Optical imaging of single biomolecules and complexes in living cells provides a useful window into cellular processes. However, the three-dimensional dynamics of most important biomolecules in living cells remains essentially uncharacterized. The precise subcellular localization of mRNA-protein complexes plays a critical role in the spatial and temporal control of gene expression, and a full understanding of the control of gene expression requires precise characterization of mRNA transport dynamics beyond the optical diffraction limit. In this paper, we describe three-dimensional tracking of single mRNA particles with 25-nm precision in the x and y dimensions and 50-nm precision in the z dimension in live budding yeast cells using a microscope with a double-helix point spread function. Two statistical methods to detect intermittently confined and directed transport were used to quantify the three-dimensional trajectories of mRNA for the first time, using ARG3 mRNA as a model. Measurements and analysis show that the dynamics of ARG3 mRNA molecules are mostly diffusive, although periods of non-Brownian confinement and directed transport are observed. The quantitative methods detailed in this paper can be broadly applied to the study of mRNA localization and the dynamics of diverse other biomolecules in a wide variety of cell types.

Three-dimensional nanoscale localization and tracking of dim single emitters can be obtained with a widefield fluorescence microscope exhibiting a double-helix point spread function (DH-PSF). We describe in detail how the localization precision quantitatively depends upon the number of photons detected and the z position of the nanoscale emitter, thereby showing a ~10 nm localization capability along x, y, and z in the limit of weak emitters. Experimental measurements are compared to Fisher information calculations of the ultimate localization precision inherent in the DH-PSF. The DH-PSF, for the first time, is used to track single quantum dots in aqueous solution and a quantum dot-labeled structure inside a living cell in three dimensions.A critical problem in nanoscale science is the need to noninvasively and from a distance determine the location of a nanoscale object in three dimensions. Even though the diffraction limit of visible light is on the order of 200 nm, the two-dimensional (2D) image of a single molecule can be fit to find the x-y position of the molecule with nanometer precision1 , 2. The localization precision scales roughly as σ/(N) 1/2 where σ is the diffraction-limited standard deviation of the microscope point spread function (PSF) and N is the number of photons detected from a point-like emitter. This "super-localization" property, well-known in many areas of science 3 , was used to great effect by many groups to track isolated point-like singlemolecule emitters in a variety of contexts 4 , 5. However, imaging a continuous structure requires a high density of fluorescent labels, and the nanoscale detail is lost because the PSFs overlap. By using photoswitchable or photoactivatable fluorophores to keep the concentration of active emitters low at any one time, super-resolution images can be extracted by successive single-molecule localizations 6 -8 . These methods were, until recently, mostly limited to 2D imaging. Unfortunately, three-dimensional (3D) position information is difficult to obtain using a diffraction-limited conventional fluorescence microscope for two reasons: (1) the PSF is symmetric about the focal plane meaning that a molecule × nm above the focal plane cannot be distinguished from a molecule × nm below the focal plane, and (2) the PSF contains little information about the axial position of the emitter for a few hundred nanometers about the focal plane 9, 10, meaning that it is quite difficult to super-localize molecules that are in the focal plane of the microscope. Recently, several solutions to these problems have been described including using astigmatism 11 -13, imaging in two different focal planes 14,10,15,16 and interferometry 17, 18 . Fast 3D fluorescent particle tracking has also been performed using confocal imaging [19][20][21] and by using two rotating laser beams to track single quantum dots 21 . We recently demonstrated that a Double-Helix Point Spread Function (DH-PSF) can be used to super-localize single molecules in three dimensions 22 . Combining...

Wide-field microscopy with a double-helix point spread function (DH-PSF) provides three-dimensional (3D) position information beyond the optical diffraction limit. We compare the theoretical localization precision for an unbiased estimator of the DH-PSF to that for 3D localization by astigmatic and biplane imaging using Fisher information analysis including pixelation and varying levels of background. The DH-PSF results in almost constant localization precision in all three dimensions for a 2 μm thick depth of field while astigmatism and biplane improve the axial localization precision over smaller axial ranges. For high signal-to-background ratio, the DH-PSF on average achieves better localization precision.

We describe the corkscrew point spread function (PSF), which can localize objects in three dimensions throughout a 3.2 µm depth of field with nanometer precision. The corkscrew PSF rotates as a function of the axial (z) position of an emitter. Fisher information calculations show that the corkscrew PSF can achieve nanometer localization precision with limited numbers of photons. We demonstrate three-dimensional super-resolution microscopy with the corkscrew PSF by imaging beads on the surface of a triangular polydimethylsiloxane (PDMS) grating. With 99,000 photons detected, the corkscrew PSF achieves a localization precision of 2.7 nm in x, 2.1 nm in y, and 5.7 nm in z.

First, we propose a new stochastic model for a spatially incoherent source in optical phenomena. The model naturally incorporates the incoherent property into the electromagnetic wave equation through a random source term. Then we propose a new numerical method based on Wiener chaos expansion (WCE) and apply it to solve the resulting stochastic wave equation. The main advantage of the WCE method is that it separates random and deterministic effects and allows the random effects to be factored out of the primary partial differential equation (PDE) very effectively. Therefore, the stochastic PDE is reduced to a set of deterministic PDEs for the coefficients of the WCE method which can be solved by conventional numerical algorithms. We solve these secondary deterministic PDEs by a finite-difference time domain (FDTD) method and demonstrate that the numerical computations based on the WCE method are considerably more efficient than the brute-force simulations. Moreover, the WCE approach does not require generation of random numbers and results in less computational errors compared to Monte Carlo simulations.