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# α-catenin switches between a slip and an asymmetric catch bond with F-actin to cooperatively regulate cell junction fluidity

## Abstract

α-catenin is a crucial protein at cell junctions that provides connection between the actin cytoskeleton and the cell membrane. At adherens junctions (AJs), α-catenin forms heterodimers with β-catenin that are believed to resist force on F-actin. Outside AJs, α-catenin forms homodimers that regulates F-actin organization and directly connect the cell membrane to the actin cytoskeleton, but their mechanosensitive properties are inherently unknown. By using ultra-fast laser tweezers we found that a single α-β-catenin heterodimer does not resist force but instead slips along F-actin in the direction of force. Conversely, the action of 5 to 10 α-β-catenin heterodimers together with force applied toward F-actin pointed end engaged a molecular switch in α-catenin, which unfolded and strongly bound F-actin as a cooperative catch bond. Similarly, an α-catenin homodimer formed an asymmetric catch bond with F-actin triggered by protein unfolding under force. Our data suggest that α-catenin clustering together with intracellular tension engage a fluid-to-solid phase transition at the membrane-cytoskeleton interface.

## Introduction

In any living cell, an array of mechanotransduction proteins responds to mechanical cues to trigger complex molecular signaling, driving cell morphology and gene expression profiles1. Among these proteins, α-catenin has been reported to act as a mechanosensor that regulates adherens junctions (AJ) in response to mechanical cues from neighboring cells in a tissue2. It is widely accepted that α-catenin in AJ binds β-catenin, which, in turn, connects to the cytoplasmatic portion of E-cadherin to constitute the cadherin-catenin complex (CCC). Since α-catenin also binds F-actin, α-catenin has been indicated as a major candidate for providing a link between the CCC and the actin cytoskeleton. This molecular link is required to provide mechanical stability to AJ. According to the most accepted model of α-catenin mechanosensitivity, tension on the actin cytoskeleton is transmitted to bound α-catenin inducing a conformational change that opens up (i.e. unfolds) its central M-domain and increases the bond strength2. α-catenin unfolding is predicted to expose cryptic binding sites for vinculin and other actin-binding proteins, which further reinforce the connection to the actin cytoskeleton. However, the role of α-catenin as the mechanical link between the CCC and F-actin has been challenged by the finding that the binding of mammalian α-catenin to F-actin and β-catenin is mutually exclusive3.

Using a Danio Rerio α-catenin4, Buckley et al. recently demonstrated that the bond between CCC and F-actin is reinforced by increasing force (i.e., it is a catch bond)5. However, it remains to be determined whether mammalian α-β-catenin heterodimers show similar catch-bond properties. Moreover, it is still unclear whether a single α-β-catenin heterodimer can bind to F-actin and bear force on it. CCCs have been recently shown to be organized in clusters in cells6. This finding raises the intriguing question whether CCC clustering and the interaction of F-actin with multiple α-catenins might be important in defining the mechanosensitivity of AJ.

Recent reports revealed the importance of α-catenin outside AJ as well, where it regulates F-actin organization7 and promotes cell adhesion and migration by providing a direct link between the cell membrane and the actin cytoskeleton8. While α-catenin forms α-β-catenin heterodimers in AJ, outside AJ α-catenin binds itself and is prevalently found as homodimers. Despite its biological relevance, no data is currently available on the force-dependence of the interaction between α-catenin homodimers and F-actin.

Finally, it is still unclear whether α-catenin unfolds during its interaction with F-actin. Using a truncated α-catenin that lacked the actin and β-catenin binding domains, Yao et al. showed that forces of about 5 pN trigger reversible unfolding of the α-catenin M-domain and allow vinculin binding9. However, there is no direct evidence that forces applied to α-β-catenin heterodimers or α-catenin homodimers through F-actin are sufficient to induce α-catenin unfolding.

Here, we use ultrafast force-clamp spectroscopy, a technique with sub-millisecond and sub-nanometer resolution based on laser tweezers10,11,12,13, to dissect the mechanosensitivity of single mammalian α-catenin homodimers and α-β-catenin heterodimers. We show that a single mammalian α-β-catenin heterodimer does not bear force on F-actin, but instead slips along F-actin in the direction of force. Conversely, the cooperative action of multiple molecules together with force applied toward F-actin pointed end triggers α-catenin unfolding and engages a catch bond with actin. Similarly, a single α-catenin homodimer forms a directionally asymmetric catch bond with F-actin triggered by α-catenin unfolding, which resists forces directed toward F-actin pointed end.

## Results

### A single α-catenin homodimer unfolds to form a catch bond with F-actin

We first analyzed single purified recombinant α-catenin homodimers (named as α-catenin below). A single actin filament was suspended between two optically trapped beads and brought in close proximity to a single α-catenin that was attached to a micron-sized glass bead on the coverslip surface (Fig. 1a, Methods section). A constant force was applied to the bead-actin-bead complex (herein named “dumbbell”) through a double force-clamp system. The applied force caused the dumbbell to move through the assay solution at a constant velocity and the force was alternated in direction to maintain the dumbbell within few hundred nanometer excursion (Fig. 1b, “1. Unbound”). When α-catenin bound to the actin filament and became loaded with the applied force, the dumbbell motion stopped within ~30 μs (Fig. 1b, “2. Bound”). The interactions between α-catenin and the actin filament were detected through the change in the dumbbell velocity and their lifetimes and position assessed under forces in the range of ± 17 pN (see Methods).

Analysis of α-catenin binding position at different forces further supports this view. Observation of position records showed that interactions occurred at single locations for forces below 3-4 pN (Fig. 1d and Supplementary Fig. 2a), whereas above 5 pN most interactions showed “steps” (red arrowheads in Fig. 1e,f and Supplementary Fig. 2b,c). This behavior raises the question whether the observed steps were a consequence of (i) α-catenin detaching and rapidly reattaching to the actin filament or (ii) α-catenin undergoing large conformational changes (i.e., unfolding) under force9, as suggested by the analysis of the interaction lifetime. The former hypothesis would imply that steps occurred only in the direction of the force, following the periodicity of the actin filament. However, analysis of step size distribution at about +5 pN showed a main peak in the force direction centered around 12.1 ± 5.8 nm and the presence of a step in the opposite direction of similar amplitude (−11.2 ± 5.1 nm) (Fig. 1h). Likewise, step size distribution at −5 pN showed steps of similar amplitude in both directions (−11.4 ± 6.4 nm and 10.3 ± 4.2 nm) (Fig. 1g). Analysis of individual records around 5 pN force display α-catenin jumping back and forth between two position levels separated by about 10–20 nm (Supplementary Fig. 2d). This indicates that, around 5 pN, α-catenin is near the equilibrium between the folded and the unfolded state and switches between the two states because of thermal noise. Our results are in very good agreement with those from Yao et al., who measured a reversible unfolding step on the α-catenin M-domain occurring around the same force and with similar amplitude9. At higher force (±11 pN), the negative step is largely reduced (Fig. 1j) or disappears (Fig. 1i and Supplementary Fig. 3), confirming the idea that force drives the unfolding of the protein in the force direction and that the reverse folding step is inhibited at large forces. Moreover, multiple steps and larger irreversible steps of about 35 and 71 nm become apparent in the position records (Fig. 1f and Supplementary Fig. 2c) and in the step distributions (Fig. 1j) in the direction of the force. These larger steps are compatible either with additional irreversible unfolding steps as observed by Yao et al. at about 12 pN9, or from rapid unbinding-rebinding of α-catenin from the actin filament, although the presence of the above mentioned two peaks in the interaction lifetime supports the former hypothesis. Moreover, possible contributions to the step size distribution due to the simultaneous binding of the two α-catenin monomers to actin are discussed in the supplementary discussion.

### A single α-β-catenin heterodimer forms a slip bond with F-actin

We then analyzed the interaction between a single α-β-catenin heterodimer and an actin filament (Fig. 2a). Plot of the event lifetime versus force showed that the bond between a single α-β-catenin heterodimer and F-actin does not bear force (Fig. 2b). The bond lifetime decreased with force, occurring in the millisecond time scale at forces around 5 pN and rapidly dropping below 1 ms for F > 7pN. We could detect such rapid interactions thanks to the unprecedented time resolution of our technique10. The lifetime vs force plot was well fitted by the Bell slip-bond model, in which the lifetime decreases exponentially with force (green lines in the log plot of Fig. 2b). Position records show that interactions of the α-β-catenin heterodimer also included “steps” (Supplementary Fig. 4). However, the step size distribution was very different than in the case of the α-catenin homodimer, displaying (i) negligible number of negative steps and (ii) strong 5.5 nm periodicity, which is characteristic of the actin monomer separation14,15 (Fig. 2c,d and Supplementary Fig. 5). The actin filament is made of two protofilaments composed by 13 monomers per helix turn, giving a periodicity of 13 × 5.5 nm = 71.5 nm. The two protofilaments are separated by a stagger of 2.75 nm, giving an apparent periodicity of 6 × 5.5 nm + 2.75 nm = 35.75 nm (Fig. 2e). Figure 2f shows the peak position, obtained from the step size distribution in Fig. 2d, against the peak number, as defined in Fig. 2d. Linear regression analysis showed that the points corresponding to peaks 0–2 and 12–14 (red squares) and those corresponding to peaks 5–10 (black squares) lied on two parallel lines with nearly identical slopes (5.56 ± 0.01 and 5.54 ± 0.04 nm/peak for the red and black lines respectively), but displaced by 2.56 ± 0.31 nm. The line slopes clearly corresponded to the distance between contiguous actin monomers, while the distance between the two lines corresponded to the protofilament stagger.

