A Semi-Markov Approach to Study a Group of Kinesin Motors.

We propose a general mathematical and computational approach to study cellular transport driven by a group of kinesin motors. It is a framework for multi-scale modeling that integrates kinetic models of single kinesin motors, including detachment and reattachment events, to study group behaviors of several motors. By formulating the problem as a semi-Markov process and applying a central limit theorem, asymptotic velocity and diffusivity can be readily calculated, which offers considerable computational advantage over Monte Carlo simulations in tasks such as parameter sensitivity analysis and model selection. We demonstrate the method with some examples. The importance of incorporating the intrinsic microscopic-level dynamics of individual motors is illustrated by showing how changes at the microscopic level propagate to the motor-cargo complex at a mesoscopic level. Particularly, we showcase an example in which changes in the second moment of single-motor characteristics gives rise to different first moment characteristics of the motor group.

Learning Biological Dynamics From Spatio-Temporal Data by Gaussian Processes.

Model discovery methods offer a promising way to understand biology from data. We propose a method to learn biological dynamics from spatio-temporal data by Gaussian processes. This approach is essentially "equation free" and hence avoids model derivation, which is often difficult due to high complexity of biological processes. By exploiting the local nature of biological processes, dynamics can be learned with data sparse in time. When the length scales (hyperparameters) of the squared exponential covariance function are tuned, they reveal key insights of the underlying process. The squared exponential covariance function also simplifies propagation of uncertainty in multi-step forecasting. After evaluating the performance of the method on synthetic data, we demonstrate a case study on real image data of E. coli colony.

A kinetic dissection of the fast and superprocessive kinesin-3 KIF1A reveals a predominant one-head-bound state during its chemomechanical cycle.

The kinesin-3 family contains the fastest and most processive motors of the three neuronal transport kinesin families, yet the sequence of states and rates of kinetic transitions that comprise the chemomechanical cycle and give rise to their unique properties are poorly understood. We used stopped-flow fluorescence spectroscopy and single-molecule motility assays to delineate the chemomechanical cycle of the kinesin-3, KIF1A. Our bacterially expressed KIF1A construct, dimerized via a kinesin-1 coiled-coil, exhibits fast velocity and superprocessivity behavior similar to WT KIF1A. We established that the KIF1A forward step is triggered by hydrolysis of ATP and not by ATP binding, meaning that KIF1A follows the same chemomechanical cycle as established for kinesin-1 and -2. The ATP-triggered half-site release rate of KIF1A was similar to the stepping rate, indicating that during stepping, rear-head detachment is an order of magnitude faster than in kinesin-1 and kinesin-2. Thus, KIF1A spends the majority of its hydrolysis cycle in a one-head-bound state. Both the ADP off-rate and the ATP on-rate at physiological ATP concentration were fast, eliminating these steps as possible rate-limiting transitions. Based on the measured run length and the relatively slow off-rate in ADP, we conclude that attachment of the tethered head is the rate-limiting transition in the KIF1A stepping cycle. Thus, KIF1A's activity can be explained by a fast rear-head detachment rate, a rate-limiting step of tethered-head attachment that follows ATP hydrolysis, and a relatively strong electrostatic interaction with the microtubule in the weakly bound post-hydrolysis state.

Renewal Reward Perspective on Linear Switching Diffusion Systems in Models of Intracellular Transport.

In many biological systems, the movement of individual agents is characterized having multiple qualitatively distinct behaviors that arise from a variety of biophysical states. For example, in cells the movement of vesicles, organelles, and other intracellular cargo is affected by their binding to and unbinding from cytoskeletal filaments such as microtubules through molecular motor proteins. A typical goal of theoretical or numerical analysis of models of such systems is to investigate effective transport properties and their dependence on model parameters. While the effective velocity of particles undergoing switching diffusion dynamics is often easily characterized in terms of the long-time fraction of time that particles spend in each state, the calculation of the effective diffusivity is more complicated because it cannot be expressed simply in terms of a statistical average of the particle transport state at one moment of time. However, it is common that these systems are regenerative, in the sense that they can be decomposed into independent cycles marked by returns to a base state. Using decompositions of this kind, we calculate effective transport properties by computing the moments of the dynamics within each cycle and then applying renewal reward theory. This method provides a useful alternative large-time analysis to direct homogenization for linear advection-reaction-diffusion partial differential equation models. Moreover, it applies to a general class of semi-Markov processes and certain stochastic differential equations that arise in models of intracellular transport. Applications of the proposed renewal reward framework are illustrated for several case studies such as mRNA transport in developing oocytes and processive cargo movement by teams of molecular motor proteins.

