Flexibility of Actin Filaments Derived from Thermal Fluctuations
1995; Elsevier BV; Volume: 270; Issue: 19 Linguagem: Inglês
10.1074/jbc.270.19.11437
ISSN1083-351X
AutoresHervé Isambert, Pascal Venier, A. C. Maggs, Abdelatif Fattoum, Ridha Kassab, Dominique Pantaloni, Marie-France Carlier,
Tópico(s)Force Microscopy Techniques and Applications
ResumoSingle actin filaments undergoing brownian movement in two dimensions were observed at 20 °C in fluorescence optical video microscopy. The persistence length (Lp) was derived from the analysis of either the cosine correlation function or the average transverse fluctuations of a series of recorded shapes of filaments assembled from rhodamine-actin. Phalloidin-stabilized filaments had a persistence length of 18 ± 1 µm, in agreement with recent observations. In the absence of phalloidin, rhodamine-labeled filaments could be observed under a variety of solution conditions once diluted in free unlabeled G-actin at the appropriate critical concentration. Such nonstabilized F-ADP-actin filaments had the same Lp of 9 ± 0.5 µm, whether they had been assembled from ATP-G-actin or from ADP-G-actin, and independently of the tightly bound divalent metal ion. In the presence of BeF3-, which mimics the γ-phosphate of ATP, F-ADP-BeF3-actin was appreciably more rigid, with Lp = 13.5 µm. Hence, newly formed F-ADP-Pi-actin filaments are more rigid than "old" F-ADP-actin filaments, a fact which has implications in actin-based motility processes.In the presence of skeletal tropomyosin and troponin, filaments were rigid (Lp = 20 ± 1 µm) in the off state (–Ca2+), and flexible (Lp = 12 µm) in the on state (+ Ca2+), consistent with the steric blocking model. In agreement with x-ray diffraction data, no appreciable difference was recorded between the off and on states using smooth muscle tropomyosin and caldesmon (Lp = 20 ± 1 µm). In conclusion, this method allows accurate measurement of small (≤15%) changes in mechanical properties of actin filaments in correlation with their biological functions. Single actin filaments undergoing brownian movement in two dimensions were observed at 20 °C in fluorescence optical video microscopy. The persistence length (Lp) was derived from the analysis of either the cosine correlation function or the average transverse fluctuations of a series of recorded shapes of filaments assembled from rhodamine-actin. Phalloidin-stabilized filaments had a persistence length of 18 ± 1 µm, in agreement with recent observations. In the absence of phalloidin, rhodamine-labeled filaments could be observed under a variety of solution conditions once diluted in free unlabeled G-actin at the appropriate critical concentration. Such nonstabilized F-ADP-actin filaments had the same Lp of 9 ± 0.5 µm, whether they had been assembled from ATP-G-actin or from ADP-G-actin, and independently of the tightly bound divalent metal ion. In the presence of BeF3-, which mimics the γ-phosphate of ATP, F-ADP-BeF3-actin was appreciably more rigid, with Lp = 13.5 µm. Hence, newly formed F-ADP-Pi-actin filaments are more rigid than "old" F-ADP-actin filaments, a fact which has implications in actin-based motility processes. In the presence of skeletal tropomyosin and troponin, filaments were rigid (Lp = 20 ± 1 µm) in the off state (–Ca2+), and flexible (Lp = 12 µm) in the on state (+ Ca2+), consistent with the steric blocking model. In agreement with x-ray diffraction data, no appreciable difference was recorded between the off and on states using smooth muscle tropomyosin and caldesmon (Lp = 20 ± 1 µm). In conclusion, this method allows accurate measurement of small (≤15%) changes in mechanical properties of actin filaments in correlation with their biological functions. Actin plays a pivotal role in cell morphology and motility. The organization of actin filaments in a meshwork in the peripheral cytoplasm is responsible for the mechanical stability of living cells (1Elson E.L. Annu. Rev. Biophys. 1988; 17: 397-430Crossref Scopus (188) Google Scholar); actin filaments, associated to the family of myosin motors, generate contractile activity and translocation of endocytic vesicles within the cell (2Cheney R.E. Mooseker M.S. Curr. Opin. Cell Biol. 1992; 4: 27-35Crossref PubMed Scopus (336) Google Scholar); finally, actin filaments are dynamic polymers whose assembly can drive cell shape changes, chemotactic movement, and cell migration (3Bray D. White J.G. Science. 