Phylogenetic estimation with partial likelihood tensors

We present an alternative method for calculating likelihoods in molecular phylogenetics. Our method is based on partial likelihood tensors, which are generalizations of partial likelihood vectors, as used in Felsenstein's approach. Exploiting a lexicographic sorting and partial likelihood tensors, it is possible to obtain significant computational savings. We show this on a range of simulated data by enumerating all numerical calculations that are required by our method and the standard approach.     

Line FRAP with the confocal laser scanning microscope for diffusion measurements in small regions of 3-D samples

We present a truly quantitative fluorescence recovery after photobleaching (FRAP) model for use with the confocal laser scanning microscope based on the photobleaching of a long line segment. The line FRAP method is developed to complement the disk FRAP method reported before. Although being more subject to the influence of noise, the line FRAP model has the advantage of a smaller bleach region, thus allowing for faster and more localized measurements of the diffusion coefficient and mobile fraction. The line FRAP model is also very well suited to examine directly the influence of the bleaching power on the effective bleaching resolution. We present the outline of the mathematical derivation, leading to a final analytical expression to calculate the fluorescence recovery. We examine the influence of the confocal aperture and the bleaching power on the measured diffusion coefficient to find the optimal experimental conditions for the line FRAP method. This will be done for R-phycoerythrin and FITC-dextrans of various molecular weights. The ability of the line FRAP method to measure correctly absolute diffusion coefficients in three-dimensional samples will be evaluated as well. Finally we show the application of the method to the simultaneous measurement of free green fluorescent protein diffusion in the cytoplasm and nucleus of living A549 cells.     

Diffusion coefficients for rigid macromolecules with irregular shapes that allow rotational-translational coupling

We consider six-dimensional diffusion and frictional tensors for a rigid macromolecule immersed in a viscous fluid at low Reynolds number. Our treatment allows for screwlike properties which couple rotational and translational movements. We show that the center of diffusion of a screwlike body can be distinct from its hydrodynamic center of reaction. Symmetry conditions which ensure coincidence are examined. The center of diffusion is found to be the point of a body with the slowest diffusive movements, while rotations about the center of reaction encounter the least average resistance. The macroscopic translational diffusion coefficient is evaluated from a perturbation analysis of the six-dimensional diffusion equation. We show that methodologies which ignore translational–rotational coupling will necessarily underestimate the diffusion rate of screwlike particles. A procedural framework is presented to calculate diffusion coefficients of complicated bodies. As an example we treat a long bent rod.     

Classification of Dynamical Diffusion States in Single Molecule Tracking Microscopy

Single molecule tracking of membrane proteins by fluorescence microscopy is a promising method to investigate dynamic processes in live cells. Translating the trajectories of proteins to biological implications, such as protein interactions, requires the classification of protein motion within the trajectories. Spatial information of protein motion may reveal where the protein interacts with cellular structures, because binding of proteins to such structures often alters their diffusion speed. For dynamic diffusion systems, we provide an analytical framework to determine in which diffusion state a molecule is residing during the course of its trajectory. We compare different methods for the quantification of motion to utilize this framework for the classification of two diffusion states (two populations with different diffusion speed). We found that a gyration quantification method and a Bayesian statistics-based method are the most accurate in diffusion-state classification for realistic experimentally obtained datasets, of which the gyration method is much less computationally demanding. After classification of the diffusion, the lifetime of the states can be determined, and images of the diffusion states can be reconstructed at high resolution. Simulations validate these applications. We apply the classification and its applications to experimental data to demonstrate the potential of this approach to obtain further insights into the dynamics of cell membrane proteins.     

Classification of Dynamical Diffusion States in Single Molecule Tracking Microscopy

Single molecule tracking of membrane proteins by fluorescence microscopy is a promising method to investigate dynamic processes in live cells. Translating the trajectories of proteins to biological implications, such as protein interactions, requires the classification of protein motion within the trajectories. Spatial information of protein motion may reveal where the protein interacts with cellular structures, because binding of proteins to such structures often alters their diffusion speed. For dynamic diffusion systems, we provide an analytical framework to determine in which diffusion state a molecule is residing during the course of its trajectory. We compare different methods for the quantification of motion to utilize this framework for the classification of two diffusion states (two populations with different diffusion speed). We found that a gyration quantification method and a Bayesian statistics-based method are the most accurate in diffusion-state classification for realistic experimentally obtained datasets, of which the gyration method is much less computationally demanding. After classification of the diffusion, the lifetime of the states can be determined, and images of the diffusion states can be reconstructed at high resolution. Simulations validate these applications. We apply the classification and its applications to experimental data to demonstrate the potential of this approach to obtain further insights into the dynamics of cell membrane proteins.     