These results strongly indicate that the observed steps are due to a single α-catenin that rapidly unbinds from an actin monomer, slips along the actin filament in the direction of the force, and rebinds to a neighbor actin monomer. Moreover, no apparent unfolding was detected. In fact, all peaks corresponded to the actin monomer separation while a negligible number of negative steps was detected. Inspection of records at any force did not show α-catenin jumping back and forth between two position levels as in the case of the α-catenin homodimer. Overall, these results indicate that a single α-β-catenin heterodimer forms a slip bond with F-actin that does not resist force. Force applied by the actin filament to the α-β-catenin complex causes a rapid slip to a neighbor actin monomer rather than inducing α-catenin unfolding and strong binding to actin. The slip-bond does not show an asymmetric response to force, as data reported in Fig. 2 were acquired as before on the same dumbbell to maintain the actin filament orientation. Possible experimental artifacts due to non-specific interaction of the actin filament with the surface and/or to the presence of a His(6x) tag in the α-catenin molecule and a GST tag in the β-catenin molecule have been carefully considered and are discussed in the supplementary methods and discussion.

### Multiple α-β-catenin heterodimers form a cooperative catch bond with F-actin

Since single mammalian α-β-catenin complexes cannot bear force on actin, we questioned whether the cooperative action of multiple complexes might resist the physiological forces that occur at AJ. We, thus, increased the α-β-catenin heterodimer concentration 10-fold to have multiple proteins interacting simultaneously with the actin filament (Fig. 3a). Similarly to what we observed using α-catenin homodimers, the bond between α-catenin and F-actin was reinforced by increasing force and responded asymmetrically with respect to the force direction, with lifetime peaks centered about 7.2 pN and 11.4 pN (Fig. 3b). Moreover, an additional peak was detected at the highest force that we could probe (17 pN). Also here, data were acquired on a single dumbbell to keep the actin filament orientation fixed and reveal asymmetries in the force-dependence. The asymmetric response to force was more pronounced here, resulting in a catch bond for force applied in one direction and a slip bond in the opposite direction (Fig. 3b and Supplementary Fig. 6). Position records were also similar to the α-catenin homodimers, mainly composed of single events at forces below 3 pN and multiple “steps” above 5 pN (Supplementary Fig. 7). Step size distribution at about −5 pN showed a main peak centered around −11.7 ± 2.9 nm in the force direction and the presence of a step in the opposite direction of similar amplitude (10.2 ± 1.8 nm) (Fig. 3c). Similarly, step size distribution at about +5 pN showed a main peak centered around 12.8 ± 3.8 nm and the presence of a negative step of similar amplitude (−12.1 ± 4.0 nm) (Fig. 3d). Analysis of individual records at force greater than 5 pN often displayed α-catenin jumping back and forth between two position levels separated by 10–20 nm (Supplementary Fig. 7). The negative step rapidly disappeared at increasing negative forces in which the bond slips, but it was still present at the highest positive forces in which the bond lifetime is maximum (Fig. 3f and Supplementary Fig. 8b).

## Discussion

Here, we provided direct evidence that a single mammalian α-catenin bound to β-catenin forms a slip bond with F-actin. The weak and rapid interactions that we observed could be detected thanks to the 10–100 μs time resolution of our setup, which is about two orders of magnitude better than conventional optical trapping assays. The limited-time resolution is likely the reason why previous reports could not detect the slip bond between a single α-β-catenin heterodimer and F-actin5, whereas they could possibly detect the action of multiple molecules, which behave as a catch bond and produce much longer interactions than single molecules under force.

Our results indicate that α-catenin binds F-actin as an unconventional slip bond that cooperatively switches into an asymmetric catch bond. A cooperative action of multiple α-catenin molecules is required because a mere sum of multiple slip bonds would result again in a slip bond. Further insight into this mechanism can be gained by analysis of the duty ratio r, i.e., the fraction of time that a molecule spends attached to the actin filament: $$r={\tau }_{{on}}/({\tau }_{{on}}+{\tau }_{{off}})$$16. When the number of α-catenin molecules that can interact with the actin filament is above $${N}_{{\min }}=1/r$$, at least one molecule is bound to actin on average and bear force on it (see Methods). The duty ratio of a single α-catenin homodimer rapidly switches from r < 0.2 for low forces (F < 5 pN) to r = 0.8–0.9 in the force range where also the lifetime is maximized (5 < F < 15 pN, Fig. 5a). Therefore, α-catenin homodimers under force remain bound to actin most of the time and an average number of 1.1–1.2 molecules can sustain force on an actin filament for a prolonged time. On the contrary, the duty ratio of a single α-β-catenin heterodimer is roughly constant to r 0.1–0.2 independently of force (Fig. 5b), whereas the cooperative action of multiple molecules is required to increase the duty ratio to r 0.7 under directional force (Fig. 5c). The r 0.1–0.2 for a single α-β-catenin heterodimer indicates that there should be about 5–10 molecules available for binding to actin to maintain continuous contact with the actin filament. Strikingly, CCC in AJ have been found to cluster in nanodomains composed by 5–6 molecules17. Our data can, thus, provide an explanation of the biological relevance of CCC clustering6,17,18,19. In fact, below 5–10 molecules the probability of finding multiple molecules bound simultaneously to an actin filament is small. The engagement of the cooperative catch bond mechanism observed here is unlikely, and AJ would not be able to bear force on actin. Previous studies indicate that α-catenin binds cooperatively to F-actin by altering the actin filament conformation20, which might be the molecular mechanism at the base of the change in the load-dependence from single to multiple α-β-catenin heterodimers that we observed. Moreover, such asymmetric cooperative load-dependence might itself (i) cause unidirectional flow of CCC clusters in cell-cell junctions21 and (ii) drive cluster formation through increased diffusional trapping mediated by α-catenin cooperativity and tension applied by myosin motors on the actin cytoskeleton6.

An indication of the average number of α-catenins that were simultaneously bound to actin in the different experiments can be obtained from a measurement of the average stiffness of the molecules bound to the dumbbell during the interactions (see methods). The single α-β-catenin heterodimer gave an average stiffness of 0.263 ± 0.006 pN/nm (error s.e.m), about half of the α-catenin homodimer (0.46 ± 0.03 pN/nm). Assuming that the stiffness of an α-catenin monomer in the homodimer and in the heterodimer is the same, this result indicates that both α-catenins in the homodimer participate to the interaction with actin for most of the time (~87% from the ratio of the homodimer stiffness to double the heterodimer stiffness). In the experiments with multiple α-β-catenin heterodimers, the average stiffness was in between the single heterodimer and the single homodimer (0.336 ± 0.007 pN/nm), suggesting that simultaneous binding of two α-catenin molecules occurred for about 64% of the time in these experiments. This result reinforces our conclusion that the catch bond behavior is a consequence of the cooperative binding of multiple α-catenin molecules to actin.

Notably, our data show that the catch-bond behavior is accompanied by a large conformational change (unfolding) of the α-catenin molecule, both in a single α-catenin homodimer and in multiple α-β-catenin heterodimers. Unfolding is not observed in a single α-β-catenin heterodimer slip bond, indicating that α-catenin unfolding provides the molecular switch that converts the link to F-actin into a catch bond. The measured unfolding step size is compatible with unfolding of the bundled α-helices in the vinculin binding domain into a linear chain of extended α-helices9. The unique ultra-fast force-clamp configuration used here allowed us to directly observe under which conditions folding/unfolding transitions of the α-catenin molecule actually occur when force is applied to the interacting actin filament, as previously hypothesized9.

Finally, the bond between actin and both α-catenin homodimers and multiple α-β-catenin heterodimers showed a strong asymmetry. The lifetime was much longer when force was applied toward the actin pointed end (Fig. 4). This result has strong biological relevance because, in cells, intracellular tension is produced toward the actin pointed end by non-muscle myosin and/or by the actin retrograde flow. The directionality of the link between α-catenin and actin can, thus, represent an important mechanism for the selective transmission of tension to adherens junctions. A similar mechanism can also be present in focal adhesions, as recent results show that α-catenin directly interacts with integrin adhesions to regulate their growth and transmission of mechanical forces into the matrix22.

