R-3 Repository :: Browsing by Author "Calderon, Christopher P."

Browsing by Author "Calderon, Christopher P."

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A Random Force is a Force, of Course, of Coarse: Decomposing Complex Enzyme Kinetics with Surrogate Models
(2009-06) Calderon, Christopher P.
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.
Analyzing Single-Molecule Manipulation Experiments
(2008-10) Calderon, Christopher P.; Harris, Nolan C.; Kiang, Ching-Hwa; Cox, Dennis D.
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 article is a methodology for using time domain based methods for extracting the effective molecular friction from single-molecule pulling data. We studied molecules composed of 8 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.
Extracting Kinetic and Stationary Distribution Information from Short MD Trajectories via a Collection of Surrogate Diffusion Models
(2008-10) Calderon, Christopher P.; Arora, Karunesh
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.
Numerical Methods for Modeling Atomistic Trajectories with Diffusion SDEs
(2008-08) Calderon, Christopher P.; Martinez, Josue G.; Carroll, Raymond J.; Sorensen, Danny C.
The stochastic dynamics of small scale systems are often not known fromﾠa prioriﾠphysical considerations. We present data-driven numerical methods which can be used to approximate the nonlinear stochastic dynamics associated with time series of system observables. Given a single time series coming from a simulation or experiment, our approach uses maximum likelihood type estimates to obtain a sequence of local stochastic differential equations. The local models coming from one times series are then patched together using a penalized spline procedure. We provide an effcient algorithm for achieving this which utilizes estimates of the local parameter covariance. We also use goodness-of-fit tests to quantitatively determine when an overdamped Langevin approxi- mation can be used to describe the data. For situations where the overdamped approximation fails, we show that other diffusive models can still be used to approximate the dynamics. In addition, we also briefly discuss how variation observed in different curves, calibrated from different time series, can provide information about "hidden" conformational degrees of freedom not explicitly included in the model and how clustering these curves can help one in learning about the effective underlying free energy surface governing the dynamics of the atomistic system. The methods presented are applied to simulations modeling forced time-dependent transport of potassium transport through a gramicidinﾠAﾠchannel, but have applicability to other forced (and unforced) systems.
P-Splines Using Derivative Information
(2009-04) Calderon, Christopher P.; Martinez, Josue G.; Carroll, Raymond J.; Sorensen, Danny C.
Time series associated with single-molecule experiments and/or simulations contain a wealth of multiscale information about complex biochemical systems. However efficiently extracting and representing useful physical information from these time series measurements can be challenging. We demonstrate how Penalized splines (P-Splines) can be useful in summarizing complex single-molecule time series data using quantities estimated from the observed data. A design matrix that simultaneously uses noisy function and derivative scatterplot information to refine function estimates using P-spline techniques is introduced. The approach is called the PuDI (P-Splines using Derivative Information) method. We show how Generalized Least Squares fits seamlessly into the PuDI method; several applications demonstrating how inclusion of uncertainty information improves the PuDI function estimates are presented. The PuDI design matrix can be used to assist scatterplot smoothing applications where both unbiased function and derivative estimates are available.
PSQR: A Stable and Efficient Penalized Spline Algorithm
(2009-05) Calderon, Christopher P.; Martinez, Josue G.; Carroll, Raymond J.; Sorensen, Danny C.
We introduce an algorithm for reliably computing quantities associated with several types of semiparametric mixed models in situations where the condition number on the random effects matrix is large. The algorithm is numerically stable and efficient. It was designed to process penalized spline (P-spline) models without making unnecessary numerical approximations. The algorithm, PSQR (P-splines via QR), is formulated in terms of QR decompositions. PSQR can treat both exactly rank deficient and ill-conditioned matrices. The latter situation often arises in large scale mixed models and/or when a P-spline is estimated using a basis with poor numerical properties, e.g. a truncated power function (TPF) basis. We provide concrete examples where unnecessary numerical approximations introduce both subtle and dramatic errors that would likely go undetected, thus demonstrating the importance of using this reliable numerical algorithm. Simulation results studying a univariate function and a longitudinal data set are used to demonstrate the algorithm. Extensions and the utility of the method in more general semiparametric regression applications are briefly discussed. MATLAB scripts demonstrating implementation are provided in the Supplemental Materials.
Quantifying DNA Melting Transitions Using Single-Molecule Force Spectroscopy
(2008-09) Calderon, Christopher P.; Chen, Wei-Hung; Lin, Kuan-Jiuh; Harris, Nolan C.; Kiang, Ching-Hwa
We stretched a DNA molecule using atomic force microscope and quantified the mechanical properties associated withﾠBandﾠSﾠforms 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.
Quantifying Multiscale Noise Sources in Single-Molecule Time Series
(2008-09) Calderon, Christopher P.; Harris, Nolan C.; Kiang, Ching-Hwa; Cox, Dennis D.
When analyzing single-molecule data, a low-dimensional set of system observables typically serve 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, e.g. instrument noise, variation induced by conformational heterogeneity, etc., (4) simulate random quantities associated with repeated experiments, (5) apply pathwise, i.e. 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.
Using Stochastic Models Calibrated from Nanosecond Non-Equilibrium Simulations to Approximate Mesoscale Information
(2008-10) Calderon, Christopher P.; Janosi, Lorant; Kosztin, Ioan
We demonstrate how the surrogate process approximation (SPA) method can be used to compute both the potential of mean force (PMF) along a reaction coordinate and the associated diffusion coefficient using a relatively small (10-20) set of bidirectional non-equilibrium trajectories coming from a complex system. Our method provides confidence bands which take the variability of the initial condition, 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 and assist in these computations. 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 this population of models are suggested. Molecular dynamics(MD) 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,ﾠ113, 118 (2009)].

Browsing by Author "Calderon, Christopher P."

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