1988; 239: 883-888Crossref PubMed Scopus (421) Google Scholar, 4Condeelis J. Annu. Rev. Cell Biol. 1993; 9: 411-444Crossref PubMed Scopus (398) Google Scholar). The flexural rigidity of actin filaments is central to the above basic functions, in which filaments have either to resist shearing or compressive forces of a few piconewtons, or to develop propulsive forces via self-assembly. The rigidity of filament networks appears regulated within a broad range by specific actin-binding proteins which either cross-link filaments in a rigid three-dimensional gel, enhance their intrinsic rigidity by bundling, or induce gel-sol transition by fragmentation (5Janmey P.A. Annu. Rev. Physiol. 1994; 56: 169-191Crossref PubMed Scopus (475) Google Scholar). Other actin-binding proteins such as profilin, thymosin β4, and capping proteins are more specifically involved in the regulation of filament assembly. Specifically, thymosin β4 allows the formation of a pool of unpolymerized actin (6Safer D. Nachmias V.T. BioEssays. 1994; 7: 473-480Crossref Scopus (91) Google Scholar); the size of this pool is controlled by the concentration of free G-actin at steady state, which is itself regulated by capping proteins and profilin (7Pantaloni D. Carlier M.-F. Cell. 1993; 75: 1007-1014Abstract Full Text PDF PubMed Scopus (460) Google Scholar, 8Carlier M.-F. Pantaloni D. Semin. Cell Biol. 1994; 5: 183-191Crossref PubMed Scopus (64) Google Scholar). It is via the concerted control of assembly dynamics and of mechanical properties that the biological functions of actin filaments are fulfilled. The mechanical properties of actin filaments are tightly correlated with the helical structure of the polymer, and it is expected that factors affecting the structure, and the thermodynamic stability, of the filament also affect its flexibility. The actin filament can be described as a two-start right-handed helix which displays appreciable variability in crossover periodicity (9Hanson J. Proc. R. Soc. Lond. Ser. B Biol. Sci. 1973; 183: 39-58Crossref PubMed Scopus (76) Google Scholar), as observed on electron micrographs, and which has been modeled in terms of cumulative angular disorder (10Egelman E.H. Francis N. DeRosier D.J. Nature. 1982; 298: 131-135Crossref PubMed Scopus (229) Google Scholar, 11Egelman E.H. DeRosier D.J. J. Mol. Biol. 1991; 217: 405-408Crossref PubMed Scopus (14) Google Scholar) or lateral slipping (12Bremer A. Millonig R.C. Sütterlin R. Engel A. Pollard T.D. Aebi U. J. Cell Biol. 1991; 115: 689-703Crossref PubMed Scopus (111) Google Scholar). The actin monomer itself is made of 4 subdomains (13Kabsch W. Mannherz H.G. Suck D. Pai E.F. Holmes K.C. Nature. 1990; 347: 37-44Crossref PubMed Scopus (1546) Google Scholar) which undergo different types of motion relative to one another, as shown by a normal mode analysis (14Tirion M.M. Ben-Avraham D. J. Mol. Biol. 1993; 230: 186-195Crossref PubMed Scopus (121) Google Scholar). The intrinsic flexibilities of all monomers cumulate to generate the collective motion of the F-actin polymer; however, a mode analysis of the filament is not available yet. The mechanical properties of the actin monomer, hence of the filament, may be regulated by ligand binding. Of particular interest is the effect of bound nucleotide. Under physiological conditions, in the presence of ATP, filaments are made of F-ADP subunits. The bound ADP, which results from ATP hydrolysis associated to filament assembly, is nonexchangeable with medium ATP. The release of Pi following ATP hydrolysis on F-actin is linked to the destabilization of actin-actin bonds in the filament (15Korn E.D. Carlier M.-F. Pantaloni D. Science. 1987; 238: 638-644Crossref PubMed Scopus (318) Google Scholar). Pi or analogs of Pi such as BeF3-, H2O, or AlF4- reconstitute a very stable F-ADP-P or F-ATP-like filament with strong actin-actin bonds (16Carlier M.-F. J. Biol. Chem. 1991; 266: 1-4Abstract Full Text PDF PubMed Google Scholar) and higher structural order (17Lepault J. Ranck J.L. Erck I. Carlier M.