Analysis of single-molecule FRET trajectories using hidden Markov modeling

The analysis of single-molecule fluorescence resonance energy transfer (FRET) trajectories has become one of significant biophysical interest. In deducing the transition rates between various states of a system for time-binned data, researchers have relied on simple, but often arbitrary methods of extracting rates from FRET trajectories. Although these methods have proven satisfactory in cases of well-separated, low-noise, two- or three-state systems, they become less reliable when applied to a system of greater complexity. We have developed an analysis scheme that casts single-molecule time-binned FRET trajectories as hidden Markov processes, allowing one to determine, based on probability alone, the most likely FRET-value distributions of states and their interconversion rates while simultaneously determining the most likely time sequence of underlying states for each trajectory. Together with a transition density plot and Bayesian information criterion we can also determine the number of different states present in a system in addition to the state-to-state transition probabilities. Here we present the algorithm and test its limitations with various simulated data and previously reported Holliday junction data. The algorithm is then applied to the analysis of the binding and dissociation of three RecA monomers on a DNA construct.     

Analysis and Interpretation of Superresolution Single-Particle Trajectories

A large number (tens of thousands) of single molecular trajectories on a cell membrane can now be collected by superresolution methods. The data contains information about the diffusive motion of molecule, proteins, or receptors and here we review methods for its recovery by statistical analysis of the data. The information includes the forces, organization of the membrane, the diffusion tensor, the long-time behavior of the trajectories, and more. To recover the long-time behavior and statistics of long trajectories, a stochastic model of their nonequilibrium motion is required. Modeling and data analysis serve extracting novel biophysical features at an unprecedented spatiotemporal resolution. The review presents data analysis, modeling, and stochastic simulations applied in particular on surface receptors evolving in neuronal cells.     

Quantifying individual heterogeneity and its influence on life‐history trajectories: different methods for different questions and contexts

Heterogeneity among individuals influences the life‐history trajectories we observe at the population level because viability selection, selective immigration and emigration processes, and ontogeny change the proportion of individuals with specific trait values with increasing age. Here, we review the two main approaches that have been proposed to account for these processes in life‐history trajectories, contrasting how they quantify ontogeny and selection, and proposing ways to overcome some of their limitations. Nearly all existing approaches to model individual heterogeneity assume either a single normal distribution or a priori known groups of individuals. Ontogenetic processes, however, can vary across individuals through variation in life‐history tactics. We show the usefulness of describing ontogenetic processes by modelling trajectories with a mixture model that focuses on heterogeneity in life‐history tactics. Additionally, most methods examine individual heterogeneity in a single trait, ignoring potential correlations among multiple traits caused by latent common sources of individual heterogeneity. We illustrate the value of using a joint modelling approach to assess the presence of a shared latent correlation and its influence on life‐history trajectories. We contrast the strengths and limitations of different methods for different research questions, and we exemplify the differences among methods using empirical data from long‐term studies of ungulates.     

Modeling inter-human movement coordination: synchronization governs joint task dynamics

Human interaction partners tend to synchronize their movements during repetitive actions such as walking. Research of inter-human coordination in purely rhythmic action tasks reveals that the observed patterns of interaction are dominated by synchronization effects. Initiated by our finding that human dyads synchronize their arm movements even in a goal-directed action task, we present a step-wise approach to a model of inter-human movement coordination. In an experiment, the hand trajectories of ten human dyads are recorded. Governed by a dynamical process of phase synchronization, the participants establish in-phase as well as anti-phase relations. The emerging relations are successfully reproduced by the attractor dynamics of coupled phase oscillators inspired by the Kuramoto model. Three different methods on transforming the motion trajectories into instantaneous phases are investigated and their influence on the model fit to the experimental data is evaluated. System identification technique allows us to estimate the model parameters, which are the coupling strength and the frequency detuning among the dyad. The stability properties of the identified model match the relations observed in the experimental data. In short, our model predicts the dynamics of inter-human movement coordination. It can directly be implemented to enrich human-robot interaction.     

MoDEL (Molecular Dynamics Extended Library): a database of atomistic molecular dynamics trajectories

More than 1700 trajectories of proteins representative of monomeric soluble structures in the protein data bank (PDB) have been obtained by means of state-of-the-art atomistic molecular dynamics simulations in near-physiological conditions. The trajectories and analyses are stored in a large data warehouse, which can be queried for dynamic information on proteins, including interactions. Here, we describe the project and the structure and contents of our database, and provide examples of how it can be used to describe the global flexibility properties of proteins. Basic analyses and trajectories stripped of solvent molecules at a reduced resolution level are available from our web server.     

