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mDia1 senses both force and torque during F-actin filament polymerization

Abstract

Formins, an important family of force-bearing actin-polymerizing factors, function as homodimers that bind with the barbed end of actin filaments through a ring-like structure assembled from dimerized FH2 domains. It has been hypothesized that force applied to formin may facilitate transition of the FH2 ring from an inhibitory closed conformation to a permissive open conformation, speeding up actin polymerization. We confirm this hypothesis for mDia1 dependent actin polymerization by stretching a single-actin filament in the absence of profilin using magnetic tweezers, and observe that increasing force from 0.5 to 10 pN can drastically speed up the actin polymerization rate. Further, we find that this force-promoted actin polymerization requires torsionally unconstrained actin filament, suggesting that mDia1 also senses torque. As actin filaments are subject to complex mechanical constraints in living cells, these results provide important insights into how formin senses these mechanical constraints and regulates actin organization accordingly.

Introduction

The actin cytoskeleton is involved in various physiological and pathological functions, such as cell migration, differentiation, embryo development, and cancer metastasis1,2. The highly dynamic organization of the actin cytoskeleton is tightly controlled by a variety of proteins that regulate actin nucleation, polymerization, de-polymerization, branching, bundling, and localization3,4. The formin family of proteins plays critical roles in promoting nucleation and regulating actin polymerization. Formin forms a homodimer through its homology FH2 domains, and these dimerized FH2 domains form a ring-like shape, which encircles an actin filament at the barbed end5. This FH2 ring slows down the recruitment of actin monomers at the barbed end compared with free barbed ends5,6. Therefore, the FH2 ring acts as a “gate”, imposing an energy barrier for adding new actin monomers to the filaments.

The N-terminus of each FH2 monomer is linked to an intrinsically disordered FH1 domain containing multiple polyproline tracks that interact with the actin-binding protein profilin with an affinity of 3.4–17.8 μM7,8. Profilin has a high affinity to actin monomers (K D ~ 0.1 μM9,10,11,12,13), therefore, the two FH1 domains of a formin homodimer enrich the local concentration of actin monomers near the vicinity of the barbed end of actin filaments, facilitating actin polymerization. Collectively, actin polymerization depends on the complex interplay between the formin FH1 domain that promotes actin polymerization and the FH2 ring that suppresses actin polymerization.

It has been believed that the inhibitory “closed” ring of the FH2 domain has to be “opened” up to allow recruitment of new actin monomers14,15. In vivo, actin filaments are subject to mechanical stretch generated by actomyosin contraction16, and the resulting tensile force is transmitted to formin in the case of formin-dependent actin polymerization15,17. It raises an appealing possibility that force may perturb the FH2 conformation and facilitate the closed-to-open transition of the FH2 ring. If this is the case, formin would be a mechanosensing protein that regulates actin polymerization in a force-dependent manner. If the closed-to-open transition of the FH2 ring is accompanied with a positive extension change Δ, then under force it will be associated with a negative potential energy change of −FΔ (the scalar F denotes the magnitude of force), which biases the probability ratio of the open state over the closed state by an exponential factor of exp(−βFΔ) (β denotes (k B T)−1. Based on this principle, Kozlov et al.18,19 predicted that force should strongly promote actin polymerization, which is hereafter referred to as the mechanical gating mechanism.

In contrast to this prediction, a recent single-molecule experiment showed that in the absence of profilin, the budding yeast formin Bni1p-dependent actin polymerization in 1.5 μM G-actin was inhibited by sub picoNewton (pN) forces: the G-actin on-rate decreased from ~13 μM−1 s−1 at zero force to <4 μM−1 s−1 at forces larger than 0.4 pN. While in the presence of 2.5 μM profilin and 1.5 μM G-actin, pN forces slightly accelerated polymerization20. Increasing force from 0 pN to near 1.5 pN, the G-actin on-rate only increased <50%. Jégou et al.15 showed that forces in the range of 1–3 pN applied to formin mDia1 in the presence of profilin strongly promoted actin polymerization, with a G-actin on-rate of ~83 μM−1 s−1 at a force of around 3 pN in the presence of 3 μM G-actin and 3 μM profilin. In the latter work, the authors did not perform experiments in the absence of profilin, therefore it remains unclear whether force promotes or inhibits actin polymerization for mDia1-mediated actin polymerization without profilin. The observation of the force inhibition of Bni1p-dependent actin polymerization by force in the absence of profilin conflicts with the predicted mechanical gating mechanism of the FH2 ring. It also raises a question whether this inhibition is specific to Bni1p or universal to other formin family proteins.

Previous structural studies have predicted that formin rotates relative to the F-actin filament around the filament axis5, and this has been demonstrated in the case of mDia1ΔN3-dependent polymerization21. In living cells actin filaments are highly cross-linked and therefore unable to freely rotate22, therefore it is the formins that undergo rotation in vivo during progressive polymerization of actin filaments. Previous studies have suggested that actin rotates clockwise relative to formin looking along the direction of elongating filament21 (alternatively, formin rotates relative to the filament in a right-hand manner looking along the reverse direction). If the relative rotation between formin at the barbed end and the superparamagnetic bead attached to actin filament is inhibited, torsion would accumulate in the filament during actin polymerization, which could possibly suppress actin polymerization. The occurrence of rapid, progressive actin polymerization in cells implies that cells employ some mechanisms to relieve the accumulated torsion during actin polymerization. The exact mechanism by which the torsion stress is relieved remains unclear.

In order to provide new insights into formin-mediated actin polymerization, we recorded the extension change of single-actin filaments under different forces using magnetic tweezers in the absence of profilin. We find that forces of a few pN applied to mDia1ΔN3 can drastically speed up the polymerization of actin filaments when they are torsionally unconstrained. Our theoretical analysis indicates that the force-induced acceleration of torsion-unconstrained actin polymerization can be explained by the reduction of energy cost needed for the closed-to-open transition of the FH2 ring, along with additional reduction of the critical concentration required of actin monomers for polymerization. Together, our results directly support the gating mechanism of the FH2 domain. In addition, the rotational slippage between the FH2 ring and the actin filament, should it happen in vivo, would require involvement of other accessory proteins. The biological implications of these findings and the possible causes of the contradiction between our results and those reported for the Bni1p-dependent actin polymerization20 are discussed.

Results

Applying constant forces to F-actin with magnetic tweezers

Tensile forces were applied to actin filaments tethered in several different ways using a transverse magnetic tweezers set-up (Fig. 1a). N-terminal GST-tagged and biotin-tagged mDia1ΔN3, indicated as GST-mDia1ΔN3 and biotin-mDia1ΔN3, respectively, were used in this study. In all cases, the mDia1ΔN3 were bound to pre-polymerized short actin filaments (with a typical length of a few μm) stabilized with phalloidin. These short actin filaments served as “seeds” for further polymerization upon introduction of G-actin solution (500 nM) (Fig. 1b).

Each mDia1ΔN3 dimer carries two tags at the N-termini of the two FH1 domains, therefore, either one of the FH1 domains or both are linked to an anti-GST–Protein A (in the case of GST-mDia1ΔN3) or to a streptavidin (in the case of biotin-mDia1ΔN3). An anti-GST provides two GST-binding sites, however, the chance that both FH1 domains in a GST-mDia1ΔN3 are bound might be limited due to potential steric interaction between the bulky GST tag (26 kDa). In contrast, a streptavidin contains four biotin-binding sites that can interact with the small biotin tag at the very end of the FH1 domain. Therefore, the chance that both FH1 domains in a biotin-mDia1ΔN3 are bound to a streptavidin should be much higher. The possibilities of single-FH1 or double-FH1 tethering are also illustrated in Fig. 1a.

Since the rotation of the superparamagnetic bead is inhibited by the magnetic tweezers, the actin filament will be rotationally constrained in the case of double-FH1 tethering, while it is rotationally unconstrained in the case of single-FH1 tethering. It was predicted5 and confirmed21 that during polymerization the FH2 domain will rotate around the axis of the actin filament. Therefore, torsion stress will build up when the relative rotation between FH2 and actin is inhibited (i.e., during double-FH1 tethering). Whether rotational constraints can suppress formin-mediated actin polymerization or whether there is a mechanism employed by formin to avoid accumulation of torsion stress have not yet been investigated. Using these differently tethered actin filaments, we investigated the effects of force dependence on mDia1ΔN3-mediated actin polymerization under rotationally unconstrained and constrained conditions. After a tether bound with a single-superparamagnetic bead was identified, force was applied and the movement of the superparamagnetic bead was recorded. In order to minimize possible bead-surface and filament-surface interactions, force was applied at a direction of 6–10° above the surface, so that the bead-surface separation increases as the filament elongates (Supplementary Note 2).