-F. J. Struct. Biol. 1994; 112: 79-91Crossref PubMed Scopus (27) Google Scholar, 18Orlova A. Egelman E.H. J. Mol. Biol. 1992; 227: 1043-1053Crossref PubMed Scopus (120) Google Scholar). Similarly, the drug phalloidin, which stabilizes F-actin, i.e. decreases the critical concentration for polymerization (19Dancker P. Low I. Hasselbach W. Wieland T. Biochim. Biophys. Acta. 1975; 400: 407-414Crossref PubMed Scopus (231) Google Scholar), decreases the angular spread in gels of oriented actin filaments (20Holmes K.C. Popp D. Gebhard W. Kabsch W. Nature. 1990; 347: 44-49Crossref PubMed Scopus (1325) Google Scholar), decreases the variability in crossover periodicity (12Bremer A. Millonig R.C. Sütterlin R. Engel A. Pollard T.D. Aebi U. J. Cell Biol. 1991; 115: 689-703Crossref PubMed Scopus (111) Google Scholar), and appears to strengthen actin-actin bonds both across and along the two-start helix (21Lorenz M. Popp D. Holmes K.C. J. Mol. Biol. 1993; 234: 826-836Crossref PubMed Scopus (445) Google Scholar). Finally, proteins binding to filaments and regulating contractility, such as tropomyosin, troponins, and caldesmon, may affect the flexibility of thin filaments as part of their function. It is therefore interesting to examine whether and how changes in thermodynamic stability of F-actin correlate with changes in filament order and in flexural rigidity. Flexural rigidity of actin has been investigated thus far using a number of techniques that proved more or less reliable and accurate. It is to be noted that from the point of view of the polymerist actin is sufficiently rigid to be classifiable as a "semi-flexible" polymer. Most classical, artificial polymers can be classified as flexible; typically, the polymer turns many times in the solution to form a relatively compact disordered coil. Actin filament is much more rigid; however, to describe its dynamics we cannot consider the system as completely stiff, simple visual observation by video microscopy in real time shows that thermal fluctuations are still important for filaments that are longer than a few microns in length. Early measurements of rigidity using electron microscopy are susceptible to artifacts due to inherent fixation/staining procedures and to the adsorption of the specimen to the grid. Rheological measurements (22Janmey P.A. Hvidt S. Peetermans J. Lamb J. Ferry J.D. Stossel T.P. Biochemistry. 1988; 27: 8218-8227Crossref PubMed Scopus (79) Google Scholar) and dynamic light scattering (23Fujima S. Adv. Biophys. 1972; 3: 1-43PubMed Google Scholar, 24Schmidt C.F. Barman M. Isenberg G. Sackman E. Macromolecules. 1989; 22: 3638-3649Crossref Scopus (175) Google Scholar) have been used to attempt to deduce the rigidity of actin. The theory of the rheology of flexible polymers (polymers in which the persistence length is small compared with the length of the polymer and also small compared with the entanglement distance in the solution) has been extremely well developed (25Doi M. Edwards S.F. The Theory of Polymer Dynamics. Clarendon Pres, Oxford1986Google Scholar). The theory of the rheological properties of semi-flexible polymers is however much less developed; attempts to simply adapt the theory of flexible polymers to the case of actin have not yet been successful. Recently it has been shown that deviations observed (26Pickenbrock Th. Sackman E. Biopolymers. 1992; 321471Crossref PubMed Scopus (26) Google Scholar) in the dynamic light scattering from the behavior of Rouse Zimm flexible chains are consistent with that expected for semiflexible chains (27Farge E. Maggs A.C. Macromolecules. 1993; 26: 5041-5044Crossref Scopus (94) Google Scholar). A detailed analysis allows one to deduce the rigidity to within 30% in a completely noninvasive manner; however, a number of delicate hydrodynamic corrections are needed to obtain reliable and consistent results (28Janmey P.A. Hvidt S. Käs J. Lerche D. Maggs A. Sackmann E. Schliwa M. Stossel T.P. J. Biol. Chem. 1994; 269: 32503-32513Abstract Full Text PDF PubMed Google Scholar). Recent developments in video-assisted optical microscopy have prompted experiments in which the fluctuations in the shape of actin filaments undergoing brownian motion in solution are observed using the fluorescence of the bound tetra-methyl rhodamine-phalloidin to visualize individual filaments. In early studies, the flexural rigidity was derived from measurements of end-to-end distance (29Takebayashi T. Morita