Fisher information matrix for single molecules with stochastic trajectories

Tracking of objects in cellular environments has become a vital tool in molecular cell biology. A particularly important example is single molecule tracking which enables the study of the motion of a molecule in cellular environments and provides quantitative information on the behavior of individual molecules in cellular environments, which were not available before through bulk studies. Here, we consider a dynamical system where the motion of an object is modeled by stochastic differential equations (SDEs), and measurements are the detected photons emitted by the moving fluorescently labeled object, which occur at discrete time points, corresponding to the arrival times of a Poisson process, in contrast to uniform time points which have been commonly used in similar dynamical systems. The measurements are distributed according to optical diffraction theory, and therefore, they would be modeled by different distributions, e.g., a Born and Wolf profile for an out-of-focus molecule. For some special circumstances, Gaussian image models have been proposed. In this paper, we introduce a stochastic framework in which we calculate the maximum likelihood estimates of the biophysical parameters of the molecular interactions, e.g., diffusion and drift coefficients. More importantly, we develop a general framework to calculate the Cram\'er-Rao lower bound (CRLB), given by the inverse of the Fisher information matrix, for the estimation of unknown parameters and use it as a benchmark in the evaluation of the standard deviation of the estimates. There exists no established method, even for Gaussian measurements, to systematically calculate the CRLB for the general motion model that we consider in this paper. We apply the developed methodology to simulated data of a molecule with linear trajectories and show that the standard deviation of the estimates matches well with the square root of the CRLB

Cramer-Rao Lower Bound for Point Based Image Registration with Heteroscedastic Error Model for Application in Single Molecule Microscopy

The Cramer-Rao lower bound for the estimation of the affine transformation parameters in a multivariate heteroscedastic errors-in-variables model is derived. The model is suitable for feature-based image registration in which both sets of control points are localized with errors whose covariance matrices vary from point to point. With focus given to the registration of fluorescence microscopy images, the Cramer-Rao lower bound for the estimation of a feature's position (e.g. of a single molecule) in a registered image is also derived. In the particular case where all covariance matrices for the localization errors are scalar multiples of a common positive definite matrix (e.g. the identity matrix), as can be assumed in fluorescence microscopy, then simplified expressions for the Cramer-Rao lower bound are given. Under certain simplifying assumptions these expressions are shown to match asymptotic distributions for a previously presented set of estimators. Theoretical results are verified with simulations and experimental data

Extensional rheology and elastic instabilities of a wormlike micellar solution in a microfluidic cross-slot device

Wormlike micellar surfactant solutions are encountered in a wide variety of important applications, including enhanced oil recovery and ink-jet printing, in which the fluids are subjected to high extensional strain rates. In this contribution we present an experimental investigation of the flow of a model wormlike micellar solution (cetyl pyridinium chloride and sodium salicylate in deionised water) in a well-defined stagnation point extensional flow field generated within a microfluidic cross-slot device. We use micro-particle image velocimetry (m-PIV) and full-field birefringence microscopy coupled with macroscopic measurements of the bulk pressure drop to make a quantitative characterization of the fluid’s rheological response over a wide range of deformation rates. The flow field in the micromachined cross-slot is first characterized for viscous flow of a Newtonian fluid, and m-PIV measurements show the flow field remains symmetric and stable up to moderately high Reynolds number, Re z 20, and nominal strain rate, _3nom z 635 s1. By contrast, in the viscoelastic micellar solution the flow field remains symmetric only for low values of the strain rate such that _3nom # lM1, where lM ¼ 2.5 s is the Maxwell relaxation time of the fluid. In this stable flow regime the fluid displays a localized and elongated birefringent strand extending along the outflow streamline from the stagnation point, and estimates of the apparent extensional viscosity can be obtained using the stressoptical rule and from the total pressure drop measured across the cross-slot channel. For moderate deformation rates (_3nom $ lM1) the flow remains steady, but becomes increasingly asymmetric with increasing flow rate, eventually achieving a steady state of complete anti-symmetry characterized by a dividing streamline and birefringent strand connecting diagonally opposite corners of the cross-slot. Eventually, as the nominal imposed deformation rate is increased further, the asymmetric divided flow becomes time dependent. These purely elastic instabilities are reminiscent of those observed in crossslot flows of polymer solutions, but seem to be strongly influenced by the effects of shear localization of the micellar fluid within the microchannels and around the re-entrant corners of the cross-slot

Criterion for purely elastic Taylor-Couette instability in the flows of shear-banding fluids

In the past twenty years, shear-banding flows have been probed by various techniques, such as rheometry, velocimetry and flow birefringence. In micellar solutions, many of the data collected exhibit unexplained spatio-temporal fluctuations. Recently, it has been suggested that those fluctuations originate from a purely elastic instability of the flow. In cylindrical Couette geometry, the instability is reminiscent of the Taylor-like instability observed in viscoelastic polymer solutions. In this letter, we describe how the criterion for purely elastic Taylor-Couette instability should be adapted to shear-banding flows. We derive three categories of shear-banding flows with curved streamlines, depending on their stability.Comment: 6 pages, 3 figure

Potential "ways of thinking" about the shear-banding phenomenon

Shear-banding is a curious but ubiquitous phenomenon occurring in soft matter. The phenomenological similarities between the shear-banding transition and phase transitions has pushed some researchers to adopt a 'thermodynamical' approach, in opposition to the more classical 'mechanical' approach to fluid flows. In this heuristic review, we describe why the apparent dichotomy between those approaches has slowly faded away over the years. To support our discussion, we give an overview of different interpretations of a single equation, the diffusive Johnson-Segalman (dJS) equation, in the context of shear-banding. We restrict ourselves to dJS, but we show that the equation can be written in various equivalent forms usually associated with opposite approaches. We first review briefly the origin of the dJS model and its initial rheological interpretation in the context of shear-banding. Then we describe the analogy between dJS and reaction-diffusion equations. In the case of anisotropic diffusion, we show how the dJS governing equations for steady shear flow are analogous to the equations of the dynamics of a particle in a quartic potential. Going beyond the existing literature, we then draw on the Lagrangian formalism to describe how the boundary conditions can have a key impact on the banding state. Finally, we reinterpret the dJS equation again and we show that a rigorous effective free energy can be constructed, in the spirit of early thermodynamic interpretations or in terms of more recent approaches exploiting the language of irreversible thermodynamics.Comment: 14 pages, 6 figures, tutorial revie

Minkowski Functionals of Abell/ACO Clusters

We determine the Minkowski functionals for a sample of Abell/ACO clusters, 401 with measured and 16 with estimated redshifts. The four Minkowski functionals (including the void probability function and the mean genus) deliver a global description of the spatial distribution of clusters on scales from $10$ to 60\hMpc with a clear geometric interpretation. Comparisons with mock catalogues of N--body simulations using different variants of the CDM model demonstrate the discriminative power of the description. The standard CDM model and the model with tilted perturbation spectrum cannot generate the Minkowski functionals of the cluster data, while a model with a cosmological constant and a model with breaking of the scale invariance of perturbations (BSI) yield compatible results.Comment: 10 pages, 13 Postscript figures, uses epsf.sty and mn.sty (included), submitted to MNRA