Based on our results, we developed a simple chemo-mechanical model to relate the sliding velocity $$v$$ of an actin filament interacting with α-catenin to the force applied to it (F), as a function of the α-catenin duty ratio (r) (see methods). When a constant force F is applied to an actin filament immersed in a viscous fluid, the filament slides at constant velocity $$F/\gamma$$ against the viscous drag of the fluid $$\gamma$$, during the fraction of time when α-catenin molecules are detached. Therefore, the sliding velocity is obtained as:

$$v=\frac{{\tau }_{{off}}}{{\tau }_{{on}}+{\tau }_{{off}}}\cdot \frac{F}{\gamma }=\left(1-r\right)\frac{F}{\gamma }\Longrightarrow F=\frac{\gamma }{1-r}v$$
(1)

This equation tells us that when the duty ratio is significantly smaller than one and independent of force, as in the case of a single α-β-catenin heterodimer (Fig. 5b), the α-catenin-actin interaction creates additional viscous drag on the actin filament, but the system behaves as a fluid ($$F\propto v$$). However, if the duty ratio gets closer to one, as it occurs when force directed toward the actin pointed end is applied to a single α-catenin homodimer (Fig. 5a) or to multiple α-β-catenin heterodimers (Fig. 5c), the velocity drops to zero and the system behaves as a solid. Therefore, α-catenin acts as a cooperative molecular switch that responds to tension to directionally regulate the fluidity of its link to actin. This result is particularly notable in light of recent findings of the occurrence of fluid-to-solid transitions in cultured epithelial monolayers23 and in-vivo24. Notably, disruption of E-cadherin clusters in living cells has been shown to increase cell-cell junction fluidity25. We thus propose that, as a consequence of α-catenin cooperative mechanosensitivity, AJ might switch from a fluid phase in the absence of CCC clustering to a solid phase when CCC become organized into clusters and tension applied by non-muscle myosin or generated by the actin retrograde flow builds up on the actin cytoskeleton.

## Methods

### Protein expression and purification

The pET28a(+) vector (Novagen) containing the full-length mouse αE-catenin was an offering from Dr. Noboru Ishiyama26. The plasmid encodes the protein fused with a 6x His-tag at the N-terminus with a linker sequence containing a thrombin cleavage site and restriction sites for NdeI and NheI (Amino acid sequence of the His tag and linker: “MGSSHHHHHHSSGLVPRGHMAS”). The plasmid was transformed in BL21 DE3 pLysS strains (Agilent). Recombinant αE-catenin was expressed via isopropyl 1-thio-β-d-galactopyranoside (IPTG,Sigma-Aldrich) induction. Cultures were grown to an OD of 0.7 and induced with 0.5 mM IPTG for 3 h at 37 °C. Protein was purified via FPLC (AKTA prime system, GE Healthcare) using HisTrap HP columns and eluted in 25 mM Tris, pH 7.5, 300 mM NaCl, 300 mM Imidazole. Size exclusion chromatography was performed in 25 mM Tris, pH 7.5, 150 mM NaCl, 1 mM DTT through with a HiLoad Superdex 200 (GE, Healthcare) as a further purification step. Monomeric and homodimeric fractions were pooled separately and purity was evaluated by SDS-PAGE gel electrophoresis. Protein concentration was measured using absorbance at 280 nm (Supplementary Fig. 9). Human β-catenin, GST tagged was bought from Sigma-Aldrich (SRP5172). β-catenin is fused with a GST tag at the N-terminus with a linker sequence containing a thrombin digestion site (amino acid sequence of the linker: “SDLVPRGS”). G-actin and biotinylated actin were bought from Cytoskeleton, Inc. (AKL99, AB07-A). Actin filaments were obtained by incubating 0.4 μg·μl−1 biotinylated actin and 0.2 μg·μl−1 actin in 50 mM KCl, 2 mM MgCl2 and 1 mM ATP in 10 mM Tris, pH 7.5 (actin polymerization buffer, Cytoskeleton, Inc. BSA02) for an hour. F-actin filaments were labeled overnight with 10 μM rhodamine-phalloidin and 20 mM DL-Dithiothreitol.

### Pull down and flow-cell assay

For the flow-cell assay, a 1% nitrocellulose-pentyl acetate solution was smeared on a coverslip and assembled to make a flow chamber27. For reactions containing β-catenin, 80 μg/ml of protein was allowed to react with the surface for 5 minutes. αE-catenin was then incubated for 5 minutes to allow binding. Unbound protein was washed away with a solution containing 20 mM MOPS pH 7.2, 50 mM KCl, 1 mM MgCl2, 0.1 mM EGTA and 1 mg/ml bovine serum albumin, following incubation with a solution containing 100 nM rhodamine phalloidin F-Actin in 20 mM MOPS pH 7.2, 50 mM KCl, 1 mM MgCl2, 0.1 mM EGTA, 1.2 µM glucose-oxidase, 0.2 µM catalase, 17 mM glucose and 20 mM DTT. Flow-cell assays were performed on an inverted fluorescence microscope (Nikon ECLIPSE TE300) equipped with a 532 nm laser (Coherent Sapphire) for rhodamine excitation (~3 mW on the sample). Images were acquired in total internal reflection configuration, through Nikon Plan Apo TIRF, 1.45 oil immersion objective on an EMCCD camera (Andor, iXon X3) with 82 nm pixel size, and 40 × 40 μm2 field of view. Integration time was 0.2 s, and EM gain 40.

For the pull-down assay of αE-catenin with F-actin, 3 µM F-actin was incubated with increasing concentrations of αE-catenin (0.05 µM–8 µM) in 20 mM MOPS pH 7.2, 50 mM KCl, 0.1 mM EGTA, 1 mM MgCl2. Binding reactions were performed at room temperature for 20 min and spun at 14000 rpm for 2 h at 4 °C. The supernatant was collected and the pellet resuspended in the original volume of assay buffer. Laemmli Sample Buffer (Sigma-Aldrich) was added to the samples and pellet and supernatant fractions were run on 4–12% BIS-Tris gels (Thermo Fisher Scientific). The gels were stained with Brilliant Blue reagent (Sigma-Aldrich) and scanned with a ChemiDocTM MP Imaging system (Bio-Rad) and analyzed by densitometry with ImageJ.

### Optical trapping experiments

Neutravidin beads were prepared as follows: 0.5% carboxylated microspheres (0.9 µm diameter, Sigma Aldrich) were incubated with 7.4 mg/ml EDC (N-(3-Dimethylaminopropyl)-N′-ethylcarbodiimide hydrochloride, Sigma-Aldrich) and 0.2 mg/ml biotin-x-cadaverine (Molecular Probes, A1594) for 30 minutes at room temperature. Biotinylated beads were then incubated with 0.1 M glycine, 0.5 mg/ml neutravidin and 0.02 mg/ml streptavidin-alexa532 or, alternatively, with 0.02 mg/ml streptavidin-alexa647 (Thermo Fisher Scientific) in the experiments with oriented dumbbells30. In the latter experiments, the correct orientation of the biotinylated actin filament was achieved by using the Bead Tailed Actin (BTA) method established by Pertici et al.31. The method takes advantage of the property of Gelsolin TL40 to cap the barbed (+end) of the actin filament. Gelsolin TL40 (N-terminal cytoplasmic gelsolin, Hypermol, Germany) was covalently bound to carboxylated microspheres (0.9 µm diameter, Sigma Aldrich) with 1-ethyl-3-(3-dimethylaminopropyl)-carbodiimide in order to bind the +end of a single actin filament. TL40-coated beads were stored in the stock solution (150 mM NaCl, 20 mM sodium phosphate buffer pH 7.4, 0.1 mM ATP, 10 mg/ml BSA, 5% (v/v) glycerol, and 3 mM NaN3) at 0 °C for about 6 months.

Silica beads (1.2 µm, diameter, Bangslabs), dispersed in 1% nitrocellulose-pentyl acetate solution, were smeared on a coverslip and allowed to dry before the flow chamber was assembled27,30. For reactions containing β-catenin, 80 μg/ml of protein was allowed to react with the surface for 5 min. αE-catenin (4 μg/ml for single molecule experiments or 40 μg/ml for multiple molecule experiments) was incubated for 5 min to form the α/β catenin complex, or in the absence of β-catenin to bind non-specifically the nitrocellulose surface. The flow chamber was then incubated with 1 mg/ml bovine serum albumin (Sigma-Aldrich) for 5 min to minimize non-specific interactions. In the experiments with non-oriented dumbbells, a solution containing the neutravidin coated-beads and the biotinylated actin was loaded into the chamber. The final reaction was composed by 0.005% neutravidin functionalized beads, 1 nM rhodamine phalloidin F-Actin in 20 mM MOPS pH 7.2, 50 mM KCl, 1 mM MgCl2, 0.1 mM EGTA, 1.2 µM glucose-oxidase, 0.2 µM catalase, 17 mM glucose and 20 mM DTT. Measurements were carried out after sealing the flow chamber with high vacuum grease. A single biotinylated actin filament was anchored through specific binding to the trapped neutravidin-coated microspheres and pre-tensioned to 3 pN. At low concentration of αE-catenin, approximately 1 in 4 silica beads showed interaction with the actin filament, providing evidence that the large majority of the beads contained at most a single catenin molecule.

Control experiments to probe non-specific interactions were performed without αE-catenin. Few tens of beads were tested in different slides and negligible non-specific interactions were detected in these conditions (supplementary discussion, section “Non-specific interactions”). Moreover, the experimental conditions assure the minimization of possible artifactual contributions to our measurements which might derive by the presence of a mixture of homodimers and heterodimers in the experiments (see supplementary discussion, section “Dimeric vs monomeric α-catenin”).

