systems}},
volume = {13},
year = {2017}
}

@Article{Sindhikara2010,
  author  = {Sindhikara, Daniel J. and Emerson, Daniel J. and Roitberg, Adrian E.},
  title   = {{Exchange Often and Properly in Replica Exchange Molecular Dynamics}},
  journal = {J. Chem. Theory Comput.},
  year    = {2010},
  volume  = {6},
  number  = {9},
  pages   = {2804--2808},
  note    = {PMID: 26616081},
  doi     = {10.1021/ct100281c},
  eprint  = {http://dx.doi.org/10.1021/ct100281c},
  Url     = {http://dx.doi.org/10.1021/ct100281c},
  issn = "1549-9618"
}

@Article{Patriksson2008,
  author    = {Patriksson, Alexandra and van der Spoel, David},
  title     = {{A temperature predictor for parallel tempering simulations}},
  journal   = {Physical Chemistry Chemical Physics},
  year      = {2008},
  volume    = {10},
  pages     = {2073--2077},
  doi       = {10.1039/b716554d},
  issue     = {15},
  publisher = {The Royal Society of Chemistry},
  Url       = {http://dx.doi.org/10.1039/b716554d},
  number = "15",
  issn = "1463-9076"
}

@Article{Abraham2008,
  author  = {Abraham, Mark J. and Gready, Jill E.},
  title   = {{Ensuring Mixing Efficiency of Replica-Exchange Molecular Dynamics Simulations}},
  journal = {J. Chem. Theory Comput.},
  year    = {2008},
  volume  = {4},
  number  = {7},
  pages   = {1119--1128},
  pmid = {26636365},
  doi     = {10.1021/ct800016r},
  eprint  = {http://dx.doi.org/10.1021/ct800016r},
  Url     = { http://dx.doi.org/10.1021/ct800016r },
  issn = "1549-9618"
}

@Article{Lyman2007a,
  author  = {Lyman, E and Zuckerman, D M},
  title   = {{On the Structural Convergence of Biomolecular Simulations by Determination of the Effective Sample Size}},
  journal = {J. Phys. Chem. B},
  year    = {2007},
  volume  = {111},
  number  = {44},
  pages   = {12876--12882},
  pmid = {PMC2538559},
  doi = {10.1021/jp073061t},
  url = {http://dx.doi.org/10.1021/jp073061t},
  issn = "1520-6106"
}

@Article{Hess2002,
  author  = {Hess, Berk},
  title   = {{Convergence of sampling in protein simulations}},
  journal = {Phys. Rev. E},
  year    = {2002},
  volume  = {65},
  number  = {3},
  pages   = {31910},
  doi     = {10.1103/PhysRevE.65.031910},
  url     = {http://dx.doi.org/10.1103/PhysRevE.65.031910},
  issn    = "1063-651X"
}

@article{Swendsen-1986,
author = {Swendsen, R H and Wang, J.-S.},
journal = {Phys. Rev. Lett.},
pages = {2607--2609},
title = {{Replica Monte Carlo Simulation of Spin-Glasses}},
volume = {57},
number = {21},
year = {1986},
doi = {10.1103/PhysRevLett.57.2607},
url = {http://dx.doi.org/10.1103/PhysRevLett.57.2607},
issn = "0031-9007"
}

@Article{Sugita1999,
  author  = {Yuji Sugita and Yuko Okamoto},
  title   = {{Replica-exchange molecular dynamics method for protein folding}},
  journal = {Chem. Phys. Lett.},
  year    = {1999},
  volume  = {314},
  number  = {1-2},
  pages   = {141--151},
  issn    = {0009-2614},
  doi     = {10.1016/S0009-2614(99)01123-9},
  url     = {http://dx.doi.org/10.1016/S0009-2614(99)01123-9}
}

@article{Okamoto-2000,
author = {Sugita, Y and Kitao, A and Okamoto, Y},
journal = {J. Chem. Phys.},
pages = {6042--6051},
title = {{Multidimensional replica-exchange method for free-energy calculations}},
volume = {113},
number = {15},
year = {2000},
doi = {10.1063/1.1308516},
url = {http://dx.doi.org/10.1063/1.1308516},
issn = "0021-9606"
}

@article{Bussi2006a,
abstract = {In this Letter we propose a new formalism to map history-dependent metadynamics in a Markovian process. We apply this formalism to model Langevin dynamics and determine the equilibrium distribution of a collection of simulations. We demonstrate that the reconstructed free energy is an unbiased estimate of the underlying free energy and analytically derive an expression for the error. The present results can be applied to other history-dependent stochastic processes, such as Wang-Landau sampling.},
author = {Bussi, Giovanni and Laio, Alessandro and Parrinello, Michele},
institution = {Computational Science, Department of Chemistry and Applied Biosciences, Eidgen{\"{o}}ssische Technische Hochschule Z{\"{u}}rich, c/o USI Campus, Via Buffi 13, CH-6900 Lugano, Switzerland. gbussi@phys.chem.ethz.ch},
journal = {Phys. Rev. Lett.},
number = {9},
pages = {90601},
pmid = {16606249},
title = {{Equilibrium free energies from nonequilibrium metadynamics}},
volume = {96},
year = {2006},
doi = {10.1103/PhysRevLett.96.090601},
url = {http://dx.doi.org/10.1103/PhysRevLett.96.090601},
issn = "0031-9007"
}