For a given superparamagnetic bead, the force F = F z applied to the bead is solely dependent on the bead-magnets separation s, where z denotes the direction of force. At the same value of s, due to heterogeneity in their magnetization, different superparamagnetic beads may have different force values, which differ from each other with a relative error of α ~ 20% among the 2.8-μm-diameter superparamagnetic beads, and α ~ 15% among the 1-μm-diameter superparamagnetic beads (Fig. 1c). Therefore, F(s) was carefully calibrated from a set of bead-magnets separations over a force range up to 20 pN. At each value of s, the forces applied to 10–15 beads were measured by recording the drifting speeds of the beads in 90% (volume fraction) glycerol solution (Methods section). The average of the force, $$\bar F(s)$$, can be fitted with a single-exponential decay function (Fig. 1c). In actin polymerization experiments, the force was determined based on the calibrating curve as $$\bar F(s) \pm \alpha \bar F(s)$$.

The change of the position of the bead along the force direction is indicative of actin elongation. Under forces of a few pN, a tether had a typical life time in the order of a couple of minutes. The breakage of a single tether resulted in sudden increase of the moving speed of the bead to the level of a free drifting bead at the same force (>10 μm s−1 at ~1 pN near the surface (Supplementary Fig. 2)). In rare cases where there are more than one tethered actins between the bead and the surface, each breakage caused a sudden change in the bead position, which could be easily distinguished from the breakage of a single tether. Using this method, we sought to quantify the force dependence of the GST-mDia1ΔN3-mediated polymerization of single-actin tethers with well-calibrated forces and minimized non-specific actin-surface or bead-surface interactions.

GST-mDia1ΔN3-mediated actin polymerization under force

In the case of GST-mDia1ΔN3-mediated actin polymerization, we found a fast elongating species (Figs. 24, red and blue solid squares in Fig. 4) and a slow elongating species (Fig. 4, red and blue hollow squares), both occupying a significant fraction. The data obtained from filaments with the barbed end tethered to glass surface through anti-GST (Fig. 4, red squares) are indistinguishable from the data obtained from actin filaments tethered to surface-immobilized NEM-HMM beads (Fig. 4, blue squares), indicating that the observed force-dependent behavior of the GST-mDia1ΔN3-mediated actin polymerization is independent on the approach used to immobilize the actin filament.

While the reason behind the existence of these two distinct elongating species is unclear, we believe that it might be related to the two different tethering cases (single-FH1 tethering or two-FH1 tethering) illustrated in Fig. 1a. The slower elongating species (Fig. 4, red and blue hollow squares) does not exhibit significant force dependence over 0–5 pN. We are particularly interested in the fast elongating species, because the elongating speed of this species is close to the speed of the protrusion of the leading edge of cells during rigidity sensing23,24,25 and this species exhibited strong force sensitivity as detailed below.

Figure 2a shows the movement of a bead attached to a fast elongating filament in the presence of 500 nM actin monomers along the direction of force at several different force levels, using GST-mDia1ΔN3 attached to the coverslip surface. The position of the bead at the beginning of the measurement was set as the origin. The data show that, at each force, the bead moved with an almost constant speed. Setting the starting of the data recorded at each force as time zero, the trajectories of the bead movement recorded at different forces can be plotted together and their speeds (i.e., the slopes) can be compared (inset). Data recorded from this example demonstrate that the actin polymerization speed increased by nearly 7-fold when force was increased from 1.3 ± 0.3 pN to 7.2 ± 1.4 pN (calibrated force at a given magnet-bead distance ±20% of relative uncertainty). In this example, force was increased from lower to higher levels. In Supplementary Fig. 3 we show that such force-dependent acceleration of actin elongation speed is independent from the force switching order. Figure 2b shows another example under the same reaction conditions. Increasing force from 3.2 ± 0.6 pN to 6.0 ± 1.2 pN resulted in a two-fold of increase in the polymerization speed. The bead movement stopped after washing away the free actin monomers, confirming that the bead movement was due to actin polymerization. Together, the results in Fig. 2 indicate the recruitment of actin monomers to the barbed end of actin filament through GST-mDia1ΔN3 is highly mechanosensitive.

Multiple independent experiments repeated at the same force under the same condition revealed a large variation in polymerization speed among different tethers. Figure 3 shows results from 17 independent experiments using different tethers, under a constant 1.9 ± 0.4 pN force in the presence of 500 nM actin monomers. The elongation speeds spread over a range from 20–127 nm s−1(7–47 subunits s−1), although two tethers produced an exceptionally fast speed that is several folds faster than the others.

Force accelerates GST-mDia1ΔN3-mediated F-actin elongation

To see how reproducible the result in Fig. 2a could be, we measured the actin polymerization speeds for multiple tethers at several forces up to 9.2 ± 1.4 pN. At forces greater than 9 pN, the tethers broke in several seconds, preventing us from collecting enough data. Figure 4 shows the averaged speeds of GST-mDia1ΔN3-mediated actin polymerization in 500 nM G-actin at different forces, both when the GST-mDia1ΔN3 was attached to the coverslip surface (red) or to the superparamagnetic bead (blue). Increasing force from 1.3 ± 0.3 pN to 9.2 ± 1.4 pN accelerated actin polymerization of the force-sensitive species by several fold, and this speed reached a plateau of ~92 subunits s−1 (250 nm s−1) at forces above 6 pN.

Biotin-mDia1ΔN3-mediated actin polymerization under force

We speculated that the emergence of the fast and slow elongating species observed in the GST-mDia1ΔN3-mediated F-actin polymerization experiments was caused by two different tethering possibilities (single-FH1 tethering or two-FH1 tethering as illustrated in Fig. 1a). We further hypothesized that the slow elongating species corresponds to two-FH1 tethering since this imposes a restriction on the relative rotation between the actin filament and the encircled FH2 ring at the barbed end. To test this possibility, we repeated the experiment using a biotin labeled mDia1ΔN3 to mediate F-actin polymerization (Fig. 1a, bottom left), where biotin-mDia1ΔN3 encircled on the barbed end of the filament was directly attached to a streptavidin-coated superparamagnetic bead. As a streptavidin has four biotin-binding states which can interact with the small biotin molecule located at the very end of the FH1 domain, we expected that two-FH1 tethering would be predominant. Consistent with this hypothesis, all the elongating filaments we observed in this experiments had speeds (Fig. 4, purple hollow squares) similar to the slow elongating species observed in the GST-mDia1ΔN3-mediated F-actin polymerization experiments.

In order to further confirm that the slow elongation was caused by the rotational constraint, we introduced a short dsDNA linker (445 bp)26 in between the biotin-mDia1ΔN3 and the superparamagnetic bead. The 5′ of one strand of the dsDNA linker was tagged with a biotin attached to the biotin-mDia1ΔN3 on the barbed end of the filament through a streptavidin, and the 5′ end of the complimentary strand was tagged with a digoxygenin attached to an anti-digoxygenin coated superparamagnetic bead. With this torsion-free dsDNA linker, the rotation between the FH2 ring of the biotin-mDia1ΔN3 and the actin filament becomes unconstrained regardless of single-FH1 or double-FH1 tethering. With the DNA linker attached, all the tethers observed in our experiments had elongation speeds similar to that of the fast species in the GST-mDia1ΔN3-mediated F-actin polymerization experiments, and this elongation could also be accelerated by increasing force (Fig. 4, brown solid squares). Together, these data suggest that the constraint on the relative rotation between the FH2 ring and the actin filament can significantly suppress actin polymerization.