In the experiments with oriented dumbbells, BTA (in a solution of 25 mM imidazole, 25 mM KCl, 4 mM MgCl2, 1 mM DTT, 3 mM NaN3) and neutravidin-coated beads were loaded into the chamber, in the same final reaction mix, and discriminated under fluorescence microscopy from the different emission spectra of the rhodamine phalloidin and alexa647, respectively. First, a BTA with an anchored actin filament was detected under 532 nm laser illumination and trapped. Second, switching to the 635 nm laser, a neutravidin-coated bead was identified and captured in the second trap. The biotinylated actin filament linked to the TL40 gelsolin coated bead was anchored through specific binding to the trapped neutravidin-coated bead and pre-tensioned to 3 pN. The dumbbell orientation with respect to the two traps was changed from one experiment to another to exclude that trap asymmetries might influence the results. The experimental conditions and control experiments for the oriented dumbbell were otherwise the same described above for the non-oriented dumbbell experiments.

### Ultrafast force-clamp spectroscopy setup

Ultra-fast force-clamp spectroscopy and the experimental apparatus are described in detail elsewhere10,30. In brief, dual optical tweezers and fluorescence microscopy were combined in a custom inverted microscope. The two laser traps were independently controlled by two acousto-optic deflectors (AODs). The forward scatter signal from the tweezers was filtered by two interferential filters before reaching the two quadrant detector photodiodes (QDPs) for particle position measurements. The applied force was calculated from the displacement of the bead (x) and the trap stiffness (k). The trap stiffness was always calibrated prior experiments for the entire range of the trap positions using a power spectrum method32. A high-magnification camera (CCD 2000X) monitored the coordinates of the pedestal bead and with the aid of a 3D piezo translator system nm-stabilization against thermal drifting and low-frequency noise was achieved33. Additionally, a 532 nm laser (Coherent Sapphire) was used for the excitation of rhodamine-labelled F-actin and streptavidin-alexa532 coated beads. A 635 nm laser (Blue Sky Research, FTEC2635-P60SF30) was used for the excitation of streptavidin-alexa647 coated beads. Both data acquisition and force-clamp feedback were controlled at 200 kHz sample rate through an FPGA board (NI-PCI-7830R) by a custom LabVIEW software.

### Data analysis

The analysis of the ultra-fast-force-clamp data was carried out by a custom MATLAB algorithm which detects catenin-actin interactions by identifying changes in the derivative of the position signal i.e., dumbbell velocity which drops to zero upon binding. Limits were imposed to ensure that false events were kept at a maximum of 1% of the total event number10,30. Step detection was performed as in Gardini et al. with small modifications12. Steps were detected from velocity changes due to proteins experiencing conformational changes or rapid detachment and reattachment to the filament. In this case, step length (>5 ms) and absolute amplitude (>5 nm) thresholds were set to limit detection of false steps due to thermal fluctuations, while step amplitude sign switch was allowed to account for both forward and backward steps. In order to distinguish if the steps were within an interaction or belonged to new interactions, maximum time interval (<10 ms) and step size (<75 nm) were set. The average stiffness of the molecule (or molecules) bound to the actin filament during the interactions was evaluated from the position variance $${\sigma }_{x}^{2}$$ during the bound events as $$k={k}_{B}T/{\sigma }_{x}^{2}$$, where $${k}_{B}$$ is the Boltzmann constant and T is the absolute temperature. Here, we assume that (i) the dumbbell is rigid and (ii) the force applied on the trapped beads is constant, i.e. the stiffness of the traps is negligible under force-clamp. The stiffness of the molecules was evaluated as the average stiffness from the experiments reported in Figs. 13 in the range 7–16 pN force and separately computed for the single α-catenin homodimer, the single α-β-catenin heterodimer, and multiple α-β-catenin heterodimers. We expect that the approximation of rigid dumbbell leads to underestimate the α-catenin molecules stiffness, mostly because of the compliance of the links between actin and beads. However, since we previously measured the stiffness of the links between biotinylated actin and neutravidinated beads to range between 2 and 6 pN/nm under similar actin pretensioning27, we evaluate the stiffness underestimation to be below 20%.

### Slip-bond model

We fitted the lifetime vs force plot of a single α-β-catenin heterodimer with a Bell-bond model in which the lifetime τ decreases exponentially with force: $$\tau ={\tau }_{0}^{\pm }{\exp }\left(-\frac{{d}_{\alpha \beta }^{\pm }F}{{k}_{B}T}\right)$$, where kB is the Boltzmann constant, T the absolute temperature, F is the absolute value of the force, τ0± and dαβ± are the unloaded lifetime and the distance parameter under positive (+) and negative (-) force respectively. The force sign is arbitrarily chosen. Fitting parameters are reported in the caption of Fig. 2b. OriginPro was used for data fitting.

### Two-step catch-bond model

We divided the plot of the lifetime vs force for the α-catenin homodimer interaction with actin into a region of low force and high force to separate the lifetime peaks (Supplementary Fig. 1a). Assuming that within each force region α-catenin can transition between two structural states, we developed a two-step kinetic model to fit each lifetime peak. The model is based on two bound states, respectively weakly (W) and strongly (S) bound to actin, and one unbound state (U) (Supplementary Fig. 1b). α-catenin can switch between state W and S and vice-versa with transition rates kWS and kSW, respectively. Detachment can occur from both states W and S with rates kWU and kSU, respectively, with kWU kSU. All rates depend exponentially on force $${k}_{{ij}}={k}_{{ij}}^{0}{\exp }\left(\frac{{d}_{{ij}}F}{{k}_{B}T}\right)$$, with distance parameters dWS, dWU, dSU > 0 and dSW < 0 when F > 0. With F < 0 signs of distance parameters are changed. Therefore, if k0SW > k0WS, α-catenin spends most of its time in state W at zero force. As the force increases, kWS increases and kSW decreases, so that, above a threshold force, state S becomes more populated than state W. The kinetic model can be solved following Nolting34 to get the measured lifetime τ(F), which depends on the detachment rates kWU and kSU from both bound states W and S and on the transition rates kWS and kSW between them. The probability density function of the bound state ([W]+[S] = 1-[U]) is:

$$f\left(t\right)=-{C}_{2}{\lambda }_{1}{{\exp }}\left(-{\lambda }_{1}t\right)-{C}_{4}{\lambda }_{2}{{\exp }}\left(-{\lambda }_{2}t\right)$$
(2)
$${C}_{2}=\frac{{k}_{{SW}}+{k}_{{WS}}+{k}_{{SU}}-{\lambda }_{1}}{{\lambda }_{1}-{\lambda }_{2}}$$
(3)
$${C}_{4}=-\frac{{k}_{{SW}}+{k}_{{WS}}+{k}_{{SU}}-{\lambda }_{2}}{{\lambda }_{1}-{\lambda }_{2}}$$
(4)
$${\lambda }_{1,2}=\frac{1}{2}\left(a\pm \sqrt{{a}^{2}-4b}\right)$$
(5)
$$a={k}_{{SW}}+{k}_{{WS}}+{k}_{{WU}}+{k}_{{SU}}$$
(6)
$$b={k}_{{SW}}{k}_{{WU}}+{k}_{{WS}}{k}_{{SU}}+{k}_{{WU}}{k}_{{SU}}$$
(7)

where C2 and C4 are calculated with initial conditions [U] = [S] = 0, [W] = 1. From the probability density function, we can calculate the average lifetime of the bound state:

$$\tau =\frac{{k}_{{SW}}+{k}_{{WS}}+{k}_{{SU}}}{{k}_{{SW}}{k}_{{WU}}+{k}_{{WS}}{k}_{{SU}}+{k}_{{WU}}{k}_{{SU}}}$$
(8)

that is used to fit experimental data (Fig. 1C and Supplementary Fig. 6). Fit parameters are represented graphically in Supplementary Figure 1c and in Supplementary Table 1. OriginPro was used for data fitting.

### Model of the interaction between α-catenin and a sliding actin filament under constant force

We develop a simple mechano-kinetic model of the interaction between N α-catenin molecules and a single actin filament. We hypothesize that the α-catenin molecules interact independently with the actin filament with an attachment rate $${k}_{{on}}=1/{\tau }_{{off}}$$, with τoff being the average lifetime of the unbound state, and a detachment rate $${k}_{{off}}=1/{\tau }_{{on}}$$, with τon being the average lifetime of the bound state (see Supplementary Fig. 10). The duty ratio

$$r=\frac{{\tau }_{{on}}}{{\tau }_{{on}}+{\tau }_{{off}}}=\frac{{k}_{{on}}}{{k}_{{on}}+{k}_{{off}}}$$
(9)

represents the fraction of time a single α-catenin molecule spends attached to actin, as well as the average fraction of α-catenin molecules bound to actin. Therefore, the average number of attached molecules is $$n=N\cdot r$$ and $${N}_{{\min }}=1/r$$ is the minimum number of molecules to assure that, on average, at least one molecule is bound to actin16. $${N}_{{\min }}$$, thus, represents the minimum number of molecules required to maintain continuous contact with the actin filament.