@article{Laio2008,
abstract = {Metadynamics is a powerful algorithm that can be used both for reconstructing the free energy and for accelerating rare events in systems described by complex Hamiltonians, at the classical or at the quantum level. In the algorithm the normal evolution of the system is biased by a history-dependent potential constructed as a sum of Gaussians centered along the trajectory followed by a suitably chosen set of collective variables. The sum of Gaussians is exploited for reconstructing iteratively an estimator of the free energy and forcing the system to escape from local minima. This review is intended to provide a comprehensive description of the algorithm, with a focus on the practical aspects that need to be addressed when one attempts to apply metadynamics to a new system: (i) the choice of the appropriate set of collective variables; (ii) the optimal choice of the metadynamics parameters and (iii) how to control the error and ensure convergence of the algorithm.},
author = {Laio, Alessandro and Gervasio, Francesco L},
doi = {10.1088/0034-4885/71/12/126601},
isbn = {0034-4885},
issn = {0034-4885},
journal = {Rep. Prog. Phys.},
number = {12},
pages = {126601},
pmid = {261185700002},
title = {{Metadynamics: a method to simulate rare events and reconstruct the free energy in biophysics, chemistry and material science}},
Url = {http://dx.doi.org/10.1088/0034-4885/71/12/126601},
volume = {71},
year = {2008}
}

@article{Darve2008,
abstract = {In free energy calculations based on thermodynamic integration, it is necessary to compute the derivatives of the free energy as a function of one ͑scalar case͒ or several ͑vector case͒ order parameters. We derive in a compact way a general formulation for evaluating these derivatives as the average of a mean force acting on the order parameters, which involves first derivatives with respect to both Cartesian coordinates and time. This is in contrast with the previously derived formulas, which require first and second derivatives of the order parameter with respect to Cartesian coordinates. As illustrated in a concrete example, the main advantage of this new formulation is the simplicity of its use, especially for complicated order parameters. It is also straightforward to implement in a molecular dynamics code, as can be seen from the pseudocode given at the end. We further discuss how the approach based on time derivatives can be combined with the adaptive biasing force method, an enhanced sampling technique that rapidly yields uniform sampling of the order parameters, and by doing so greatly improves the efficiency of free energy calculations. Using the backbone dihedral angles ⌽ and ⌿ in N-acetylalanyl-NЈ-methylamide as a numerical example, we present a technique to reconstruct the free energy from its derivatives, a calculation that presents some difficulties in the vector case because of the statistical errors affecting the derivatives.},
author = {Darve, Eric and Rodr{\'{i}}guez-G{\'{o}}mez, David and Pohorille, Andrew},
doi = {10.1063/1.2829861},
url = {http://dx.doi.org/10.1063/1.2829861},
isbn = {0021-9606},
issn = {0021-9606},
journal = {J. Chem. Phys.},
number = {14},
pages = {144120},
pmid = {18412436},
title = {{Adaptive biasing force method for scalar and vector free energy calculations}},
volume = {128},
year = {2008}
}

@article{Comer2015,
abstract = {In the host of numerical schemes devised to calculate free energy differences by way of geometric transformations, the adaptive biasing force algorithm has emerged as a promising route to map complex free-energy landscapes. It relies upon the simple concept that as a simulation progresses, a continuously updated biasing force is added to the equations of motion, such that in the long-time limit it yields a Hamiltonian devoid of an average force acting along the transition coordinate of interest. This means that sampling proceeds uniformly on a flat free-energy surface, thus providing reliable free-energy estimates. Much of the appeal of the algorithm to the practitioner is in its physically intuitive underlying ideas and the absence of any requirements for prior knowledge about free-energy landscapes. Since its inception in 2001, the adaptive biasing force scheme has been the subject of considerable attention, from in-depth mathematical analysis of convergence properties to novel developments and extensions. The method has also been successfully applied to many challenging problems in chemistry and biology. In this contribution, the method is presented in a comprehensive, self-contained fashion, discussing with a critical eye its properties, applicability, and inherent limitations, as well as introducing novel extensions. Through free-energy calculations of prototypical molecular systems, many methodological aspects are examined, from stratification strategies to overcoming the so-called hidden barriers in orthogonal space, relevant not only to the adaptive biasing force algorithm but also to other importance-sampling schemes. On the basis of the discussions in this paper, a number of good practices for improving the efficiency and reliability of the computed free-energy differences are proposed.},
author = {Comer, Jeffrey and Gumbart, James C. and H{\'{e}}nin, J{\'{e}}r{\^{o}}me and Lelievre, Tony and Pohorille, Andrew and Chipot, Christophe},
doi = {10.1021/jp506633n},
url = {http://dx.doi.org/10.1021/jp506633n},
isbn = {1520-6106},
issn = {1520-6106},
journal = {J. Phys. Chem. B},
number = {3},
pages = {1129--1151},
pmid = {25247823},
title = {{The adaptive biasing force method: Everything you always wanted to know but were afraid to ask}},
volume = {119},
year = {2014}
}