Y. Oosawa F. Biochim. Biophys. Acta. 1977; 492: 357-363Crossref PubMed Scopus (78) Google Scholar); however, this analysis is more appropriate for flexible than for semiflexible polymers. More precise values of the flexural rigidity were recently determined using either a mode analysis (30Gittes F. Mickey B. Nettleton J. Howard J. J. Cell Biol. 1993; 120: 923-934Crossref PubMed Scopus (1420) Google Scholar) or the cosine correlation function (31Ott A. Magnasco M. Simon A. Libchaber A. Phys. Rev. E. 1994; 48: R 1642Crossref Scopus (283) Google Scholar). In both cases, a persistence length of 17–18 µm was found for phalloidin-stabilized F-actin. Because we thought that phalloidin might change the mechanical properties of actin, we have designed appropriate experiments to measure the flexural rigidity of native filaments without phalloidin and investigate how it can be regulated by bound nucleotide, associated proteins, and ionic conditions. A novel analysis method, using the average two-dimensional transverse fluctuations, was found to be a convenient alternative to the cosine correlation function to derive the persistence length of semiflexible chains. The values found for the persistence length are in agreement with recently published data for phalloidin-decorated F-actin; however, we show that in the absence of phalloidin actin filaments are 2-fold more flexible (Lp = 9–10 µm). Filaments are also 50% stiffer in the F-ADP-P state than in the F-ADP state. Finally, the flexibility is regulated by tropomyosin and troponin in a Ca2+-dependent fashion. ATP, ADP, Ap5A, 1The abbreviations used are: Ap5A, P1,P5-di(adenosine 5′)-penta-phosphate; NHSR, 5-(and 6)-carboxytetramethylrhodamine succinimidyl ester; PIPES, 1,4-piperazinediethanesulfonic acid. EGTA, dithiothreitol, troponins, subtilisin Carls-berg (type VIII, Catalog No. P5380), and phalloidin were from Sigma. Tetramethylrhodamine-labeled phalloidin and 5-(and 6)-carboxytetramethylrhodamine succinimidyl ester (NHSR) were from Molecular Probes. Hexokinase, catalase, and glucose oxidase were from Boehringer, beryllium sulfate and aluminum nitrate (Gold label) were from Aldrich. All other chemicals were Merck analytical grade. Actin was purified from rabbit muscle acetone powder (32Spudich J.A. Watt S. J. Biol. Chem. 1971; 246: 4866-4871Abstract Full Text PDF PubMed Google Scholar) and isolated as CaATP-G-actin by Sephadex G-200 chromatography in G buffer (5 mM Tris Cl–, pH 7.5, 0.2 mM dithiothreitol, 0.2 mM ATP, 0.1 mm CaCl2, 0.01% NaN3) (33McLean-Fletcher S. Pollard T.D. Biochem. Biophys. Res. Commun. 1980; 96: 18-27Crossref PubMed Scopus (357) Google Scholar). G-actin was cleaved by subtilisin between Met47 and Gly48 essentially as described (34Schwyter D. Phillips M. Reisler E. Biochemistry. 1989; 28: 5889-5895Crossref PubMed Scopus (88) Google Scholar). Briefly, G-actin in G buffer was incubated with 0.05–0.1 unit/ml subtilisin for 20 min at 20 °C. These conditions ensured completion of the subtilisin cleavage, yielding a single homogeneous 35-kDa product and no lower molecular weight polypeptides in gel electrophoresis. Actin was fluorescently labeled in the F-actin state using NHSR. The procedure previously described (35Kellogg D.R. Mitchison T.J. Alberts B.M. Development. 1988; 103: 675-686Crossref PubMed Google Scholar) yielded a labeled material containing on average 3.5–4.5 lysines modified per actin molecule, which heavily interfered with its polymerization properties. A lower labeling ratio of 0.8–1.2 on average was obtained by incubating F-actin (2.5 ml, 40–60 µm) in 50 mM PIPES, pH 6.8, 50 mM KCl, 0.1 mM CaCl2, 0.2 mM ATP with 0.3 mM NHSR (20-fold less than in Ref. 35Kellogg D.R. Mitchison T.J. Alberts B.M. Development. 