Mechanism of mDia1-mediated actin polymerization under force

Our results suggest that the mDia1ΔN3-mediated actin polymerization is strongly sensitive to applied force in the scale of a few pN, when under rotationally unconstrained conditions. In this section we analyze a physical model that may explain this observation (Fig. 5). For simplicity, the model will be based on biotin-mDia1ΔN3-mediated actin polymerization with a torsionally unconstrained DNA linker, where both FH1 domains are linked to a streptavidin and are subject to force. We consider a simple two-state model where the FH2 ring can either be in the closed state or in the open state, and assume that actin monomers can associate with or dissociate from the barbed end of an actin filament only when the FH2 domain is in the permissive opened state denoted by • (Fig. 5a–c). A force-dependent net actin polymerization speed along the force direction, v = v z, can be written as:

$$v(F,c) = \frac{{\delta \tilde k _{\mathrm{a}}^ \bullet (0){\rm e}^{\beta F\delta }}}{{\left( {{\rm e}^{\beta \varepsilon } + {\rm e}^{\beta F\Delta /2}} \right)}}\left( {c - c_*^{ \bullet {\mathrm{0}}}{\rm e}^{ - \beta F\delta }} \right),$$
(1)

where β = (k B T)−1, k B is the Boltzmann constant and T is the temperature. $$\epsilon$$ denotes the energy cost of the transition from the closed state to the open state of the FH2 ring. δ ~ 2.7 nm is the filament extension increase by adding a new actin monomer to the barbed end, which is half of the size of one actin monomer (~5.4 nm)15. $$c_*^{ \bullet {\mathrm{0}}}$$ is the critical concentration of mDia1 associated barbed end of actin filament at zero force when the FH2 ring is in the open conformation.

v(F, c)/δ is the number of subunits recruited to the barbed end of the filament.

Equation (1) predicts that force speeds up v(F, c) until saturation if δ = Δ, or force speeds up v(F, c) over some force range then suppresses the polymerization at larger forces if δ < Δ. In addition, it predicts a force-dependent concentration of $$c_*^ \bullet (F) = c_*^{ \bullet {\mathrm{0}}}{\rm e}^{ - \beta F\delta }$$, consistent with that predicted in ref. 18. Here we emphasize that this is a general result from actin elongation under force, which is independent on assumptions made for the FH2 ring. The details of the derivation of Eq. (1) and the assumptions can be found in the Supplementary Note 1.

The equation contains four model parameters: $$c_*^{ \bullet {\mathrm{0}}}$$, Δ, $$\epsilon$$, and $$k_{\mathrm{a}}^ \bullet (0)$$. The critical concentration has been reported to be ~100 nM for the barbed end of naked actin filament27 and ~400 nM for mDia1 associated actin filament28. Therefore, $$c_*^{ \bullet {\mathrm{0}}}$$ should be somewhere between the two values. When $$c_*^{ \bullet {\mathrm{0}}}$$ is set as 400 nM, v(F, c)/δ is able to reasonably fit the averaged speeds of fast elongating filaments obtained from both GST-mDia1ΔN3 and biotin-mDia1ΔN3 with DNA linker at different forces (black solid line in Fig. 4), with the best fitting values for the remaining three parameters: Δ ≈ 5.36 nm, $$\epsilon$$ ≈ 1.21 k B T and $$k_{\mathrm{a}}^ \bullet (0) \approx 0.20$$ nM−1 s−1. When $$c_*^{ \bullet {\mathrm{0}}}$$ is set as 100 nM, the best fitting parameters become Δ ≈ 5.49 nm, $$\epsilon$$ ≈ 1.8 k B T and $$k_{\mathrm{a}}^ \bullet (0) \approx 0.23$$ nM−1 s−1. These results show that the predicted actin elongation rate has a very weak dependence on the value of $$c_*^{ \bullet {\mathrm{0}}}$$ in the range of 100–400 nM.

Discussion

In summary, we show that mDia1ΔN3-mediated actin polymerization speed can be drastically accelerated by forces of a few pN, but only when the relative rotation between the FH2 ring and the encircled filament is not restricted. When such rotation is restricted, polymerization still occurred, but with an average speed at a few pN nearly 10-fold slower than when rotation was unconstrained, and polymerization rate also lost the sensitivity to the applied force. Similar force-dependent actin polymerization activity was also observed for Bni1(FH1-FH2)p (a.a.1227–1766, Supplementary Fig. 4), suggesting that this force dependence is likely conserved among different formins.

The actin elongation speed for the rotationally unconstrained filament could reach 102 subunits per second at forces of a few pN in 500 nM G-actin monomers. This result implies that the G-actin on-rate was in the order of 102 μM−1 s−1, which is >10 folds faster than the value (~11 μM−1 s−1) measured for association of G-actin to the free barbed end of actin filament in the absence of force and profilin29. On-rate faster than ~11 μM−1 s−1 was also reported in other single-molecule experiments. The experiment by Jégou et al.15 reported a net on-rate of 80 μM−1 s−1 at a force of around 3 pN for mDia1-mediated actin polymerization in the presence of profilin. Another experiment reported that CLIP-170 (a microtubule-binding protein) could bind tightly to mDia1, and the CLIP-170–mDia1 complex accelerates actin polymerization with a rate 18 times faster than free barbed end (~200 μM−1 s−1) in the absence of force30. Together, these experiments demonstrate that the on-rate of G-actin to the barbed end of actin filaments can be accelerated by multiple external factors such as tension, profilin, or other actin-binding proteins.

The force-dependent acceleration of the rotationally unconstrained filament was observed in the absence of profilin, which indicates that the FH2 homodimer encircled at the barbed end of the actin filament must be responsible for the force dependence of actin polymerization. Our theoretical analysis shows that the experimental results can be explained based on a force-dependent competition between association and dissociation of actin monomers at the barbed end of the actin filament that is encircled by a ring-like FH2 homodimer. Force biases the competition toward association by two energy saving factors, namely (1) reducing the critical concentration of net polymerization compared to the case without applied force, and (2) releasing the inhibition from the closed state of the FH2 ring.

The first factor arised as a natural result from the fact that recruitment of new actin monomers must lead to elongation along the force direction. The second factor results from the mechanical gating hypothesis of the FH2 ring, and in this case force contributes to a bias towards the open state of the ring. Our derivation is based on a reasonable assumption that there is a deformation between the closed and the open state, resulting in a positive extension change Δ along the force direction during a closed-to-open transition of the FH2 ring. Acceleration of actin polymerization based on mechanical deformation of the FH2 ring was first proposed by Kozlov et al 18. The derivation of the force-dependent actin polymerization speed in this study compliments that work by providing an analytical solution that can be directly compared with the single-molecule manipulation experiments.

Prior to this work, Courtemanche et al.20 investigated the effects of force on budding yeast formin Bni1p-dependent actin polymerization. In the absence of profilin, the authors found that force suppressed actin polymerization at sub pN forces, which contradicts to the observations in our study obtained based on using formin mDiaΔN3 and Bni1(FH1-FH2)p (a.a.1227–1766). In their experiments, biotin-tagged Bni1p molecules were anchored to streptavidin molecules on a lipid bilayer embedded with rigid diffusion barriers. Force was applied by flow-stretching of the filament. In flow, the filament is rotated around the anchoring point toward the surface, which might press the filament against the barrier, causing a rotational constraint that might explain this discrepancy.

A previous study by Jégou et al.15 using mDia1 shows much less extent of the force-dependent acceleration of actin elongation than in this study. However, that experiment was carried out in the presence of profilin, which is different from our experiments done in the absence of profilin. Therefore, the difference can be explained by the presence/absence of profilin in solution. A more recent study by Kubota et al.31 reported force-dependent acceleration of mDia1ΔN3-mediated actin polymerization for both ATP-G-actin and ADP-G-actin, in the presence or absence of profilin. Under the condition of ATP-G-actin in the absence of profilin, they reported a very weak force dependence of actin polymerization (~11 subunits per second at ~4 pN) similar to the slower species observed in our experiments (Fig. 4, hollow squares). In that work, the experiments were carried out using dual-trap optical tweezers, where the bead-to-bead distance was increased manually to maintain a roughly constant force. The rate of bead-to-bead distance increase was interpreted as the actin elongation speed at the corresponding forces. Such measurement would depend on how fast the bead-to-bead distance was changed. Based on the Supplementary Fig. 2 in ref. 31, the typical rate of change of the bead-to-bead distance was in the range of 20–30 nm s−1. At this speed, only the slower species could build forces of a few pN that were detected in experiments. In contrast, forces built in the faster species would be less than 1 pN (Fig. 4, solid squares), which might be the reason why it was not detected in that work.

