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Shape and Length of Myosin Heads and Radial Compressive Forces in Muscle
J. theor. Biol. (1997) 184, 133–138
Shape and Length of Myosin Heads and Radial Compressive Forces in Muscle J-E M Z M
Ecole Centrale de Paris, Laboratoire de Biologie, Grande Voie des Vignes, 92295 Chaˆtenay-Malabry Ce´dex, France
(Received on 2 October 1995, Accepted in revised form on 1 August 1996)
Irrespective of the exact shape and length of myosin heads, it is shown that radial compressive forces occur in contracting muscle, which tend to bring the myofilaments closer to each other. This is in contradiction with common observation: a muscle that shortens swells laterally. The presence of these forces is model-dependent and they appear only in the case of the traditional concepts of force generation. Here we present an overview and a historical account of the problem. 7 1997 Academic Press Limited
Introduction In his News and Views article published in Nature, Squire (1988) commented on the elegant experimental results obtained by Curmi et al. (1988). These authors have shown that S1 (the proteolytic subfragment of myosin equivalent to the head) does not change its general shape, when attached to actin in the rigor state, as compared with the case where it is free in solution. It is widely accepted that the ‘‘rigor position’’ (i.e. the 45° position) corresponds to the end of the power stroke, while the 90° position corresponds to the beginning of the stroke and is more or less comparable to S1 free in solution. As pointed out by Squire (1988), these experimental results impose ‘‘severe constraints on the structural events that might be involved in the contractile cycle’’. Among these constraints, we would like to raise the problem of the existence of transverse compressive forces between the thin and thick filaments, induced by rotation of the heads during contraction. As far as we know, the first authors who presented this fact were Elliott (1974) and Dragomir et al. (1976), who wrote: ‘‘to the force producing sliding and therefore *Postal address: De´partement de Biologie Cellulaire et Mole´culaire, Service de Biophysique des Prote´ines et des Membranes, CE-Saclay, 91191 Gif-sur-Yvette Ce´dex, France 0022–5193/97/020133 + 06 $25.00/0/jt960268
shortening of the sarcomere a transverse force is added that tends to bring the myofilaments closer’’. Morel (1985a) discussed this problem in depth and confirmed the existence of such radial compressive forces. However, in the absence of data on the shape of S1 during the power stroke, the existence of this force could be dismissed. In the light of the experimental results of Curmi et al. (1988), we think this problem should be re-examined.
Length of Myosin Heads and Related Problems It is widely agreed that the length of the globular rigid part of the myosin head in solution is around 12 nm (e.g. Mendelson & Kretzschmar, 1980; Garrigos et al., 1983; Bachouchi et al., 1985; Morel et al., 1992). It is also agreed that the length of a myosin head attached to actin is around 13 nm (e.g. Seymour & O’Brien, 1985; Milligan & Flicker, 1987; Curmi et al., 1988). In the case of Curmi et al. (1988), we calculated the maximum chord as follows. Let us call RG the radius of gyration of S1, which is 13.5 nm, p the axial ratio of the equivalent prolate and Fmax its length. We have p = (Fmax/2R)3/2 with R = (3MrV/4pNA )1/3 7 1997 Academic Press Limited
.-. .
134
where NA = Avogardro’s number, Mr = 127 000, V = 0.743 ml/g and RG = [(p 2 + 2)/5]1/2(Fmax/2p). We deduce that RG = R{[(Fmax/2R)3 + 2]/5}1/2 (Fmax/2R) − 1/2, from which we deduce Fmax 1 13 nm. Thus, we suggest a length for the globular part of the myosin head of 112.5 nm (for a discussion on the shape and length of S1, see Morel et al., 1992). An elementary calculation presented by Bachouchi & Morel (1989) indicates that, in the case of rabbit psoas, the closest surface-to-surface distance between actin and myosin filaments at the slack length is also 112.5 nm. As for the actin-binding site on the myosin head, it is
(a) m
5n
12.
S2
A
A
H
10.5 nm
H (b) m
5n
S2
12.
A
A
‘‘located from 2 nm to 6 nm from the tip of S1’’ (Wakabayashi et al., 1988; approximately confirmed by Rayment et al., 1993a). Here, it may be assumed that the myosin heads bind to actin as shown in Fig. 1. We conclude that the mean apparent length of a myosin head along the plane passing through the filament axes is around 8.2 nm, when the head attaches to actin (90° position). The surface-tosurface spacing between the thin and thick filaments being 112.5 nm and assuming a length of 1 40 nm for the S2 part of myosin, this S2 moiety becomes tilted by an angle a, where a = sin − 1 (12.5 − 8.2)/40 = 6° upon attachment of the head to actin. It is highly probable that S2 possessed a non-negligible elasticity coefficient in the radial direction and this tilting of S2 very likely explains the well known phenomena of latency relaxation and elongation (minute decrease in the resting tension and minute increase in sarcomere length at the beginning of contraction; e.g. Mulieri, 1972; Haugen & Sten-Knudsen, 1976). In fact, rotation of S2 induces the appearance of a small elastic restoring force that tends to bring the myofilaments closer to each other; a limited reduction in the axis-to-axis spacing between the myofilaments occurs, and since the constant volume relationship holds (Morel, 1985a), a slight sarcomere lengthening should be observed. As for the resting tension at the slack length, this tension probably results, at least partly, from electrostatic repulsive forces between the negatively charged myofilaments. An increase in the attractive forces, described above, results in an apparent decrease in the electrostatic repulsive forces and, therefore, in a limited decrease in the resting tension.