1988; 103: 675-686Crossref PubMed Google Scholar) at 20 °C. The labeling reaction was monitored spectrophotometrically in a 0.2-cm path cuvette using the 16% decrease in absorbance at 567 nm that occurs upon reaction of lysines with NHSR. Using a Δ∊667 = –40 00 m–1 cm–1 derived from the difference spectra of NHSR after versus before reaction with excess lysine, the reaction was stopped by addition of 50 mM lysine when the decrease in absorbance at 567 nm indicated that 2 lysines per actin had reacted, i.e. routinely at time 40–50 min. The time course of the NHSR reaction showed no evidence for rapid specific reaction of 1 lysine; rather, the slow monotonous absorbance decrease was consistent with simultaneous titration of several residues. The labeled F-actin was pelleted by centrifugation at 400,000 × g for 40 min at 20 °C in a TL100 Beckman ultracentrifuge, resuspended in 2.5 ml of G buffer, and chromatographed over Sephadex G-25 PD-10 to eliminate free dye. The labeled actin contained at this point 0.8–1.2 mol of rhodamine incorporated on average per mol of actin. This ratio did not appreciably change (less than 5%) after one cycle of polymerization followed by resuspension in G buffer. After one night dialysis against G-buffer and centrifugation at 400,000 × g, at 4 °C for 40 min, the supernatant (rhodamine-G-actin) was drop-frozen in liquid nitrogen in fractions of 20 µl and stored at –80 °C. Subtilisin-actin was labeled with NHSR using the same procedure. It was found that labeled actin was not as efficiently cleaved by subtilisin as unlabeled actin; hence, the labeling was routinely carried out after the cleavage reaction. Hence, we cannot absolutely guarantee that the distribution of rhodamine-labeled lysines was the same on uncleaved and cleaved actins. We preferred to label actin lysines rather than Cys374, to avoid interference with the binding of F-actin binding proteins such as caldesmon. ADP-G-actin (either unlabeled or rhodamine-Iabeled) was prepared according to the recently updated procedure (36Pollard T.D. Goldberg I. Schwarz W.H. J. Biol. Chem. 1992; 267: 20339-20345Abstract Full Text PDF PubMed Google Scholar) using hexokinase. ATP-G-actin (50 µm in G buffer containing 10–20 µm ATP and supplemented with 0.2 mM EGTA and 100 µm MgCl2) was incubated on ice for 1 h with 30 units/ml hexokinase and 2.5 mM glucose. The solution was then supplemented with 50 µm ADP and 10 µm Ap5A to inhibit myokinase activity (37Pantaloni D. Carlier M.-F. Coué M. Lal A. Brenner S. Korn E.D. J. Biol. Chem. 1984; 259: 6274-6283Abstract Full Text PDF PubMed Google Scholar). The ADP-G-actin was kept at 0 °C and used within the next 4 h. To polymerize actin from ADP-G-actin, the above solution of ADP-rhodamine-G-actin (50 µm, 100 µl) was brought to 20 °C and 0.1 m KCl and 1 mM MgCl2 were added. Polymerization was very slow due to the poor ability of ADP-actin to nucleate (38Carlier M.-F. Pantaloni D. Korn E.D. J. Biol. Chem. 1985; 260: 6565-6571Abstract Full Text PDF PubMed Google Scholar), and the polymerization process was accelerated by fragmentation of filaments by pipetting every 5 min (~3 times) until viscosity developed in the solution. The solution of rhodamine-Iabeled F-actin at equilibrium in ADP was kept at 20 °C and used within the next 3–4 h. The CaATP-G-actin in G buffer was converted into MgATP-G-actin by simultaneous additions of 0.2 mM EGTA and an amount of MgCl2 equal to 1 molar eq of G-actin plus 20 µm excess. Divalent cation-free G-actin was prepared as described (39) by adding 2 mm EDTA to a solution of G-actin in buffer G containing 2 mm ATP. Divalent cation-free actin (which contains ATP instead of metal-ATP in the nucleotide site) was polymerized by addition of 0.1 m KCl after a 2-min incubation with EDTA. Rabbit skeletal muscle tropomyosin was prepared as described (40) and purified by hydroxylapatite chromatography (41). Smooth muscle tropomyosin (42) and caldesmon (43) were purified from fresh turkey gizzards. Actin filaments were visualized by the fluorescence of either rhodamine-phalloidin or rhodamine-actin itself. To observe a single filament of 2000–4000 actin subunits, i.e. 5.5–11-µm length in a 50 × 50 µm field (corresponding to a 50 × 50 × 2 (µm)3 volume of solution), the final F-actin concentration must be 10–20 nm, which is an order of magnitude lower than the critical concentration for actin assembly in physiological ionic conditions. Phalloidin greatly stabilizes filaments, i.e. it causes a decrease in critical concentration to a value much lower than 10–20 mm (19). Hence, F-actin assembled at 40 µm in the presence of 1 molar eq of phalloidin for 2 h can be clearly seen without fluorescent background after 2000-fold dilution in polymerization buffer. In the absence of phalloidin, filaments are stable in the presence of monomer at the critical concentration (typically 0.1–0.15 