In this corrigendum, we corrected the potential function U(Φ, σ) t`hat used to derive the Schrödinger equation for a one-dimensional harmonic oscillator in [Chaos, Solitons and Fractals 103 (2017) 567–570]. We also corrected several typos in the equations in the paper and the supplementary information. These corrections do not alter the conclusion of the article.

General relativity and quantum mechanism are two separate rules of modern physics explaining how nature works. Both theories are accurate, but the direct connection between two theories was not yet clarified. Recently, researchers blur the line between classical and quantum physics by connecting chaos and entanglement equation. Here, we showed the Duan's extended HM theory, which has the solution of the general relativity, can also have the solutions of the classic chaos equations and even the solution of Schrödinger equation in quantum physics, suggesting the extended theory of harmonic maps may act as a universal theory of physics.

The dynamic personalities and structural heterogeneity of proteins are essential for proper functioning. Structural determination of dynamic/heterogeneous proteins is limited by conventional approaches of X-ray and electron microscopy (EM) of single-particle reconstruction that require an average from thousands to millions different molecules. Cryo-electron tomography (cryoET) is an approach to determine three-dimensional (3D) reconstruction of a single and unique biological object such as bacteria and cells, by imaging the object from a series of tilting angles. However, cconventional reconstruction methods use large-size whole-micrographs that are limited by reconstruction resolution (lower than 20 Å), especially for small and low-symmetric molecule (<400 kDa). In this study, we demonstrated the adverse effects from image distortion and the measuring tilt-errors (including tilt-axis and tilt-angle errors) both play a major role in limiting the reconstruction resolution. Therefore, we developed a "focused electron tomography reconstruction" (FETR) algorithm to improve the resolution by decreasing the reconstructing image size so that it contains only a single-instance protein. FETR can tolerate certain levels of image-distortion and measuring tilt-errors, and can also precisely determine the translational parameters via an iterative refinement process that contains a series of automatically generated dynamic filters and masks. To describe this method, a set of simulated cryoET images was employed; to validate this approach, the real experimental images from negative-staining and cryoET were used. Since this approach can obtain the structure of a single-instance molecule/particle, we named it individual-particle electron tomography (IPET) as a new robust strategy/approach that does not require a pre-given initial model, class averaging of multiple molecules or an extended ordered lattice, but can tolerate small tilt-errors for high-resolution single "snapshot" molecule structure determination. Thus, FETR/IPET provides a completely new opportunity for a single-molecule structure determination, and could be used to study the dynamic character and equilibrium fluctuation of macromolecules.

One potential pathway to find an ultimate rule governing our universe is to hunt for a connection among the fundamental equations in physics. Recently, Ren et al. reported that the harmonic maps with potential introduced by Duan, named extended harmonic mapping (EHM), connect the equations of general relativity, chaos and quantum mechanics via a universal geodesic equation. The equation, expressed as Euler-Lagrange equations on the Riemannian manifold, was obtained from the principle of least action. Here, we further demonstrate that more than ten fundamental equations, including that  of classical mechanics, fluid physics, statistical physics, astrophysics, quantum physics and general relativity, can be connected by the same universal geodesic equation. The connection sketches a family tree of the physics equations, and their intrinsic connections reflect an alternative ultimate rule of our universe, i.e., the principle of least action on a Finsler manifold.

Structural determination of proteins is rather challenging for proteins with molecular masses between 40 - 200 kDa. Considering that more than half of natural proteins have a molecular mass between 40 - 200 kDa, a robust and high-throughput method with a nanometer resolution capability is needed. Negative staining (NS) electron microscopy (EM) is an easy, rapid, and qualitative approach which has frequently been used in research laboratories to examine protein structure and protein-protein interactions. Unfortunately, conventional NS protocols often generate structural artifacts on proteins, especially with lipoproteins that usually form presenting rouleaux artifacts. By using images of lipoproteins from cryo-electron microscopy (cryo-EM) as a standard, the key parameters in NS specimen preparation conditions were recently screened and reported as the optimized NS protocol (OpNS), a modified conventional NS protocol. Artifacts like rouleaux can be greatly limited by OpNS, additionally providing high contrast along with reasonably high-resolution (near 1 nm) images of small and asymmetric proteins. These high-resolution and high contrast images are even favorable for an individual protein (a single object, no average) 3D reconstruction, such as a 160 kDa antibody, through the method of electron tomography. Moreover, OpNS can be a high-throughput tool to examine hundreds of samples of small proteins. For example, the previously published mechanism of 53 kDa cholesteryl ester transfer protein (CETP) involved the screening and imaging of hundreds of samples. Considering cryo-EM rarely successfully images proteins less than 200 kDa has yet to publish any study involving screening over one hundred sample conditions, it is fair to call OpNS a high-throughput method for studying small proteins. Hopefully the OpNS protocol presented here can be a useful tool to push the boundaries of EM and accelerate EM studies into small protein structure, dynamics and mechanisms.

Purpose of review

Cholesteryl ester transfer proteins (CETP) regulate plasma cholesterol levels by transferring cholesteryl esters (CEs) among lipoproteins. Lipoprotein cholesterol levels correlate with the risk factors for atherosclerotic cardiovascular disease (ASCVD). This article reviews recent research on CETP structure, lipid transfer mechanism, and its inhibition.