When a constant force F is applied to the actin filament and no molecule is attached to it, the filament slides at constant velocity v* against viscous drag: $${v}^{* }=F/\gamma$$, where γ is the viscous drag coefficient of the filament in the buffer solution or cytosol. When one or more α-catenin molecules attach to actin, the filament velocity drops to zero, as we observe in our experiments under constant force. The filament, thus, moves only during the fraction of time when α-catenin molecules are detached, and the sliding velocity is obtained as:

$$v=\frac{{\tau }_{{off}}\cdot {v}^{* }}{{\tau }_{{on}}+{\tau }_{{off}}}=\left(1-r\right)\frac{F}{\gamma }$$
(10)

### Reporting summary

Further information on research design is available in the Nature Research Reporting Summary linked to this article.

## Data availability

The data that support this study are available from the corresponding author upon reasonable request. The data generated in this study – duration and step size of single interactions that were used to generate lifetime plots (Figs. 1c, 2b, 3b, 4b, c, Supplementary Fig. 1a, 6), duty ratio plots (Fig. 5d–f), and step size distributions (Figs. 1g–j, 2c, d, 3c–f, Supplementary Fig. 3, 5, 8), as well as raw data of densitometry analysis in pull-down experiments (Supplementary Fig. 11 and 12) – are provided in the Source Data file. Source data are provided with this paper.

## Code availability

The software code used for this study is available on request from the corresponding authors.

## References

1. Hoffman, B. D., Grashoff, C. & Schwartz, M. A. Dynamic molecular processes mediate cellular mechanotransduction. Nature 475, 316–323 (2011).

2. Maiden, S. L. & Hardin, J. The secret life of alpha-catenin: moonlighting in morphogenesis. J. Cell Biol. 195, 543–552 (2011).

3. Yamada, S., Pokutta, S., Drees, F., Weis, W. I. & Nelson, W. J. Deconstructing the Cadherin-catenin-actin complex. Cell 123, 889–901 (2005).

4. Miller, P. W. et al. Danio rerio αE-catenin is a monomeric F-actin binding protein with distinct properties from Mus musculus αE-catenin. J. Biol. Chem. 288, 22324–22332 (2013).

5. Buckley, C. D. et al. The minimal cadherin-catenin complex binds to actin filaments under force. Science 346, 1254211–1254211 (2014).

6. Yap, A. S., Gomez, G. A. & Parton, R. G. Adherens junctions revisualized: Organizing cadherins as nanoassemblies. Dev. Cell 35, 12–20 (2015).

7. Rimm, D. L., Koslov, E. R., Kebriaei, P., Cianci, C. D. & Morrow, J. S. α1(E)-catenin is an actin-binding and -bundling protein mediating the attachment of F-actin to the membrane adhesion complex. Proc. Natl. Acad. Sci. USA. 92, 8813–8817 (1995).

8. Wood, M. N. et al. α-Catenin homodimers are recruited to phosphoinositide-activated membranes to promote adhesion. J. Cell Biol. 216, 3767–3783 (2017).

9. Yao, M. et al. Force-dependent conformational switch of α-catenin controls vinculin binding. Nat. Commun. 5, 4525 (2014).

10. Capitanio, M. et al. Ultrafast force-clamp spectroscopy of single molecules reveals load dependence of myosin working stroke. Nat. Methods 9, 1013–1019 (2012).

11. Tempestini, A. et al. Sliding of a single lac repressor protein along DNA is tuned by DNA sequence and molecular switching. Nucleic Acids Res. 46, 5001–5011 (2018).

12. Gardini, L. et al. Dissecting myosin-5B mechanosensitivity and calcium regulation at the single molecule level. Nat. Commun. 9, 2844 (2018).

13. Woody, M. S., Winkelmann, D. A., Capitanio, M., Ostap, E. M. & Goldman, Y. E. Single molecule mechanics resolves the earliest events in force generation by cardiac myosin. Elife 8, e49266 (2019).

14. Holmes, K. C., Popp, D., Gebhard, W. & Kabsch, W. Atomic model of the actin filament. Nature 347, 44–49 (1990).

15. Steffen, W., Smith, D., Simmons, R. & Sleep, J. Mapping the actin filament with myosin. Proc. Natl Acad. Sci. U. S. A. 98, 14949–14954 (2001).

16. Howard, J. Mechanics of motor proteins and the cytoskeleton. (Sinauer Associates, Inc. Publisher, 2001).

17. Wu, Y., Kanchanawong, P. & Zaidel-Bar, R. Actin-delimited adhesion-independent clustering of E-Cadherin forms the nanoscale building blocks of adherens junctions. Dev. Cell 32, 139–154 (2015).

18. Truong Quang, B.-A., Mani, M., Markova, O., Lecuit, T. & Lenne, P.-F. Principles of E-cadherin supramolecular organization in vivo. Curr. Biol. 23, 2197–2207 (2013).

19. Anup Padmanabhan, A., Ting Ong, H., Zaidel-Bar Correspondence, R., Padmanabhan, A. & Zaidel-Bar, R. Non-junctional E-cadherin clusters regulate the actomyosin cortex in the C. elegans zygote. Curr. Biol. 27, 103–112 (2017).

20. Hansen, S. D. et al. αE-catenin actin-binding domain alters actin filament conformation and regulates binding of nucleation and disassembly factors. Mol. Biol. Cell 24, 3710–3720 (2013).

21. Kametani, Y. & Takeichi, M. Basal-to-apical cadherin flow at cell junctions. Nat. Cell Biol. 9, 92–98 (2007).

22. Mukherjee, A., Nadjar-Boger, E., Sheetz, M. P. & Wolfenson, H. α-catenin facilitates mechanosensing and rigidity-dependent growth by linking integrin adhesions to F-actin. bioRxiv 2021.01.21.427565 (2021). https://doi.org/10.1101/2021.01.21.427565

23. Park, J.-A. et al. Unjamming and cell shape in the asthmatic airway epithelium. Nat. Mater. 14, 1040–1048 (2015).

24. Mongera, A. et al. A fluid-to-solid jamming transition underlies vertebrate body axis elongation. Nature 561, 401–405 (2018).

25. Strale, P.-O. et al. The formation of ordered nanoclusters controls cadherin anchoring to actin and cell–cell contact fluidity. J. Cell Biol. 210, 333–346 (2015).

26. Ishiyama, N. et al. An autoinhibited structure of α-catenin and its implications for vinculin recruitment to adherens junctions. J. Biol. Chem. 288, 15913–15925 (2013).

27. Capitanio, M. et al. Two independent mechanical events in the interaction cycle of skeletal muscle myosin with actin. Proc. Natl Acad. Sci. USA 103, 87–92 (2006).

28. Lapetina, S. & Gil-Henn, H. A guide to simple, direct, and quantitative in vitro binding assays. J. Biol. Methods 4, e62 (2017).

29. Pertici, I. et al. A myosin II nanomachine mimicking the striated muscle. Nat. Commun. 9, 3532 (2018).

30. Gardini, L., Tempestini, A., Pavone, F. S., Capitanio, M. High-speed optical tweezers for the study of single molecular motors. in Molecular Motors: Methods and Protocols, Second Edition (ed. Lavelle, C.) (Methods in Molecular Biology, Springer US).

31. Pertici, I., Bianchi, G., Bongini, L., Lombardi, V. & Bianco, P. A myosin II-based nanomachine devised for the study of Ca2+-dependent mechanisms of muscle regulation. Int. J. Mol. Sci. 21, 7372 (2020).

32. Capitanio, M. et al. Calibration of optical tweezers with differential interference contrast signals. Rev. Sci. Instrum. 73, 1687–1696 (2002).

33. Capitanio, M., Cicchi, R. & Pavone, F. S. Position control and optical manipulation for nanotechnology applications. Eur. Phys. J. B. 46, 1–8 (2005).

34. Nolting, B. Protein Folding Kinetics - Biophysical Methods. (Springer, 1999).

## Acknowledgements

We thank Dr. Noboru Ishiyama for the pET28a vector containing mouse αE-catenin. This work was supported by the European Union’s Horizon 2020 research and innovation program under grant agreement no 871124 Laserlab-Europe, by the Italian Ministry of University and Research (FIRB “Futuro in Ricerca” 2013 grant n. RBFR13V4M2), the University of Florence grant MURPASS, and by Ente Cassa di Risparmio di Firenze. A.V. Kashchuk was supported by the Human Frontier Science Program Cross-Disciplinary Fellowship LT008/2020-C.

## Author information

Authors

### Contributions

M.C. designed the research and supervised the experiments, C.A. expressed and purified α-catenin constructs, C.A. set up and performed optical trapping experiments, G.B. and A.V.K performed experiments on the oriented dumbbell, M.S. analyzed data, M.S., L.G. and M.C. developed the step detection analysis, G.B. performed pull-down experiments, C.A. and L.G performed flow-cell experiments, I.P. and P.B. setup the actin-tailed beads for the oriented dumbbell experiments, M.C., M.S., C.A. wrote the paper, M.S. prepared the figures, M.C., M.S., C.A., L.G., and F.S.P. revised the paper.

