Christopher P. Calderon

Quantifying DNA melting transitions using single-molecule force spectroscopy

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.

Analyzing Single-Molecule Manipulation Experiments

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. Copyright

Fitting Effective Diffusion Models to Data Associated with a “Glassy" Potential: Estimation, Classical Inference Procedures, and Some Heuristics

A variety of researchers have successfully obtained the parameters of low-dimensional diffusion models using the data that results from atomistic simulations. This raises a variety of questions about efficient estimation, goodness-of-fit tests, and confidence interval estimation. Long time series data and knowledge of the transition density associated with the assumed process model can be of great assistance in obtaining answers to the aforementioned questions. Unfortunately, a closed-form expression for the transition density is rarely available. Estimation and inference are carried out using approximated transition densities (the primary interest is in how Ait-Sahalias transition density expansions perform in various multiscale contexts). A heuristic strategy for estimating a nonlinear diffusion which does not require one to assume the exact functional form of the diffusion coefficients is also presented. The final part of this study explores how the above findings can be used to help construct a diffu...

Using stochastic models calibrated from nanosecond nonequilibrium simulations to approximate mesoscale information

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)].

Quantifying Transient 3D Dynamical Phenomena of Single mRNA Particles in Live Yeast Cell Measurements

Single-particle tracking (SPT) has been extensively used to obtain information about diffusion and directed motion in a wide range of biological applications. Recently, new methods have appeared for obtaining precise (10s of nm) spatial information in three dimensions (3D) with high temporal resolution (measurements obtained every 4 ms), which promise to more accurately sense the true dynamical behavior in the natural 3D cellular environment. Despite the quantitative 3D tracking information, the range of mathematical methods for extracting information about the underlying system has been limited mostly to mean-squared displacement analysis and other techniques not accounting for complex 3D kinetic interactions. There is a great need for new analysis tools aiming to more fully extract the biological information content from in vivo SPT measurements. High-resolution SPT experimental data has enormous potential to objectively scrutinize various proposed mechanistic schemes arising from theoretical biophysics and cell biology. At the same time, methods for rigorously checking the statistical consistency of both model assumptions and estimated parameters against observed experimental data (i.e., goodness-of-fit tests) have not received great attention. We demonstrate methods enabling (1) estimation of the parameters of 3D stochastic differential equation (SDE) models of the underlying dynamics given only one trajectory; and (2) construction of hypothesis tests checking the consistency of the fitted model with the observed trajectory so that extracted parameters are not overinterpreted (the tools are applicable to linear or nonlinear SDEs calibrated from nonstationary time series data). The approach is demonstrated on high-resolution 3D trajectories of single ARG3 mRNA particles in yeast cells in order to show the power of the methods in detecting signatures of transient directed transport. The methods presented are generally relevant to a wide variety of 2D and 3D SPT tracking applications.

Inferring Latent States and Refining Force Estimates via Hierarchical Dirichlet Process Modeling in Single Particle Tracking Experiments.

Understanding the basis for intracellular motion is critical as the field moves toward a deeper understanding of the relation between Brownian forces, molecular crowding, and anisotropic (or isotropic) energetic forcing. Effective forces and other parameters used to summarize molecular motion change over time in live cells due to latent state changes, e.g., changes induced by dynamic micro-environments, photobleaching, and other heterogeneity inherent in biological processes. This study discusses limitations in currently popular analysis methods (e.g., mean square displacement-based analyses) and how new techniques can be used to systematically analyze Single Particle Tracking (SPT) data experiencing abrupt state changes in time or space. The approach is to track GFP tagged chromatids in metaphase in live yeast cells and quantitatively probe the effective forces resulting from dynamic interactions that reflect the sum of a number of physical phenomena. State changes can be induced by various sources including: microtubule dynamics exerting force through the centromere, thermal polymer fluctuations, and DNA-based molecular machines including polymerases and protein exchange complexes such as chaperones and chromatin remodeling complexes. Simulations aiming to show the relevance of the approach to more general SPT data analyses are also studied. Refined force estimates are obtained by adopting and modifying a nonparametric Bayesian modeling technique, the Hierarchical Dirichlet Process Switching Linear Dynamical System (HDP-SLDS), for SPT applications. The HDP-SLDS method shows promise in systematically identifying dynamical regime changes induced by unobserved state changes when the number of underlying states is unknown in advance (a common problem in SPT applications). We expand on the relevance of the HDP-SLDS approach, review the relevant background of Hierarchical Dirichlet Processes, show how to map discrete time HDP-SLDS models to classic SPT models, and discuss limitations of the approach. In addition, we demonstrate new computational techniques for tuning hyperparameters and for checking the statistical consistency of model assumptions directly against individual experimental trajectories; the techniques circumvent the need for “ground-truth” and/or subjective information.

Robust hypothesis tests for detecting statistical evidence of two-dimensional and three-dimensional interactions in single-molecule measurements.

Experimental advances have improved the two- (2D) and three-dimensional (3D) spatial resolution that can be extracted from in vivo single-molecule measurements. This enables researchers to quantitatively infer the magnitude and directionality of forces experienced by biomolecules in their native environment. Situations where such force information is relevant range from mitosis to directed transport of protein cargo along cytoskeletal structures. Models commonly applied to quantify single-molecule dynamics assume that effective forces and velocity in the x,y (or x,y,z) directions are statistically independent, but this assumption is physically unrealistic in many situations. We present a hypothesis testing approach capable of determining if there is evidence of statistical dependence between positional coordinates in experimentally measured trajectories; if the hypothesis of independence between spatial coordinates is rejected, then a new model accounting for 2D (3D) interactions can and should be considered. Our hypothesis testing technique is robust, meaning it can detect interactions, even if the noise statistics are not well captured by the model. The approach is demonstrated on control simulations and on experimental data (directed transport of intraflagellar transport protein 88 homolog in the primary cilium).

A new computational method for ambiguity assessment of solutions to assignment problems arising in target tracking

We present a new approach to estimating the probability of each association in a 2D assignment problem defined by likelihood ratios. Our method divides the set of feasible hypotheses into clusters, and converts a collection of hypotheses into a collection of clusters containing them, reducing the variance of the estimate. Simulations show that our method often generates substantially more accurate probability estimates in less time than traditional methods. Our method can obtain reasonably accurate probabilities of association based on only the input cost matrix and single best candidate solution, eliminating the need for a K-best solution method or MCMC sampling.

Data-driven approach to decomposing complex enzyme kinetics with surrogate models

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.

Erratum: P-Splines Using Derivative Information

This is a correction to the authors article [Multiscale Model. Simul., 8 (2010), pp. 1562–1580].