Journal of Muscle Research and Cell Motility 13, 590--607 (1992) REVIEW

Myosin step size: estimates from motility assays and shortening muscle KEVIN

BURTON*

MRC Muscle and Cell Motility Unit, King's College London, 26-29 Drury Lane, London WC2B 5RL, United Kingdom Received 27 May 1992; accepted 11 June 1992

Introduction In attempting to understand the mechanism of the movement of filaments by molecular motors, several questions arise. How fast can the filament be moved by one or an ensemble of motors? What is the amount of movement produced by the motor while attached to a single binding site on the filament? What proportion of the biochemical cycle is composed of attached states which can produce movement or force? To emphasize the relationship of biochemistry to movement, the last question can be restated: How much filament sliding results from the action of a given motor during one cycle of nucleotide hydrolysis? This quantity, which following current usage will be referred to as 'step size', is to a molecular motor what the number of ions transported per ATP is to systems such as mitochondrial electron transport or the Ca2+-ATPase of sarcoplasmic reticulum (Alexandre et al., 1978; Hasselbach, 1978; Berman, 1982). For the motile system composed of myosin and F-actin powered by the hydrolysis of ATP, a natural working hypothesis was that the filament sliding which occurs while myosin is bound to a single actin site is coupled to the splitting of ATP on a one to one basis (A. F. Huxley, 1957; H. E. Huxley, i964). The step size of myosin has generally been thought to be no more than a few tens of nanometres for several reasons. In developing the sliding-filament-crossbridge model of muscle contraction, it was reasonable to suppose that a repetitive interaction between crossbridge and actin sites would occur at the spatial periodicity of the actin filament, which is 5.5 nm in axial monomer spacing and about 36 nm in helical repeat distance (Hanson & Huxley, 1955; H. E. Huxley, 1964). In the model in which movement is produced by a rotation of the myosin head (H. E. Huxley, 1969), a limit *To whom correspondence should be addressed at: Center for Light Microscope Imaging, Carnegie Mellon University, 4400 Fifth Avenue, Pittsburgh, PA 15213, USA.

0142-4319 9

1992 Chapman & Hall

of twice the length of the head, or about 40 nm, would apply. For heads working in the filament lattice of a sarcomere, the limit is generally thought to be less (about 30 nm if the head swings through an angle of _+45% An estimate of a small step size also came from modelling the force-velocity and energetic properties of shortening muscle (A. F. Huxley, 1957). Using this model and a crossbridge detachment rate obtained from the loss of tension after an isometric twitch, A. F. Huxley estimated the distance over which a crossbridge could generate force for shortening to be about 16 nm. More direct evidence has come from mechanical studies in which rapid changes in either force or length are applied to single muscle fibres during isometric contraction and the resultant transients in length or force, respectively, are monitored. In the case of length steps (A. F. Huxley & Simmons, 1971), the fibre is able to rapidly take up the shortening and regain force for steps of less than about 14 nm of filament sliding (i.e. 14 nm per half sarcomere (hs)) (Ford et al., 1977). With larger steps force goes to zero and there is a delay while, it is presumed, crossbridges detach from the initial actin site and reattach to another site farther along the filament. Huxley and Simmons interpreted their results to indicate the size of a working stroke during which a myosin head can exert positive strain on an actin site. In modelling data from the converse experiment where load is suddenly reduced and transients in velocity result (Civian & Podolsky, 1966), Podolsky and Nolan (1971) suggested that crossbridges could attach and produce force in the shortening direction over a length of about 11 nm. This value of between 10 and 20 nm for the step size is well within the structural limit imposed by the size of the myosin head. More recently, H. E. Huxley and Kress (1985) have suggested that the working stroke might be even smaller based on a consideration of evidence from electron paramagnetic resonance studies that about 20% of crossbridges are strongly attached during isometric contraction. This value can be used to calculate the force

M y o s i n step size

591

per head and assuming I ATP per step, a value of 4 nm results for very slow shortening. Interest in the coupling of A T P to m o v e m e n t between actin and myosin has increased with recent studies in which the sliding of fluorescently-labelled actin filaments has been visualized in shortening myofibrils (Yanagida et al., 1985) and in in vitro motility assays (Kron & Spudich, 1986; Harada et al., 1990; T o y o s h i m a et al., 1990). Estimates of the amount of filament sliding per A T P hydrolyzed in the range of > 60 nm to > 200 nm have been made by Yanagida and colleagues and these would seem to require some modification of ideas about the mechanism of crossbridge action or the coupling of A T P hydrolysis to it. However, a step size of between 5 and 28 n m - - m o r e consistent with conventional crossbridge t h e o r y - - h a s been obtained b y Spudich and colleagues using a similar assay system. This controversy has p r o v o k e d a number of workers from different areas of muscle and motility research to investigate the problem. In addition to studies of in vitro motility, these include experimental studies of muscle and a c t o - m y o s i n biochemistry (Taylor, 1989; Higuchi & Goldman, 1991) and muscle mechanics (Brenner, 1991; Burton & Simmons, 1991; Lombardi et a]., 1992). Several theoretical analyses have also been published recently (Oosawa & Hayashi, 1986; Mitsui & Ohshima, 1988; Vale & Oosawa, 1990; Pate & Cooke, 1991; Tawada & Sekimoto, 1991) as well as reviews of the step size problem and in vitro motility

assays (H. E. Huxley, 1990; Irving, 1991; Sellers & Homsher, 1991; Yanagida et al., 1991). The purpose of this review is twofold. The first is to assess estimates of step size obtained from motility assays as well as from shortening muscle and to consider sources of error in each of these. The second is to consider the mechanism of 'stepping' during rapid filament sliding and evidence for properties consistent with large steps. Much of this evidence comes from studies of the biochemistry, energetics, mechanics, and structure of shortening muscle.