For the fast elongating filaments obtained under rotationally unconstrained conditions, we observed a large, several-fold variation in the polymerization speeds among different tethers under the same actin monomer concentration and force. We reason that the intrinsic uncertainty of force calibration by approximately ~20% may be one of the causes of this variation of polymerization speed (Supplementary Note 3). In fact, the highly sensitive force dependence (Fig. 4, black solid curve) predicted by Eq. (1) results in significant variation of force-dependent polymerization speeds when a ~20% force uncertainty is considered. However, this factor alone is insufficient to explain the level of variation in the polymerization speeds observed in experiments (Supplementary Fig. 1). Another possibility is that the streptavidin-coated superparamagnetic bead might slide on the actin filament, or in the case of actin filament trapped by NEM-HMM-coated beads, the filament might slide over the NEM-HMM coated beads. This possibility requires stochastic dissociation of actin filament from a streptavidin or NEM-HMM-coated bead, which must be re-caught by the same (or another) streptavdin or NEM-HMM molecule on the same bead. Although we cannot completely exclude this possibility, such events must be rare since once detached, the filament would drift away at a speed >10 μm s−1 in our force range (Supplementary Fig. 2). In addition, should such slippage events happen, one would expect large stepwise jumps in extension, which were not observed in our experiments. Therefore, this large variation in polymerization speeds is likely due to other factors, which warrants future studies.

The capability of the FH2 homodimer of mDia1 to sense force enables it to accelerate actin polymerization by a few folds upon a few pN force increase, as suggested by this work. Forces of a few pN are in a range that can be produced by a single-myosin motor32 and by microtubule protrusion33,34. Besides force generated by motor molecules, large force-bearing proteins that contain a tandem repeats of subdomains, which are often present as components in the force transmission pathway in cytoskeleton network, can also buffer force over a similar range due to stochastic unfolding and refolding of the domains under force35. Therefore, the force range in which the mDia1-mediated actin polymerization shows mechanosensitivity is highly physiologically relevant. Accelerated actin polymerization in response to actomyosin contraction can act as a mechanical absorber, preventing fast accumulation of tension that may disrupt the filament.

Our results show that the force-dependent mDia1-mediated actin polymerization requires unconstrained relative rotation between the FH2 ring and the actin filament it encircles. It is likely that in living cells mDia1 is linked to other cellular structures through flexible adapter proteins that can rotate freely (as proposed in ref. 21). Importantly, formin can bind to cytoplasm membrane indirectly through other adapter proteins. For example, mammalian formin-1 and formin-like 2 can bind α-catenin36,37, while α-catenin is known to form a heterodimer with β-catenin and together they bind the intracellular tail of the E-cadherin at cell–cell adherence junction38. Due to the fluidity of lipid membrane, membrane anchored formin should be able to rotate; therefore the actin filament polymerized from such membrane anchored formin is likely torsionally unconstrained.

In our experiment, the method of anchoring mDia1 to the surface also results in the FH1 domains being under force. The FH1 domain provides binding sites for profilin through its polyproline tracts, and this interaction is also likely to be mechanically regulated. Since our study focuses solely on the understanding of the mechanosensitivity of the FH2 homodimer, profilin was not introduced in order to avoid complications from the possible mechanosensitive interaction between FH1 domains and profilin/actin complex. To fully understand the mechanosensitive regulation of actin polymerization by mDia1, the effects of force on the activities of both FH2 and FH1 domains and the interplay between the two domains under force must be studied. Therefore, our ongoing studies will aim to quantify the affinity between FH1 and profilin/actin complex and the transfer of profilin/actin complexes from the FH1 domains to the barbed end of actin filaments encircled by FH2 homodimers when the FH1 domains are subject to force.

Methods

Materials

Actin was purified from rabbit muscle, stored in G-Ca buffer(2 mM Tris, pH 8.0, 0.5 mM DTT, 0.2 mM ATP, 0.1 mM CaCl2, 0.01% sodium azide, 10% glycerol) at −80 °C. Formin mDia1ΔN3(GST-avi-FH1-FH2) was expressed in Rosetta cells and purified, stored in buffer(50 mM Tris pH 7.0, 50 mM KCl, 1 mM EGTA, 1 mM DTT, 0.01% sodium azide, 10% glycerol) at −80 °C. Formin Bni1(FH1-FH2)p (GST-FH1-FH2, a.a. 1227–1766) was expressed in Rosette 2 cells and purified, stored in buffer (25 mM Tris pH 7.4, 0.5 mM EDTA, 500 mM NaCl, 4 mM DTT) at −80 °C. Biotinylated-mDia1ΔN3 was obtained by BirA biotin ligase reaction, and the GST tag was digested by thrombin before BirA biotin ligation. HMM was kindly provided from the lab of Dr. James Sellers, NIH. HMM was treated with NEM and stopped with DTT.

Sample preparation

Four different tethering methods were used in our experiments (Fig. 1a), which required different preparations of coverslip, seed actin filaments, and superparamagnetic beads. These preparations are described in detail in Supplementary Methods.

Experimental procedure

In order to perform the measurements, the seeding actin filaments were incubated in the channel for 10 min to allow actin filaments seeds to bind to the surface. After removal of free actin filaments by washing the channel with KMEI buffer (10 mM imidazole, 50 mM KCl, 1 mM MgCl2, 1 mM EGTA, 400 μM ATP, 0.5 mM DTT), superparamagnetic beads were introduced into the channel, and incubated for 5 min to allow them to bind to actin filaments. Finally, 500 nM G-actin was flowed into the channel in KMEI buffer for actin elongation. Force was applied to the tether through the attached superparamagnetic beads. The elongation of the filament was indicated by the movement of the tethered superparamagnetic bead in real time.

Magnetic tweezers

The transverse magnetic tweezers using in this study was made in house, and controlled as described in ref. 39. This set-up can apply force to a superparamagnetic bead attached to the filament by a pair of permanent magnets placed outside the reaction chamber. Force can be generated with a small angle above the focal plane (Supplementary Note 2). The stage was mounted on a step motor controlled with a MP285 manipulator (Shutter), which can move in 3-dimensions with stepping accuracy of 40 nm. When the superparamagnetic bead attached to a filament moved off the viewing area, the stage could be moved back so the elongation of the filament could be continuously tracked. The superparamagnetic beads were imaged with 50X magnification at a sampling rate of 100 Hz, with its position determined by the centroid.

Force calibration

For a given superparamagnetic bead, the force applied to the bead F(s), depends on the magnet-bead separation s. At a given magnet-bead separation, different superparamagnetic beads may feel a different force, F i (s), due to different levels of magnitude of magnetization. The force ratio of two beads, F 2(s)/F 1(s), equals the ratio of the maximal magnetization of the two beads, $$M_1^0{\mathrm{/}}M_2^0$$. For the M270 superparamagnetic Dynal beads (Invitrogen) used in this study, we found that the histogram of the pairwise force ratio built from 100 beads, $$\alpha _{ij} = F_i(s){\mathrm{/}}F_j(s) = M_i^0{\mathrm{/}}M_j^0$$, when peaked at 1 has a standard deviation of σ = 0.2. These calibration results suggest that for any given bead, the force can be estimated as: $$F_i(s) = (1 + \sigma )\bar F(s)$$, where $$\bar F(s)$$ is the average force over many (>20) beads at the same magnet-bead separation s. In other words, for any bead we can estimate the force by $$\bar F(s)$$ with a relative uncertainty of ~20%. $$\bar F(s)$$ was obtained by recording the drift speed of beads in 90% (volume fraction) glycerol water solution at 25°, with an expected viscosity of η = 0.21 Pa·s40. During the measurements, the beads were at least 5 μm above the surface to avoid hydrodynamic coupling between the bead and the surface. Therefore its drag coefficient can be calculated with the Stokes law ξ = 6πηr, where r = 1.4 or 0.5 μm is the radius of the beads. Drift speeds v i (s) were recorded for more than 20 beads indexed by i at various values of magnet-bead separation, and forces were obtained by F i (s) = ξv i (s).

Solution exchange

A 100 μm thick polymer (OF-134-V2, Mypolymers) membrane with an array of rectangular through holes (0.2 mm × 2 mm) was used for eliminating flow perturbation during solution exchange, which was slightly revised from the 50 μm-diameter circular hole array described in our previous publication41, in order to increase the chance to find tethers in the holes.