A Metabolomic Approach for Predicting Diurnal Changes in Cortisol.

Introduction: The dysregulation of cortisol secretion has been associated with a number of mental health and mood disorders. However, diagnostics for mental health and mood disorders are behavioral and lack biological contexts. Objectives: The goal of this work is to identify volatile metabolites capable of predicting changes in total urinary cortisol across the diurnal cycle for long-term stress monitoring in psychological disorders. Methods: We applied comprehensive two-dimensional gas chromatography coupled with time-of-flight mass spectrometry to sample the urinary volatile metabolome using an untargeted approach across three time points in a single day for 60 subjects. Results: The finalized multiple regression model includes 14 volatile metabolites and 7 interaction terms. A review of the selected metabolites suggests pyrrole, 6-methyl-5-hepten-2-one and 1-iodo-2-methylundecane may originate from endogenous metabolic mechanisms influenced by glucocorticoid signaling mechanisms. Conclusion: This analysis demonstrated the feasibility of using specific volatile metabolites for the prediction of secreted cortisol across time.

Effective behavior of cooperative and nonidentical molecular motors.

Analytical formulas for effective drift, diffusivity, run times, and run lengths are derived for an intracellular transport system consisting of a cargo attached to two cooperative but not identical molecular motors (for example, kinesin-1 and kinesin-2) which can each attach and detach from a microtubule. The dynamics of the motor and cargo in each phase are governed by stochastic differential equations, and the switching rates depend on the spatial configuration of the motor and cargo. This system is analyzed in a limit where the detached motors have faster dynamics than the cargo, which in turn has faster dynamics than the attached motors. The attachment and detachment rates are also taken to be slow relative to the spatial dynamics. Through an application of iterated stochastic averaging to this system, and the use of renewal-reward theory to stitch together the progress within each switching phase, we obtain explicit analytical expressions for the effective drift, diffusivity, and processivity of the motor-cargo system. Our approach accounts in particular for jumps in motor-cargo position that occur during attachment and detachment events, as the cargo tracking variable makes a rapid adjustment due to the averaged fast scales. The asymptotic formulas are in generally good agreement with direct stochastic simulations of the detailed model based on experimental parameters for various pairings of kinesin-1 and kinesin-2 under assisting, hindering, or no load.

Estimation and prediction for a mechanistic model of measles transmission using particle filtering and maximum likelihood estimation.

Disease incidence reported directly within health systems frequently reflects a partial observation relative to the true incidence in the population. State-space models present a general framework for inferring both the dynamics of infectious disease processes and the unobserved burden of disease in the population. Here, we present a state-space model of measles transmission and vaccine-based interventions at the country-level and a particle filter-based estimation procedure. Our dynamic transmission model builds on previous work by incorporating population age-structure to allow explicit representation of age-targeted vaccine interventions. We illustrate the performance of estimators of model parameters and predictions of unobserved states on simulated data from two dynamic models: one on the annual time-scale of observations and one on the biweekly time-scale of the epidemiological dynamics. We show that our model results in approximately unbiased estimates of unobserved burden and the underreporting rate. We further illustrate the performance of the fitted model for prediction of future disease burden in the next one to 15 years.

Insights into Kinesin-1 Stepping from Simulations and Tracking of Gold Nanoparticle-Labeled Motors.