### Corresponding author

Correspondence to M. Capitanio.

## Ethics declarations

### Competing interests

The authors declare no competing interests.

## Peer review

### Peer review information

Nature Communications thanks the anonymous reviewers for their contribution to the peer review of this work. Peer reviewer reports are available.

Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

## Rights and permissions

Reprints and Permissions

Arbore, C., Sergides, M., Gardini, L. et al. α-catenin switches between a slip and an asymmetric catch bond with F-actin to cooperatively regulate cell junction fluidity. Nat Commun 13, 1146 (2022). https://doi.org/10.1038/s41467-022-28779-7

• Accepted:

• Published:

• DOI: https://doi.org/10.1038/s41467-022-28779-7


text:
# Matrix Chain Multiplication

Time Limit: 1000MS

Memory Limit: 65536K

## Description

Suppose you have to evaluate an expression like A*B*C*D*E where A,B,C,D and E are matrices. Since matrix multiplication is associative, the order in which multiplications are performed is arbitrary. However, the number of elementary multiplications needed strongly depends on the evaluation order you choose. For example, let A be a 50*10 matrix, B a 10*20 matrix and C a 20*5 matrix. There are two different strategies to compute A*B*C, namely (A*B)*C and A*(B*C). The first one takes 15000 elementary multiplications, but the second one only 3500. Your job is to write a program that determines the number of elementary multiplications needed for a given evaluation strategy.

## Input

Input consists of two parts: a list of matrices and a list of expressions. The first line of the input file contains one integer n (1 <= n <= 26), representing the number of matrices in the first part. The next n lines each contain one capital letter, specifying the name of the matrix, and two integers, specifying the number of rows and columns of the matrix. The second part of the input file strictly adheres to the following syntax (given in EBNF):
SecondPart = Line { Line }
Line       = Expression
Expression = Matrix | "(" Expression Expression ")"
Matrix     = "A" | "B" | "C" | ... | "X" | "Y" | "Z"

## Output

For each expression found in the second part of the input file, print one line containing the word "error" if evaluation of the expression leads to an error due to non-matching matrices. Otherwise print one line containing the number of elementary multiplications needed to evaluate the expression in the way specified by the parentheses.

## Sample Input

9
A 50 10
B 10 20
C 20 5
D 30 35
E 35 15
F 15 5
G 5 10
H 10 20
I 20 25
A
B
C
(AA)
(AB)
(AC)
(A(BC))
((AB)C)
(((((DE)F)G)H)I)
(D(E(F(G(HI)))))
((D(EF))((GH)I))

## Sample Output

0
0
0
error
10000
error
3500
15000
40500
47500
15125

Ulm Local 1996

format: plain


text:
## What is Better: GMM of Two Gaussians or Two Clusters With One Gaussian?

In this report, we provide a theoretical discussion on temporal data cluster analysis: does the data come from one source or two sources; is it better to cluster the data into two clusters or leave it as one cluster. Here we analyse only the simplest case: when the data comes from two symmetric Gaussian probability-density-functions (pdfs), i.e., with same variance and same absolute value of the mean, with the same prior probability per Gaussian. The data consists of segments with an a-priori known segment length. It will be shown that if the data belongs to two different Gaussian models, the likelihood of two clusters is always higher or equal than the one of a GMM with two Gaussians for any mean, variance, and segment length. If the data belongs to the GMM, the likelihood of two clusters might be either higher or less than the GMM one.

Year:
2002
Publisher:
Martigny, Switzerland, IDIAP
Keywords:
Laboratories:


text:
# Properties

 Label 53361.bm Number of curves 4 Conductor 53361 CM no Rank 0 Graph

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("53361.bm1")

sage: E.isogeny_class()

## Elliptic curves in class 53361.bm

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
53361.bm1 53361bj4 [1, -1, 0, -7818498, -8404372971] [2] 1658880
53361.bm2 53361bj2 [1, -1, 0, -614763, -58125600] [2, 2] 829440
53361.bm3 53361bj1 [1, -1, 0, -347958, 78425199] [2] 414720 $$\Gamma_0(N)$$-optimal
53361.bm4 53361bj3 [1, -1, 0, 2320092, -454331025] [2] 1658880

## Rank

sage: E.rank()

The elliptic curves in class 53361.bm have rank $$0$$.

## Modular form 53361.2.a.bm

sage: E.q_eigenform(10)

$$q + q^{2} - q^{4} - 2q^{5} - 3q^{8} - 2q^{10} - 2q^{13} - q^{16} + 2q^{17} + O(q^{20})$$

## Isogeny matrix

sage: E.isogeny_class().matrix()

The $$i,j$$ entry is the smallest degree of a cyclic isogeny between the $$i$$-th and $$j$$-th curve in the isogeny class, in the LMFDB numbering.

$$\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)$$

## Isogeny graph

sage: E.isogeny_graph().plot(edge_labels=True)

The vertices are labelled with LMFDB labels.


text:
# Wumpus

Time Limit: 2 Seconds

Memory Limit: 65536 KB

## Description

One day Leon finds a very classic game called Wumpus.The game is as follow.
Once an agent fell into a cave. The legend said that in this cave lived a kind of monster called Wumpus, and there were horrible pits which could lead to death everywhere. However, there were also a huge amount of gold in the cave. The agent must be careful and sensitive so that he could grab all of the gold and climb out of the cave safely.

The cave can be regarded as a n*n board. In each square there could be a Wumpus, a pit, a brick of gold, or nothing. The agent would be at position (0,0) at first and headed right.(As the picture below)

For each step, there are six possible movements including going forward, turning left, turning right, shooting, grabbing the gold, and climbing out of the cave. If the agent steps into a square containing a pit or Wumpus, he will die. When the agent shoots, the Wumpus in front of him will die. The goal of the agent is to grab all of the gold and return to the starting position and climb out(it's OK if any Wumpus is still living).When a brick of gold is grabbed successfully, you will gain 1000 points. For each step you take, you will lose 10 points.

Your job is to help him compute the highest point he can possibly get.

For the purpose of simplification, we suppose that there is only one brick of gold and the agent cannot shoot the Wumpus.

If there is a pit at (0, 0), the agent dies immediately. There will not be a Wumpus at (0, 0).

## Input

There are multiple cases. The first line will contain one integer k that indicates the number of cases.

For each case:
The first line will contain one integer n (n <= 20).
The following lines will contain three integers, each line shows a position of an object. The first one indicates the type of the object. 1 for Wumpus, 2 for pit and 3 for gold. Then the next two integers show the x and y coordinates of the object.
The input end with -1 -1 -1. (It is guaranteed that no two things appear in one position.)

## Output

The output contains one line with one integer, which is the highest point Leon could possibly get. If he cannot finish the game with a non-negative score, print "-1".

## Sample Input

2
3
1 1 1
2 2 0
3 2 2
-1 -1 -1
3
1 1 1
3 2 2
-1 -1 -1

## Sample Output

850
870

## Hint

For the sample 1, the following steps are taken:
turn left, forward, forward, turn right, forward, forward, grab, turn left, turn left, forward, forward, turn left, forward, forward, climb.
There are in all 15 steps, so the final score is 840. For the sample 2 , the path is as follow:

None


text:
Last edited by Zolosida
Wednesday, August 5, 2020 | History

2 edition of Beta decay found in the catalog.

Beta decay

Otto Kofoed-Hansen

# Beta decay

## by Otto Kofoed-Hansen

Written in English

Subjects:
• Beta rays,

• Edition Notes

Classifications The Physical Object Statement [by O.M. Kofoed-Hansen and C.J. Christensen] Series Handbuch der Physik ;, Bd. 41:2 Contributions Christensen, C. J., joint author. LC Classifications QC21 .H327 Bd. 41. t. 2 Pagination 117 p. Number of Pages 117 Open Library OL5874008M LC Control Number 63002092 OCLC/WorldCa 12896940

Beta Decay is a type of radioactive decay in which a proton is transformed into a neutron or vice versa inside the nucleus of the radioactive sample. Processes like this and alpha decay allow the nucleus of the radioactive sample to get as close as possible to the optimum neutron/ proton ratio. While doing so, the nucleus emits a beta particle.   texts All Books All Texts latest This Just In Smithsonian Libraries FEDLINK (US) Genealogy Lincoln Collection. National Emergency Library. Top Beta decay by Wu, C. S. (Chien-shiung), Publication date Topics Beta decay Publisher New York: Interscience Publishers CollectionPages:

Additional Physical Format: Online version: Wu, Chien-shiung. Beta decay. New York, Interscience Publishers [] (OCoLC) Document Type: Book. Beta Decay []. Among the three types of radioactivity, the and rays were easily explained. The emission of particle is kind of lop-sided fission reaction, when an initial nucleus spontaneously decays in two fragments one of which is the nucleus He (i.e. particle). The rays are electromagnetic quanta (photons) emitted by a nuclear system when it transits from one .