Data availability

The authors declare that all data supporting the findings of this study are available within the article and its Supplementary Information or from the corresponding author upon reasonable request.

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Drees, F., Pokutta, S., Yamada, S., Nelson, W. J. & Weis, W. I. α-catenin is a molecular switch that binds e-cadherin-β-catenin and regulates actin-filament assembly. Cell 123, 903–915 (2005).

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Zhang, X., Chen, H., Fu, H., Doyle, P. S. & Yan, J. Two distinct overstretched dna structures revealed by single-molecule thermodynamics measurements. Proc. Natl Acad. Sci. USA 109, 8103–8108 (2012).

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Cheng, N.-S. Formula for the viscosity of a glycerol-water mixture. Ind. Eng. Chem. Res. 47, 3285–3288 (2008).

41. 41.

Le, S. et al. Disturbance-free rapid solution exchange for magnetic tweezers single-molecule studies. Nucleic Acids Res. 43, e113–e113 (2015).

Acknowledgements

We thank the protein expression facility of the Mechanobiology Institute for protein purification. The GST-tagged Bni1(FH1-FH2)p plasmid was kindly provided by David Kover (The University of Chicago). This work is funded by the National Research Foundation (NRF), Prime Minister’s Office, Singapore under its NRF Investigatorship Programme (NRF Investigatorship Award No. NRF-NRFI2016-03) (to J.Y.), and grants from the National Research Foundation through the Mechanobiology Institute Singapore (to J.Y., A.B. and M.S.).

Author information

Authors

Contributions

J.Y. and A.B. conceived the research. M.Y. performed the experiments. M.Y., J.Y., A.K.E. and S.L. analyzed the data. M.Y. and J.Y. wrote the manuscript. C.L. and R.K. and J.Y. constructed the transverse magnetic tweezers set-up. J.Y., A.K.E. and M.M.K. developed the theoretical model. X.Y. contributed to early development of the filament polymerization and the anti-GST/protein A coverslip surface for GST-mDia1ΔN3-mediated actin polymerization. S.L., M.Y. and J.Y. developed protocols of NEM-HMM anchoring of actin filaments and surface coating of superparamagnetic beads (anti-GST/protein A coated bead or anti-DIG coated bead with DNA linker). Z.Z. and X.Y. purified the actin monomers and mDia1ΔN3. J.Y., A.B. and M.S. supervised the research.

Corresponding authors

Correspondence to Alexander Bershadsky or Jie Yan.

Ethics declarations

Competing interests

The authors declare no competing financial interests.

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

Rights and permissions

Reprints and Permissions

Yu, M., Yuan, X., Lu, C. et al. mDia1 senses both force and torque during F-actin filament polymerization. Nat Commun 8, 1650 (2017). https://doi.org/10.1038/s41467-017-01745-4

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### Topic: IDE 0013: error compiling (WProgram.h stlib.h) (Read 1 time)previous topic - next topic

#### mtz

##### Feb 12, 2009, 11:38 am
I have a sketch that works with the IDE 0011 (Windows Vista), I've just installed the new version (IDE 0013) but I can't compile because it gives me an error.

This is the error:

In file included from C:\Program Files\arduino-0013\hardware\cores\arduino/WProgram.h:4,
c:/program files/arduino-0013/hardware/tools/avr/lib/gcc/../../avr/include/stdlib.h:111: error: expected unqualified-id before 'int'
c:/program files/arduino-0013/hardware/tools/avr/lib/gcc/../../avr/include/stdlib.h:111: error: expected `)' before 'int'
c:/program files/arduino-0013/hardware/tools/avr/lib/gcc/../../avr/include/stdlib.h:111: error: expected `)' before 'int'
In file included from C:\Program Files\arduino-0013\hardware\cores\arduino/WProgram.h:6,

Can anybody help me?

mtz

#### halley

#1
##### Feb 12, 2009, 01:36 pm
On that line in avr's stdlib.h is the declaration for int abs(int).

Did you mix up the files with prior versions of the IDE?  Did you call the abs() function with an argument that was cast to integer with this syntax:  int(x) ?  Do any tabs of your sketch end in the .c filename instead of .cpp?

#### mtz

#2
##### Feb 12, 2009, 02:26 pm
No, I don't. Only I put my own librarys...
In any library nor skecth I use abs() function...

Maybe I've files ended in .c in the librarys...

#### mikalhart

#3
##### Feb 12, 2009, 06:30 pmLast Edit: Feb 12, 2009, 06:30 pm by mikalhart Reason: 1
No, the problem here is caused by the fact that in wiring.h we still for some reason define our own version of abs.

I'm glad we removed all the #define int... etc. from 0012, but I really think we should also remove the definitions for macros that are also #defined elsewhere, especially since stdlib.h seems to be automatically included these days.

Code: [Select]
`#undef abs`

at the top of your sketch right after all the #include lines.

Mikal

#### mtz

#4
##### Feb 12, 2009, 08:48 pm
Yeah...

You have been faster than me... I found it too...

Thanks a lot...

#### wjoojoo

#5
##### Mar 01, 2009, 12:24 am
I found the exact same problem when I compiled a sketch I'm working on to control some servo motors with the Servo Library I downloaded from the "Contributed Libraries" section of the arduino website.

When I add the "#undef abs" line underneath my #include lines I get the same error message except now it says 'double' instead of 'int'.

I am so confused!

#### enzo

#6
##### Mar 11, 2009, 11:37 pm
I had exactly the same problem as you wjoojoo and someone on IRC told me to add #include <WProgram.h> at the right beginning of my sketch, and it works now

#### GrahamC

#7
##### Mar 12, 2009, 11:57 am
Yupp - just come across exactly the same problem on 0014.  Adding [font=Courier]#include <WProgram.h>[/font] before any other #includes fixes it...

#### kdogzxc

#8
##### Mar 16, 2009, 08:33 pm
i think i am having the same problem as wjoojoo. i have some servos that i want to control with the servotimetimer1 library but it is giving me hell. i have tried all the stuff suggested in old posts.

initially it was giving me the
In file included from C:\Program Files\arduino-0013\hardware\cores\arduino/WProgram.h:4,
c:/program files/arduino-0013/hardware/tools/avr/lib/gcc/../../avr/include/stdlib.h:111: error: expected unqualified-id before 'int'
c:/program files/arduino-0013/hardware/tools/avr/lib/gcc/../../avr/include/stdlib.h:111: error: expected `)' before 'int'
c:/program files/arduino-0013/hardware/tools/avr/lib/gcc/../../avr/include/stdlib.h:111: error: expected `)' before 'int'
In file included from C:\Program Files\arduino-0013\hardware\cores\arduino/WProgram.h:6,

i #included the <WProgram.h> and this allowed it to compile but it still will not control my servos. so i went on to changes:
TIMSK0 &= ~(_BV(TICIE1) | _BV(OCIE1A) | _BV(OCIE1B) | _BV(TOIE1) );
to:
TIMSK0 &= ~(_BV(OCIE1A) | _BV(OCIE1B) | _BV(TOIE1) );
and then to:
TIMSK0 &= ~( _BV(ICIE1) | _BV(OCIE1A) | _BV(OCIE1B) | _BV(TOIE1) );

and still nothing

any help to get this library working or info on how to control some servos more accurately than the servo library can do would be great.

#### mem

#9
##### Mar 16, 2009, 08:44 pmLast Edit: Mar 16, 2009, 08:45 pm by mem Reason: 1
Quote
i have some servos that i want to control with the servotimetimer1 library

why not use the Servo library distributed with Arduino 0012 and later,
It looks functionally the same to me.

#### kdogzxc

#10
##### Mar 16, 2009, 08:48 pm
you can only control a servo with accuracy of a degree. I want to be able to turn a servo half a degree if i wanted to.

#### mem

#11
##### Mar 16, 2009, 09:28 pmLast Edit: Mar 16, 2009, 09:31 pm by mem Reason: 1
Quote
you can only control a servo with accuracy of a degree. I want to be able to turn a servo half a degree if i wanted to.
perhaps you are looking at a different library, my copy of ServoTimer1 has the same capabilities (and limitations) as Servo.

BTW, half a degree is about 0.3% of the movement of the servo. I would be (pleasantly) surprised if the pot and feedback circuit of a servo was half that accurate.