High-resolution tracking of gold nanoparticle-labeled proteins has emerged as a powerful technique for measuring the structural kinetics of processive enzymes and other biomacromolecules. These techniques use point spread function (PSF) fitting methods borrowed from single-molecule fluorescence imaging to determine molecular positions below the diffraction limit. However, compared to fluorescence, gold nanoparticle tracking experiments are performed at significantly higher frame rates and utilize much larger probes. In the current work, we use Brownian dynamics simulations of nanoparticle-labeled proteins to investigate the regimes in which the fundamental assumptions of PSF fitting hold and where they begin to break down. We find that because gold nanoparticles undergo tethered diffusion around their anchor point, PSF fitting cannot be extended to arbitrarily fast frame rates. Instead, camera exposure times that allow the nanoparticle to fully populate its stationary positional distribution achieve a spatial averaging that increases fitting precision. We furthermore find that changes in the rotational freedom of the tagged protein can lead to artifactual translations in the fitted particle position. Finally, we apply these lessons to dissect a standing controversy in the kinesin field over the structure of a dimer in the ATP waiting state. Combining new experiments with simulations, we determine that the rear kinesin head in the ATP waiting state is unbound but not displaced from its previous microtubule binding site and that apparent differences in separately published reports were simply due to differences in the gold nanoparticle attachment position. Our results highlight the importance of gold conjugation decisions and imaging parameters to high-resolution tracking results and will serve as a useful guide for the design of future gold nanoparticle tracking experiments.

Monitoring changes in the healthy female metabolome across the menstrual cycle using GC × GC-TOFMS.

Urinary metabolomics offers a non-invasive means of obtaining information about the system-wide biological health of a patient. Untargeted metabolomics approaches using one-dimensional gas chromatography (GC) are limited due to the chemical complexity of urine, which poorly detects co-eluting low-abundance analytes. Metabolite detection and identification can be improved by applying comprehensive two-dimensional GC, allowing for the discovery of additional viable biomarkers of disease. In this work, we applied comprehensive two-dimensional GC coupled with time-of-flight mass spectrometry (GC × GC-TOFMS) to the analysis of urine samples collected daily across 28-days from 10 healthy female subjects for a personalized approach to female reproductive health monitoring. Through this analysis, we identified 935 unique volatile metabolites. Two statistical methods, a modified T-statistic and Wilcoxon Rank Sum, were applied to analyze differences in metabolome abundance on ovulation days as compared to non-ovulation days. Four metabolites (2-pentanone, 3-penten-2-one, carbon disulfide, acetone) were identified as statistically significant by the modified T-statistic but not the Rank Sum, after a false-discovery rate of 0.1 was set using a Benjamini-Hochberg correction. Subsequent analyses by boxplot indicated that the putative volatile metabolic biomarkers for fertility are expressed in increased or decreased abundance in urine on the day of ovulation. Individual analysis of metabolome expression across 28-days revealed some subject-specific features, which suggest a potential for long-term, personalized fertility monitoring using metabolomics.

Load-dependent detachment kinetics plays a key role in bidirectional cargo transport by kinesin and dynein.

Bidirectional cargo transport along microtubules is carried out by opposing teams of kinesin and dynein motors. Despite considerable study, the factors that determine whether these competing teams achieve net anterograde or retrograde transport in cells remain unclear. The goal of this work is to use stochastic simulations of bidirectional transport to determine the motor properties that most strongly determine overall cargo velocity and directionality. Simulations were carried out based on published optical tweezer characterization of kinesin-1 and kinesin-2, and for available data for cytoplasmic dynein and the dynein-dynactin-BicD2 (DDB) complex. By varying dynein parameters and analyzing cargo trajectories, we find that net cargo transport is predicted to depend minimally on the dynein stall force, but strongly on dynein load-dependent detachment kinetics. In simulations, dynein is dominated by kinesin-1, but DDB and kinesin-1 are evenly matched, recapitulating recent experimental work. Kinesin-2 competes less well against dynein and DDB, and overall, load-dependent motor detachment is the property that most determines a motor's ability to compete in bidirectional transport. It follows that the most effective intracellular regulators of bidirectional transport are predicted to be those that alter motor detachment kinetics rather than motor velocity or stall force.

Analysis of single particle diffusion with transient binding using particle filtering.

Diffusion with transient binding occurs in a variety of biophysical processes, including movement of transmembrane proteins, T cell adhesion, and caging in colloidal fluids. We model diffusion with transient binding as a Brownian particle undergoing Markovian switching between free diffusion when unbound and diffusion in a quadratic potential centered around a binding site when bound. Assuming the binding site is the last position of the particle in the unbound state and Gaussian observational error obscures the true position of the particle, we use particle filtering to predict when the particle is bound and to locate the binding sites. Maximum likelihood estimators of diffusion coefficients, state transition probabilities, and the spring constant in the bound state are computed with a stochastic Expectation-Maximization (EM) algorithm.