Intracellular battlegrounds: conflict and cooperation between transposable elements

Transposable elements (TEs) are genomic parasites that amplify their own representation on hosts' chromosomes by inserting into new positions. It is traditionally thought that their copy number is regulated by purifying selection that eliminates hosts with higher than average TE abundance. Here, we stress that selection due to beneficial or harmful interactions between TEs introduces a whole new dimension, with implications for TE evolutionary trajectories and TE loads on hosts. This framework poses new questions requiring conceptual and experimental advances. Considering primarily Drosophila data, we make a case for within host selection on TEs by thinking expansively about the lifecycle of several TE families.     

Expectation-Maximization Binary Clustering for Behavioural Annotation

The growing capacity to process and store animal tracks has spurred the development of new methods to segment animal trajectories into elementary units of movement. Key challenges for movement trajectory segmentation are to (i) minimize the need of supervision, (ii) reduce computational costs, (iii) minimize the need of prior assumptions (e.g. simple parametrizations), and (iv) capture biologically meaningful semantics, useful across a broad range of species. We introduce the Expectation-Maximization binary Clustering (EMbC), a general purpose, unsupervised approach to multivariate data clustering. The EMbC is a variant of the Expectation-Maximization Clustering (EMC), a clustering algorithm based on the maximum likelihood estimation of a Gaussian mixture model. This is an iterative algorithm with a closed form step solution and hence a reasonable computational cost. The method looks for a good compromise between statistical soundness and ease and generality of use (by minimizing prior assumptions and favouring the semantic interpretation of the final clustering). Here we focus on the suitability of the EMbC algorithm for behavioural annotation of movement data. We show and discuss the EMbC outputs in both simulated trajectories and empirical movement trajectories including different species and different tracking methodologies. We use synthetic trajectories to assess the performance of EMbC compared to classic EMC and Hidden Markov Models. Empirical trajectories allow us to explore the robustness of the EMbC to data loss and data inaccuracies, and assess the relationship between EMbC output and expert label assignments. Additionally, we suggest a smoothing procedure to account for temporal correlations among labels, and a proper visualization of the output for movement trajectories. Our algorithm is available as an R-package with a set of complementary functions to ease the analysis.     

Effects of tumors on inhaled pharmacologic drugs

Lung carcinomas are now the most common form of cancer. Clinical data suggest that tumors are found preferentially in upper airways, perhaps specifically at carina within bifurcations. The disease can be treated by aerosolized pharmacologic drugs. To enhance their efficacies site-specific drugs must be deposited selectively. Since inhaled particles are transported by air, flow patterns will naturally affect their trajectories. Therefore, in Part I of a systematic investigation, we focused on tumor-induced effects on airstreams, in Part II (the following article [p. 245]), particle trajectories were determined. To facilitate the targeted delivery of inhaled drugs, we simulated bifurcations with tumors on carinas using a commercial computational fluid dynamics (CFD) software package (FIDAP) with a Cray T90 supercomputer and studied effects of tumor sizes and ventilatory parameters on localized flow patterns. Critical tumor sizes existed; e.g., tumors had dominant effects when r/R > or = 0.8 for bifurcation 3-4 and r/R > or = 0.6 for bifurcation 7-8 (r = tumor radius and R = airway radius). The findings suggest that computer modeling is a means to integrate alterations to airway structures caused by diseases into aerosol therapy protocols.     

Bayesian-Estimated Hierarchical HMMs Enable Robust Analysis of Single-Molecule Kinetic Heterogeneity

Single-molecule kinetic experiments allow the reaction trajectories of individual biomolecules to be directly observed, eliminating the effects of population averaging and providing a powerful approach for elucidating the kinetic mechanisms of biomolecular processes. A major challenge to the analysis and interpretation of these experiments, however, is the kinetic heterogeneity that almost universally complicates the recorded single-molecule signal versus time trajectories (i.e., signal trajectories). Such heterogeneity manifests as changes and/or differences in the transition rates that are observed within individual signal trajectories or across a population of signal trajectories. Because characterizing kinetic heterogeneity can provide critical mechanistic information, we have developed a computational method that effectively and comprehensively enables such analysis. To this end, we have developed a computational algorithm and software program, hFRET, that uses the variational approximation for Bayesian inference to estimate the parameters of a hierarchical hidden Markov model, thereby enabling robust identification and characterization of kinetic heterogeneity. Using simulated signal trajectories, we demonstrate the ability of hFRET to accurately and precisely characterize kinetic heterogeneity. In addition, we use hFRET to analyze experimentally recorded signal trajectories reporting on the conformational dynamics of ribosomal pre-translocation (PRE) complexes. The results of our analyses demonstrate that PRE complexes exhibit kinetic heterogeneity, reveal the physical origins of this heterogeneity, and allow us to expand the current model of PRE complex dynamics. The methods described here can be applied to signal trajectories generated using any type of signal and can be easily extended to the analysis of signal trajectories exhibiting more complex kinetic behaviors. Moreover, variations of our approach can be easily developed to integrate kinetic data obtained from different experimental constructs and/or from molecular dynamics simulations of a biomolecule of interest.     