#### kdogzxc

#12
##### Mar 16, 2009, 09:56 pm
im looking at the servotimetimer1 library and i thought that that would allow you to enter the pulse you want rather than the degree you want. if i could enter the pules i want im am only limited by the servo like you said. but if i can only enter the degree i am limited by the library. like if 0 degrees is a 600ms pulse and 1 degree is 610, i want to be bale to enter the 605 that servotimetimer1 allows not only the 1 or 0 that servo allows.

#### mem

#13
##### Mar 16, 2009, 10:14 pm
Can you post a link to the ServoTime1 library that allows input in microseconds

#### kdogzxc

#14
##### Mar 16, 2009, 10:22 pm
http://www.arduino.cc/playground/ComponentLib/Servotimetimer1

this is the one that i cannot get to work, the normal servo library works fine.

Go Up

format: plain


text:
# Do you ever before need to do any kind of normal upkeep?

As an example, de - fragmenting, disc cleaning, or things like that?

0
2019-12-02 03:09:33
Source Share

You intend to make normal back-ups of very important records, that is one of the most vital "regular maintenance" job there is. Yet certainly you are currently doing this ... ; -)

Furthermore, you can make use of applications like Computer Janitor to clean up unnecessary collections, old bits, etc

In most instances fragmentation is not actually a large concern, and also regardless it will not damage you as high as it carried out in the old days. The documents that are more than likely to get fragmented are possibly log documents (which actually isn't all that vital ; if you require them split seconds are not what you respect) and also extra as a whole when tiny portions of information get contacted numerous documents in parallel. I would not bother with this however, as long as you do not see filesystem performance weaken severely (my major desktop computer was mounted in 2007 and also updated since, and also I do not see any kind of performance concerns, also if at times both dividings have actually been fairly complete).

0
2019-12-03 05:06:20
Source

Not actually. There is no demand to defrag the disks, due to the fact that they do not get fragmented unless you have like 99% of the disk complete. There is no demand for a cleaning, unless you are actually short on room. Performance is not a concern, like in Windows. There is additionally no demand to cleaning the computer system registry, due to the fact that Ubuntu does not have one.

Nonetheless, some applications could require upkeep. As an example, Firefox data sources could come to be puffed up with time, so you need to vacuum them to maintain the performance high. Any kind of application that makes use of SQLite documents can deal with the very same concern.

If you intend to do cleaning, see CCleaner equivalent?.

What I do to stay clear of throwing away room is to constantly cleaning the proper cache. This is what I make use of when updating plans :

sudo apt-get clean
sudo apt-get update
sudo apt-get autoremove

The first command cleaning the cache, the 2nd upgrade the software program resources, the 3rd upgrades essential plans and also the last one uninstall unneeded software program.

0
2019-12-03 05:06:07
Source

While you do not have to do normal cleanings, there might be times you intend to cleaning unneeded things to make some even more disk room and also lighten the system.

Examine Cleaning up a Ubuntu GNU/Linux system at UbuntuGeek.
Update : additionally How do I free up disk space? below (as DrKenobi refers).

If you make use of firefox, there are addons like Vacuum Places Improved to maximize data source.

I additionally favor developing a tmpfs point for the whole firefox account - - yet that would certainly refer individual choice. As an added benefit, this aids me keep an account back-up - - simply in instance.

0
2019-12-03 05:01:48
Source

For a regular Ubuntu customer, below is what you can do :

• make use of the Computer Janitor (System > Administration) to cleaning old plans and also all make use of the apt-get command line device to cleaning up extra plans
• the Trash (lower appropriate edge of the screen) to cleaning removed documents
• hand-operated upkeep of your residence (/ residence/ ~ username ) directory site

Other than that, you do not require to defragment your disk drive, actually. When it comes to spywares and also all, I never ever had any kind of trouble until now (for a regular customer).

0
2019-12-03 05:01:38
Source


text:
# IUCN is also called as

$\begin{array}{1 1}(a)\;\text{Man & Biosphere program}\\(b)\;\text{World conservation union} \\(c)\;\text{World conservation consortium}\\(d)\;\text{World wide conservation union}\end{array}$

IUCN is also called as world conservation union
Hence (b) is the correct answer.


text:
# Why does the sky of Mars appear blue in this video of pictures sent back by the Chinese rover?

Is there any explanation for the blue sky of Mars shown in this video of photos transmitted by the Chinese rover?

Zhurong Rover is driving on Mars sending back new images (Chinese Tianwen-1 mission) - May 22, 2021

Despite my comments about the sky not always being red, I think there is a simpler explanation.

### This video was made by a "content" company on a monetized YouTube channel. The screenshot is simply a colorized version of a black and white image

Note that the notes below the video do not provide the source of the images. You can't check the source of their data, they offer it only in the form they want you to see.

from the "Front Obstacle Avoidance Cam B" shown later between about 00:39 and 00:43 in the video.

Edit: The original color analysis I had here is not conclusive as @Ruslan points out so I'll replace it. It's still viewable in the edit history. Instead, while not conclusive I'll compare a cropped bit of Ruslan's suggested PIA19070 with a cropped bit of the (likely) colorized image screenshot from the video.

### Don't trust random monetized channels!

• What scatterplots would you expect from a faithfully-colored image? The one from NASA with blue sky also has two-part scatterplots: i.stack.imgur.com/XVwb7.png. Though, it seems it's also been manipulated to not-quite-real colors: "The color has been approximately white-balanced to resemble how the scene would appear under daytime lighting conditions on Earth". May 23, 2021 at 9:12
• @Ruslan the suspicious thing is not so much that the scatterplot is “two-part”, but that it's two pretty straight lines both pointing to (0,0). (Plus some noise and clipping.) In the scatterplots you show, both of the regions are more irregular / cloud-shaped, which is more characteristic of actual physical light scattering around. May 24, 2021 at 16:08
• @Ruslan that's a really good point and after seeing your results, mine don't really look very convincing. So I'll delete them (still viewable in edit history) and replace with a simple comparison of a bit of PIA19070 with the screenshot. It's not meant to be convincing by it's own. The key part of the answer is that the black and white image from Obstacle Avoidance Camera B at 00:41 is the same as the colorized image shown at the beginning of the video. Thanks for your comment and analysis!
– uhoh
May 24, 2021 at 19:38
• A spec sheet stating that that camera is b/w would be conclusive (maaaybe the youtuber had secret access to the original color image). But were it a color camera, it's hardly likely that image-conscious China would degrade its images to b/w before publishing them. May 24, 2021 at 22:17
• @CamilleGoudeseune There are some good avenues to pursue in answer(s) to Where can I find access to information provided by CNSA and Chinese scientists and reputable science authors? Knock yourself out! :-)
– uhoh
May 24, 2021 at 22:25

Why wouldn't the sky appear that way? The sky color has little to do with the color of the land. It is to do with scattering of light by particles in the air as we are looking away from the planet. Think about it, why isn't the Earth's sky green here in the Northwest where there is endless forest? Read the NASA article below for why the sky is "blue". When we see the planet Mars from a distance, then it is reflected light from the surface of the planet which is predominately red, and we are viewing towards the planet. Similarly, the Earth likes blue.

Reference Del Genio, Anthony D. (2003) Why is the sky blue? NASA. https://www.scientificamerican.com/article/why-is-the-sky-blue/

• What color is the sky on Mars?. It's all relative. Is it about the wavelengths of light detected or the limitations of the physiology of the human eye?
– Fred
May 25, 2021 at 9:34
• The sky is blue because of Rayleigh Scattering, which happens to a degree relative to the thickness of the atmosphere. Given that Mars's atmosphere is much thinner than Earth's we would expect that effect to be much less pronounced May 25, 2021 at 19:38

The sky of Mars is blue like the one of Earth. The ESA picture you can admire below the text shows unequivocally that the sky of the Red Planet is much like the one above the Sahara desert.


text:
# Systems Of Linear Equations (Simultaneous Linear Equations)

Suppose you buy a book for an amount where you give the seller $20 and they return$8. Can we know the price of the book? Of course, the matter is very simple, we can express this problem with a simple equation , where is the price of the book (One Unknown).

But now suppose you bought 3 books and 9 notebooks for \$120. Can we know the price of one book and the price of one notebook? Of course not, this time it became a little more complicated. We can express this problem by the equation where is the price of the book and is the price of the book.

To solve this problem, we need another equation with the same unknowns as the first equation. For example .