An attraction-repulsion point process model for respiratory syncytial virus infections.

How is the progression of a virus influenced by properties intrinsic to individual cells? We address this question by studying the susceptibility of cells infected with two strains of the human respiratory syncytial virus (RSV-A and RSV-B) in an in vitro experiment. Spatial patterns of infected cells give us insight into how local conditions influence susceptibility to the virus. We observe a complicated attraction and repulsion behavior, a tendency for infected cells to lump together or remain apart. We develop a new spatial point process model to describe this behavior. Inference on spatial point processes is difficult because the likelihood functions of these models contain intractable normalizing constants; we adapt an MCMC algorithm called double Metropolis-Hastings to overcome this computational challenge. Our methods are computationally efficient even for large point patterns consisting of over 10,000 points. We illustrate the application of our model and inferential approach to simulated data examples and fit our model to various RSV experiments. Because our model parameters are easy to interpret, we are able to draw meaningful scientific conclusions from the fitted models.

Estimating Velocity for Processive Motor Proteins with Random Detachment.

We show that, for a wide range of models, the empirical velocity of processive motor proteins has a limiting Pearson type VII distribution with finite mean but infinite variance. We develop maximum likelihood inference for this Pearson type VII distribution. In two simulation studies, we compare the performance of our MLE with the performance of standard Student's t-based inference. The studies show that incorrectly assuming normality (1) can lead to imprecise inference regarding motor velocity in the one-sample case, and (2) can significantly reduce power in the two-sample case. These results should be of interest to experimentalists who wish to engineer motors possessing specific functional characteristics.

Asymptotic analysis of microtubule-based transport by multiple identical molecular motors.

We describe a system of stochastic differential equations (SDEs) which model the interaction between processive molecular motors, such as kinesin and dynein, and the biomolecular cargo they tow as part of microtubule-based intracellular transport. We show that the classical experimental environment fits within a parameter regime which is qualitatively distinct from conditions one expects to find in living cells. Through an asymptotic analysis of our system of SDEs, we develop a means for applying in vitro observations of the nonlinear response by motors to forces induced on the attached cargo to make analytical predictions for two parameter regimes that have thus far eluded direct experimental observation: (1) highly viscous in vivo transport and (2) dynamics when multiple identical motors are attached to the cargo and microtubule.

Assessment of the 2010 global measles mortality reduction goal: results from a model of surveillance data.

BACKGROUND: In 2008 all WHO member states endorsed a target of 90% reduction in measles mortality by 2010 over 2000 levels. We developed a model to estimate progress made towards this goal. METHODS: We constructed a state-space model with population and immunisation coverage estimates and reported surveillance data to estimate annual national measles cases, distributed across age classes. We estimated deaths by applying age-specific and country-specific case-fatality ratios to estimated cases in each age-country class. FINDINGS: Estimated global measles mortality decreased 74% from 535,300 deaths (95% CI 347,200-976,400) in 2000 to 139,300 (71,200-447,800) in 2010. Measles mortality was reduced by more than three-quarters in all WHO regions except the WHO southeast Asia region. India accounted for 47% of estimated measles mortality in 2010, and the WHO African region accounted for 36%. INTERPRETATION: Despite rapid progress in measles control from 2000 to 2007, delayed implementation of accelerated disease control in India and continued outbreaks in Africa stalled momentum towards the 2010 global measles mortality reduction goal. Intensified control measures and renewed political and financial commitment are needed to achieve mortality reduction targets and lay the foundation for future global eradication of measles. FUNDING: US Centers for Disease Control and Prevention (PMS 5U66/IP000161).

Kinesins with extended neck linkers: a chemomechanical model for variable-length stepping.

We develop a stochastic model for variable-length stepping of kinesins engineered with extended neck linkers. This requires that we consider the separation in microtubule binding sites between the heads of the motor at the beginning of a step. We show that this separation is stationary and can be included in the calculation of standard experimental quantities. We also develop a corresponding matrix computational framework for conducting computer experiments. Our matrix approach is more efficient computationally than large-scale Monte Carlo simulation. This efficiency greatly eases sensitivity analysis, an important feature when there is considerable uncertainty in the physical parameters of the system. We demonstrate the application and effectiveness of our approach by showing that the worm-like chain model for the neck linker can explain recently published experimental data. While we have focused on a particular scenario for kinesins, these methods could also be applied to myosin and other processive motors.