6 nm F. 1. Morel (1984) has shown that the most probable spacing between two consecutive binding sites on the thin filaments corresponds to the diameter of an actin monomer (A 1 6 nm). This is experimentally confirmed by the fact that, under rigor conditions, myosin binds stoichiometrically to F-actin. Therefore, two extreme relative positions of the actin sites and the myosin heads can be considered. (a) The actin site is exactly facing the myosin binding site. There are some distortions in the myosin head, H, but the values suggested here are probably good approximates. (b) The actin binding site is perpendicular to the plane passing through the filament axes. The distortions of the heads are limited and the values given here are also probably good approximates. Note that the site on actin can also be located, as indicated by the symbol (w). In this case the head can attach on the other actin monomer at the site indicated by the symbol (w) (here we assume that the sites on the thin filaments are diametrically opposed and that the heads can undergo a 180° rotation with respect to the horizontal dashed axis, which is in agreement with the conclusion of Toyoshima et al. (1989). Thus, (a) and (b) represent the two extreme positions. All the intermediary positions are possible. The mean length between the thin filament surface and the S1/S2 joint, along the horizontal dashed line is 1(6.0 + 10.5)/2 1 8.2 nm.
What Happens During Contraction? It is widely accepted that the heads adopt the 45° position during contraction. Therefore, the above calculated length of 8.2 nm becomes 8.2 cos 45° = 5.8 nm, along the plane perpendicular to the filament axes. Under these conditions, the mean tilting angle of S2 becomes b = sin − 1 (12.5 − 5.8)/40 = 10° It is explicitely assumed that the rotation of the head is associated with a longitudinal stretching of S2 (Huxley & Simmons, 1971). Therefore this ‘‘stretching force’’ exerted on S2 can be resolved into two components: a sliding force (about 3 kg/cm2) and also a ‘‘collapsing’’ force representing about 3 tan 10° = 0.5 kg/cm2, which is considerable and which
induces a decrease in the myofilament spacing. This conclusion, deduced from the usual concepts as they stand and recent experimental data, is in contradiction with the common observation: under contraction conditions, the muscle swells laterally. The problem is even more dramatic in shortened or osmotically swollen fibres (in these cases the angle b increases and the radial compressive force increases). Note that there is no consensus as for the length of the myosin heads (Craig, 1985; Mendelson, 1985; Craig et al., 1986) and a value of 19 nm is also widely accepted (e.g. Offer & Elliott; 1978; Garrigos et al., 1992). Nevertheless, according to the model of Offer & Elliott (1978), the arrangement of the heads on the actin filaments (see their fig. 4) results in an apparent length along the plane of the filaments’ axes around 12.5 nm and the problem raised above still remains. However, in the present case, the apparent length along the plane passing through the filament axes becomes (rotation of 45°) 12.5 cos 45° = 8.8 nm and the angle b' = sin − 1 (12.5 − 8.8)/40 = 5° The ‘‘collapsing’’ tension reduces to 3 tan 5° = 0.3 kg/ cm2, which remains considerable. Katayama (1989) has found that S1 attached to F-actin in the presence of ATP (situation comparable to contraction) is globular and 18 nm long. If this finding is confirmed, the apparent length of a myosin head along the plane passing through the filament axes would be 18.2 × 8/12.5 = 5.2 nm. Upon rotation by 45°, this length becomes 5.2 cos 45° = 3.7 nm and the tilting angle of S2 becomes b0 = sin − 1 (12.5 − 3.7)/40 = 13° which leads to a considerable ‘‘collapsing’’ tension of 3 tan 13° = 0.7 kg/cm2. Squire (1988) has selected some models that seem to be compatible with the results of Curmi et al. (1988). However, in these models the S2 part of myosin is considerably stretched and the problem of radial compressive forces is not solved. At this juncture, it may be argued that there are too many uncertainties in the assumptions that led to the estimates of the tilt angles of S2. For example, it may be stressed that we do not know precisely how S2 extends from the backbone of the thick filament. However, as far as we know, the model of Huxley & Simmons (1971) is used by most investigators in the muscle field and, as we recall above, this model takes into account a longitudinal stretching of S2 and also a tilt angle. We must also recall here the work of Morel (1985a), in which he assumed that, at the beginning of contraction S2 could be stuck to the
