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Phloem-feeding insects feed on plant phloem using their stylets. While ingesting phloem sap, these insects secrete saliva to circumvent plant defenses. Previous studies have shown that, to facilitate their feeding, many phloem-feeding insects can elicit the salicylic acid- (SA-) signaling pathway and thus suppress effective jasmonic acid defenses. However, the molecular basis for the regulation of the plant’s defense by phloem-feeding insects remains largely unknown. Here, we show that Bt56, a whitefly-secreted low molecular weight salivary protein, is highly expressed in the whitefly primary salivary gland and is delivered into host plants during feeding. Overexpression of the Bt56 gene in planta promotes susceptibility of tobacco to the whitefly and elicits the SA-signaling pathway. In contrast, silencing the whitefly Bt56 gene significantly decreases whitefly performance on host plants and interrupts whitefly phloem feeding with whiteflies losing the ability to activate the SA pathway. Protein-protein interaction assays show that the Bt56 protein directly interacts with a tobacco KNOTTED 1-like homeobox transcription factor that decreases whitefly performance and suppresses whitefly-induced SA accumulation. The Bt56 orthologous genes are highly conserved but differentially expressed in different species of whiteflies. In conclusion, Bt56 is a key salivary effector that promotes whitefly performance by eliciting salicylic acid-signaling pathway.

We prove that every smoothly embedded surface in a 4-manifold can be isotoped to be in bridge position with respect to a given trisection of the ambient 4-manifold; that is, after isotopy, the surface meets components of the trisection in trivial disks or arcs. Such a decomposition, which we call a generalized bridge trisection, extends the authors’ definition of bridge trisections for surfaces in S⁴. Using this construction, we give diagrammatic representations called shadow diagrams for knotted surfaces in 4-manifolds. We also provide a low-complexity classification for these structures and describe several examples, including the important case of complex curves inside ℂℙ². Using these examples, we prove that there exist exotic 4-manifolds with (g, 0)—trisections for certain values of g. We conclude by sketching a conjectural uniqueness result that would provide a complete diagrammatic calculus for studying knotted surfaces through their shadow diagrams.

We classify topological 4-manifolds with boundary and fundamental group Z, under some assumptions on the boundary. We apply this to classify surfaces in simply-connected 4-manifolds with S3 boundary, where the fundamental group of the surface complement is Z. We then compare these homeomorphism classifications with the smooth setting. For manifolds, we show that every Hermitian form over Z[t±1] arises as the equivariant intersection form of a pair of exotic smooth 4-manifolds with boundary and fundamental group Z. For surfaces we have a similar result, and in particular we show that every 2-handlebody with S3 boundary contains a pair of exotic discs.

Spontaneous folding of a polypeptide chain into a knotted structure remains one of the most puzzling and fascinating features of protein folding. The folding of knotted proteins is on the timescale of minutes and thus hard to reproduce with atomistic simulations that have been able to reproduce features of ultrafast folding in great detail. Furthermore, it is generally not possible to control the topology of the unfolded state. Single-molecule force spectroscopy is an ideal tool for overcoming this problem: by variation of pulling directions, we controlled the knotting topology of the unfolded state of the 5₂-knotted protein ubiquitin C-terminal hydrolase isoenzyme L1 (UCH-L1) and have therefore been able to quantify the influence of knotting on its folding rate. Here, we provide direct evidence that a threading event associated with formation of either a 3₁ or 5₂ knot, or a step closely associated with it, significantly slows down the folding of UCH-L1. The results of the optical tweezers experiments highlight the complex nature of the folding pathway, many additional intermediate structures being detected that cannot be resolved by intrinsic fluorescence. Mechanical stretching of knotted proteins is also of importance for understanding the possible implications of knots in proteins for cellular degradation. Compared with a simple 3₁ knot, we measure a significantly larger size for the 5₂ knot in the unfolded state that can be further tightened with higher forces. Our results highlight the potential difficulties in degrading a 5₂ knot compared with a 3₁ knot.

For every integer g 2 we construct 3-dimensional genus- g 1-handlebodies smoothly embedded in S^4 with the same boundary, and which are defined by the same cut systems of their boundary, yet which are not isotopic rel. boundary via any locally flat isotopy even when their interiors are pushed into B^5 . This proves a conjecture of Budney–Gabai for genus at least 2.