Bayesian Decision Tree for the Classification of the Mode of Motion in Single-Molecule Trajectories

Membrane proteins move in heterogeneous environments with spatially (sometimes temporally) varying friction and with biochemical interactions with various partners. It is important to reliably distinguish different modes of motion to improve our knowledge of the membrane architecture and to understand the nature of interactions between membrane proteins and their environments. Here, we present an analysis technique for single molecule tracking (SMT) trajectories that can determine the preferred model of motion that best matches observed trajectories. The method is based on Bayesian inference to calculate the posteriori probability of an observed trajectory according to a certain model. Information theory criteria, such as the Bayesian information criterion (BIC), the Akaike information criterion (AIC), and modified AIC (AICc), are used to select the preferred model. The considered group of models includes free Brownian motion, and confined motion in 2nd or 4th order potentials. We determine the best information criteria for classifying trajectories. We tested its limits through simulations matching large sets of experimental conditions and we built a decision tree. This decision tree first uses the BIC to distinguish between free Brownian motion and confined motion. In a second step, it classifies the confining potential further using the AIC. We apply the method to experimental Clostridium Perfingens -toxin (CPT) receptor trajectories to show that these receptors are confined by a spring-like potential. An adaptation of this technique was applied on a sliding window in the temporal dimension along the trajectory. We applied this adaptation to experimental CPT trajectories that lose confinement due to disaggregation of confining domains. This new technique adds another dimension to the discussion of SMT data. The mode of motion of a receptor might hold more biologically relevant information than the diffusion coefficient or domain size and may be a better tool to classify and compare different SMT experiments.     

Improved Coral Population Estimation Reveals Trends at Multiple Scales on Australia’s Great Barrier Reef

The global erosion of biodiversity presents unique challenges for identifying major changes in population dynamics, establishing their causes, and managing and conserving affected ecosystems at broad spatial scales. Adaptive learning approaches connecting different spatial scales through the transfer of hierarchical information are powerful tools to address such challenges. Here, we use a Semi-Parametric Bayesian Hierarchical (SPa-BaH) model to estimate coral cover trajectories using 16 years of a broad-scale survey on Australia’s Great Barrier Reef (GBR). The spatiotemporal variability of coral populations has been considered by separating three-tiered spatial scales and allowing for alternating phases of increasing and decreasing in the estimation of their trajectories. Model estimates revealed coral cover trajectories that were highly variable according to location but that fairly consistently declined at a regional spatial scale. Notwithstanding this general trend, individual reefs within subregions in the central part of the GBR often displayed different trajectory types between sites separated by only a few hundred meters. These coral dynamics were also associated with reduced recovery rates in the Cairns and Swain subregions. Our study highlights the importance of accounting for local variation in coral cover when estimating the spatiotemporal trends in coral cover trajectories, in this case, at the GBR scale. By retaining information at different hierarchical spatial scales, our SPa-BaH model supports better estimation of large-scale coral cover trajectories. The quantitative approaches developed here can also be applied to other species with complex dynamics thereby enhancing estimations of their trajectories at local- and larger-scales and options for their management.     

Global and national trends,gaps, and opportunities in documenting and monitoring species distributions

Conserving and managing biodiversity in the face of ongoing global change requires sufficient evidence to assess status and trends of species distributions. Here, we propose novel indicators of biodiversity data coverage and sampling effectiveness and analyze national trajectories in closing spatiotemporal knowledge gaps for terrestrial vertebrates (1950 to 2019). Despite a rapid rise in data coverage, particularly in the last 2 decades, strong geographic and taxonomic biases persist. For some taxa and regions, a tremendous growth in records failed to directly translate into newfound knowledge due to a sharp decline in sampling effectiveness. However, we found that a nation’s coverage was stronger for species for which it holds greater stewardship. As countries under the post-2020 Global Biodiversity Framework renew their commitments to an improved, rigorous biodiversity knowledge base, our findings highlight opportunities for international collaboration to close critical information gaps.

