Time-ordering and a generalized Magnus expansion

Both the classical time-ordering and the Magnus expansion are well-known in the context of linear initial value problems. Motivated by the noncommutativity between time-ordering and time derivation, and related problems raised recently in statistical physics, we introduce a generalization of the Magnus expansion. Whereas the classical expansion computes the logarithm of the evolution operator of a linear differential equation, our generalization addresses the same problem, including however directly a non-trivial initial condition. As a by-product we recover a variant of the time ordering operation, known as T*-ordering. Eventually, placing our results in the general context of Rota-Baxter algebras permits us to present them in a more natural algebraic setting. It encompasses, for example, the case where one considers linear difference equations instead of linear differential equations

Elastin-like Polypeptide Linkers for Single-Molecule Force Spectroscopy

Single-molecule force spectroscopy (SMFS) is by now well established as a standard technique in biophysics and mechanobiology. In recent years, the technique has benefitted greatly from new approaches to bioconjugation of proteins to surfaces. Indeed, optimized immobilization strategies for biomolecules and refined purification schemes are being steadily adapted and improved, which in turn has enhanced data quality. In many previously reported SMFS studies, poly(ethylene glycol) (PEG) was used to anchor molecules of interest to surfaces and/or cantilever tips. The limitation, however, is that PEG exhibits a well-known trans-trans-gauche to all-trans transition, which results in marked deviation from standard polymer elasticity models such as the worm-like chain, particularly at elevated forces. As a result, the assignment of unfolding events to protein domains based on their corresponding amino acid chain lengths is significantly obscured. Here, we provide a solution to this problem by implementing unstructured elastin-like polypeptides as linkers to replace PEG. We investigate the suitability of tailored elastin-like polypeptides linkers and perform direct comparisons to PEG, focusing on attributes that are critical for single-molecule force experiments such as linker length, monodispersity, and bioorthogonal conjugation tags. Our results demonstrate that by avoiding the ambiguous elastic response of mixed PEG/peptide systems and instead building the molecular mechanical systems with only a single bond type with uniform elastic properties, we improve data quality and facilitate data analysis and interpretation in force spectroscopy experiments. The use of all-peptide linkers allows alternative approaches for precisely defining elastic properties of proteins linked to surfaces

Monodisperse measurement of the biotin-streptavidin interaction strength in a well-defined pulling geometry

The widely used interaction of the homotetramer streptavidin with the small molecule biotin has been intensively studied by force spectroscopy and has become a model system for receptor ligand interaction. However, streptavidin's tetravalency results in diverse force propagation pathways through the different binding interfaces. This multiplicity gives rise to polydisperse force spectroscopy data. Here, we present an engineered monovalent streptavidin tetramer with a single cysteine in its functional subunit that allows for site-specific immobilization of the molecule, orthogonal to biotin binding. Functionality of streptavidin and its binding properties for biotin remain unaffected. We thus created a stable and reliable molecular anchor with a unique high-affinity binding site for biotinylated molecules or nanoparticles, which we expect to be useful for many single-molecule applications. To characterize the mechanical properties of the bond between biotin and our monovalent streptavidin, we performed force spectroscopy experiments using an atomic force microscope. We were able to conduct measurements at the single-molecule level with 1: 1-stoichiometry and a well-defined geometry, in which force exclusively propagates through a single subunit of the streptavidin tetramer. For different force loading rates, we obtained narrow force distributions of the bond rupture forces ranging from 200 pN at 1,500 pN/s to 230 pN at 110,000 pN/s. The data are in very good agreement with the standard Bell-Evans model with a single potential barrier at Delta x(0) = 0.38 nm and a zero-force off-rate k(off,0) in the 10(-6) s(-1) range

Quasi-K\"ahler groups, 3-manifold groups, and formality

In this note, we address the following question: Which 1-formal groups occur as fundamental groups of both quasi-K\"ahler manifolds and closed, connected, orientable 3-manifolds. We classify all such groups, at the level of Malcev completions, and compute their coranks. Dropping the assumption on realizability by 3-manifolds, we show that the corank equals the isotropy index of the cup-product map in degree one. Finally, we examine the formality properties of smooth affine surfaces and quasi-homogeneous isolated surface singularities. In the latter case, we describe explicitly the positive-dimensional components of the first characteristic variety for the associated singularity link.Comment: 18 pages; accepted for publication in Mathematische Zeitschrif