ATP-dependent proteases translocate proteins through a narrow pore for their controlled destruction. However, how a protein substrate containing a knotted topology affects this process remains unknown. Here, we characterized the effects of the trefoil-knotted protein MJ0366 from Methanocaldococcus jannaschii on the operation of the ClpXP protease from Escherichia coli. ClpXP completely degrades MJ0366 when pulling from the C-terminal ssrA-tag. However, when a GFP moiety is appended to the N terminus of MJ0366, ClpXP releases intact GFP with a 47-residue tail. The extended length of this tail suggests that ClpXP tightens the trefoil knot against GFP, which prevents GFP unfolding. Interestingly, if the linker between the knot core of MJ0366 and GFP is longer than 36 residues, ClpXP tightens and translocates the knot before it reaches GFP, enabling the complete unfolding and degradation of the substrate. These observations suggest that a knot-induced stall during degradation of multidomain proteins by AAA proteases may constitute a novel mechanism to produce partially degraded products with potentially new functions.

The meridional rank conjecture asks whether the bridge number of a knot in $S^3$ is equal to the minimal number of meridians needed to generate the fundamental group of its complement. In this paper, we investigate the analogous conjecture for knotted spheres in $S^4$ . Towards this end, we give a construction to produce classical knots with quotients sending meridians to elements of any finite order in Coxeter groups and alternating groups, which detect their meridional ranks. We establish the equality of bridge number and meridional rank for these knots and knotted spheres obtained from them by twist-spinning. On the other hand, we show that the meridional rank of knotted spheres is not additive under connected sum, so that either bridge number also collapses, or meridional rank is not equal to bridge number for knotted spheres.

Nematic braids are reconfigurable knots and links formed by the disclination loops that entangle colloidal particles dispersed in a nematic liquid crystal. We focus on entangled nematic disclinations in thin twisted nematic layers stabilized by 2D arrays of colloidal particles that can be controlled with laser tweezers. We take the experimentally assembled structures and demonstrate the correspondence of the knot invariants, constructed graphs, and surfaces associated with the disclination loop to the physically observable features specific to the geometry at hand. The nematic nature of the medium adds additional topological parameters to the conventional results of knot theory, which couple with the knot topology and introduce order into the phase diagram of possible structures. The crystalline order allows the simplified construction of the Jones polynomial and medial graphs, and the steps in the construction algorithm are mirrored in the physics of liquid crystals.

In this work, we investigate the topological properties of knotted defects in smectic liquid crystals. Our story begins with screw dislocations, whose radial surface structure can be smoothly accommodated on S 3 for fibred knots by using the corresponding knot fibration. To understand how a smectic texture may take on a screw dislocation in the shape of a knot without a fibration, we study first knotted edge defects. Unlike screw defects, knotted edge dislocations force singular points in the system for any non-trivial knot. We provide a lower bound on the number of such point defects required for a given edge dislocation knot and draw an analogy between the point defect structure of knotted edge dislocations and that of focal conic domains. By showing that edge dislocations, too, are sensitive to knot fibredness, we reinterpret the so-called Morse–Novikov points required for non-fibred screw dislocation knots as analogous smectic defects. Our methods are then applied to + 1 / 2 and negative-charge disclinations in the smectic phase, furthering the analogy between knotted smectic defects and focal conic domains and uncovering an intricate relationship between point and line defects in smectic liquid crystals. The connection between smectic defects and knot theory not only unravels the uniquely topological knotting of smectic defects but also provides a mathematical and experimental playground for modern questions in knot and Morse–Novikov theory.

AlphaFold is a groundbreaking deep learning tool for protein structure prediction. It achieved remarkable accuracy in modeling many 3D structures while taking as the user input only the known amino acid sequence of proteins in question. Intriguingly though, in the early steps of each individual structure prediction procedure, AlphaFold does not respect topological barriers that, in real proteins, result from the reciprocal impermeability of polypeptide chains. This study aims to investigate how this failure to respect topological barriers affects AlphaFold predictions with respect to the topology of protein chains. We focus on such classes of proteins that, during their natural folding, reproducibly form the same knot type on their linear polypeptide chain, as revealed by their crystallographic analysis. We use partially artificial test constructs in which the mutual non-permeability of polypeptide chains should not permit the formation of complex composite knots during natural protein folding. We find that despite the formal impossibility that the protein folding process could produce such knots, AlphaFold predicts these proteins to form complex composite knots. Our study underscores the necessity for cautious interpretation and further validation of topological features in protein structures predicted by AlphaFold.