µm for ATP-actin and 1.5–2 µm for ADP-actin, under physiological conditions). Hence, in the absence of phalloidin, filaments assembled from rhodamine-actin (Mg- or Ca-actin or divalent cation free-actin) at 40–60 µm could be observed once diluted into polymerization buffer containing unlabeled G-actin at the appropriate critical concentration. This hybrid system (labeled polymer/unlabeled monomer) was necessary to avoid the background of fluorescent monomer and sufficient to keep the fluorescent filaments stable and allow their observation. Filament turnover due to monomer-polymer exchange is known to be extremely slow so that, within the 15 min following dilution of rhodamine-F-actin into unlabeled G-actin at the critical concentration, filaments remain fluorescent virtually all along their length. In fact, the loss of label from the ends did not cause any appreciable change in observable length over several minutes, for ~10-µm-long filaments, in agreement with the well known slow turnover of actin filaments (36, 44–46). Note that the same experimental design (fluorescent filaments diluted in unlabeled G-actin at the critical concentration) would also allow us to run actomyosin motility assays with native filaments in the absence of phalloidin. Individual filaments were observed at a final concentration of ~20 nm polymerized actin in the indicated buffer. 1 mm Dithiothreitol, 100 µg/ml catalase, 10 mm glucose, and 30 µg/ml glucose oxidase were added to all buffers prior to observation, to reduce photobleaching (47). The brownian motion was restricted to two dimensions by placing the 3-µl sample between a slide and a 22 × 22 mm coverslip, both precoated with bovine serum albumin (1 mg/ml). The F-actin solution was gently flattened between the two glass surfaces to avoid breakage of filaments by shearing. Excess solution was removed, and the edges of the coverslip were sealed to the slide with Valap. The thickness of the solution was estimated to 2 µm by measuring the difference in focus between the two internal glass surfaces. Observations were made at 20 °C on a Reichert Polyvar microscope equipped with a plan apochromat-100 oil immersion objective (1.3 numerical aperture) and a G2 Reichert filter. A 6-watt argon laser was used as a high intensity source. An attached acousticooptic modulator of the frequency and duration of illumination (<1 ms) allowed to collect images of objects undergoing rapid movement under conditions minimizing the irradiation of the sample. The beam was guided through a multimodal optic fiber submitted to a 100-Hz mechanical vibration to average out speckle occurring while the image is formed in the camera and obtain a homogeneous light distribution. Images were detected at 50 frames/s using a silicon-intensified camera (Lhesa, LHL 4046 sensitivity: 10–6 lux) connected to a Hamamatsu Argus 10 image processor and recorded in SVHS format. Images of filaments up to 20 µm in length were digitized manually collecting typically 20 to 25 points for a filament of 10–15 µm. Although this manual digitization is more time-consuming than the automatic skeletonization procedure, it proved to be necessary when images are not highly contrasted. This situation occurs with filaments assembled from rhodamine-actin, because rhodamine bound to actin has a lower quantum yield than rhodamine bound to phalloidin. 10 to 15 clean images of a given filament were collected every 6 ± 1 s. A total of ~100 images were collected for a given experiment. The experiment was duplicated, leading to a second pool of ~100 images. The two pools of data were analyzed separately and checked for reproducibility, then the data were cumulated, giving a total of 200 digitized filaments analyzed globally. Two different methods have been used to derive the rigidity of actin filaments from the digitized images. The energy Uf of a semiflexible chain of length L confined to two dimensions is related to the curvature (dθ/ds) in the following manner (48Landau L.D. Lifshitz E.M. Statistical Physics.3rd Ed. Pergamon Press, Oxford1980Google Scholar).Uf=K2∫0L(dθds)2dsEq. 1 where K is the flexural rigidity, related to the persistence length Lp by the following equationLp=KkBTEq. 