## Define System of Linear Equations

### Note:

Every algebraic expression is written as:

is called a system of linear equations (In some references it is also called Simultaneous Linear Equations).

Whereas and are real numbers.

The solution of the system is the ordered pair .

## Graphical Interpretation of a System of Linear Equations

the algebraic expression is an equation of a straight line that can be written in the form:. So, the system of two linear equations give us the equations of two straight lines (graphes) in a cartesian coordinate system. the relative positions of these two lines express the solution of the system:

1. The graphs intersect at exactly one point, which gives the (single) ordered-pair solution of the system. The system is consistent and the equations are independent.

2. The graphs are parallel lines, so there is no solution and the solution set is ∅. The system is inconsistent and the equations are independent.

3. The graphs are identical (lines coincide), and there are an infinitely many solutions. The system is consistent and the equations are dependent.

## Solving Systems of Linear Equations

To solve a linear system of two equations, we have three different methods:

### Graphical Solution

Write the equations of linear systems in the form of , and then represent the lines in the coordinate plane. As we explained earlier, these two lines can be intersecting, parallel, or coincident.

We will try to take an example for each of the three cases.

#### Example1:

Let's solve the following system of linear equations:

We rewrite the two equations as follows in order to make it easier to represent them in the coordinate plane.

From the graph, we notice that the two lines intersect at the point with coordinates .

So the solution of the system is the ordered-pair .

The ordered-pair that means and .

Check the solution

We substitute and in the two equations:

#### Example2:

Let's solve the following system of linear equations:

We rewrite the two equations as follows in order to make it easier to represent them in the coordinate plane.

From the graph, we notice that the two lines are parallel. Then there is no solution to the system.

### Note:

We say that two lines are parallel if they have the same slope.

#### Example3:

Let's solve the following system of linear equations:

We rewrite the two equations as follows in order to make it easier to represent them in the coordinate plane.

From the graph, we notice that the two lines are identical. Then there is infinitely many solutions.

### Note:

We say that two lines are coincide if they have the same slope and the same ordinate at the origin.

### Algebraic Solution: Substitution Method

In a system of two linear equations with two variables, the substitution method involves converting the system to one equation in one variable by an appropriate substitution.

Solving Linear Systems by Substitution

### Note:

1. Find an expression for one variable in terms of the other. (If one of the equations is already in this form, you can skip this step.)

2. Substitute the expression found in step 1 into the other equation of the system. This will excludes one variable.

3. Solve the equation containing one variable found in step 2.

4. Back-substitute the value found in step 3 into one of the original equations of the system. Simplify and find the value of the remaining variable.

5. Check the solution obtained in both of the system’s given equations.

#### Example4:

Solve the following system by the substitution method:

#### Solution:

Before solving, it is advisable to number the two equations:

#### Step1: Find an expression for one variable in terms of the other

We choose one of the unknowns in one of the two equations, write it in terms of the other unknown. If possible, solve for a variable whose coefficient is 1 or - 1 to avoid working with fractions as much as possible.

we choose the equation (2) and write in terms of because the coeffiecient of is .

#### Step2: Substitute the expression found in step 1 into the other equation of the system

Substitute in the equation (1):

The variable has been eliminated.

#### Step3: Solve the equation containing one variable found in step 2

We simplify and solve the equation (3):

#### Step4: Back-substitute

Now in Equation (1), we substitute into the value 4.

The solution of the system is the ordered pair .

#### Step5: Check the solution

Chek the solution in the both equations (1) and (2).

For equation (1):

For equation (2):

True statements result when the solution is substituted in both equations, confirming that the solution set is .

### Algebraic Solution: Elimination Method

In a system of two linear equations with two variables, the Elimination Method involves finding the equivalent system which the coefficients of one of the two variables must be additive inverses.

Solving Linear Systems by Elimination

### Note:

1. Choose the variable to be eliminated. (The coefficient is preferred 1 or -1 if exists)
2. We multiply all the terms of one equation (or both in some cases) by a constant that will get the variable chosen in step 1 to add up to zero.
3. Add the two equations together, in order to eliminate a variable, and then solve the simultaneous equation.
4. Back-substitute the value found in step 3 into one of the original equations of the system. Simplify and find the value of the remaining variable.
5. Check the solution obtained in both of the system’s given equations.

#### Example5:

Solve the following system by the elimination method:

#### Solution:

Before solving, it is advisable to number the two equations:

#### Step1: Choose the variable to be eliminated

For example, we can choose to eliminat the variable because the coefficient is in the equation(1).

#### Step2: Find the eequivalent system

The coefficient of the variable in the equation (2) is 2, then we must multiply all terms of the equation (1) by 2 to get the additive inverses( and ).

The equivalent system is:

#### Step3: Adding the equations of the equivalent system

We add the equations (3) and (4) side by side:

#### Step4: Back-substitute

Now in Equation (1), we substitute into the value -1.

The solution of the system is the ordered pair .

#### Step5: Check the solution

Chek the solution in the both equations (1) and (2).

For equation (1):

For equation (2):

True statements result when the solution is substituted in both equations, confirming that the solution set is .


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# Two photon cross section

1. Jun 28, 2012

### Mr Boom

Hello. I was hoping someone could help me understand some units. In several papers, I have seen the two photon LIF signal written as:

S = n*(Ω/4π)*T*G*σ*(E/hv)^2*η*a

where the units are

n (density of measured atom): #/cm3
Ω/4π (solid angle fraction): unitless
T (optical transmission fraction): unitless
G (statistical factor): unitless
E/hv (total photons): photons2 to photons4
η (detector conversion): unitless

So that leaves σ, which I've only seen defined in cm4*s. However, that leaves cm on top as well as square photons, which doesn't make sense. What am I not getting?

2. Jun 29, 2012

### M Quack

E should be an energy

hv should proabably be $h \nu = \hbar \omega$ which is also an energy, so the fraction is unitless.

3. Jun 29, 2012

### Mr Boom

Yes, the fraction has no units. It's just a large number. Thanks.

Any idea about the x-section units?

4. Jun 29, 2012

### M Quack

Normally cross sections are given in (length)^2, e.g. barn

I am not quite sure what units S is supposed to have.

5. Jun 29, 2012

### Mr Boom

The units of S can be photons, counts, or electrons depending if you put those conversions in. This is a two photon cross section so the tabulated values I see are cm4*s. Not sure where the seconds come from unless it is supposed to be multiplied by a power rather than energy.


text:
# 2552. Fractal Snowflakes

A snowflake of size 1 looks like this.

+

A snowflake of size 2 looks like this.

v

+<

^

Snowflakes of a larger size n are composed of a center (the ‘+’ character) and 4 branches, each a snowflake of size n-1, but each with only 3 branches. The branch closest to the center of the big snowflake is replaced by a string of hyphens (‘-‘) or pipes (‘|’), depending on whether it is horizontal or vertical. See the sample output for snowflakes of size 3 and 4. Your task is to write a program that can draw snowflakes.

### 输入格式

The first line of input gives the number of cases, N. N test cases follow. Each one is an integer n on a line by itself.

1 ≤ N ≤ 10,1 ≤ n ≤ 7.

### 输出格式

For each test case, print a line “Case #x:”, where x is the number of the test case, followed by the snowflake. Make sure that none of the output lines contains any trailing whitespace characters.

### 样例

Input
4
1
2
3
4

Output
Case #1:
+
Case #2:
v
>+<
^
Case #3:
v
>+<
v | v
>+-+-+<
^ | ^
>+<
^
Case #4:
v
>+<
v | v
>+-+-+<
v ^ | ^ v
>+<  |  >+<
v |   |   | v
>+-+---+---+-+<
^ |   |   | ^
>+<  |  >+<
^ v | v ^
>+-+-+<
^ | ^
>+<
^

2 人解决，10 人已尝试。

2 份提交通过，共有 22 份提交。

9.7 EMB 奖励。


text:
# Thread: Set is Subspace

1. ## Set is Subspace

I need to determine whether all vectors (a,b,c) for which a+b+c=0 are a subspace of R^3.

I believe it is under scalar multiplication:
k(a,b,c)= (ka, kb, kc)
(ka + kb +kc) = 0
let a= 1, b=2, c=-3, k=5
(5(1) + 5(2) + 5(-3)) = 0
0 = 0
So therefore this is closed under scalar multiplication.

I think I'm correct in that, but how do I find out addition?