A tethered ligand assay to probe SARS-CoV-2:ACE2 interactions

Severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) infections are initiated by attachment of the receptor-binding domain (RBD) on the viral Spike protein to angiotensin-converting enzyme-2 (ACE2) on human host cells. This critical first step occurs in dynamic environments, where external forces act on the binding partners and avidity effects play an important role, creating an urgent need for assays that can quantitate SARS-CoV-2 interactions with ACE2 under mechanical load. Here, we introduce a tethered ligand assay that comprises the RBD and the ACE2 ectodomain joined by a flexible peptide linker. Using magnetic tweezers and atomic force spectroscopy as highly complementary single-molecule force spectroscopy techniques, we investigate the RBD:ACE2 interaction over the whole physiologically relevant force range. We combine the experimental results with steered molecular dynamics simulations and observe and assign fully consistent unbinding and unfolding events across the three techniques, enabling us to establish ACE2 unfolding as a molecular fingerprint. Measuring at forces of 2 to 5 pN, we quantify the force dependence and kinetics of the RBD:ACE2 bond in equilibrium. We show that the SARS-CoV-2 RBD:ACE2 interaction has higher mechanical stability, larger binding free energy, and a lower dissociation rate compared to SARS-CoV-1, which helps to rationalize the different infection patterns of the two viruses. By studying how free ACE2 outcompetes tethered ACE2, we show that our assay is sensitive to prevention of bond formation by external binders. We expect our results to provide a way to investigate the roles of viral mutations and blocking agents for targeted pharmaceutical intervention

A Tethered Ligand Assay to Probe SARS-CoV-2:ACE2 Interactions

SARS-CoV-2 infections are initiated by attachment of the receptor-binding domain (RBD) on the viral Spike protein to angiotensin-converting enzyme-2 (ACE2) on human host cells. This critical first step occurs in dynamic environments, where external forces act on the binding partners and multivalent interactions play critical roles, creating an urgent need for assays that can quantitate SARS-CoV-2 interactions with ACE2 under mechanical load and in defined geometries. Here, we introduce a tethered ligand assay that comprises the RBD and the ACE2 ectodomain joined by a flexible peptide linker. Using magnetic tweezers and atomic force spectroscopy as highly complementary single-molecule force spectroscopy techniques, we investigate the RBD:ACE2 interaction over the whole physiologically relevant force range. We combine the experimental results with steered molecular dynamics simulations and observe and assign fully consistent unbinding and unfolding events across the three techniques, enabling us to establish ACE2 unfolding as a molecular fingerprint. Measuring at forces of 2-5 pN, we quantify the force dependence and kinetics of the RBD:ACE2 bond in equilibrium. We show that the SARS-CoV-2 RBD:ACE2 interaction has higher mechanical stability, larger binding free energy, and a lower dissociation rate in comparison to SARS-CoV-1, which helps to rationalize the different infection patterns of the two viruses. By studying how free ACE2 outcompetes tethered ACE2, we show that our assay is sensitive to prevention of bond formation by external binders. We expect our results to provide a novel way to investigate the roles of mutations and blocking agents for targeted pharmaceutical intervention.N

A tethered ligand assay to probe SARS-CoV-2:ACE2 interactions

Severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) infections are initiated by attachment of the receptor-binding domain (RBD) on the viral Spike protein to angiotensin-converting enzyme-2 (ACE2) on human host cells. This critical first step occurs in dynamic environments, where external forces act on the binding partners and avidity effects play an important role, creating an urgent need for assays that can quantitate SARS-CoV-2 interactions with ACE2 under mechanical load. Here, we introduce a tethered ligand assay that comprises the RBD and the ACE2 ectodomain joined by a flexible peptide linker. Using magnetic tweezers and atomic force spectroscopy as highly complementary single-molecule force spectroscopy techniques, we investigate the RBD:ACE2 interaction over the whole physiologically relevant force range. We combine the experimental results with steered molecular dynamics simulations and observe and assign fully consistent unbinding and unfolding events across the three techniques, enabling us to establish ACE2 unfolding as a molecular fingerprint. Measuring at forces of 2 to 5 pN, we quantify the force dependence and kinetics of the RBD:ACE2 bond in equilibrium. We show that the SARS-CoV-2 RBD:ACE2 interaction has higher mechanical stability, larger binding free energy, and a lower dissociation rate compared to SARS-CoV-1, which helps to rationalize the different infection patterns of the two viruses. By studying how free ACE2 outcompetes tethered ACE2, we show that our assay is sensitive to prevention of bond formation by external binders. We expect our results to provide a way to investigate the roles of viral mutations and blocking agents for targeted pharmaceutical intervention.This study was supported by German Research Foundation Projects 386143268 and 111166240, a Human Frontier Science ProgramCross Disciplinary Fellowship (LT000395/2020C) and European Molecular Biology Organization Non-Stipendiary long-term fellowship (ALTF 1047-2019) to L.F.M., and the Physics Department of LMU Munich. R.C.B. and P.S.F.C.G. are supported by start-up funds provided by Auburn University, and D.L. acknowledges support from the Spanish Ministry of Science, Innovation and Universities for the Spanish State Research Agency Retos Grant RTI2018- 099318-B-I00, cofunded by the European Regional Development Fund.Peer reviewe