Conserving and managing biodiversity in the face of ongoing global change requires sufficient evidence to assess status and trends of species distributions. This study analyzes national trajectories in closing spatiotemporal knowledge gaps for terrestrial vertebrates (1950-2019) based on novel indicators of data coverage and sampling effectiveness.     

Evolution of extreme ontogenetic allometric diversity and heterochrony in pythons,a clade of giant and dwarf snakes

Ontogenetic allometry, how species change with size through their lives, and heterochony, a decoupling between shape, size, and age, are major contributors to biological diversity. However, macroevolutionary allometric and heterochronic trends remain poorly understood because previous studies have focused on small groups of closely related species. Here, we focus on testing hypotheses about the evolution of allometry and how allometry and heterochrony drive morphological diversification at the level of an entire species‐rich and diverse clade. Pythons are a useful system due to their remarkably diverse and well‐adapted phenotypes and extreme size disparity. We collected detailed phenotype data on 40 of the 44 species of python from 1191 specimens. We used a suite of analyses to test for shifts in allometric trajectories that modify morphological diversity. Heterochrony is the main driver of initial divergence within python clades, and shifts in the slopes of allometric trajectories make exploration of novel phenotypes possible later in divergence history. We found that allometric coefficients are highly evolvable and there is an association between ontogenetic allometry and ecology, suggesting that allometry is both labile and adaptive rather than a constraint on possible phenotypes.     

Going beyond Clustering in MD Trajectory Analysis: An Application to Villin Headpiece Folding

Recent advances in computing technology have enabled microsecond long all-atom molecular dynamics (MD) simulations of biological systems. Methods that can distill the salient features of such large trajectories are now urgently needed. Conventional clustering methods used to analyze MD trajectories suffer from various setbacks, namely (i) they are not data driven, (ii) they are unstable to noise and changes in cut-off parameters such as cluster radius and cluster number, and (iii) they do not reduce the dimensionality of the trajectories, and hence are unsuitable for finding collective coordinates. We advocate the application of principal component analysis (PCA) and a non-metric multidimensional scaling (nMDS) method to reduce MD trajectories and overcome the drawbacks of clustering. To illustrate the superiority of nMDS over other methods in reducing data and reproducing salient features, we analyze three complete villin headpiece folding trajectories. Our analysis suggests that the folding process of the villin headpiece is structurally heterogeneous.     

A model-independent test for the presence of regulatory equilibrium and non-random structure in island species trajectories

Aims To test a key prevision of the dynamic equilibrium theory of island biogeography, namely that changes in species numbers on islands over time (hereafter, species trajectories) are equilibrial, and to characterize aspects of the dynamical properties of species change over time using a model‐independent test. Methods We tested for regulatory equilibrium and non‐random structure in species numbers through time by comparing observed correlation coefficients at lag‐k for species trajectories from four true islands and two habitat islands. First, we estimated the shape of the autocorrelation function for each observed species trajectory by calculating correlation coefficients of the observed data between pairs of values N_t?k and N_t separated by lag‐k (k = 1, 2, …, N ? 1). Second, we tested the observed correlation coefficients at each lag against a distribution of correlation coefficients generated by randomly ordering observed numbers in the species trajectories. Results The patterns of autocorrelation functions for all but one of the observed species trajectories did not exhibit evidence of regulatory equilibrium, and, in fact, closely matched what would be expected from a non‐stationary or ‘random walk’ process. The majority of the correlation coefficients generated from the observed species trajectories did not deviate significantly from correlation coefficients produced by the randomized trajectories. However, there was strong evidence of unusual positive autocorrelation at small time lags for birds on islands measured annually (2‐ to 4‐year lags) and for arthropods on islands measured weekly (7‐ to 8‐week lags), suggesting some degree of structure in change in species richness over time. Main conclusions The autocorrelation function patterns for all but one of the observed species trajectories showed various forms of non‐stationarity. These types of patterns suggest that the numbers of species through time gradually wandered away from their initial sizes. Our model‐independent test of individual correlation coefficients revealed significant structure in the observed species trajectories. These trajectories appear to be non‐random at relatively short lag intervals, indicating a process with short memory.     

