Fisher information metric for the Langevin equation and least informative models of continuous stochastic dynamics

Single-Molecule Imaging of Protein Interactions and Dynamics

Live-cell single-molecule fluorescence imaging has become a powerful analytical tool to investigate cellular processes that are not accessible to conventional biochemical approaches. This has greatly enriched our understanding of the behaviors of single biomolecules in their native environments and their roles in cellular events. Here, we review recent advances in fluorescence-based single-molecule bioimaging of proteins in living cells. We begin with practical considerations of the design of single-molecule fluorescence imaging experiments such as the choice of imaging modalities, fluorescent probes, and labeling methods. We then describe analytical observables from single-molecule data and the associated molecular parameters along with examples of live-cell single-molecule studies. Lastly, we discuss computational algorithms developed for single-molecule data analysis.

Prior-Apprised Unsupervised Learning of Subpixel Curvilinear Features in Low Signal/Noise Images

Many biophysical problems involve molecular and nanoscale targets moving next to a curvilinear track, e.g., a cytosolic cargo transported by motor proteins moving along a microtubule. For this type of problem, fluorescence imaging is usually the primary tool of choice. There is, however, an ∼20-fold mismatch between target-localization precision and track-imaging resolution such that questions requiring high-fidelity definition of the target's track remain inaccessible. On the other hand, if the contextual image of the tracks can be refined to a level comparable to that of the target, many intuitive yet mechanistically important issues can begin to be addressed. This work demonstrates that it is possible to statistically infer, to subpixel precision, curvilinear features in a low signal/noise image. This is achieved by a framework that consists of three stages: the Hessian-based feature enhancement, the subimage feature sampling and registration, and the statistical learning of the underlying curvilinear structure using a new, to our knowledge, method developed here for inferring the principal curves. In each stage, the descriptive prior information that the features come from curvilinear elements is explicitly taken into account. It is fully automated without user supervision, which is distinctly different from approaches that require user seeding or well-defined training data sets. Computer simulations of realistic images are used to investigate the performance of the framework and its implementation. The characterization results suggest that curvilinear features are refined to the same order of precision as that of the target and that the bootstrap confidence intervals from the analysis allow an estimate for the statistical bounds of the simulated "true" curve. Also shown are analyses of experimental images from three different microscopy modalities: two-photon laser-scanning microscopy, epifluorescence microscopy, and total internal reflection fluorescence microscopy. The practical application of this prior-apprised unsupervised learning framework as well as its potential outlook are discussed.

Statistical Learning of Discrete States in Time Series

Time series obtained from time-dependent experiments contain rich information on kinetics and dynamics of the system under investigation. This work describes an unsupervised learning framework, along with the derivation of the necessary analytical expressions, for the analysis of Gaussian-distributed time series that exhibit discrete states. After the time series has been partitioned into segments in a model-free manner using the previously developed change-point (CP) method, this protocol starts with an agglomerative hierarchical clustering algorithm to classify the detected segments into possible states. The initial state clustering is further refined using an expectation-maximization (EM) procedure, and the number of states is determined by a Bayesian information criterion (BIC). Also introduced here is an achievement scalarization function, usually seen in artificial intelligence literature, for quantitatively assessing the performance of state determination. The statistical learning framework, which is comprised of three stages, detection of signal change, clustering, and number-of-state determination, was thoroughly characterized using simulated trajectories with random intensity segments that have no underlying kinetics, and its performance was critically evaluated. The application to experimental data is also demonstrated. The results suggested that this general framework, the implementation of which is based on firm theoretical foundations and does not require the imposition of any kinetics model, is powerful in determining the number of states, the parameters contained in each state, as well as the associated statistical significance.

Compound Molecular Logic in Accessing the Active Site of Mycobacterium tuberculosis Protein Tyrosine Phosphatase B

Protein tyrosine phosphatase B (PtpB) from Mycobacterium tuberculosis (Mtb) extends the bacteria's survival in hosts and hence is a potential target for Mtb-specific drugs. To study how Mtb-specific sequence insertions in PtpB may regulate access to its active site through large-amplitude conformational changes, we performed free-energy calculations using an all-atom explicit solvent model. Corroborated by biochemical assays, the results show that PtpB's active site is controlled via an "either/or" compound conformational gating mechanism, an unexpected discovery that Mtb has evolved to bestow a single enzyme with such intricate logical operations. In addition to providing unprecedented insights for its active-site surroundings, the findings also suggest new ways of inactivating PtpB.