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backbone (a = 0°; extension of S1 up to 12.5 nm long or more). In this case, another explanation would be necessary for the latency relaxation and elongation (e.g. extension of S1 would generate a minute elastic restoring force on the thin filaments). Anyhow, during rotation of S1, S2 becomes unstuck from the backbone and the tilting angle is now b1 = sin − 1 [12.5 (1-cos 45°)/40] = 11° not very different from the other angles. Moreover, under hypotonic conditions, S2 cannot remain stuck to the filament core and the problem is the same as above. How to Try to Solve the Problem of the Radial Compressive Forces The data presented by Curmi et al. (1988) are of major importance since they raise more difficulties than expected by those researching muscle contraction mechanisms. It will be necessary, in the near future, to solve this crucial problem of the transverse compressive forces, which might be a stumbling block for the usual concepts of muscle contraction. We believe that an elegant proposal for solving the problem is very likely to introduce electrostatic repulsive forces between the myofilaments during contraction (Elliott et al., 1970; Dragomir et al., 1976; Morel et al., 1976; Morel 1985b). In fact, an increase in these repulsive forces, resulting from attachment of the myosin heads to actin, leads to an automatic balancing of the radial compressive forces. If it is assumed that the electrostatic repulsive forces are comparable to the intensity of the ‘‘collapsing’’ forces, the problem is merely solved. If it is proposed that the repulsive forces are considerably higher than the radial compressive forces, the problem is also solved, but the electrostatic models must be considered with great attention (for a discussion on this subject see Morel, 1985a). If it is suggested that repulsive forces are only slightly higher than collapsing forces, hybrid models can be considered, in which part of the contraction results from rotation of the heads and another part from electrostatic repulsions. In any case, as underlined above, we think the best solution for solving the problem is probably to remember that electrostatic repulsive forces exists between the myofilaments (e.g. Morel, 1985b and references therein), in order to, at least, counterbalance the transverse compressive forces resulting unavoidably from rotation of the heads. The problem of the radial forces on the myofilament lattice, exerted by the crossbridges, has been studied by Schoenberg (1980) and more recently
136
.-. .
by Bachouchi & Morel (1989), but not exactly in the way studied here. In the present paper, we confirm that, in the light of the data of Curmi et al. (1988), the traditional concepts lead automatically to the existence of collapsing forces. At this juncture, the problem is to know experimentally whether transverse compressive forces exist in contracting living fibres. We suggest studying the cross-section of intact fibres, under isometric conditions, at various osmolarities. For example, when the external osmolarity decreases, the interfilament spacing increases and the electrostatic repulsive forces decrease. By contrast, always under isometric conditions, the collapsing forces (if they exist) increase. The angle b increases as do the radial compressive forces. Under such conditions, it could be expected that, for a critical amount of swelling of the fibre, the transverse compression becomes predominant and the interfilament spacing abruptly decreases. It is clear that such a phenomenon would be related to the existence of collapsing forces, i.e. it is dependent upon the model. In the case of the traditional concepts one may expect the phenomenon described above. In the case of the electrostatic swelling models and the hybrids models, such a phenomenon would not occur. Up-to-date Position of the Problem and Conclusion Brenner & Yu (1991) studied experimentally the radial forces in skinned fibres of rabbit psoas muscle. First, they found that these forces are compressive at lattice spacings of d10 q 34 nm and expansive at d10 Q 34 nm. Second, they found that the elementary radial force is 11.3 pN/myosin, i.e. 10.65 pN/head. The force per head is 18 pN (Morel, 1991). Therefore, the ratio between the radial and the axial forces is 18%, not 1100% as claimed by the authors, who took an axial force 11 pN/myosin (see Merah & Morel, 1993). Morel (1985a) claimed that X-ray diffraction on skinned fibres to determine d10 would lead to artifactual results, related to a disappearance of the thick filaments at the periphery of skinned fibres. Thus, we consider that the results of Brenner & Yu (1991) may be re-examined and that they do not systematically support the existence of collapsing forces in skinned fibres. Such forces, if they would have been genuine, should also appear in living fibres, which is not the case (see below). A full discussion of this possible artifact was given by Morel (1985a). Recently, a new model was proposed in which the problem of the radial compressive forces was neither evoked nor solved (Rayment et al., 1993b). Moreover, we wonder whether crystallized methylated heads are equivalent to heads in vivo, since