2 where kB is the Boltzmann constant and T the absolute temperature. The correlation length is so-called because it expresses the distance over which the filament bends due to thermal fluctuations. One can show that in a system confined to two dimensions, the tangential directions along the filament are correlated to the persistence length as follows: = =e-|s|/2LpEq. 3 where is the average correlation function of the tangential directions θ, measured at each point along the curvilinear abscissa S. The angular brackets denote an average over all configurations. Thus, the filament loses memory of its initial direction θ(0) in a distance comparable to Lp. The rigidity of actin filaments was measured by fitting the average of the correlation function of the digitized shapes to this functional form. As a cross-check, we derived a second independent value of the persistence length from the measurement of average transverse fluctuations <[D(s)]2>. The rationale for using this parameter is that, in contrast to average longitudinal fluctuations used in earlier works, the transverse fluctuations are highly sensitive to changes in Lp for semiflexible filaments, which are of a length comparable to Lp. This point is illustrated on Fig. 1 where a filament undergoes a small fluctuation, in which the distance of a point on abscissa s from the tangent at the origin varies from D1 to D2, while the corresponding longitudinal distance r does not vary much (26). The average quadratic distance can be calculated by noting that for any configuration: =2∫0s∫s″s ds′ds″Eq. 4 Since θ(s″) and θ(s′-s″) are independent variables, Equation 4 can be written =2∫0s∫s″s ds′ds″Eq. 5 Substituting the result (Equation 3) in Equation 5, we find after some calculation that =Lp2[2sLp+163e(-s/2Lp)-13e(-2s/Lp)-5]Eq. 6 Note that for s«Lp, ~ (s3/3 Lp) as expected (27), and for s » Lp, ~ 2 Lp s. The discretized filaments were interpolated by a third order Bezier spline (49). The density of guide points in the spline was varied between 1 and 2 points per µm, to verify that the derived value of Lp was invariant and independent of the exact fitting procedure. Each fitted curve was then analyzed to find the functions C(s) and [D(s)]2. Averaging over many filaments allowed us to fit the average functions and <[D(s)]2> to the functional forms 3 and 6. Using a density of points along the filament outside the selected optimum range introduces artifacts. If the plotting point density is too high, the resulting artificial roughness makes the filament look more flexible at short distances while at long distances the apparent longer curvilinear distance would imply a higher stiffness. The same kind of artifact (unphysical roughness) would arise from the use of an automatic skeletonization if the plotting point density is too high (e.g. connected pixels) and might be responsible for the reported apparent wave vector-dependent bending stiffness of freely flickering filaments (50Kaes J. Strey H. Baermann M. Sackman E. Europhys. Lett. 1993; 21: 865-870Crossref Google Scholar). On the other hand, if the plotting point density is exceedingly low, the converse effect takes place. Consideration of these artifacts enabled us to adjust the point density to obtain the best fit to theoretical curves consistent with a gaussian bending model. The reliability of different methods for measuring the flexural rigidity can be expressed in terms of relative sensitivity to stiffness, which can be evaluated as a function of s:w(s)=s ⋅d dLpEq. 7 where is the mean quantity which is measured. Highest values of w(s) correspond to the best sensitivity. The value of w at s = (Lp/2) is 0.35 for the mean transverse fluctuations , and 0.5 for In , while a much lower sensitivity of 0.04 is obtained for the mean square end-to-end distance. In this work, the same sets of data have been analyzed with the two methods (cosine correlation function and mean transverse fluctuations), and the values of the persistence length have been compared. Fig. 2 shows typical video microscopy images of filaments visualized either by tetramethylrhodamine-phalloidin decoration (a to d) or by the fluorescence of rhodamine-actin in the absence of unlabeled phalloidin (e to h). The analysis of the data using either the correlation function or the mean transverse fluctuations is shown in Fig. 3. The
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