Atomic structure of a single large biomolecule from diffraction patterns of random orientations

The short and intense pulses of the new X-ray free electron lasers, now operational or under construction, may make possible diffraction experiments on single molecule-sized objects with high resolution, before radiation damage destroys the sample. In a single molecule imaging (SMI) experiment thousands of diffraction patterns of single molecules with random orientations are recorded. One of the most challenging problems of SMI is how to assemble these noisy patterns of unknown orientations into a consistent single set of diffraction data. Here we present a new method which can solve the orientation problem of SMI efficiently even for large biological molecules and in the presence of noise. We show on simulated diffraction patterns of a large protein molecule, how the orientations of the patterns can be found and the structure to atomic resolution can be solved. The concept of our algorithm could be also applied to experiments where images of an object are recorded in unknown orientations and/or positions like in cryoEM or tomography.     

Transport properties of particles with segmental flexibility. I. Hydrodynamic resistance and diffusion coefficients of a freely hinged particle

Expressions are derived for the hydrodynamic resistance tensor and the diffusion tensor of a particle consisting of two rigid subunits connected by a free hinge. No restrictions are placed on the shapes of the subunits. The resistance tensor is obtained by using two independent approaches: first, from the Rayleigh dissipation function and, second, from an examination of the generalized forces for the appropriate seven-dimensional coordinate system. For the derivation of the generalized Einstein equation connecting the diffusion and resistance tensors, the Brownian motion is treated as a stochastic process. That derivation is based on the assumption that the restoring force for bending is negligible, and the Einstein relation holds instantaneously only if that assumption is true. The relationship between these tensors and the macroscopically observable parameters is discussed, and it is shown that the separate measurement of resistance and diffusion coefficients can be used to detect macromolecular flexibility. One example is treated, the diffusion of a particle composed of two long rods joined at a free hinge. Those calculations are carried out with the first-order assumption of negligible hydrodynamic interactions between the subunits. For the hinged rod, the bending degree of freedom produces a 34% increase in the translational diffusion coefficient over that of a stiff rod of the same total length, while the rotational diffusion coefficient about the axis perpendicular to the plane of bending is increased by 125%.     

Single particle tracking. Analysis of diffusion and flow in two-dimensional systems.

Analysis of the trajectories of small particles at high spatial and temporal resolution using video enhanced contrast microscopy provides a powerful approach to characterizing the mechanisms of particle motion in living cells and in other systems. We present here the theoretical basis for the analysis of these trajectories for particles undergoing random diffusion and/or systematic transport at uniform velocity in two-dimensional systems. The single particle tracking method, based on observations of the trajectories of individual particles, is compared with methods that characterize the motions of a large collection of particles such as fluorescence photobleaching recovery. Determination of diffusion coefficients or transport velocities either from correlation of positions or of velocities of the particles is discussed. A result of practical importance is an analysis of the dependence of the expected statistical uncertainty of these determinations on the number of position measurements. This provides a way of judging the accuracy of the diffusion coefficients and transport velocities obtained using this approach.     

Hydrodynamic resistance and diffusion coefficients of a freely hinged rod

The seven-dimensional hydrodynamic resistance and diffusion tensors are evaluated for a rod which is freely hinged at its center and immersed in a viscous fluid. The hydrodynamic resistance tensor is first determined at the hinge, then transformed to other points and inverted to obtain the diffusion tensor. Hydrodynamic interactions between rod halves are neglected, which is asymptotically correct for long rods. In the long-rod limit, the diffusion coefficient characterizing translations over macroscopic distances is decreased by 3–6% from that for a rigid straight rod of same total length, while the average end-over-end rotational diffusion coefficient for each rod half is increased 4.67 times.     