methylation ‘‘causes a complete loss of in vitro motility of actin filaments over HMM [heavy meromyosin]’’ (Phan et al., 1994). In any event, the length of 16.5 nm found by Rayment et al. (1993a) lies between the two extreme lengths studied here (12 nm and 19 nm) and the transverse compressive forces would also exist in the model of Rayment et al. (1993b). However, due to the absence of ‘‘functionality’’ of the methylated heads, it is difficult to discuss both the existence and the amount of the transverse compressive forces in the new model. Nevertheless, the structure presented by Rayment et al. (1993a) seems to corroborate the traditional structures and the collapsing forces probably also occur in the model of Rayment et al. (1993b). In fact, these authors discuss their model in terms of the experimental data obtained by Curmi et al. (1988). Recently, Elliott & Worthington (1994) have presented the snap-back theory, which is certainly a very interesting alternative to the traditional swinging crossbridge models. Due to the unavoidable electrostatic repulsive forces between the rotating heads and the myofilaments, the problem of collapsing forces is very likely solved in this new model. Bagni et al. (1994) have studied the variations in lattice spacing in intact fibres from the frog using X-ray diffraction. Under contraction conditions, these authors have found the existence of small compressive forces, under particular experimental conditions. They found that the collapsing force was around 74 pN/thick filament. For the frog, it is well known that the thick filaments are three-stranded. Moreover, there are 108 crowns/filament (Morel, 1985a) and 2 heads/myosin molecule. Thus, the apparent transverse compressive force per myosin head is 74 pN/108 × 3 × 2 = 0.11 pN. The axial force per myosin head is around 8 pN (Morel, 1991; Merah & Morel, 1993) and the radial compressive force represents only about 1.4% of the axial force, which is extremely low. We think one must be cautious with these experiments for two main reasons. (1) Neering et al. (1991) have found that the cross-section of a given fibre, both at rest and in contraction, is far from being uniform and that, under contraction conditions, a mean increase of around 25% in the cross-sectional area is observed. We deduce that it is possible that, for given sections of a contracting fibre, a reduction in diameter could be observed, while for other sections a swelling could be observed. In any case, the results of Bagni et al. (1994) and Neering et al. (1991) are in contradiction. (2) We have observed (unpublished results) that, under conditions mimicking the contraction–relaxation cycle, synthetic
myosin filaments exhibit a particularly interesting behaviour, which we call ‘‘respiration’’. Under ‘‘contraction’’ conditions, the backbone of synthetic filaments swells by 35% in diameter and this phenomenon reverses on ‘‘relaxation’’. We have not studied the filament respiration vs. the osmotic pressure, but it is highly probable that the value of 35% could vary with this pressure. The major consequence of this respiration phenomenon would be a misleading interpretation of measurement of the lattice spacing by means of X-say diffraction. Therefore, the experiments of Bagni et al. (1994) would have been interpreted in a different way if the authors had been aware of the particular behaviour of the thick filaments. In any case, as already pointed out by Morel (1985a), the results obtained by means of X-ray diffraction always appear to be different to, and possibly in contradiction with, results obtained by other techniques. The problem of the interfilament spacing and/or the cross-sectional area is to know whether X-ray experiments are better suited or not. This is an open question.
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G, M., M, J. E. & G T, J. (1983). Reinvestigation of the shape and state of hydration of the myosin subfragment 1 monomer in solution. Biochemistry 22, 4961–4969. G, M., M, S., V, P. & B, J. (1992). Structure of the myosin head in solution and the effect of the light chain 2 removal. Biophys. J. 63, 1462–1470. H, P. & S-K, O. (1976). Sarcomere lengthening and tension drop in the latent period of isolated frog skeletal muscle fibers. J. Gen. Physiol. 