@article{Darve2001,
abstract = {A new, general formula that connects the derivatives of the free energy along the selected, generalized coordinates of the system with the instantaneous force acting on these coordinates is derived. The instantaneous force is defined as the force acting on the coordinate of interest so that when it is subtracted from the equations of motion the acceleration along this coordinate is zero. The formula applies to simulations in which the selected coordinates are either unconstrained or constrained to fixed values. It is shown that in the latter case the formula reduces to the expression previously derived by den Otter and Briels [Mol. Phys. 98, 773 (2000)]. If simulations are carried out without constraining the coordinates of interest, the formula leads to a new method for calculating the free energy changes along these coordinates. This method is tested in two examples — rotation around the C–C bond of 1,2-dichloroethane immersed in water and transfer of fluoromethane across the water-hexane interface. The calculated free energies are compared with those obtained by two commonly used methods. One of them relies on determining the probability density function of finding the system at different values of the selected coordinate and the other requires calculating the average force at discrete locations along this coordinate in a series of constrained simulations. The free energies calculated by these three methods are in excellent agreement. The relative advantages of each method are discussed.},
author = {Darve, Eric and Pohorille, Andrew},
doi = {10.1063/1.1410978},
isbn = {0021-9606},
issn = {0021-9606},
journal = {J. Chem. Phys.},
number = {20},
pages = {9169--9183},
pmid = {172129300010},
title = {{Calculating free energies using average force}},
url = {http://dx.doi.org/10.1063/1.1410978},
volume = {115},
year = {2001}
}

@Book{Tibshirani1998,
  title     = {{An Introduction to the Bootstrap}},
  publisher = {Chapman and Hall/CRC},
  year      = {1998},
  author    = {Efron, B. and Tibshirani, R. J.},
  address   = {Boca Raton},
  owner     = {agrossf},
  timestamp = {2006.05.12},
}

@Article{Flyvbjerg-1989,
  author    = {Flyvbjerg, H. and Petersen, H. G.},
  title     = {Error estimates on averages of correlated data},
  journal   = {J. Chem. Phys.},
  year      = {1989},
  volume    = {91},
  number    = {1},
  pages     = {461--466},
  owner     = {agrossf},
  timestamp = {2006.05.13},
  doi       = {10.1063/1.457480},
  url       = {http://dx.doi.org/10.1063/1.457480},
  issn      = {0021-9606}
}

@Misc{LOOS,
  author    = {Romo, T. D. and Grossfield, A.},
  title     = {{LOOS}: A lightweight object-oriented software library. Version 2.3.2},
  year      = {2017},
  comment   = {Version 2.3.2},
  owner     = {alan},
  timestamp = {2011.01.06},
  url       = {https://github.com/GrossfieldLab/loos},
}

@Article{LOOS-JCC,
  author      = {Romo, Tod D. and Leioatts, Nicholas and Grossfield, Alan},
  title       = {Lightweight object oriented structure analysis: tools for building tools to analyze molecular dynamics simulations.},
  journal     = {J. Comput. Chem.},
  year        = {2014},
  volume      = {35},
  number      = {32},
  pages       = {2305--2318},
  abstract    = {LOOS (Lightweight Object Oriented Structure-analysis) is a C++ library designed to facilitate making novel tools for analyzing molecular dynamics simulations by abstracting out the repetitive tasks, allowing developers to focus on the scientifically relevant part of the problem. LOOS supports input using the native file formats of most common biomolecular simulation packages, including CHARMM, NAMD, Amber, Tinker, and Gromacs. A dynamic atom selection language based on the C expression syntax is included and is easily accessible to the tool-writer. In addition, LOOS is bundled with over 140 prebuilt tools, including suites of tools for analyzing simulation convergence, three-dimensional histograms, and elastic network models. Through modern C++ design, LOOS is both simple to develop with (requiring knowledge of only four core classes and a few utility functions) and is easily extensible. A python interface to the core classes is also provided, further facilitating tool development.},
  doi         = {10.1002/jcc.23753},
  institution = {Department of Biochemistry and Biophysics, University of Rochester Medical Center, Rochester, New York, 14642.},
  language    = {eng},
  medline-pst = {ppublish},
  owner       = {alan},
  pmid        = {25327784},
  timestamp   = {2015.05.21},
  Url         = {http://dx.doi.org/10.1002/jcc.23753},
  issn        = {0192-8651}
}