Genetic determinants of heel bone properties: genome-wide association meta-analysis and replication in the GEFOS/GENOMOS consortium

Quantitative ultrasound of the heel captures heel bone properties that independently predict fracture risk and, with bone mineral density (BMD) assessed by X-ray (DXA), may be convenient alternatives for evaluating osteoporosis and fracture risk. We performed a meta-analysis of genome-wide association (GWA) studies to assess the genetic determinants of heel broadband ultrasound attenuation (BUA; n = 14 260), velocity of sound (VOS; n = 15 514) and BMD (n = 4566) in 13 discovery cohorts. Independent replication involved seven cohorts with GWA data (in silico n = 11 452) and new genotyping in 15 cohorts (de novo n = 24 902). In combined random effects, meta-analysis of the discovery and replication cohorts, nine single nucleotide polymorphisms (SNPs) had genome-wide significant (P < 5 × 10(-8)) associations with heel bone properties. Alongside SNPs within or near previously identified osteoporosis susceptibility genes including ESR1 (6q25.1: rs4869739, rs3020331, rs2982552), SPTBN1 (2p16.2: rs11898505), RSPO3 (6q22.33: rs7741021), WNT16 (7q31.31: rs2908007), DKK1 (10q21.1: rs7902708) and GPATCH1 (19q13.11: rs10416265), we identified a new locus on chromosome 11q14.2 (rs597319 close to TMEM135, a gene recently linked to osteoblastogenesis and longevity) significantly associated with both BUA and VOS (P < 8.23 × 10(-14)). In meta-analyses involving 25 cohorts with up to 14 985 fracture cases, six of 10 SNPs associated with heel bone properties at P < 5 × 10(-6) also had the expected direction of association with any fracture (P < 0.05), including three SNPs with P < 0.005: 6q22.33 (rs7741021), 7q31.31 (rs2908007) and 10q21.1 (rs7902708). In conclusion, this GWA study reveals the effect of several genes common to central DXA-derived BMD and heel ultrasound/DXA measures and points to a new genetic locus with potential implications for better understanding of osteoporosis pathophysiology

Functional Stability of Unliganded Envelope Glycoprotein Spikes among Isolates of Human Immunodeficiency Virus Type 1 (HIV-1)

The HIV-1 envelope glycoprotein (Env) spike is challenging to study at the molecular level, due in part to its genetic variability, structural heterogeneity and lability. However, the extent of lability in Env function, particularly for primary isolates across clades, has not been explored. Here, we probe stability of function for variant Envs of a range of isolates from chronic and acute infection, and from clades A, B and C, all on a constant virus backbone. Stability is elucidated in terms of the sensitivity of isolate infectivity to destabilizing conditions. A heat-gradient assay was used to determine T90 values, the temperature at which HIV-1 infectivity is decreased by 90% in 1 h, which ranged between ∼40 to 49°C (n = 34). For select Envs (n = 10), the half-lives of infectivity decay at 37°C were also determined and these correlated significantly with the T90 (p = 0.029), though two ‘outliers’ were identified. Specificity in functional Env stability was also evident. For example, Env variant HIV-1ADA was found to be labile to heat, 37°C decay, and guanidinium hydrochloride but not to urea or extremes of pH, when compared to its thermostable counterpart, HIV-1JR-CSF. Blue native PAGE analyses revealed that Env-dependent viral inactivation preceded complete dissociation of Env trimers. The viral membrane and membrane-proximal external region (MPER) of gp41 were also shown to be important for maintaining trimer stability at physiological temperature. Overall, our results indicate that primary HIV-1 Envs can have diverse sensitivities to functional inactivation in vitro, including at physiological temperature, and suggest that parameters of functional Env stability may be helpful in the study and optimization of native Env mimetics and vaccines