Beta decay occurs when, in a nucleus with too many protons or too many neutrons, one of the protons or neutrons is transformed into the other. In beta minus decay, as shown in Fig. a, a neutron decays into a proton, an electron, and an antineutrino: n Æ p + e-+—n. In beta plus decay, shown in Fig. b, a proton decays into. beta decay. In beta decay. In positron emission, also called positive beta decay (β +-decay), a proton in the parent nucleus decays into a neutron that remains in the daughter nucleus, and the nucleus emits a neutrino and a positron, which is a positive particle like an ordinary electron in mass but Read More; radioactivity classifications.

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Online shopping from a great selection at Books Store. Controversy and Consensus: Nuclear Beta Decay – (Science Networks. Beta Decay Hardcover – January 1, by S.A. Moszkowski (Author), C.S.

Wu (Author) See all formats and editions Hide other formats and editionsAuthors: C.S. Wu, S.A. Moszkowski. 3-decay interaction allowed transitions angle angular correlation angular distribution angular momentum annihilation antineutrino approximation asymmetry atomic axis beta decay boson C.V.C.

theory calculated capture rate Chap charge conjugation circular polarization circularly combination components correction Coulomb coupling constants cross section Decay scheme. The Theory of Beta-Decay covers the formulas, theories, probabilities, and spectra of beta-decay. This book is divided into 2 parts compassing 12 chapters, and starts with the introduction to the neutrino and the quantum theoretical background, explaining the basic phenomenon of beta-decay and the emission of electrons.

The Theory of Beta-Decay covers the formulas, theories, probabilities, and spectra of beta-decay. This book is divided into 2 parts compassing 12 chapters, and starts with the introduction to the Beta decay book and the quantum theoretical background, explaining the basic phenomenon of beta-decay and the emission of Edition: 1.

Beta Decay. By C. Wu and S. Moszkowski. (Interscience Monographs and Texts in Physics and Astronomy, Vol. ) Pp. xiv + (New York and London: Interscience Cited by: 3. A particle physics picture of β−-decay is given in Figure Figure The particle physics view of β−-decay.

In this case, the Beta decay book force is carried by the intermediate vector boson, the W−. In the case of β−-decay, the weak force is carried by the intermediate vector boson, the W+, the antiparticle to the W−.File Size: KB. 2 alpha!decay,!angular!momentumplays!a!crucial!role!in!understanding!the!process.!Let!us.

consider!the!simplest!formof!tonandthe!File Size: 2MB. Beta Decay. Beta decay is somewhat more complex than alpha decay is. These points present a simplified view of what beta decay actually is: 1) A neutron inside the nucleus of an atom breaks down, changing into a proton.

2) It emits an electron and an antineutrino (more on this later), both of which go zooming off into space. Beta decay, any of three processes of radioactive disintegration by which some unstable atomic nuclei spontaneously dissipate excess energy and undergo a change of one unit of positive charge without any change in mass number.

The three processes are electron emission, positron (positive electron) emission, and electron capture. The second type of beta decay is less common than the first.

It is $$\beta^+$$ decay. Certain nuclides decay by the emission of a positive electron. This is antielectron or positron decay (Figure $$\PageIndex{5}$$). Figure $$\PageIndex{5}$$: $$\beta^+$$ decay is the emission of a positron that eventually finds an electron to annihilate, characteristically producing gammas in.

Beta decay converts a neutron to a proton and emits a high-energy electron, producing a daughter nucleus with the same mass number as the parent and an atomic number that is higher by 1. Positron emission is the opposite of beta decay and converts a proton to a neutron plus a positron.

Positron emission does not change the mass number of the. Beta decay is the loss of an electron from the nucleus of an atom. In Beta decay, a high-energy electron (called a beta particle) is emitted from a neutron in the nucleus of a radioactive atom.

That neutron may be thought of as a combination of a beta particle (negative charge) with a proton (positive charge). The loss of the negatively charged beta particle leaves behind a. A Fermi transition or a Gamow–Teller transition are types of nuclear beta decay determined by changes in angular momentum or spin.

In the Fermi transition, the spins of the emitted particles are antiparallel, coupling to =, so the angular momentum of the initial and final angular momentum states of the nucleus are unchanged (=).This is in contrast to a Gamow-Teller transition, where. Beta Decay.

likes 1 talking about this. This is our Dark Fairytale ; lowers: Beta decays tend to allow the nucleus to approach the optimal proton/neutron ratio. When there are too many neutrons related to the protons, negative beta decay occurs; when there are too many protons related to the neutrons, positive beta decay takes place.

This book has been divided into four chapters Radioactivity and Isotopes, X-particles, Bdecay, Y Radiations. This book is very helpful for the students of.

colorado lab answers beta decay librarydoc65 PDF may not make exciting reading, but phet colorado lab answers beta decay librarydoc65 is packed with valuable instructions, information and warnings.

We also have many ebooks and user guide is also related with phet colorado lab answers beta decayFile Size: 64KB.

Krane’s book ﬁrst. This reading is supplementary to that, and the subsection ordering will mirror that of Krane’s, at least until further notice.

The Radioactive Decay Law Exponential decay law Consider a system of particles, N 0 in number at time, t= 0. Each of these particles has an independent, but equal probability of decay per File Size: KB.

In this graduate-level text, Harry J. Lipkin presents simply, but without oversimplification, the aspects of beta decay that can be understood without reference to the formal theory; that is, the reactions that follow directly from conservation laws and elementary quantum :.

Beta decay (cont) • The second type of beta decay is β+ (positron) decay. • In this decay, ΔZ = -1, ΔN =+1, ΔA =0, i.e., a nuclear proton changes into a nuclear neutron with the emission of a positron, β+, and an electron neutrino, ν e • An example of this decay is • Like β- decay, in β+ decay, the decay energy is shared between theFile Size: 8MB.Beta decay is one process that unstable atoms can use to become more stable.

There are two types of beta decay, beta-minus and beta-plus. During beta-minus decay, a neutron in an atom's nucleus turns into a proton, an electron and an antineutrino.

The electron and antineutrino fly away from the nucleus.Beta decay is radioactive decay (see radioactivity) by emission of a beta process proceeds via the weak interaction and includes all nuclear decays in which the mass number, A, remains constant and the atomic number, Z, changes by one unit.

There are three main types of beta decay.


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# Classifier-Free Guidance (CFG) Scale

The Classifier-Free Guidance Scale, or “CFG Scale”, is a number (typically somewhere between 7.0 to 13.0) that’s described as controlling how much influence your input prompt has over the resulting generation.

It’s easy to misinterpret that explanation, though, and to expect the wrong thing from this parameter, so let’s look at CFG scale in more detail.

## What Does “Guidance” Mean?

Stable Diffusion generates art by gradually “restoring” a noisy image into a piece of artwork. (It operates under the assumption that there’s a painting hiding under all that noise, and it’s trying to uncover it). It does this gradually over a number of steps–making small adjustments to the image each time.

To decide what improvements to make to the image at each step, it looks at the noisy image and tries to puzzle out what it’s looking at.

For example, the below image is still pretty rough, but you and I can clearly see that it’s supposed to be some kind of alien or monster playing a guitar, and we could fix it up if we had the artistic skill.

Of course, in addition to looking at the image, it’s also being guided by your description of the image to generate.

Making visual sense of the below image is really hard without knowing that it’s supposed to be a set of stairs leading down into the ocean!

This is what “guidance” refers to in “Classifier-Free Guidance”–the image generation being guided by the text description.

And the CFG “Scale” refers to the ability to increase or decrease the amount of influence the text description has on the image generation.

This can sometimes improve the quality of the generated result. In the below example of “Bob Ross riding a dragon”, it’s not till a scale of 13 that we get something reasonable.

## What Does “Classifier-Free” Mean?

And what the heck does “Classifier-Free” mean? The inclusion of that term is unfortunate, I think, because it’s just a reference to an older technique for guiding the image generation that’s no longer relevant. It’s similar to if, instead of “Electric Cars”, we called them “Gasoline-Free Cars”. 🤦♂

For more technical readers, here’s my understanding of the classifier technique: Instead of generating form a prompt, you could only specify a category of object to generate, like “dog”, “cat”, “car”, “plane” (i.e., the ImageNet categories), and the model used a standard image classifier to evaluate the progress and help supervise the diffusion process.

## What to Expect from CFG Scale

When you’re having trouble with a generation, it’s tempting to try getting more and more specific in your description, and to feel like you’re dealing with an obstinate child. You think, “surely the model understands what I’m asking for, and it’s just being stubborn and not listening to me”, so you crank the CFG hoping it will start obeying you.

In reality, if it doesn’t seem to be understanding the intent of your prompt (even after you’ve generated lots of examples and used various wordings) then it’s probably just beyond the model’s current abilities.

In the below example, the prompt subject is “A painting of a horse with eight legs, standing in an apocalyptic wasteland”. I really like the seed, but upping the CFG scale doesn’t seem to do anything to increase the number of legs!

The best approach I’m aware of currently for exercising more control is to use image-2-image generation.

## Another Source of Variety

In practice, I think it’s best to simply view the CFG scale as another way to vary the results of your generation.

Once I’ve found a prompt and seed that I like, I like to use the technique of generating a grid of images to explore different combinations of CFG values and step counts, as in the below example (click the image to view a full resolution version).