68, 247–265. H, A. F. & S, R. M. (1971). Proposed mechanism of force generation in striated muscle. Nature 233, 533–538. K, E. (1989). The effects of various nucleotides on the structure of actin-attached myosin subfragment 1 studied by quick-freeze deep etch electron microscopy. J. Biochem. (Tokyo) 106, 751–770. M, Z. & M, J. E. (1993). Isometric tension exerted by a myofibril of the frog, at 0°C: Geometrical considerations. J. Muscle Res. Cell Motility 14, 552–553. M, R. A. (1985). Length of myosin subfragment 1. Nature 318, 20. M, R. A. & K, K. M. (1980). Structure of myosin subfragment 1 from low-angle X-ray scattering. Biochemistry 19, 4103–4108. M, R. A. & F, P. F. (1987). Structural relationship of actin, myosin and tropomyosin revealed by cryo-electron microscopy. J. Cell. Biol. 105, 29–39. M, J. E. (1984). A theoretical and semi-empirical approach of muscle mechanics and energetics, independent of themechanical role of the crossbridges. Part I. Mechanics of a single fibre. Prog. Biophys. Mol. Biol. 44, 47–71. M, J. E. (1985a). Models of muscle contraction and cell motility: A comparative study of the usual concepts and the swelling theories. Prog. Biophys. Mol. Biol. 46, 97–126. M, J. E. (1985b). Discussion on the state of water in the myofilament lattice and other biological systems, based on the fact that the usual concepts of colloid stability can not explain the stability of the myofilament lattice. J. theor. Biol. 112, 847–858. M, J. E. (1991). The force exerted per myosin head in a muscle fibre is 8 pN. Consequence on the validity of the traditional concepts of force generation. J. theor. Biol. 151, 285–288. M, J. E., B, N. & M, Z. (1992). Shape and length of myosin heads. J. theor. Biol. 156, 73–90. M, J. E., P-H¨¨, I. & G, M. P. (1976). Muscular contraction and cytoplasmic streaming: A new general hypothesis. J. theor. Biol. 62, 17–51. M, L. A. (1972). The dependence of the latency relaxation on sarcomere length and other characteristics of isolated muscle fibres. J. Physiol. (Lond.) 223, 333–354. N, I. R., Q, L. A., M, V. A. & T, S. R. (1991). Nonuniform volume changes during muscle contraction. Biophys. J. 59, 926–932. O, G. & E, A. (1978). Can a myosin molecule attach to two actin filaments? Nature 271, 325–329. P, B. C., C, P., M, C., R, E. & M, A. (1994). Extensively methylated myosin subfragment-1: Examination of local structure, interactions with nucleotides and actin, and ligand-induced conformational changes. Biochemistry 33, 11286–11295. R, I., R, W. R., S-B , K., S, R., T, D. R., B, M. M., et al. (1993a). Three-dimensional structure of myosin subfragment-1: A molecular motor. Science 261, 50–58. R, I., H, H. M., W, M., Y, C. B., L, M., H, K. C. & M, R. A. (1993b). Structure of the actin-myosin complex and its implication for muscle contraction. Science 261, 58–65.
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Shape and Length of Myosin Heads and Radial Compressive Forces in Muscle J-E M Z M
Ecole Centrale de Paris, Laboratoire de Biologie, Grande Voie des Vignes, 92295 Chaˆtenay-Malabry Ce´dex, France
(Received on 2 October 1995, Accepted in revised form on 1 August 1996)
Irrespective of the exact shape and length of myosin heads, it is shown that radial compressive forces occur in contracting muscle, which tend to bring the myofilaments closer to each other. This is in contradiction with common observation: a muscle that shortens swells laterally. The presence of these forces is model-dependent and they appear only in the case of the traditional concepts of force generation. Here we present an overview and a historical account of the problem. 7 1997 Academic Press Limited
Introduction In his News and Views article published in Nature, Squire (1988) commented on the elegant experimental results obtained by Curmi et al. (1988). These authors have shown that S1 (the proteolytic subfragment of myosin equivalent to the head) does not change its general shape, when attached to actin in the rigor state, as compared with the case where it is free in solution. It is widely accepted that the ‘‘rigor position’’ (i.e. the 45° position) corresponds to the end of the power stroke, while the 90° position corresponds to the beginning of the stroke and is more or less comparable to S1 free in solution. As pointed out by Squire (1988), these experimental results impose ‘‘severe constraints on the structural events that might be involved in the contractile cycle’’. Among these constraints, we would like to raise the problem of the existence of transverse compressive forces between the thin and thick filaments, induced by rotation of the heads during contraction. As far as we know, the first authors who presented this fact were Elliott (1974) and Dragomir et al. (1976), who wrote: ‘‘to the force producing sliding and therefore *Postal address: De´partement de Biologie Cellulaire et Mole´culaire, Service de Biophysique des Prote´ines et des Membranes, CE-Saclay, 91191 Gif-sur-Yvette Ce´dex, France 0022–5193/97/020133 + 06 $25.00/0/jt960268
shortening of the sarcomere a transverse force is added that tends to bring the myofilaments closer’’. Morel (1985a) discussed this problem in depth and confirmed the existence of such radial compressive forces. However, in the absence of data on the shape of S1 during the power stroke, the existence of this force could be dismissed. In the light of the experimental results of Curmi et al. (1988), we think this problem should be re-examined.