Time-varying functional regression for predicting remaining lifetime distributions from longitudinal trajectories

A recurring objective in longitudinal studies on aging and longevity has been the investigation of the relationship between age-at-death and current values of a longitudinal covariate trajectory that quantifies reproductive or other behavioral activity. We propose a novel technique for predicting age-at-death distributions for situations where an entire covariate history is included in the predictor. The predictor trajectories up to current time are represented by time-varying functional principal component scores, which are continuously updated as time progresses and are considered to be time-varying predictor variables that are entered into a class of time-varying functional regression models that we propose. We demonstrate for biodemographic data how these methods can be applied to obtain predictions for age-at-death and estimates of remaining lifetime distributions, including estimates of quantiles and of prediction intervals for remaining lifetime. Estimates and predictions are obtained for individual subjects, based on their observed behavioral trajectories, and include a dimension-reduction step that is implemented by projecting on a single index. The proposed techniques are illustrated with data on longitudinal daily egg-laying for female medflies, predicting remaining lifetime and age-at-death distributions from individual event histories observed up to current time.     

Black spaces,black places: Strategic assimilation and identity construction in middle-class suburbia

While white ethnics and immigrants of colour have been studied in terms of their attempts to assimilate into the American mainstream, sociologists assume that ongoing racial discrimination obviates the need for an extensive examination of the actual assimilation trajectories of middle-class blacks. Many middle-class blacks travel from the black to the white world rather than existing exclusively in one racially distinct environment. Yet, we do not fully understand how middle-class blacks conceptualize their own integration into American society. Drawing on data collected through in-depth interviews with middle-class blacks and ethnographic research in a white and a black suburb, I establish the link between an affinity for black spaces and the alternative assimilation trajectories of middle-class blacks. I find that middle-class blacks engage in a variant of segmented assimilation, privileging the black world as a site for socializing even if they live in a white suburb. This selective pattern of assimilation, what I term strategic assimilation, suggests that this population of middle-class blacks does not perceive itself as permanently constrained to the bottom rung of a racial hierarchy.     

Adaptation of Drosophila to a novel laboratory environment reveals temporally heterogeneous trajectories of selected alleles

The genomic basis of adaptation to novel environments is a fundamental problem in evolutionary biology that has gained additional importance in the light of the recent global change discussion. Here, we combined laboratory natural selection (experimental evolution) in Drosophila melanogaster with genome‐wide next generation sequencing of DNA pools (Pool‐Seq) to identify alleles that are favourable in a novel laboratory environment and traced their trajectories during the adaptive process. Already after 15 generations, we identified a pronounced genomic response to selection, with almost 5000 single nucleotide polymorphisms (SNP; genome‐wide false discovery rates < 0.005%) deviating from neutral expectation. Importantly, the evolutionary trajectories of the selected alleles were heterogeneous, with the alleles falling into two distinct classes: (i) alleles that continuously rise in frequency; and (ii) alleles that at first increase rapidly but whose frequencies then reach a plateau. Our data thus suggest that the genomic response to selection can involve a large number of selected SNPs that show unexpectedly complex evolutionary trajectories, possibly due to nonadditive effects.     

CODEX,a neural network approach to explore signaling dynamics landscapes

Current studies of cell signaling dynamics that use live cell fluorescent biosensors routinely yield thousands of single‐cell, heterogeneous, multi‐dimensional trajectories. Typically, the extraction of relevant information from time series data relies on predefined, human‐interpretable features. Without a priori knowledge of the system, the predefined features may fail to cover the entire spectrum of dynamics. Here we present CODEX, a data‐driven approach based on convolutional neural networks (CNNs) that identifies patterns in time series. It does not require a priori information about the biological system and the insights into the data are built through explanations of the CNNs'' predictions. CODEX provides several views of the data: visualization of all the single‐cell trajectories in a low‐dimensional space, identification of prototypic trajectories, and extraction of distinctive motifs. We demonstrate how CODEX can provide new insights into ERK and Akt signaling in response to various growth factors, and we recapitulate findings in p53 and TGFβ‐SMAD2 signaling.     

Virtual trajectory and stiffness ellipse during multijoint arm movement predicted by neural inverse models

We predict the virtual trajectories and stiffness ellipses during multijoint arm movements by computer simulations. A two-link manipulator with four single-joint muscles and two double-joint muscles is used as a model of the human arm. Physical parameters of the model are derived from several experimental data. Among them, special emphasis is put on low values of the dynamic hand stiffness recently measured during single joint and multijoint movements. The feedback-error-learning scheme to acquire the inverse dynamics model and the inverse statics model is utilized for this prediction. The virtual trajectories are much more complex than the actual trajectories. This indicates that planning the virtual trajectory is as difficult as solving the inverse dynamics problem for medium and fast movements, and simply falsifies the advocated computational advantage of the virtual trajectory control hypothesis. Thus, we conclude that learning inverse models is essential even in the virtual trajectory control framework. Finally, we propose a new computational model to learn the complicated shape of the virtual trajectories by integrating the virtual trajectory control and the feedback-error-learning scheme.