Spatially resolved fluorescence correlation spectroscopy using a spinning disk confocal microscope

We develop an extension of fluorescence correlation spectroscopy (FCS) using a spinning disk confocal microscope. This approach can spatially map diffusion coefficients or flow velocities at up to approximately 10(5) independent locations simultaneously. Commercially available cameras with frame rates of 1000 Hz allow FCS measurements of systems with diffusion coefficients D~10(-7) cm(2)/s or smaller. This speed is adequate to measure small microspheres (200-nm diameter) diffusing in water, or hindered diffusion of macromolecules in complex media (e.g., tumors, cell nuclei, or the extracellular matrix). There have been a number of recent extensions to FCS based on laser scanning microscopy. Spinning disk confocal microscopy, however, has the potential for significantly higher speed at high spatial resolution. We show how to account for a pixel size effect encountered with spinning disk confocal FCS that is not present in standard or scanning FCS, and we introduce a new method to correct for photobleaching. Finally, we apply spinning disk confocal FCS to microspheres diffusing in Type I collagen, which show complex spatially varying diffusion caused by hydrodynamic and steric interactions with the collagen matrix.     

A methodology for registration of a histological slide and in vivo MRI volume based on optimizing mutual information

We present a method for registering histology and in vivo imaging that requires minimal microtoming and is automatic following the user's initialization. In this demonstration, we register a single hematoxylin-and-eosin-stained histological slide of a coronal section of a rat brain harboring a 9L gliosarcoma with an in vivo 7T MR image volume of the same brain. Because the spatial resolution of the in vivo MRI is limited, we add the step of obtaining a high spatial resolution, ex vivo MRI in situ for intermediate registration. The approach taken was to maximize mutual information in order to optimize the registration between all pairings of image data whether the sources are MRI, tissue block photograph, or stained sample photograph. The warping interpolant used was thin plate splines with the appropriate basis function for either 2-D or 3-D applications. All registrations were implemented by user initialization of the approximate pose between the two data sets, followed by automatic optimization based on maximizing mutual information. Only the higher quality anatomical images were used in the registration process; however, the spatial transformation was directly applied to a quantitative diffusion image. Quantitative diffusion maps from the registered location appeared highly correlated with the H&E slide. Overall, this approach provides a robust method for coregistration of in vivo images with histological sections and will have broad applications in the field of functional and molecular imaging.     

Bayesian regularization of diffusion tensor images

Diffusion tensor imaging (DTI) is a powerful tool in the study of the course of nerve fiber bundles in the human brain. Using DTI, the local fiber orientation in each image voxel can be described by a diffusion tensor which is constructed from local measurements of diffusion coefficients along several directions. The measured diffusion coefficients and thereby the diffusion tensors are subject to noise, leading to possibly flawed representations of the 3-dimensional (3D) fiber bundles. In this paper, we develop a Bayesian procedure for regularizing the diffusion tensor field, fully utilizing the available 3D information of fiber orientation. The use of the procedure is exemplified on synthetic and in vivo data.     

Image blurring by thermal diffusion in the observation of hydrated biomolecules with soft X-ray microscopy.

A simple model is presented for the estimation of image blurring in X-ray microscopy of biological specimens in a hydrated environment. The model is essentially based on thermal diffusion of an object to be imaged. The degree of image blurring by diffusion depends on the following situations of the object. The object is free from, is tightly fixed to, or is partially connected to the surrounding structures. The proper imaging time required to achieve a given resolution in X-ray microscopy of biological structures was estimated with the present method. The results suggest that imaging time shorter than 3 msec (free) to 1.4 sec (tightly fixed) is required for the observation of a cell (30 microns in diameter) at the resolution of 100 nm. The model is also applicable to a fragmented object caused by imaging X-rays.     

Angular smoothing and radial regularization of ODF fields: Application on deterministic crossing fiber tractography

The advent of high angular resolution diffusion imaging (HARDI) has opened up new perspectives for the delineation of crossing and branching fiber pathways. However, image acquisition under clinical conditions with limited measurement time faces the problem of poor spatial and angular resolution and the technique’s high susceptibility to noise. In this paper we present a straightforward spatial filter for ODF fields that uses the data-inherent structural information around a voxel as part of a directionally selective method for angular smoothing and radial regularization (ASRR). Especially in regions where fibers cross (multimodal voxels), the method allows us to reduce noise, improve the accuracy of ODF diffusion peaks, and strengthen signals of non-dominant fibers. Moreover, we propose a dynamic scheme in which regularization is applied only to ODFs classified as multimodal. The approach is quantitatively evaluated on synthetic datasets of various configurations. With an in vivo dataset of a human subject, measured under clinical imaging conditions, we demonstrate the method’s ability to improve tractography of non-dominant transcallosal fiber pathways and the long fibers of the superior longitudinal fasciculus.     