Length of Myosin Heads and Related Problems It is widely agreed that the length of the globular rigid part of the myosin head in solution is around 12 nm (e.g. Mendelson & Kretzschmar, 1980; Garrigos et al., 1983; Bachouchi et al., 1985; Morel et al., 1992). It is also agreed that the length of a myosin head attached to actin is around 13 nm (e.g. Seymour & O’Brien, 1985; Milligan & Flicker, 1987; Curmi et al., 1988). In the case of Curmi et al. (1988), we calculated the maximum chord as follows. Let us call RG the radius of gyration of S1, which is 13.5 nm, p the axial ratio of the equivalent prolate and Fmax its length. We have p = (Fmax/2R)3/2 with R = (3MrV/4pNA )1/3 7 1997 Academic Press Limited
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134
where NA = Avogardro’s number, Mr = 127 000, V = 0.743 ml/g and RG = [(p 2 + 2)/5]1/2(Fmax/2p). We deduce that RG = R{[(Fmax/2R)3 + 2]/5}1/2 (Fmax/2R) − 1/2, from which we deduce Fmax 1 13 nm. Thus, we suggest a length for the globular part of the myosin head of 112.5 nm (for a discussion on the shape and length of S1, see Morel et al., 1992). An elementary calculation presented by Bachouchi & Morel (1989) indicates that, in the case of rabbit psoas, the closest surface-to-surface distance between actin and myosin filaments at the slack length is also 112.5 nm. As for the actin-binding site on the myosin head, it is
(a) m
5n
12.
S2
A
A
H
10.5 nm
H (b) m
5n
S2
12.
A
A
‘‘located from 2 nm to 6 nm from the tip of S1’’ (Wakabayashi et al., 1988; approximately confirmed by Rayment et al., 1993a). Here, it may be assumed that the myosin heads bind to actin as shown in Fig. 1. We conclude that the mean apparent length of a myosin head along the plane passing through the filament axes is around 8.2 nm, when the head attaches to actin (90° position). The surface-tosurface spacing between the thin and thick filaments being 112.5 nm and assuming a length of 1 40 nm for the S2 part of myosin, this S2 moiety becomes tilted by an angle a, where a = sin − 1 (12.5 − 8.2)/40 = 6° upon attachment of the head to actin. It is highly probable that S2 possessed a non-negligible elasticity coefficient in the radial direction and this tilting of S2 very likely explains the well known phenomena of latency relaxation and elongation (minute decrease in the resting tension and minute increase in sarcomere length at the beginning of contraction; e.g. Mulieri, 1972; Haugen & Sten-Knudsen, 1976). In fact, rotation of S2 induces the appearance of a small elastic restoring force that tends to bring the myofilaments closer to each other; a limited reduction in the axis-to-axis spacing between the myofilaments occurs, and since the constant volume relationship holds (Morel, 1985a), a slight sarcomere lengthening should be observed. As for the resting tension at the slack length, this tension probably results, at least partly, from electrostatic repulsive forces between the negatively charged myofilaments. An increase in the attractive forces, described above, results in an apparent decrease in the electrostatic repulsive forces and, therefore, in a limited decrease in the resting tension.
6 nm F. 1. Morel (1984) has shown that the most probable spacing between two consecutive binding sites on the thin filaments corresponds to the diameter of an actin monomer (A 1 6 nm). This is experimentally confirmed by the fact that, under rigor conditions, myosin binds stoichiometrically to F-actin. Therefore, two extreme relative positions of the actin sites and the myosin heads can be considered. (a) The actin site is exactly facing the myosin binding site. There are some distortions in the myosin head, H, but the values suggested here are probably good approximates. (b) The actin binding site is perpendicular to the plane passing through the filament axes. The distortions of the heads are limited and the values given here are also probably good approximates. Note that the site on actin can also be located, as indicated by the symbol (w). In this case the head can attach on the other actin monomer at the site indicated by the symbol (w) (here we assume that the sites on the thin filaments are diametrically opposed and that the heads can undergo a 180° rotation with respect to the horizontal dashed axis, which is in agreement with the conclusion of Toyoshima et al. (1989). Thus, (a) and (b) represent the two extreme positions. All the intermediary positions are possible. The mean length between the thin filament surface and the S1/S2 joint, along the horizontal dashed line is 1(6.0 + 10.5)/2 1 8.2 nm.