FYI, Automatic1111 has this feature in the “scripts” section:

(I’ll have to provide a tutorial on Auto1111 at some point!)

## How CFG Scale Works

The remainder of this article is probably most interesting to more technical readers interested in understanding the implementation, as well as some insight into why the technique is not all that effective in practice.

When you give Stable Diffusion a prompt to generate, it actually generates two images in parallel–one guided by your prompt, and one not (technically, the second image is guided by an empty prompt).

The difference between the two is considered to be the influence of the prompt, and we scale that influence up or down by multiplying it with the CFG scale.

### Two Artists

To understand the intent of this, imagine you took a talented artist named “Tim” and cloned him. We’ll call his clone “Ted”. (We’re doing this so that Tim and Ted have identical minds).

We ask both Tim and Ted to restore this image, but only Ted gets to know the description.

Tim is going to make adjustments purely based on what he can see in the image. “I think this looks like an astronaut, so I’m going to take things in that direction.”

But Ted is going to use a combination of what he sees, and the description he’s been given. “I can tell there’s a person standing there, and it looks like an astronaut, but the prompt says it’s a guitarist, so I’m going to go in that direction.”

(I made Tim and Ted clones so that they have an identical visual interpretation of the image).

Here’s where the scaling happens… At each “iteration” of working on the image, we can look at the difference between Tim and Ted’s suggested changes, and lean more heavily to one or the other.

In practice, we always lean towards Ted’s suggestion more, and the scaling factor just determines by how much.

If Ted (who has the description) doesn’t really understand the prompt the way you’ve written it, or isn’t familiar with your subject matter, or maybe just doesn’t have the skill to create it (SD seems to struggle with more complicated imagery), then amplifying Ted’s influence on the result isn’t going to solve the problem.

I have to imagine that setting the scale to, e.g., 1,000 would just “break” things and give you garbage. The tools limit you to more reasonable values, though. Auto1111 goes up to 30:

It’s absolutely worth experimenting with, though!

### The Original Purpose of CFG

In fact, the real purpose of the CFG parameter is that, In the witches brew of math that was used to train stable diffusion, apparently this guidance scaling technique was critical for getting good results during training.

Stable Diffusion was trained with this scaling factor set to some value (I’ve tried to find out what exact value was used, but so far no luck! The closest guess I’ve got is that the CompVis library sets the cfg to 7.5 by default), and this greatly improved how well the model performed on the training task.

Then, the ability to tweak this parameter at “inference” time (i.e., when we’re using SD to generate art) is actually just a secondary benefit.

## Appendix: Examples

Prompt: “bob ross riding a dragon, model pose, ultra realistic, concept art, intricate details, highly detailed, photorealistic, octane render, 8 k, unreal engine. art by artgerm and greg rutkowski and alphonse mucha”,
Negative Prompt: “”, Seed: 1442287716, Euler A, 30 steps, 512x512

Prompt: “A painting of a horse with eight legs, standing in an apocalyptic wasteland, trending on art station, by greg rutkowski”
Negative Prompt: “jump, jumping, leaping”, Steps: 20, Sampler: Euler a, CFG scale: 13, Seed: 2405405571, Size: 512x512

Prompt: "full face epic portrait, male wizard with glowing eyes, elden ring, matte painting concept art, midjourney, beautifully backlit, swirly vibrant color lines, majestic, cinematic aesthetic, smooth, intricate, 8 k, by ilya kuvshinov, artgerm, darius zawadzki and zdizslaw beksinski ",
Negative Prompt: ""
Seed: 1718067705, Sampler: Euler, Size: 512x704, varied steps & cfg


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# How to divide a dataset for training and testing when the features and targets are in two different files?

I am trying to divide a dataset into training dataset and testing dataset for multi-label classification. The datset I am working on is this one. It is divided into a file which contains the features and another file which contains the targets. They look like this below.

This is the image about the features.

This is the image about the targets.

I intend to use this dataset for multilabel classification. I am following this tutorial . Here the dataset looks like this.

The dataset that I am working on has 17203824 samples and 58255 different and unique labels in the target file. So to follow the tutorial what I intend to create is a new numpy 2d array with 17203824 rows and 58255 columns where appropriate indices will be marked with 1. I am able to create it. But when I try to populate with 1s in the appropriate indices, I am getting an error. Its says that I don't have enough memory. My code is given below.

questions = pd.read_csv("/kaggle/input/stacklite/questions.csv")
d = {v: i[0] for i, v in np.ndenumerate(question_tags["Tag"].unique())}

y = np.zeros([questions.shape[0], len(question_tags["Tag"].unique())], dtype = int)

for k in question_tags["Tag"]:
j = d[k]
for i, l in enumerate(y):
y[i][j] = 1

I suggest you look at some of the common python libraries that will convert a column value into labels. Many of these functions have been around a while, and are optimized to use less memory and/or run faster. For example, you can use what is called "One Hot Encoding" from get_dummies() in pandas or "LableEncoder" from sklearn.

Here is a good reference with many methods to try, depending on your needs.

https://pbpython.com/categorical-encoding.html

Here is a sample of how word embedding works. Each word in this example is reduced to 2 values (x and y)

• Thank you for the article. It has very enlightening. I have tried get_dummies. print(pd.get_dummies(question_tags, columns=["Tag"]).head()). I am getting MemoryError: Unable to allocate 2.68 TiB for an array with shape (50576842, 58254) and data type uint8. Jun 13, 2020 at 8:35
• Ok. Think you have too many labels for one hot encoding to be useful or even successful on a normal desktop computer. The data will be too sparse for most purposes. In general, is there any similarity between the labels that can be used, or do you need the labels to be unique? Any grouping you can do would make this simpler to implement. Jun 13, 2020 at 11:26
• If this is the case, you can try word embedding to decrease the number of columns for the encoding. This web site has a good explanation of this technique. towardsdatascience.com/…. This is using the similarity between words to decrease the number of dimensions needed. Jun 13, 2020 at 11:28
• Another idea is to use sampling, but need to be careful since you have so many labels that you don't exclude some of them. However, it may be okay if you miss a few rare target values, depending on your goal. Jun 13, 2020 at 12:57

You can load the features in a variable (X) and the 2nd column of targets in another variable (y). Then use train_test_split available in sklearn library.

X_train,X_test,y_train,y_test = train_test_split(X,y, test_size=__)

BTW you need to preprocess the data first. There are a lot of NAN values, id and date probably has no value in predicting the target.

• won't it be a problem that X_train and y_train have different row numbers? Jun 12, 2020 at 15:01
• You need them to have equal number of rows. A dataset provided by kaggle should already have that though. Jun 12, 2020 at 20:38
• This is a multilabel classification. If you look at the first two images that I posted again, then you will see that sample 4 has 4 different targets. So y_train has far more rows than x_train. Jun 13, 2020 at 8:21


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# Error propagation problem

1. Feb 25, 2016

### tjosan

1. The problem statement, all variables and given/known data
Hello,

I have the following operation that I want to perform:

$$f=\frac{\bar{X}}{100-\sum \bar{Y}_j}*K$$
$\bar{X}$ and $\bar{Y}$ are averages with variances $S_{X}^2$ and $S_{Y_j}^2$ and $K$ is a constant.

How will the error propagate?

2. Relevant equations
According to Wikipedia:
(1) $f=a\bar{A} \Rightarrow S_f^2=a^2S_f^2$ where $a$ is a constant.
(2) $f=\bar{A}\bar{B} \Rightarrow S_f^2=S_A^2+S_B^2$
(3) $f=\frac{\bar{A}}{\bar{B}} \Rightarrow S_f^2=f^2\left(\frac{S_A^2}{A^2}+\frac{S_B^2}{B^2}\right)$

3. The attempt at a solution
So then the error of the nominator will be $S_{X}^2$
Only looking at the denominator i will have: $100-\sum S_{Y_j}^2$
Using the third and first equation will then yield:

$$S_f^2=f^2 \left(\frac{S_{X}^2}{\bar{X}^2} + \frac{100-\sum S_{Y_j}^2}{(100-\sum \bar{Y}_j)^2} \right)K^2$$

Where $K^2$ comes from the first equation.

I am a little bit confused though. Is this correct?

Thanks.

Edit: Covariance=0

2. Feb 25, 2016

### Ray Vickson

No, it is not correct: the squared error in $100 - \sum Y_j$ is not $100 - \sum S_{Y_j}^2$. For one thing, the '100' is a constant that has no error; for another thing, the $Y_i$ squared errors should not be subtracted from anything.

3. Feb 26, 2016

### tjosan

$$S_f^2=f^2 \left(\frac{S_{X}^2}{\bar{X}^2} + \frac{\sum S_{Y_j}^2}{(100-\sum \bar{Y}_j)^2} \right)K^2$$

Thanks.

4. Feb 26, 2016

### tjosan

$$S_f^2=\left(\frac{\bar{X}}{100-\sum \bar{Y}_j}\right)^2 \left(\frac{S_{X}^2}{\bar{X}^2} + \frac{\sum S_{Y_j}^2}{(100-\sum \bar{Y}_j)^2} \right)K^2$$