@article{Yang2004,
abstract = {A method is proposed for improving the accuracy and efficiency of free energy simulations. The essential idea is that the convergence of the relevant measure (e.g., the free energy derivative in thermodynamic integration) is monitored in the reverse direction starting from the last frame of the trajectory, instead of the usual approach, which begins with the first frame and goes in the forward direction. This simple change in the use of the simulation data makes it straightforward to eliminate the contamination of the averages by contributions from the equilibrating region. A statistical criterion is introduced for distinguishing the equilibrated (production) region from the equilibrating region. The proposed method, called reverse cumulative averaging, is illustrated by its application to the well-studied case of the alchemical free energy simulation of ethane to methanol.},
author = {Yang, Wei and Bitetti-Putzer, Ryan and Karplus, Martin},
doi = {10.1063/1.1638996},
url = {http://dx.doi.org/10.1063/1.1638996},
isbn = {0021-9606},
issn = {0021-9606},
journal = {J. Chem. Phys.},
number = {6},
pages = {2618--2628},
pmid = {15268405},
title = {{Free energy simulations: Use of reverse cumulative averaging to determine the equilibrated region and the time required for convergence}},
volume = {120},
year = {2004}
}

@Article{Chodera-2016,
  author        = {Chodera, John D},
  title         = {A simple method for automated equilibration detection in molecular simulations},
  journal       = {J. Chem. Theory Comput.},
  year          = {2016},
  volume        = {12},
  number        = {4},
  pages         = {1799--1805},
  __markedentry = {[alan:6]},
  owner         = {alan},
  publisher     = {ACS Publications},
  timestamp     = {2017.09.21},
  doi           = {10.1021/acs.jctc.5b00784},
  url           = {http://dx.doi.org/10.1021/acs.jctc.5b00784},
  issn          = {1549-9618}
}

@article{Roe2013,
  author = {Roe, Daniel R. and Cheatham, Thomas E.},
  title = {{PTRAJ and CPPTRAJ: Software for Processing and Analysis of Molecular Dynamics Trajectory Data}},
  journal = {J. Chem. Theory Comput.},
  volume = {9},
  number = {7},
  pages = {3084--3095},
  year = {2013},
  doi = {10.1021/ct400341p},
  pmid ={26583988},
  URL = {http://dx.doi.org/10.1021/ct400341p},
  eprint = { http://dx.doi.org/10.1021/ct400341p },
  issn = {1549-9618}
}

@article{patrone1,
title = "Uncertainty quantification in molecular dynamics studies of the glass transition temperature ",
journal = "Polymer ",
volume = "87",
pages = "246--259",
year = "2016",
issn = "0032-3861",
doi = "10.1016/j.polymer.2016.01.074",
url = "http://dx.doi.org/10.1016/j.polymer.2016.01.074",
author = "Paul N. Patrone and Andrew Dienstfrey and Andrea R. Browning and Samuel Tucker and Stephen Christensen"
}

@Article{patrone3,
author="Boettinger, W. J.
and Williams, M. E.
and Moon, K.-W.
and McFadden, G. B.
and Patrone, P. N.
and Perepezko, J. H.",
title="Interdiffusion in the Ni-Re System: Evaluation of Uncertainties",
journal="J Phase Equilib. Diff.",
year="2017",
day="01",
volume="38",
number="5",
pages="750--763",
abstract="Diffusion couple experiments between Ni and Re at 1200 and 1350 {\textdegree}C were performed. These experiments established the limits of the two-phase FCC + HCP region. No intermediate phase was observed at these temperatures. Composition-dependent interdiffusion coefficients and associated uncertainties were estimated by three methods. The first employed fitting of the penetration curves in conjunction with the Sauer-Freise (SF) method. The second method employed a numerical solution of the Boltzmann-Matano ordinary differential equation for composition-dependent interdiffusion coefficient functions whose parameters were optimized by a least squares fitting to the data. Discrepancies between the results of these methods indicate typical uncertainties in experimental determination of diffusion coefficients. To further assess such discrepancies, a third method was employed to perform an uncertainty quantification of the diffusion coefficients via a statistical analysis based on the SF method.",
issn="1547-7037",
doi="10.1007/s11669-017-0562-7",
Url="http://dx.doi.org/10.1007/s11669-017-0562-7"
}

@article{patrone2,
title = "Estimating yield-strain via deformation-recovery simulations",
journal = "Polymer",
volume = "116",
pages = "295--303",
year = "2017",
issn = "0032-3861",
doi = "10.1016/j.polymer.2017.03.046",
url = "http://dx.doi.org/10.1016/j.polymer.2017.03.046",
author = "Paul N. Patrone and Samuel Tucker and Andrew Dienstfrey",
keywords = "Yield strain",
keywords = "Molecular dynamics",
keywords = "Uncertainty quantification"
}