What Happens During Contraction? It is widely accepted that the heads adopt the 45° position during contraction. Therefore, the above calculated length of 8.2 nm becomes 8.2 cos 45° = 5.8 nm, along the plane perpendicular to the filament axes. Under these conditions, the mean tilting angle of S2 becomes b = sin − 1 (12.5 − 5.8)/40 = 10° It is explicitely assumed that the rotation of the head is associated with a longitudinal stretching of S2 (Huxley & Simmons, 1971). Therefore this ‘‘stretching force’’ exerted on S2 can be resolved into two components: a sliding force (about 3 kg/cm2) and also a ‘‘collapsing’’ force representing about 3 tan 10° = 0.5 kg/cm2, which is considerable and which
induces a decrease in the myofilament spacing. This conclusion, deduced from the usual concepts as they stand and recent experimental data, is in contradiction with the common observation: under contraction conditions, the muscle swells laterally. The problem is even more dramatic in shortened or osmotically swollen fibres (in these cases the angle b increases and the radial compressive force increases). Note that there is no consensus as for the length of the myosin heads (Craig, 1985; Mendelson, 1985; Craig et al., 1986) and a value of 19 nm is also widely accepted (e.g. Offer & Elliott; 1978; Garrigos et al., 1992). Nevertheless, according to the model of Offer & Elliott (1978), the arrangement of the heads on the actin filaments (see their fig. 4) results in an apparent length along the plane of the filaments’ axes around 12.5 nm and the problem raised above still remains. However, in the present case, the apparent length along the plane passing through the filament axes becomes (rotation of 45°) 12.5 cos 45° = 8.8 nm and the angle b' = sin − 1 (12.5 − 8.8)/40 = 5° The ‘‘collapsing’’ tension reduces to 3 tan 5° = 0.3 kg/ cm2, which remains considerable. Katayama (1989) has found that S1 attached to F-actin in the presence of ATP (situation comparable to contraction) is globular and 18 nm long. If this finding is confirmed, the apparent length of a myosin head along the plane passing through the filament axes would be 18.2 × 8/12.5 = 5.2 nm. Upon rotation by 45°, this length becomes 5.2 cos 45° = 3.7 nm and the tilting angle of S2 becomes b0 = sin − 1 (12.5 − 3.7)/40 = 13° which leads to a considerable ‘‘collapsing’’ tension of 3 tan 13° = 0.7 kg/cm2. Squire (1988) has selected some models that seem to be compatible with the results of Curmi et al. (1988). However, in these models the S2 part of myosin is considerably stretched and the problem of radial compressive forces is not solved. At this juncture, it may be argued that there are too many uncertainties in the assumptions that led to the estimates of the tilt angles of S2. For example, it may be stressed that we do not know precisely how S2 extends from the backbone of the thick filament. However, as far as we know, the model of Huxley & Simmons (1971) is used by most investigators in the muscle field and, as we recall above, this model takes into account a longitudinal stretching of S2 and also a tilt angle. We must also recall here the work of Morel (1985a), in which he assumed that, at the beginning of contraction S2 could be stuck to the
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backbone (a = 0°; extension of S1 up to 12.5 nm long or more). In this case, another explanation would be necessary for the latency relaxation and elongation (e.g. extension of S1 would generate a minute elastic restoring force on the thin filaments). Anyhow, during rotation of S1, S2 becomes unstuck from the backbone and the tilting angle is now b1 = sin − 1 [12.5 (1-cos 45°)/40] = 11° not very different from the other angles. Moreover, under hypotonic conditions, S2 cannot remain stuck to the filament core and the problem is the same as above. How to Try to Solve the Problem of the Radial Compressive Forces The data presented by Curmi et al. (1988) are of major importance since they raise more difficulties than expected by those researching muscle contraction mechanisms. It will be necessary, in the near future, to solve this crucial problem of the transverse compressive forces, which might be a stumbling block for the usual concepts of muscle contraction. We believe that an elegant proposal for solving the problem is very likely to introduce electrostatic repulsive forces between the myofilaments during contraction (Elliott et al., 1970; Dragomir et al., 1976; Morel et al., 1976; Morel 1985b). In fact, an increase in these repulsive forces, resulting from attachment of the myosin heads to actin, leads to an automatic balancing of the radial compressive forces. If it is assumed that the electrostatic repulsive forces are comparable to the intensity of the ‘‘collapsing’’ forces, the problem is merely solved. If it is proposed that the repulsive forces are considerably higher than the radial compressive forces, the problem is also solved, but the electrostatic models must be considered with great attention (for a discussion on this subject see Morel, 1985a). If it is suggested that repulsive forces are only slightly higher than collapsing forces, hybrid models can be considered, in which part of the contraction results from rotation of the heads and another part from electrostatic repulsions. In any case, as underlined above, we think the best solution for solving the problem is probably to remember that electrostatic repulsive forces exists between the myofilaments (e.g. Morel, 1985b and references therein), in order to, at least, counterbalance the transverse compressive forces resulting unavoidably from rotation of the heads. The problem of the radial forces on the myofilament lattice, exerted by the crossbridges, has been studied by Schoenberg (1980) and more recently