A mixture model for quantum dot images of kinesin motor assays.

We introduce a nearly automatic procedure to locate and count the quantum dots in images of kinesin motor assays. Our procedure employs an approximate likelihood estimator based on a two-component mixture model for the image data; the first component has a normal distribution, and the other component is distributed as a normal random variable plus an exponential random variable. The normal component has an unknown variance, which we model as a function of the mean. We use B-splines to estimate the variance function during a training run on a suitable image, and the estimate is used to process subsequent images. Parameter estimates are generated for each image along with estimates of standard errors, and the number of dots in the image is determined using an information criterion and likelihood ratio tests. Realistic simulations show that our procedure is robust and that it leads to accurate estimates, both of parameters and of standard errors.

A matrix computational approach to kinesin neck linker extension.

Kinesin stepping requires both tethered diffusion of the free head and conformational changes driven by the chemical state of the motor. We present a numerical method using matrix representations of approximating Markov chains and renewal theory to compute important experimental quantities for models that include both tethered diffusion and chemical transitions. Explicitly modeling the tethered diffusion allows for exploration of the model under perturbation of the neck linker; comparisons are made between the computed models and in vitro assays.

Monte Carlo analysis of neck linker extension in kinesin molecular motors.

Kinesin stepping is thought to involve both concerted conformational changes and diffusive movement, but the relative roles played by these two processes are not clear. The neck linker docking model is widely accepted in the field, but the remainder of the step--diffusion of the tethered head to the next binding site--is often assumed to occur rapidly with little mechanical resistance. Here, we investigate the effect of tethering by the neck linker on the diffusive movement of the kinesin head, and focus on the predicted behavior of motors with naturally or artificially extended neck linker domains. The kinesin chemomechanical cycle was modeled using a discrete-state Markov chain to describe chemical transitions. Brownian dynamics were used to model the tethered diffusion of the free head, incorporating resistive forces from the neck linker and a position-dependent microtubule binding rate. The Brownian dynamics and chemomechanical cycle were coupled to model processive runs consisting of many 8 nm steps. Three mechanical models of the neck linker were investigated: Constant Stiffness (a simple spring), Increasing Stiffness (analogous to a Worm-Like Chain), and Reflecting (negligible stiffness up to a limiting contour length). Motor velocities and run lengths from simulated paths were compared to experimental results from Kinesin-1 and a mutant containing an extended neck linker domain. When tethered by an increasingly stiff spring, the head is predicted to spend an unrealistically short amount of time within the binding zone, and extending the neck is predicted to increase both the velocity and processivity, contrary to experiments. These results suggest that the Worm-Like Chain is not an adequate model for the flexible neck linker domain. The model can be reconciled with experimental data if the neck linker is either much more compliant or much stiffer than generally assumed, or if weak kinesin-microtubule interactions stabilize the diffusing head near its binding site.

Exploratory Spatial Analysis of in vitro Respiratory Syncytial Virus Co-infections.

The cell response to virus infection and virus perturbation of that response is dynamic and is reflected by changes in cell susceptibility to infection. In this study, we evaluated the response of human epithelial cells to sequential infections with human respiratory syncytial virus strains A2 and B to determine if a primary infection with one strain will impact the ability of cells to be infected with the second as a function of virus strain and time elapsed between the two exposures. Infected cells were visualized with fluorescent markers, and location of all cells in the tissue culture well were identified using imaging software. We employed tools from spatial statistics to investigate the likelihood of a cell being infected given its proximity to a cell infected with either the homologous or heterologous virus. We used point processes, K-functions, and simulation procedures designed to account for specific features of our data when assessing spatial associations. Our results suggest that intrinsic cell properties increase susceptibility of cells to infection, more so for RSV-B than for RSV-A. Further, we provide evidence that the primary infection can decrease susceptibility of cells to the heterologous challenge virus but only at the 16 h time point evaluated in this study. Our research effort highlights the merits of integrating empirical and statistical approaches to gain greater insight on in vitro dynamics of virus-host interactions.