@article{Kolafa1986,
  author = { Jiří Kolafa },
  title = {Autocorrelations and subseries averages in Monte Carlo Simulations},
  journal = {Mol. Phys.},
  volume = {59},
  number = {5},
  pages = {1035--1042},
  year  = {1986},
  publisher = {Taylor \& Francis},
  doi = {10.1080/00268978600102561},
  url = {http://dx.doi.org/10.1080/00268978600102561},
  issn = {0026-8976}
}

@Book{FrenkelSmit2002,
  author = {Daan Frenkel and Berend Smit},
  title = {Understanding Molecular Simulation: From Algorithms to Applications},
  publisher = {Academic Press},
  year = {2002},
  address = {New York}
}

@article{Friedberg1970,
  author = {R. Friedberg and J. E. Cameron},
  title = {Test of the Monte Carlo Method: Fast Simulation of a Small Ising Lattice},
  journal = {J. Chem. Phys.},
  volume = {52},
  number = {12},
  pages = {6049--6058},
  year = {1970},
  doi = {10.1063/1.1672907},
  url = {http://dx.doi.org/10.1063/1.1672907},
  issn = {0021-9606}
}

@Book{Taylor1997,
  author = {John R. Taylor},
  title = {An Introduction to Error Analysis},
  publisher = {University Science Books},
  year = {1997},
  address = {Sausalito, California},
}

@book{Rasmussen,
 author = {Rasmussen, Carl Edward and Williams, Christopher K. I.},
 title = {Gaussian Processes for Machine Learning (Adaptive Computation and Machine Learning)},
 year = {2005},
 isbn = {026218253X},
 publisher = {The MIT Press},
} 

@Article{Boettinger2017,
author="Boettinger, W. J.
and Williams, M. E.
and Moon, K.-W.
and McFadden, G. B.
and Patrone, P. N.
and Perepezko, J. H.",
title="Interdiffusion in the Ni-Re System: Evaluation of Uncertainties",
journal="J. Phase Equilib. Diff.",
year="2017",
day="01",
volume="38",
number="5",
pages="750--763",
issn="1547-7037",
doi="10.1007/s11669-017-0562-7",
url="http://dx.doi.org/10.1007/s11669-017-0562-7"
}

@article{Henriksen2013,
author = {Henriksen, Niel M. and Roe, Daniel R. and Cheatham, Thomas E.},
title = {{Reliable Oligonucleotide Conformational Ensemble Generation in Explicit Solvent for Force Field Assessment Using Reservoir Replica Exchange Molecular Dynamics Simulations}},
journal = {J. Phys. Chem. B},
volume = {117},
number = {15},
pages = {4014--4027},
year = {2013},
doi = {10.1021/jp400530e},
pmid ={23477537},
URL = {http://dx.doi.org/10.1021/jp400530e},
eprint = { http://dx.doi.org/10.1021/jp400530e },
issn = {1520-6106},
}

@article{Roe2014,
author = {Roe, Daniel R. and Bergonzo, Christina and Cheatham, Thomas E.},
title = {{Evaluation of Enhanced Sampling Provided by Accelerated Molecular Dynamics with Hamiltonian Replica Exchange Methods}},
journal = {J. Phys. Chem. B},
volume = {118},
number = {13},
pages = {3543--3552},
year = {2014},
doi = {10.1021/jp4125099},
issn = {1520-6106},
pmid = {24625009},
URL = {http://dx.doi.org/10.1021/jp4125099},
eprint = { http://dx.doi.org/10.1021/jp4125099 }
}

@article{Zuckerman2011,
abstract = {Equilibrium sampling of biomolecules remains an unmet challenge after more than 30 years of atomistic simulation. Efforts to enhance sampling capability, which are reviewed here, range from the development of new algorithms to parallelization to novel uses of hardware. Special focus is placed on classifying algorithms--most of which are underpinned by a few key ideas--in order to understand their fundamental strengths and limitations. Although algorithms have proliferated, progress resulting from novel hardware use appears to be more clear-cut than from algorithms alone, due partly to the lack of widely used sampling measures.},
annote = {PMC Exempt - invited review},
author = {Zuckerman, Daniel M},
doi = {10.1146/annurev-biophys-042910-155255},
institution = {Department of Computational and Systems Biology, University of Pittsburgh, Pittsburgh, PA 15213, USA. ddmmzz@pitt.edu},
journal = {Annu. Rev. Biophys.},
keywords = {Algorithms; Animals; Biopolymers,Biological; Models,Chemical; Models,Statistical; Sample Size; Thermodynamics,chemistry/metabolism; Computer Simulation; Humans},
pages = {41--62},
number = {1},
pmid = {21370970},
title = {{Equilibrium sampling in biomolecular simulations.}},
Url = {http://dx.doi.org/10.1146/annurev-biophys-042910-155255},
volume = {40},
year = {2011},
issn = {1936-122X},
}