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by Bachouchi & Morel (1989), but not exactly in the way studied here. In the present paper, we confirm that, in the light of the data of Curmi et al. (1988), the traditional concepts lead automatically to the existence of collapsing forces. At this juncture, the problem is to know experimentally whether transverse compressive forces exist in contracting living fibres. We suggest studying the cross-section of intact fibres, under isometric conditions, at various osmolarities. For example, when the external osmolarity decreases, the interfilament spacing increases and the electrostatic repulsive forces decrease. By contrast, always under isometric conditions, the collapsing forces (if they exist) increase. The angle b increases as do the radial compressive forces. Under such conditions, it could be expected that, for a critical amount of swelling of the fibre, the transverse compression becomes predominant and the interfilament spacing abruptly decreases. It is clear that such a phenomenon would be related to the existence of collapsing forces, i.e. it is dependent upon the model. In the case of the traditional concepts one may expect the phenomenon described above. In the case of the electrostatic swelling models and the hybrids models, such a phenomenon would not occur. Up-to-date Position of the Problem and Conclusion Brenner & Yu (1991) studied experimentally the radial forces in skinned fibres of rabbit psoas muscle. First, they found that these forces are compressive at lattice spacings of d10 q 34 nm and expansive at d10 Q 34 nm. Second, they found that the elementary radial force is 11.3 pN/myosin, i.e. 10.65 pN/head. The force per head is 18 pN (Morel, 1991). Therefore, the ratio between the radial and the axial forces is 18%, not 1100% as claimed by the authors, who took an axial force 11 pN/myosin (see Merah & Morel, 1993). Morel (1985a) claimed that X-ray diffraction on skinned fibres to determine d10 would lead to artifactual results, related to a disappearance of the thick filaments at the periphery of skinned fibres. Thus, we consider that the results of Brenner & Yu (1991) may be re-examined and that they do not systematically support the existence of collapsing forces in skinned fibres. Such forces, if they would have been genuine, should also appear in living fibres, which is not the case (see below). A full discussion of this possible artifact was given by Morel (1985a). Recently, a new model was proposed in which the problem of the radial compressive forces was neither evoked nor solved (Rayment et al., 1993b). Moreover, we wonder whether crystallized methylated heads are equivalent to heads in vivo, since
methylation ‘‘causes a complete loss of in vitro motility of actin filaments over HMM [heavy meromyosin]’’ (Phan et al., 1994). In any event, the length of 16.5 nm found by Rayment et al. (1993a) lies between the two extreme lengths studied here (12 nm and 19 nm) and the transverse compressive forces would also exist in the model of Rayment et al. (1993b). However, due to the absence of ‘‘functionality’’ of the methylated heads, it is difficult to discuss both the existence and the amount of the transverse compressive forces in the new model. Nevertheless, the structure presented by Rayment et al. (1993a) seems to corroborate the traditional structures and the collapsing forces probably also occur in the model of Rayment et al. (1993b). In fact, these authors discuss their model in terms of the experimental data obtained by Curmi et al. (1988). Recently, Elliott & Worthington (1994) have presented the snap-back theory, which is certainly a very interesting alternative to the traditional swinging crossbridge models. Due to the unavoidable electrostatic repulsive forces between the rotating heads and the myofilaments, the problem of collapsing forces is very likely solved in this new model. Bagni et al. (1994) have studied the variations in lattice spacing in intact fibres from the frog using X-ray diffraction. Under contraction conditions, these authors have found the existence of small compressive forces, under particular experimental conditions. They found that the collapsing force was around 74 pN/thick filament. For the frog, it is well known that the thick filaments are three-stranded. Moreover, there are 108 crowns/filament (Morel, 1985a) and 2 heads/myosin molecule. Thus, the apparent transverse compressive force per myosin head is 74 pN/108 × 3 × 2 = 0.11 pN. The axial force per myosin head is around 8 pN (Morel, 1991; Merah & Morel, 1993) and the radial compressive force represents only about 1.4% of the axial force, which is extremely low. We think one must be cautious with these experiments for two main reasons. (1) Neering et al. (1991) have found that the cross-section of a given fibre, both at rest and in contraction, is far from being uniform and that, under contraction conditions, a mean increase of around 25% in the cross-sectional area is observed. We deduce that it is possible that, for given sections of a contracting fibre, a reduction in diameter could be observed, while for other sections a swelling could be observed. In any case, the results of Bagni et al. (1994) and Neering et al. (1991) are in contradiction. (2) We have observed (unpublished results) that, under conditions mimicking the contraction–relaxation cycle, synthetic
myosin filaments exhibit a particularly interesting behaviour, which we call ‘‘respiration’’. Under ‘‘contraction’’ conditions, the backbone of synthetic filaments swells by 35% in diameter and this phenomenon reverses on ‘‘relaxation’’. We have not studied the filament respiration vs. the osmotic pressure, but it is highly probable that the value of 35% could vary with this pressure. The major consequence of this respiration phenomenon would be a misleading interpretation of measurement of the lattice spacing by means of X-say diffraction. Therefore, the experiments of Bagni et al. (1994) would have been interpreted in a different way if the authors had been aware of the particular behaviour of the thick filaments. In any case, as already pointed out by Morel (1985a), the results obtained by means of X-ray diffraction always appear to be different to, and possibly in contradiction with, results obtained by other techniques. The problem of the interfilament spacing and/or the cross-sectional area is to know whether X-ray experiments are better suited or not. This is an open question.
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