Recent findings

Genetic deficiency in CETP is associated with a low plasma level of low-density lipoprotein cholesterol (LDL-C) and a profoundly elevated plasma level of high-density lipoprotein cholesterol (HDL-C), which correlates with a lower risk of atherosclerotic cardiovascular disease (ASCVD). However, a very high concentration of HDL-C also correlates with increased ASCVD mortality. Considering that the elevated CETP activity is a major determinant of the atherogenic dyslipidemia, i.e., pro-atherogenic reductions in HDL and LDL particle size, inhibition of CETP emerged as a promising pharmacological target during the past two decades. CETP inhibitors, including torcetrapib, dalcetrapib, evacetrapib, anacetrapib and obicetrapib, were designed and evaluated in phase III clinical trials for the treatment of ASCVD or dyslipidemia. Although these inhibitors increase in plasma HDL-C levels and/or reduce LDL-C levels, the poor efficacy against ASCVD ended interest in CETP as an anti-ASCVD target. Nevertheless, interest in CETP and the molecular mechanism by which it inhibits CE transfer among lipoproteins persisted. Insights into the structural-based CETP-lipoprotein interactions can unravel CETP inhibition machinery, which can hopefully guide the design of more effective CETP inhibitors that combat ASCVD. Individual-molecule 3D structures of CETP bound to lipoproteins provide a model for understanding the mechanism by which CETP mediates lipid transfer and which in turn, guide the rational design of new anti-ASCVD therapeutics.

To enable fusion between biological membranes, t-SNAREs and v-SNARE present in opposing bilayers, interact and assemble in a circular configuration forming ring-complexes, which establish continuity between the opposing membranes, in presence of calcium ions. The size of a t-/v-SNARE ring complex is dictated by the curvature of the opposing membrane. Hence smaller vesicles form small SNARE-ring complexes, as opposed to large vesicles. Neuronal communication depends on the fusion of 40-50 nm in diameter membrane-bound synaptic vesicles containing neurotransmitters at the nerve terminal. At the presynaptic membrane, 12-17 nm in diameter cup-shaped neuronal porosomes are present where synaptic vesicles transiently dock and fuse. Studies demonstrate the presence of SNAREs at the porosome base. Atomic force microscopy (AFM), electron microscopy (EM), and electron density measurement studies demonstrate that at the porosome base, where synaptic vesicles dock and transiently fuse, proteins, possibly comprised of t-SNAREs, are found assembled in a ring conformation. To further determine the structure and arrangement of the neuronal t-/v-SNARE complex, 50 nm t-and v-SNARE proteoliposomes were mixed, allowing t-SNARE-vesicles to interact with v-SNARE vesicles, followed by detergent solubilization and imaging of the resultant t-/v-SNARE complexes formed using both AFM and EM. Our results demonstrate formation of 6-7 nm membrane-directed self-assembled t-/v-SNARE ring complexes, similar to, but twice as large as the ring structures present at the base of neuronal porosomes. The smaller SNARE ring at the porosome base may reflect the 3-4 nm base diameter, where 40-50 nm in diameter v-SNARE-associated synaptic vesicle transiently dock and fuse to release neurotransmitters.

Three-dimensional (3D) structural analysis is essential to understand the relationship between the structure and function of an object. Many analytical techniques, such as X-ray diffraction, neutron spectroscopy, and electron microscopy imaging, are used to provide structural information. Transmission electron microscopy (TEM), one of the most popular analytic tools, has been widely used for structural analysis in both physical and biological sciences for many decades, in which 3D objects are projected into two-dimensional (2D) images. In many cases, 2D-projection images are insufficient to understand the relationship between the 3D structure and the function of nanoscale objects. Electron tomography (ET) is a technique that retrieves 3D structural information from a tilt series of 2D projections, and is gradually becoming a mature technology with sub-nanometer resolution. Distinct methods to overcome sample-based limitations have been separately developed in both physical and biological science, although they share some basic concepts of ET. This review discusses the common basis for 3D characterization, and specifies difficulties and solutions regarding both hard and soft materials research. It is hoped that novel solutions based on current state-of-the-art techniques for advanced applications in hybrid matter systems can be motivated.

To understand the dynamic structure-function relationship of soft- and biomolecules, the determination of the three-dimensional (3D) structure of each individual molecule (nonaveraged structure) in its native state is sought-after. Cryo-electron tomography (cryo-ET) is a unique tool for imaging an individual object from a series of tilted views. However, due to radiation damage from the incident electron beam, the tolerable electron dose limits image contrast and the signal-to-noise ratio (SNR) of the data, preventing the 3D structure determination of individual molecules, especially at high-resolution. Although recently developed technologies and techniques, such as the direct electron detector, phase plate, and computational algorithms, can partially improve image contrast/SNR at the same electron dose, the high-resolution structure, such as tertiary structure of individual molecules, has not yet been resolved. Here, we review the cryo-electron microscopy (cryo-EM) and cryo-ET experimental parameters to discuss how these parameters affect the extent of radiation damage. This discussion can guide us in optimizing the experimental strategy to increase the imaging dose or improve image SNR without increasing the radiation damage. With a higher dose, a higher image contrast/SNR can be achieved, which is crucial for individual-molecule 3D structure. With 3D structures determined from an ensemble of individual molecules in different conformations, the molecular mechanism through their biochemical reactions, such as self-folding or synthesis, can be elucidated in a straightforward manner.