@article{Huber-1996,
author = {Huber, G A and Kim, S},
journal = {Biophys. J.},
pages = {97--110},
title = {{Weighted-ensemble Brownian dynamics simulations for protein association reactions}},
volume = {70},
number = {1},
year = {1996},
doi = {10.1016/S0006-3495(96)79552-8},
url = {http://dx.doi.org/10.1016/S0006-3495(96)79552-8},
issn = {0006-3495}
}

@article{Zhang2010a,
abstract = {The "weighted ensemble" method, introduced by Huber and Kim [Biophys. J. 70, 97 (1996)], is one of a handful of rigorous approaches to path sampling of rare events. Expanding earlier discussions, we show that the technique is statistically exact for a wide class of Markovian and non-Markovian dynamics. The derivation is based on standard path-integral (path probability) ideas, but recasts the weighted-ensemble approach as simple "resampling" in path space. Similar reasoning indicates that arbitrary nonstatic binning procedures, which merely guide the resampling process, are also valid. Numerical examples confirm the claims, including the use of bins which can adaptively find the target state in a simple model.},
pmid = {PMC2830257},
author = {Zhang, Bin W and Jasnow, David and Zuckerman, Daniel M},
doi = {10.1063/1.3306345},
institution = {Department of Computational Biology, School of Medicine, University of Pittsburgh, Pennsylvania 15260, USA.},
journal = {J. Chem. Phys.},
keywords = {Biological; Models,Chemical; Protein Binding,Computational Biology; Models,genetics; Protein Folding; Stochastic Processes},
number = {5},
pages = {054107},
pmid = {20136305},
title = {{The "weighted ensemble" path sampling method is statistically exact for a broad class of stochastic processes and binning procedures.}},
Url = {http://dx.doi.org/10.1063/1.3306345},
volume = {132},
year = {2010},
issn = {0021-9606}
}

@article{Suarez2014,
abstract = {Equilibrium formally can be represented as an ensemble of uncoupled systems undergoing unbiased dynamics in which detailed balance is maintained. Many nonequilibrium processes can be described by suitable subsets of the equilibrium ensemble. Here, we employ the "weighted ensemble" (WE) simulation protocol [Huber and Kim, Biophys. J. 1996, 70, 97-110] to generate equilibrium trajectory ensembles and extract nonequilibrium subsets for computing kinetic quantities. States do not need to be chosen in advance. The procedure formally allows estimation of kinetic rates between arbitrary states chosen after the simulation, along with their equilibrium populations. We also describe a related history-dependent matrix procedure for estimating equilibrium and nonequilibrium observables when phase space has been divided into arbitrary non-Markovian regions, whether in WE or ordinary simulation. In this proof-of-principle study, these methods are successfully applied and validated on two molecular systems: explicitly solvated methane association and the implicitly solvated Ala4 peptide. We comment on challenges remaining in WE calculations.},
author = {Su{\'{a}}rez, Ernesto and Lettieri, Steven and Zwier, Matthew C and Stringer, Carsen A and Subramanian, Sundar Raman and Chong, Lillian T and Zuckerman, Daniel M},
doi = {10.1021/ct401065r},
institution = {Department of Computational and Systems Biology, University of Pittsburgh , 4200 Fifth Ave, Pittsburgh, Pennsylvania 15260, United States.},
journal = {J. Chem. Theory Comput.},
number = {7},
pages = {2658--2667},
pmid = {25246856},
title = {{Simultaneous Computation of Dynamical and Equilibrium Information Using a Weighted Ensemble of Trajectories.}},
Url = {http://dx.doi.org/10.1021/ct401065r},
volume = {10},
year = {2014},
issn = {1549-9618}
}

@article{Bhatt2010a,
pmid = {PMC2912933},
author = {Bhatt, Divesh and Zhang, Bin W and Zuckerman, Daniel M},
journal = {J. Chem. Phys.},
pages = {014110},
title = {{Steady state via weighted ensemble path sampling}},
volume = {133},
number = {1},
year = {2010},
doi = {10.1063/1.3456985},
url = {http://dx.doi.org/10.1063/1.3456985},
issn = {0021-9606}
}


@TechReport{JCGM:GUM2008,
  author = {JCGM},
  title = {JCGM 100: Evaluation of measurement data - Guide to the expression of uncertainty in measurement},
  institution =  {Joint Committee for Guides in Metrology},
  year = {2008},
  url = {https://www.bipm.org/utils/common/documents/jcgm/JCGM_100_2008_E.pdf},
  Comment  = {Accessed 2 April 2018},
}

@TechReport{JCGM:VIM2012,
  author = {JCGM},
  title = {JCGM 200: International vocabulary of metrology - Basic and general concepts and associated terms (VIM)},
  institution =  {Joint Committee for Guides in Metrology},
  year = {2012},
  url = {https://www.bipm.org/utils/common/documents/jcgm/JCGM_200_2012.pdf},
  Comment  = {Accessed 2 April 2018},
}

@article{Pitera2014,
author = {Pitera, Jed W.},
title = {{Expected Distributions of Root-Mean-Square Positional Deviations in Proteins}},
journal = {J. Phys. Chem. B},
volume = {118},
number = {24},
pages = {6526--6530},
year = {2014},
doi = {10.1021/jp412776d},
pmid = {24655018},
URL = {http://dx.doi.org/10.1021/jp412776d},
eprint = { http://dx.doi.org/10.1021/jp412776d },
issn = {1520-6106}
}

@article{Bergonzo2014,
author = {Bergonzo, Christina and Henriksen, Niel M. and Roe, Daniel R. and Swails, Jason M. and Roitberg, Adrian E. and Cheatham, Thomas E.},
title = {Multidimensional Replica Exchange Molecular Dynamics Yields a Converged Ensemble of an {RNA} Tetranucleotide},
journal = {J. Chem. Theory Comput.},
volume = {10},
number = {1},
pages = {492--499},
year = {2014},
doi = {10.1021/ct400862k},
pmid = {24453949},
URL = {http://dx.doi.org/10.1021/ct400862k},
eprint = { http://dx.doi.org/10.1021/ct400862k },
issn = {1549-9618}
}


@Article{Quenouille_Notes_1956,
  Title                    = {Notes on Bias in Estimation},
  Author                   = {Quenouille, M. H.},
  Journal                  = {Biometrika},
  Year                     = {1956},
  Pages                    = {353--360},
  Volume                   = {43},
  number                   = {3-4},
  Doi                      = {10.1093/biomet/43.3-4.353},
  Url                      = {http://dx.doi.org/10.1093/biomet/43.3-4.353},
  Owner                    = {dsideriu},
  Timestamp                = {2017.12.15},
  issn                     = {0006-3444}
}

@Article{Quenouille_Approximate_1949,
  Title                    = {Approximate Tests of Correlation in Time-Series},
  Author                   = {Quenouille, M. H.},
  Journal                  = {J. Roy. Stat. Soc. B Met.},
  Year                     = {1949},
  Pages                    = {68--84},
  Volume                   = {11},
  Number                   = {1},
  url                      = {http://www.jstor.org/stable/2983696},
  Owner                    = {dsideriu},
  Timestamp                = {2017.12.15}
}

@Article{Tukey_Bias_1958,
  Title                    = {Bias and confidence in not quite large samples (abstract)},
  Author                   = {Tukey, J. W.},
  Journal                  = {Ann. Math. Stat.},
  Year                     = {1958},
  Pages                    = {614--623},
  Volume                   = {29},
  Number                   = {2},
  Doi                      = {10.1214/aoms/1177706647},
  Url                      = {http://dx.doi.org/10.1214/aoms/1177706647},
  Owner                    = {dsideriu},
  Timestamp                = {2017.12.15}
}

@article{Merchant2011,
author = {Merchant, Bonnie A. and Madura, Jeffry D.},
doi = {10.1016/B978-0-444-53835-2.00003-1},
journal = {Annu. Rep. Comput. Chem.},
pages = {67--87},
title = {{A Review of Coarse-Grained Molecular Dynamics Techniques to Access Extended Spatial and Temporal Scales in Biomolecular Simulations}},
Url = {http://dx.doi.org/10.1016/B978-0-444-53835-2.00003-1},
volume = {7},
year = {2011},
issn = {1574-1400}
}

@article{Kmiecik2016,
author = {Kmiecik, Sebastian and Gront, Dominik and Kolinski, Michal and Wieteska, Lukasz and Dawid, Aleksandra Elzbieta and Kolinski, Andrzej},
title = {{Coarse-Grained Protein Models and Their Applications}},
journal = {Chem. Rev.},
volume = {116},
number = {14},
pages = {7898--7936},
year = {2016},
doi = {10.1021/acs.chemrev.6b00163},
URL = {http://dx.doi.org/10.1021/acs.chemrev.6b00163},
issn = {0009-2665}
}

@Book{NIST_Sematech_eHandbook,
  Title                    = {NIST/SEMATECH e-Handbook of Statistical Methods},
  Editor                   = {Croarkin, Carroll and Tobias, Paul},
  Publisher                = {National Institute of Standards and Technology},
  Url                      = {http://www.itl.nist.gov/div898/handbook/},
  Year                     = {2012},
  Comment                  = {Accessed 15 February 2018},
}

@article{Okur2006,
author = {Okur, Asim and Wickstrom, Lauren and Layten, Melinda and Geney, Raphäel and Song, Kun and Hornak, Viktor and Simmerling, Carlos},
title = {Improved Efficiency of Replica Exchange Simulations through Use of a Hybrid Explicit/Implicit Solvation Model},
journal = {J. Chem. Theory Comput.},
volume = {2},
number = {2},
pages = {420--433},
year = {2006},
doi = {10.1021/ct050196z},
pmid = {26626529},
URL = {http://dx.doi.org/10.1021/ct050196z},
eprint = { http://dx.doi.org/10.1021/ct050196z },
issn = {1549-9618}
}

@article{Shao2007,
author = {Shao, Jianyin and Tanner, Stephen W. and Thompson, Nephi and Cheatham, Thomas E.},
title = {Clustering Molecular Dynamics Trajectories: 1. Characterizing the Performance of Different Clustering Algorithms},
journal = {J. Chem. Theory Comput.},
volume = {3},
number = {6},
pages = {2312--2334},
year = {2007},
doi = {10.1021/ct700119m},
pmid = {26636222},
URL = {http://dx.doi.org/10.1021/ct700119m},
eprint = { http://dx.doi.org/10.1021/ct700119m },
issn = {1549-9618}
}

@Book{NIST_SRSW,
  Title                    = {{NIST} {S}tandard {R}eference {S}imulation {W}ebsite, NIST Standard Reference Database Number 173},
  Editor                   = {Shen, V. K. and Siderius, D. W. and Krekelberg, W. P. and Hatch, H. W.},
  Publisher                = {National Institute of Standards and Technology},
  Year                     = {2006},
  Address                  = {Gaithersburg, MD, 20899},
  Comment                  = {Accessed 27 March 2018},
  Url                      = {http://dx.doi.org/10.18434/T4M88Q},
  doi                   = {10.18434/T4M88Q},
}

@article{Huber1994,
author = {Huber, T. and Torda, A. E. and van Gunsteren, W. F.},
journal = {J. Comput.-Aided Mol. Des.},
number = {6},
pages = {695--708},
title = {{Local elevation: A method for improving the searching properties of molecular dynamics simulation}},
volume = {8},
year = {1994},
doi = {10.1007/BF00124016}
}

@article{Grubmueller1995,
  title = {{Predicting slow structural transitions in macromolecular systems: Conformational flooding}},
  author = {Grubm\"uller, Helmut},
  journal = {Phys. Rev. E},
  volume = {52},
  issue = {3},
  pages = {2893--2906},
  year = {1995},
  month = {Sep},
  publisher = {American Physical Society},
  doi = {10.1103/PhysRevE.52.2893},
  url = {https://link.aps.org/doi/10.1103/PhysRevE.52.2893}
}

@book{smith2013uncertainty,
  title={Uncertainty Quantification: Theory, Implementation, and Applications},
  author={Smith, R.C.},
  isbn={9781611973228},
  series={Computational Science and Engineering},
  url={https://books.google.com/books?id=Tc1GAgAAQBAJ},
  year={2013},
  publisher={SIAM}
}


@article{SMC,
author = {Patrone,Paul N.  and Rosch,Thomas W. },
title = {Beyond histograms: Efficiently estimating radial distribution functions via spectral Monte Carlo},
journal = {J. Chem. Phys.},
volume = {146},
number = {9},
pages = {094107},
year = {2017},
doi = {10.1063/1.4977516}
}


@article{acfnoise,
author = {Kim,Kang-Sahn  and Han,Myung Hoon  and Kim,Changho  and Li,Zhen  and Karniadakis,George Em  and Lee,Eok Kyun },
title = {Nature of intrinsic uncertainties in equilibrium molecular dynamics estimation of shear viscosity for simple and complex fluids},
journal = {J. Chem. Phys.},
volume = {149},
number = {4},
pages = {044510},
year = {2018},
doi = {10.1063/1.5035119}
}

@Article{Gowers_Automated_2018,
  author    = {Gowers, R. J. and Farmahini, A. H. and Friedrich, D. and Sarkisov, L.},
  title     = {Automated analysis and benchmarking of {GCMC} simulation programs in application to gas adsorption},
  journal   = {Mol. Simul.},
  year      = {2017},
  volume    = {44},
  issue     = {4},
  pages     = {309--321},
  doi       = {10.1080/08927022.2017.1375492},
  owner     = {dsideriu},
  timestamp = {2018.09.26},
}

@InCollection{Janke_Statistical_2002,
  author    = {Janke, W.},
  title     = {Statistical analysis of simulations: data correlations and error estimation},
  booktitle = {Quantum simulations of complex many-body systems: from theory to algorithms},
  year      = {2002},
  editor    = {J. Grotendorst and D. Marx and A. Muramatsu},
  owner     = {dsideriu},
  timestamp = {2018.09.26},
  publisher = {John von Neumann Institute for Computing},
  address   = {J\"ulich},
  pages     = {423--445}
}

@Article{Shirts_Statistically_2008,
  author    = {Shirts, M. R. and Chodera, J. D.},
  title     = {Statistically optimal analysis of samples from multiple equilibrium states},
  journal   = {J. Chem. Phys.},
  year      = {2008},
  volume    = {129},
  issue     = {12},
  pages     = {124105},
  doi       = {10.1063/1.2978177},
  owner     = {dsideriu},
  timestamp = {2018.09.26},
}