VAMP2 and synaptotagmin mobility in chromaffin granule membranes: implications for regulated exocytosis

Granule-plasma membrane docking and fusion can only occur when proteins that enable these reactions are present at the granule-plasma membrane contact. Thus, the mobility of granule membrane proteins may influence docking and membrane fusion. We measured the mobility of vesicle associated membrane protein 2 (VAMP2), synaptotagmin 1 (Syt1), and synaptotagmin 7 (Syt7) in chromaffin granule membranes in living chromaffin cells. We used a method that is not limited by standard optical resolution. A bright flash of strongly decaying evanescent field produced by total internal reflection was used to photobleach GFP-labeled proteins in the granule membrane. Fluorescence recovery occurs as unbleached protein in the granule membrane distal from the glass interface diffuses into the more bleached proximal regions, enabling the measurement of diffusion coefficients. We found that VAMP2-EGFP and Syt7-EGFP are mobile with a diffusion coefficient of ∼3 × 10⁻¹⁰ cm²/s. Syt1-EGFP mobility was below the detection limit. Utilizing these diffusion parameters, we estimated the time required for these proteins to arrive at docking and nascent fusion sites to be many tens of milliseconds. Our analyses raise the possibility that the diffusion characteristics of VAMP2 and Syt proteins could be a factor that influences the rate of exocytosis.     

Measurement of 15N relaxation in deuterated amide groups in proteins using direct nitrogen detection

15N chemical shielding tensors contain useful structural information, and their knowledge is essential for accurate analysis of protein backbone dynamics. The anisotropic component (CSA) of 15N chemical shielding can be obtained from 15N relaxation measurements in solution. However, the predominant contribution to nitrogen relaxation from 15N-(1)H dipolar coupling in amide groups limits the sensitivity of these measurements to the actual CSA values. Here we present nitrogen-detected NMR experiments for measuring 15N relaxation in deuterated amide groups in proteins, where the dipolar contribution to 15N relaxation is significantly reduced by the deuteration. Under these conditions nitrogen spin relaxation becomes a sensitive probe for variations in 15N chemical shielding tensors. Using the nitrogen direct-detection experiments we measured the rates of longitudinal and transverse 15N relaxation for backbone amides in protein G in D(2)O at 11.7 T. The measured relaxation rates are validated by comparing the overall rotational diffusion tensor obtained from these data with that from the conventional 15N relaxation measurements in H(2)O. This analysis revealed a 17-24 degree angle between the NH-bond and the unique axis of the 15N chemical shielding tensor.     

Solid-state NMR spin diffusion for measurement of membrane-bound peptide structure: gramicidin A

A recently developed solid-state NMR method for measurement of depths in membrane systems is applied to gramicidin A, a membrane-bound peptide of known structure, to investigate the potential of this method. (15)N-detected, (1)H spin diffusion experiments demonstrate the resolution of the technique by measuring the 4-5 A depth differences between three (15)N-labeled backbone sites (Trp13, Val7, Gly2) in gramicidin A. We also show that (13)C-detected, (1)H spin diffusion experiments on unlabeled gramicidin A are sufficient to discriminate between the end-to-end dimer and double-helix structures of gramicidin A. Thus, spin diffusion solid-state NMR experiments can provide a simple approach, which does not require labeled samples, for testing structural models of membrane-bound peptides.     

Analysis of kinetics in noisy systems: application to single molecule tethered particle motion

In the tethered particle motion method the length of a DNA molecule is monitored by measuring the range of diffusion of a microsphere tethered to the surface of a microscope coverslip through the DNA molecule itself. Looping of DNA (induced by binding of a specific protein) can be detected with this method and the kinetics of the looping/unlooping processes can be measured at the single molecule level. The microsphere's position variance represents the experimental variable reporting on the polymer length. Therefore, data windowing is required to obtain position variance from raw position data. Due to the characteristic diffusion time of the microsphere, the low-pass filtering required to attain a good signal/noise ratio (S/N) in the discrimination of looped versus unlooped state impacts significantly the measurement's time resolution. Here we present a method for measuring lifetimes based on half-amplitude thresholding and then correcting the kinetic measurements, taking into account low S/N (leading to false events) and limited time resolution (leading to missed events). This method allows an accurate and unbiased estimation of the kinetic parameters under investigation, independently of the choice of the window used for variance calculation, with potential applications to other single molecule measurements with low S/N.