2. Originally Posted by Hellreaver
I need to determine whether all vectors (a,b,c) for which a+b+c=0 are a subspace of R^3.

I believe it is under scalar multiplication:
k(a,b,c)= (ka, kb, kc)
(ka + kb +kc) = 0
let a= 1, b=2, c=-3, k=5
(5(1) + 5(2) + 5(-3)) = 0
0 = 0
So therefore this is closed under scalar multiplication.

I think I'm correct in that, but how do I find out addition?
You just need to check two things. First, it is closed under vector addition. Second, it is closed under scalar multiplication. You were correct in showing it is closed under scalar multiplication. Now let $\bold{a} = (a_1,a_2,a_3)$ and $\bold{b} = (b_1,b_2,b_3)$. Therefore, $a_1+a_2+a_3=0$ and $b_1+b_2+b_3 = 0$. Then $\bold{a}+\bold{b} = (a_1+b_1,a_2+b_2,a_3+b_3)$. And $(a_1+b_1)+(a_2+b_2) + (a_3+b_3)$ can be rearranged as $(a_1+a_2+a_3)+(b_1+b_2+b_3) = 0 + 0 = 0$. Thus, it is closed under vector addition.

3. Awesome, thanks. I was really unsure about the addition, and I'm happy I have the scalar multiplication part. Thanks a bunch.


text:
# Prove that the lim f(x) = 1 using the $\epsilon-\delta$ definition

Show directly (from the $\epsilon-\delta$ definition ) that

$$\lim_{x\to-1}\frac{x^{2}-3}{x-1} = 1$$

Attempt

Using $$\left\lvert \frac{x^{2}-3}{x-1} - 1 \right\rvert = \left\lvert \frac{x^{2}-x-2}{x-1} \right\rvert = \left\lvert \frac{(x-2)(x+1)}{x-1} \right\rvert$$

and $|x-2|\leq |x - 1 - 1 | \leq |x-1|+|-1| = |x-1|+1 \leq \delta + 1 = 2$ (if $\delta \leq$ 1)

and $|x-1| = |x+1-2| \geq |-2| - |x+1| > 2 - \delta \geq 1$ (if |x+1| < $\delta$)

hence $$\left\lvert \frac{x^{2}-3}{x-1} - 1 \right\rvert = \left\lvert \frac{x^{2}-x-2}{x-1} \right\rvert = \left\lvert \frac{(x-2)(x+1)}{x-1} \right\rvert < 2 \delta \leq \epsilon$$

so $\delta = \frac{\epsilon}{2}$

and picking $\delta$ = min(1,$\frac{\epsilon}{2}$)

hence $$\forall \epsilon>0 \exists \delta = \min(1,\frac{\epsilon}{2}) > 0 : \forall 0 < \lvert x--1\rvert < \delta = \min(1,\frac{\epsilon}{2}) \implies \left\lvert\frac{x^{2}-3}{x-1} - 1 \right\rvert = \left\lvert \frac{x^{2}-x-2}{x-1} \right\rvert = \left\lvert \frac{(x-2)(x+1)}{x-1} \right\rvert < 2 \delta \leq \epsilon$$

hence

$$\lim_{x\to-1}(\frac{x^{2}-3}{x-1}) = 1$$

• There was an error. Since $x\to -1$, we are seeking a $\delta(\epsilon)>0$, with $|x+1|<\delta$. Following your analysis, we can write $|x-2|=|x+1-3|\le |x+1|+3<\delta +3\le 4$ if $\delta \le 1$. Oct 26, 2015 at 15:46
• What is the question ?
– user65203
Oct 26, 2015 at 15:57

I find it easier to make simplifying assumptions first. You have an expression $| { (x-2)(x+1) \over x-1 } |$ which you would like to bound near $x=-1$. Start by bounding $\delta$ by, say, $1$ (that is $|x+1| < 1$), in which case you have $|x-1| > 1$, and $|x-2| < 4$. Then you have $| { (x-2)(x+1) \over x-1 } | \le 4 |x+1|$.
Now choose $\epsilon>0$ and let $\delta = \min (1, {1 \over 4} \epsilon)$ to get the desired result.


text:
# Solving for least square coefficients of power law

1. Oct 14, 2011

### enfield

$$y_i=A{x_i}^b$$

When I solve for A two different ways I am getting different answers..so somewhere I'm doing something wrong. If someone could point out where I would be grateful :).

Using logs:

$$y_i=A{x_i}^b$$
$$ln(y_i)=ln(A)+b*ln(x_i)$$
$$ln(y_i)-(ln(A)+b*ln(x_i))=r_i$$ for least squares we want to minimize: $$S=\sum_i^{n}{r_i}^2$$ which means the gradient has to be zero. I only care about finding A right now so I only have to deal with the partial with respect to A:
$$\frac{\partial S}{\partial A}\sum_i^{n}{r_i}^2 =2 \sum_i^{n}{r_i}\frac{\partial r_i}{\partial A}= 2\sum_i^{n}(ln(y_i)-ln(A)-b*ln(x_i))\frac{1}{A}=0$$

The numerator inside the sum has to be zero, and we can ignore the 2, so:

$$\sum_i^{n}(ln(y_i)-ln(A)-b*ln(x_i))=0$$
$$\sum_i^{n}ln(y_i)-n*ln(A)-b\sum_i^{n}ln(x_i)=0$$
$$ln(A)=\frac{\sum_i^{n}ln(y_i)-b\sum_i^{n}ln(x_i)}{n}$$

(this is the derivation that I think is correct).
But when i solve for A without taking the logs of each side first I get something else:

$$y_i - A{x_i}^b = r_i$$

$$\frac{\partial S}{\partial A}\sum_i^{n}{r_i}^2 =2 \sum_i^{n}{r_i}\frac{\partial r_i}{\partial A}= 2\sum_i^{n}(y_i - A{x_i}^b)*-{x_i}^b=0$$

$$\sum_i^{n}(-{x_i}{y_i} + A{x_i}^{2b})=0$$

$$-\sum_i^{n}{x_i}{y_i}+A\sum_i^{n}{x_i}^{2b}=0$$
$$A=\frac{\sum_i^{n}{x_i}{y_i}}{\sum_i^{n}{x_i}^{2b}}$$

And if you take the ln of it to compare it with what we got before you get:

$$ln(A)= ln(\frac{\sum_i^{n}{x_i}{y_i}}{\sum_i^{n}{x_i}^{2b}}) =ln(\sum_i^{n}{x_i}{y_i})-ln(\sum_i^{n}{x_i}^{2b})$$

Which is not the same as:

$$ln(A)=\frac{\sum_i^{n}ln(y_i)-b\sum_i^{n}ln(x_i)}{n}$$

as far as I can tell...

2. Oct 14, 2011

### hotvette

That's because you solved two different problems.

Minimizing $\sum[ln(y_i)-ln(A)-b*ln(x_i)]^2$ is not necessarily the same thing as minimizing $\sum(y_i - A{x_i}^b)^2$. The 2nd method will result in a lower residual unless the data happen to fit the curve exactly (i.e. zero residual), in which case both will yield the same result.

Last edited: Oct 14, 2011
3. Oct 14, 2011

### enfield

Ah, thanks so much. Yeah, I had thought they were maybe different problems, but I couldn't see exactly how.

When you do least squares with the log of the data and function you are fitting to the data you won't get the same coefficients for the function that you would if you just did least squares with the unmodified data/function.

Okay, that makes sense. Does lower residual mean lower mean squared error, so better fit?

4. Oct 14, 2011

### hotvette

.

Yes. Two different words for the same thing.


text:
Article Contents
Article Contents

# Gaussian estimates for a heat equation on a network

• We consider a diffusion problem on a network on whose nodes we impose Dirichlet and generalized, non-local Kirchhoff-type conditions. We prove well-posedness of the associated initial value problem, and we exploit the theory of sub-Markovian and ultracontractive semigroups in order to obtain upper Gaussian estimates for the integral kernel. We conclude that the same diffusion problem is governed by an analytic semigroup acting on all $L^p$-type spaces as well as on suitable spaces of continuous functions. Stability and spectral issues are also discussed. As an application we discuss a system of semilinear equations on a network related to potential transmission problems arising in neurobiology.
Mathematics Subject Classification: Primary: 47D06, 47A07; Secondary 90B10.

 Citation: