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Echeverria I, Amzel LM. Estimation of Free-Energy Differences from Computed Work Distributions: An Application of Jarzynski’s Equality. J Phys Chem B 2012; 116:10986-95. [DOI: 10.1021/jp300527q] [Citation(s) in RCA: 13] [Impact Index Per Article: 1.1] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/28/2022]
Affiliation(s)
- Ignacia Echeverria
- Department of Biophysics
and Biophysical Chemistry,
School of Medicine, Johns Hopkins University, Baltimore, Maryland 21205, United States
| | - L. Mario Amzel
- Department of Biophysics
and Biophysical Chemistry,
School of Medicine, Johns Hopkins University, Baltimore, Maryland 21205, United States
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2
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Calderon CP. Estimation and Inference of Diffusion Coefficients in Complex Biomolecular Environments. J Chem Theory Comput 2011; 7:280-90. [DOI: 10.1021/ct1004966] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/29/2022]
Affiliation(s)
- Christopher P. Calderon
- Department of Computational and Applied Mathematics, Rice University, Houston, Texas 77005-1892, United States
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Nicolini P, Procacci P, Chelli R. Hummer and Szabo-like Potential of Mean Force Estimator for Bidirectional Nonequilibrium Pulling Experiments/Simulations. J Phys Chem B 2010; 114:9546-54. [DOI: 10.1021/jp102263y] [Citation(s) in RCA: 18] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/28/2022]
Affiliation(s)
- Paolo Nicolini
- Dipartimento di Chimica, Università di Firenze, Via della Lastruccia 3, I-50019 Sesto Fiorentino, Italy, European Laboratory for Nonlinear Spectroscopy (LENS), Via Nello Carrara 1, I-50019 Sesto Fiorentino, Italy, and Centro Interdipartimentale per lo Studio delle Dinamiche Complesse (CSDC), Via Sansone 1, I-50019 Sesto Fiorentino, Italy
| | - Piero Procacci
- Dipartimento di Chimica, Università di Firenze, Via della Lastruccia 3, I-50019 Sesto Fiorentino, Italy, European Laboratory for Nonlinear Spectroscopy (LENS), Via Nello Carrara 1, I-50019 Sesto Fiorentino, Italy, and Centro Interdipartimentale per lo Studio delle Dinamiche Complesse (CSDC), Via Sansone 1, I-50019 Sesto Fiorentino, Italy
| | - Riccardo Chelli
- Dipartimento di Chimica, Università di Firenze, Via della Lastruccia 3, I-50019 Sesto Fiorentino, Italy, European Laboratory for Nonlinear Spectroscopy (LENS), Via Nello Carrara 1, I-50019 Sesto Fiorentino, Italy, and Centro Interdipartimentale per lo Studio delle Dinamiche Complesse (CSDC), Via Sansone 1, I-50019 Sesto Fiorentino, Italy
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Echeverria I, Amzel LM. Helix propensities calculations for amino acids in alanine based peptides using Jarzynski's equality. Proteins 2010; 78:1302-10. [PMID: 20014024 DOI: 10.1002/prot.22649] [Citation(s) in RCA: 20] [Impact Index Per Article: 1.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/07/2022]
Abstract
Jarzynski's equality (Jarzynski, Phys Rev E 1997; 56:5018 and Jarzynski, Phys Rev Lett 1997; 78:2690) relates equilibrium free energy differences between two states A and B to the work done when the system is driven repeatedly and irreversibly from an equilibrium state A to equilibrium state B. We present calculations of helix propensities using a novel procedure based on this equality. In particular, a work probability distribution is built based on combinations of multi-step trajectories that give representative work distributions without requiring computing an unreasonable large number of trajectories between states. A small number of trajectories (15) are used to construct a distribution that contains millions of work values. This distribution is used to calculate DeltaG(AB) using Jarsynski's equation. To apply and test this method, we used as a model system a dodeca-alanine helix, analyzing its extension using mechanical force. This helix was used as the basis of a host guest system in which two of the 12 residues are substituted by some other amino acid (as the guests). The differences between the unfolding free energies of the substituted peptides and the all-alanine peptide provided values for DeltaDeltaG that can be interpreted as the helix propensities of each amino acid. Results show good correlation with the experimental measurements of Baldwin and coworkers (Chakrabartty et al., Protein Sci 1994; 3:843-852).
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Affiliation(s)
- Ignacia Echeverria
- Department of Biophysics and Biophysical Chemistry, Johns Hopkins University, School of Medicine, Baltimore, Maryland 21205, USA
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Calderon CP. Detection of Subtle Dynamical Changes Induced by Unresolved “Conformational Coordinates” in Single-Molecule Trajectories via Goodness-of-Fit Tests. J Phys Chem B 2010; 114:3242-53. [DOI: 10.1021/jp911124z] [Citation(s) in RCA: 8] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/29/2022]
Affiliation(s)
- Christopher P. Calderon
- High Performance Computing Research Department, Lawrence Berkeley National Laboratory, Berkeley, California 94720
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CALDERON CHRISTOPHERP, MARTINEZ JOSUEG, CARROLL RAYMONDJ, SORENSEN DANNYC. P-SPLINES USING DERIVATIVE INFORMATION. MULTISCALE MODELING & SIMULATION : A SIAM INTERDISCIPLINARY JOURNAL 2010; 8:1562-1580. [PMID: 21691592 PMCID: PMC3117255 DOI: 10.1137/090768102] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/30/2023]
Abstract
Time series associated with single-molecule experiments and/or simulations contain a wealth of multiscale information about complex biomolecular systems. We demonstrate how a collection of Penalized-splines (P-splines) can be useful in quantitatively summarizing such data. In this work, functions estimated using P-splines are associated with stochastic differential equations (SDEs). It is shown how quantities estimated in a single SDE summarize fast-scale phenomena, whereas variation between curves associated with different SDEs partially reflects noise induced by motion evolving on a slower time scale. P-splines assist in "semiparametrically" estimating nonlinear SDEs in situations where a time-dependent external force is applied to a single-molecule system. The P-splines introduced simultaneously use function and derivative scatterplot information to refine curve estimates. We refer to the approach as the PuDI (P-splines using Derivative Information) method. It is shown how generalized least squares ideas fit seamlessly into the PuDI method. Applications demonstrating how utilizing uncertainty information/approximations along with generalized least squares techniques improve PuDI fits are presented. Although the primary application here is in estimating nonlinear SDEs, the PuDI method is applicable to situations where both unbiased function and derivative estimates are available.
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Affiliation(s)
| | - JOSUE G. MARTINEZ
- Department of Statistics, Texas A&M University, College Station, TX 77843
| | - RAYMOND J. CARROLL
- Department of Statistics, Texas A&M University, College Station, TX 77843
| | - DANNY C. SORENSEN
- Department of Computational and Applied Mathematics, Rice University, Houston, TX 77005
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Calderon CP. Data-driven approach to decomposing complex enzyme kinetics with surrogate models. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2009; 80:061118. [PMID: 20365129 DOI: 10.1103/physreve.80.061118] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/29/2009] [Indexed: 05/29/2023]
Abstract
The temporal autocorrelation (AC) function associated with monitoring order parameters characterizing conformational fluctuations of an enzyme is analyzed using a collection of surrogate models. The surrogates considered are phenomenological stochastic differential equation (SDE) models. It is demonstrated how an ensemble of such surrogate models, each surrogate being calibrated from a single trajectory, indirectly contains information about unresolved conformational degrees of freedom. This ensemble can be used to construct complex temporal ACs associated with a "non-Markovian" process. The ensemble of surrogates approach allows researchers to consider models more flexible than a mixture of exponentials to describe relaxation times and at the same time gain physical information about the system. The relevance of this type of analysis to matching single-molecule experiments to computer simulations and how more complex stochastic processes can emerge from a mixture of simpler processes is also discussed. The ideas are illustrated on a toy SDE model and on molecular-dynamics simulations of the enzyme dihydrofolate reductase.
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Affiliation(s)
- Christopher P Calderon
- Department of Computational and Applied Mathematics, Rice University, Houston, Texas 77005, USA.
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Calderon CP, Harris NC, Kiang CH, Cox DD. Analyzing single-molecule manipulation experiments. J Mol Recognit 2009; 22:356-62. [PMID: 19479747 DOI: 10.1002/jmr.959] [Citation(s) in RCA: 17] [Impact Index Per Article: 1.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/10/2022]
Abstract
Single-molecule manipulation studies can provide quantitative information about the physical properties of complex biological molecules without ensemble artifacts obscuring the measurements. We demonstrate computational techniques which aim at more fully utilizing the wealth of information contained in noisy experimental time series. The "noise" comes from multiple sources e.g., inherent thermal motion, instrument measurement error, etc. The primary focus of this paper is a methodology that uses time domain based methods to extract the effective molecular friction from single-molecule pulling data. We studied molecules composed of eight tandem repeat titin I27 domains, but the modeling approaches have applicability to other single-molecule mechanical studies. The merits and challenges associated with applying such a computational approach to existing single-molecule manipulation data are also discussed.
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Affiliation(s)
- Christopher P Calderon
- Department of Computational and Applied Mathematics, Rice University, Houston, TX 77005, USA.
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Calderon CP, Janosi L, Kosztin I. Using stochastic models calibrated from nanosecond nonequilibrium simulations to approximate mesoscale information. J Chem Phys 2009; 130:144908. [PMID: 19368472 PMCID: PMC2832035 DOI: 10.1063/1.3106225] [Citation(s) in RCA: 15] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/12/2008] [Accepted: 02/23/2009] [Indexed: 11/14/2022] Open
Abstract
We demonstrate how the surrogate process approximation (SPA) method can be used to compute both the potential of mean force along a reaction coordinate and the associated diffusion coefficient using a relatively small number (10-20) of bidirectional nonequilibrium trajectories coming from a complex system. Our method provides confidence bands which take the variability of the initial configuration of the high-dimensional system, continuous nature of the work paths, and thermal fluctuations into account. Maximum-likelihood-type methods are used to estimate a stochastic differential equation (SDE) approximating the dynamics. For each observed time series, we estimate a new SDE resulting in a collection of SPA models. The physical significance of the collection of SPA models is discussed and methods for exploiting information in the population of estimated SPA models are demonstrated and suggested. Molecular dynamics simulations of potassium ion dynamics inside a gramicidin A channel are used to demonstrate the methodology, although SPA-type modeling has also proven useful in analyzing single-molecule experimental time series [J. Phys. Chem. B 113, 118 (2009)].
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Affiliation(s)
- Christopher P Calderon
- Department of Computational and Applied Mathematics, Rice University, Houston, Texas 77005, USA.
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Calderon CP, Chen WH, Lin KJ, Harris NC, Kiang CH. Quantifying DNA melting transitions using single-molecule force spectroscopy. JOURNAL OF PHYSICS. CONDENSED MATTER : AN INSTITUTE OF PHYSICS JOURNAL 2009; 21:34114. [PMID: 20046972 PMCID: PMC2684650 DOI: 10.1088/0953-8984/21/3/034114] [Citation(s) in RCA: 25] [Impact Index Per Article: 1.7] [Reference Citation Analysis] [Abstract] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/26/2023]
Abstract
We stretched a DNA molecule using atomic force microscope and quantified the mechanical properties associated with B and S forms of double-stranded DNA (dsDNA), molten DNA, and single-stranded DNA (ssDNA). We also fit overdamped diffusion models to the AFM time series and used these models to extract additional kinetic information about the system. Our analysis provides additional evidence supporting the view that S-DNA is a stable intermediate encountered during dsDNA melting by mechanical force. In addition, we demonstrated that the estimated diffusion models can detect dynamical signatures of conformational degrees of freedom not directly observed in experiments.
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Calderon CP, Harris NC, Kiang CH, Cox DD. Quantifying multiscale noise sources in single-molecule time series. J Phys Chem B 2009; 113:138-48. [PMID: 19072043 PMCID: PMC2682735 DOI: 10.1021/jp807908c] [Citation(s) in RCA: 20] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/28/2022]
Abstract
When analyzing single-molecule data, a low-dimensional set of system observables typically serves as the observational data. We calibrate stochastic dynamical models from time series that record such observables. Numerical techniques for quantifying noise from multiple time scales in a single trajectory, including experimental instrument and inherent thermal noise, are demonstrated. The techniques are applied to study time series coming from both simulations and experiments associated with the nonequilibrium mechanical unfolding of titin's I27 domain. The estimated models can be used for several purposes, (1) detect dynamical signatures of "rare events" by analyzing the effective diffusion and force as a function of the monitored observable, (2) quantify the influence that conformational degrees of freedom, which are typically difficult to directly monitor experimentally, have on the dynamics of the monitored observable, (3) quantitatively compare the inherent thermal noise to other noise sources, for example, instrument noise, variation induced by conformational heterogeneity, and so forth, (4) simulate random quantities associated with repeated experiments, and (5) apply pathwise, that is, trajectory-wise, hypothesis tests to assess the goodness-of-fit of the models and even detect conformational transitions in noisy signals. These items are all illustrated with several examples.
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Affiliation(s)
- Christopher P Calderon
- Department of Computational & Applied Mathematics, Rice University, Houston, Texas 77005-1892, USA.
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Calderon CP, Arora K. Extracting Kinetic and Stationary Distribution Information from Short MD Trajectories via a Collection of Surrogate Diffusion Models. J Chem Theory Comput 2009; 5:47-58. [PMID: 20046947 PMCID: PMC2739417 DOI: 10.1021/ct800282a] [Citation(s) in RCA: 8] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/29/2022]
Abstract
Low-dimensional stochastic models can summarize dynamical information and make long time predictions associated with observables of complex atomistic systems. Maximum likelihood based techniques for estimating low-dimensional surrogate diffusion models from relatively short time series are presented. It is found that a heterogeneous population of slowly evolving conformational degrees of freedom modulates the dynamics. This underlying heterogeneity results in a collection of estimated low-dimensional diffusion models. Numerical techniques for exploiting this finding to approximate skewed histograms associated with the simulation are presented. In addition, statistical tests are also used to assess the validity of the models and determine physically relevant sampling information, e.g. the maximum sampling frequency at which one can discretely sample from an atomistic time series and have a surrogate diffusion model pass goodness-of-fit tests. The information extracted from such analyses can possibly be used to assist umbrella sampling computations as well as help in approximating effective diffusion coefficients. The techniques are demonstrated on simulations of Adenylate Kinase.
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Affiliation(s)
- Christopher P. Calderon
- Department of Statistics and Department of Computational and Applied Mathematics, Rice University, Houston, TX 77005-1892, USA
| | - Karunesh Arora
- Department of Chemistry, Biophysics Program, University of Michigan, 930 North University Avenue, Ann Arbor, MI 48109 USA
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