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We present a new proof of the caloric smoothing related to the fractional Gauss-Weierstrass semi-group in Triebel-Lizorkin spaces. This property will be used to prove existence and uniqueness of mild and strong solutions of the Cauchy problem for a fractional nonlinear heat equation.

Introduced more than fifty years ago, dynamic light scattering is routinely used to determine the size distribution of colloidal suspensions, as well as of macromolecules in solution, such as proteins, nucleic acids, and their complexes. More recently, differential dynamic microscopy has been proposed as a way to perform dynamic light scattering experiments with a microscope, with much less stringent constraints in terms of cleanliness of the optical surfaces, but a potentially lower sensitivity due to the use of camera-based detectors. In this work, we push bright-field differential dynamic microscopy beyond known limits and show it to be sufficiently sensitive to size small macromolecules in diluted solutions. By considering solutions of three different proteins (Bovine Serum Albumin, Lysozyme, and Pepsin), we accurately determine the diffusion coefficient and hydrodynamic radius of both single proteins and small protein aggregates down to concentrations of a few milligrams per milliliter. In addition, we present preliminary results showing unexplored potential for the determination of virial coefficients. Our results are in excellent agreement with the ones obtained in parallel with a state-of-the-art commercial dynamic light scattering setup, showing that differential dynamic microscopy represents a valuable alternative for rapid, label-free protein sizing with an optical microscope.

The paper is concerned with the Cauchy problem for a semi-linear hyperdissipative heat equation in Besov and Triebel-Lizorkin spaces which is related to the generalized Gauss-Weierstrass semi-group via Duhamel's principle. Using caloric smoothing properties of the semi-group we prove existence and uniqueness of mild and strong solutions which are local in time. Moreover, we study well-posedness of the problem.

Optoplasmonic methods capable of single protein detection so far rely on analyte immobilization in order to facilitate detection [1-6]. These detection schemes, even if they facilitate transient single-molecule detection [7,8] via consequent formation and cleavage of chemical bonds, typically exhibit time resolutions on the order of milliseconds. The need for analyte immobilisation is a direct consequence of the minuscule dimensions of plasmonic near fields typically providing sub-attolitre-sized detection volumes which in turn demand sub-microsecond temporal resolution for the direct detection of proteins in motion. Here we show that such temporal resolution can indeed be achieved. We demonstrate the observation of single proteins as small as Hemoglobin (molecular weight: 64 kDa) as they traverse plasmonic near fields of gold nanorods and interact with their surface, all while maintaining signal-to-noise ratios larger than 5 and an unprecedented temporal resolution well below microseconds. This method enables the label-free observation of single-molecule dynamics on previously unaccessible timescales., Comment: 15 pages, 7 figures

Optical detection of individual nanometer-sized analytes, virus particles, and protein molecules holds great promise for understanding and control of biological samples and healthcare applications. As fluorescent labels impose restrictions on detection bandwidth and require lengthy and invasive processes, label-free optical techniques are highly desirable. Powerful label-free optical methods have recently emerged, such as interferometric scattering microscopy, plasmonic nanoparticle-based assays and microcavity-based assays. Although highly sensitive, these methods are so far restricted to integration times in excess of microseconds. This often imposes a requirement to impede analyte motion during these periods via specific molecular tethers, unspecific adsorption or confining arrangements. Here we introduce an optical technique capable of transforming gold nanorods commonly used as photostable labels into highly localized high-speed probes. Our method provides a time resolution well below microseconds. This mitigates the requirement for molecular tethers and allows us to detect single untethered nanoparticles in Brownian motion traversing sub atto-liter sensing volumes. Our method opens a novel gateway for the investigation of highly localized and highly dynamic nanoscale systems and constitutes a first step towards the label-free recognition of single untethered proteins., Comment: Manuscript and suppl. information are on 30 pages and possess 16 figures

Monitoring the kinetics and conformational dynamics of single enzymes is crucial in order to better understand their biological functions as these motions and structural dynamics are usually unsynchronized among the molecules. Detecting the enzyme-reactant interactions and associated conformational changes of the enzyme on a single molecule basis, however, remain as a challenge with established optical techniques due to the commonly required labeling of the reactants or the enzyme itself. The labeling process is usually non-trivial and the labels themselves might skew the physical properties of the enzyme. Here we demonstrate an optical, label-free method capable of observing enzymatic interactions and the associated conformational changes on the single molecule level. We monitor polymerase/DNA interactions via the strong near-field enhancement provided by plasmonic nanorods resonantly coupled to whispering gallery modes in microcavities. Specifically, we employ two different recognition schemes: one in which the kinetics of polymerase/DNA interactions are probed in the vicinity of DNA-functionalized nanorods, and the other in which these interactions are probed via the magnitude of conformational changes in the polymerase molecules immobilized on nanorods. In both approaches we find that low and high polymerase activities can be clearly discerned via their characteristic signal amplitude and signal length distributions. Furthermore, the thermodynamic study of the monitored interactions suggests the occurrence of DNA polymerization. This work constitutes a proof-of-concept study of enzymatic activities via plasmonically enhanced microcavities and establishes an alternative and label-free method capable of investigating structural changes in single molecules.

We study two different methods for inferring the parameters of a spheroid distribution from planar sections of a stationary spatial system of spheroids: one method first unfolds non-parametrically the joint size-shape-orientation distribution of the observable ellipses in the plane into the joint size-shape-orientation distribution of the spheroids followed by a maximum likelihood estimation of the parameters; the second method directly estimates these parameters based on statistics of the observable ellipses using a quasi-likelihood approach. As an application we consider a metal-matrix composite with ceramic particles as reinforcing inclusions, model the inclusions as prolate spheroids and estimate the parameters of their distribution from planar sections., Comment: 20 pages, 5 figures (including 16 images)