Extraction of Protein Conformational Modes from Distance Distributions Using Structurally Imputed Bayesian Data Augmentation

Protein conformational changes are known to play important roles in assorted biochemical and biological processes. Driven by thermal motions of surrounding solvent molecules, such a structural remodeling often occurs stochastically. Yet, regardless of how random the conformational reconfiguration may appear, it could in principle be described by a linear combination of a set of orthogonal modes which, in turn, are contained in the intramolecular distance distributions. The central challenge is how to obtain the distribution. This contribution proposes a Bayesian data-augmentation scheme to extract the predominant modes from only few distance distributions, be they from computational sampling or directly from experiments such as single-molecule Förster-type resonance energy transfer (smFRET). The inference of the complete protein structure from insufficient data was recognized as isomorphic to the missing-data problem in Bayesian statistical learning. Using smFRET data as an example, the missing coordinates were deduced, given protein structural constraints and multiple but limited number of smFRET distances; the Boltzmann weighing of each inferred protein structure was then evaluated using computational modeling to numerically construct the posterior density for the global protein conformation. The conformational modes were then determined from the iteratively converged overall conformational distribution using principal component analysis. Two examples were presented to illustrate these basic ideas as well as their practical implementation. The scheme described herein was based on the theory behind the powerful Tanner-Wang algorithm that guarantees convergence to the true posterior density. However, instead of assuming a mathematical model to calculate the likelihood as in conventional statistical inference, here the protein structure was treated as a statistical parameter and was imputed from the numerical likelihood function based on structural information, a probability model-free method. The framework put forth here is anticipated to be generally applicable, offering a new way to articulate protein conformational changes in a quantifiable manner.

Accuracy of maximum likelihood estimates of a two-state model in single-molecule FRET

Photon sequences from single-molecule Förster resonance energy transfer (FRET) experiments can be analyzed using a maximum likelihood method. Parameters of the underlying kinetic model (FRET efficiencies of the states and transition rates between conformational states) are obtained by maximizing the appropriate likelihood function. In addition, the errors (uncertainties) of the extracted parameters can be obtained from the curvature of the likelihood function at the maximum. We study the standard deviations of the parameters of a two-state model obtained from photon sequences with recorded colors and arrival times. The standard deviations can be obtained analytically in a special case when the FRET efficiencies of the states are 0 and 1 and in the limiting cases of fast and slow conformational dynamics. These results are compared with the results of numerical simulations. The accuracy and, therefore, the ability to predict model parameters depend on how fast the transition rates are compared to the photon count rate. In the limit of slow transitions, the key parameters that determine the accuracy are the number of transitions between the states and the number of independent photon sequences. In the fast transition limit, the accuracy is determined by the small fraction of photons that are correlated with their neighbors. The relative standard deviation of the relaxation rate has a "chevron" shape as a function of the transition rate in the log-log scale. The location of the minimum of this function dramatically depends on how well the FRET efficiencies of the states are separated.

Perspective: Reaches of chemical physics in biology

Chemical physics as a discipline contributes many experimental tools, algorithms, and fundamental theoretical models that can be applied to biological problems. This is especially true now as the molecular level and the systems level descriptions begin to connect, and multi-scale approaches are being developed to solve cutting edge problems in biology. In some cases, the concepts and tools got their start in non-biological fields, and migrated over, such as the idea of glassy landscapes, fluorescence spectroscopy, or master equation approaches. In other cases, the tools were specifically developed with biological physics applications in mind, such as modeling of single molecule trajectories or super-resolution laser techniques. In this introduction to the special topic section on chemical physics of biological systems, we consider a wide range of contributions, all the way from the molecular level, to molecular assemblies, chemical physics of the cell, and finally systems-level approaches, based on the contributions to this special issue. Chemical physicists can look forward to an exciting future where computational tools, analytical models, and new instrumentation will push the boundaries of biological inquiry.