Estimates of step size Estimates of step size from motility assays are summarized in Table 1. Several approaches have been used to calculate these values (Fig. 1). The amount of filament sliding resulting from the action of a single myosin head during one cycle of ATP hydrolysis can be calculated: (1)

D = tsVs

where D is the step size, ts is the amount of time per ATPase cycle spent in 'stroking' states which m o v e the filament, and V~ is the velocity of sliding during this period. To do this calculation, it is necessary to obtain ts and V~ from measurable quantities. In a simple model, V~ can be taken as the velocity when there is always at least one myosin head attached (Harada et a]., 1990; T o y o s h i m a eta[., 1990; Uyeda et al., 1990). For motility

Table 1. D estimated from motility assays A TPase

Reference

Preparation

Harada and Yanagida (1988) Harada eta]. (1990)$

Thick and thin filaments Myosin and actin filaments

Toyoshima eta]. (1990)$ Uyeda et al. (1990) Uyeda et a]. (1991):1: :l:

Myosin and thin filaments HMM and actin filaments

Temperature (~

23

MLF (nm)

--

f

M L F -I, Head 1 (s 1) ts/t c

--

8

n/N

V D (~m s 1)(nm)

--

1/5

5

130

0.62 0.70 0.76 0.66

5.5

0.61

2/2 2/3 2/6 2/2 1/N 2/2

155 261w 577w 68 51"* 2O4

--

1/N

4.6

22

40

44

30

40

220

22 15 7.3 110

22

45

44

22

30

150

600

--

--

13

0.05

~

7.4

--

--

31

0.023

--

0.17

--

11.2 8.3

7.8** 28

*MLF = minimum length filament; f = proportion of heads attached; N = number of heads available to interact with filament; n = average number attached; t s - working stroke time; t c= cycle time; V = measured velocity; D - step size. SNearest neighbour distances: 16 nm, myosin (Harada et al.); 32 nm, HMM (Toyoshima et aI.). w = 3 gives internal consistency where ts/t c ~ n/N; N - 6 calculated from 44 s i. MLF i and 7.3 s -I. head 1 given in Harada et al. (1990). ,jN varied; for comparison to Toyoshima et al. (1990), who used about 1000 HMM per gmz, N would be about 60 heads per gm filament at that density using the 'band' model of Uyeda et al. (1990; their Table I). **D calculated using Equations 1 and 3 assuming n - 1 . :l: ~V = presumed unitary velocity and f calculated from Vo-: 7.4 s i.

Its}

592

n/V,

BURTON

t~Vs

=

average velocity and presumably includes periods when the filament is being driven as well as when it is only undergoing Brownian motion. In this case, tc is used directly and:

] [, -- D

t~-td

tcV1

J

D = tcV~

Fig. 1. Equations used to calculate step size. D = step size. t s= time in stroking states, V~= stroking velocity, tc= ATPase cycle time, VI = unitary velocity, f = fraction of bridges attached in stroking states, n=number of heads attached to minimum-length filament, Uf= ATP per filament per s. t d = time in detached states.

assays, this is the case when filament length and density of heads on the surface are above minimum values (Harada et al., 1990; Uyeda et al., 1990). An estimate of is requires knowledge of the ATPase rate during sliding and the number or fraction of cycling heads attached at any instant. Using an estimate of the actin-activated ATPase activity per head: ts --ftc

(2)

where f is the fraction of the ATPase cycle spent in the stroking states and tc is the cycle time (1/ATPase). Equations I and 2 were used to calculate D in the studies of Harada and Yanagida (1988) and Uyeda and colleagues (1990) (Table 1). An equivalent calculation of ts can be made if a value for the ATPase per filament is available with an estimate of the number of attached heads (Yanagida et aI., 1985; Harada et al., 1990; Toyoshima et al., 1990):

ts=n/U;

(3)

where n is the average number of heads attached at any instant and Uf is ATP.s-~.filament 2. Note that Uf=N/(tc) where N = total number of cycling heads and taking f = n/N, Equation 3 is seen to be equivalent to Equation 2. Although the data of Harada and colleagues (1990) allow an estimate of ts by Equation 3, they approached this problem by expressing ts as the difference of tc and ta, where ta is the time in states which do not produce force or movement. The fd was taken to be the sum of the time required for the step during which ATP is split plus the time to enter the stroking states. The rate of the latter step ( k o N ) was estimated based on the minimum length filament which remains attached to the surface, its velocity, and the average number of attached heads required to prevent diffusion of the filament from the surface. Both tc and koN were initially calculated on a per filament basis. For calculation of D, these were converted to a per head basis using assumed values for the number of heads available to interact with the filament (N, see Table 1). A more direct approach to estimating D is to measure the velocity of sliding from the action of a single head ('unitary velocity'; Uyeda et al., 199I). This is a time

(4)

where V~ is the unitary velocity. In the experiments of Uyeda and colleagues (1991), the minimum motor unit was the two-headed subfragment of myosin, heavy meromyosin (HMM), and hence the measured unitary velocity was taken to be twice that of a single head, assuming the two heads act independently. The following paragraphs discuss the considerable degree of uncertainty in these estimates of step size. It is frequently the case that small differences in individual quantities compound one another to produce large discrepancies in D. As mentioned at the outset, the results from the laboratories of Yanagida and Spudich are at variance. Any attempt to reconcile these results is beset with several difficulties. First, several approaches have been used to calculate D and experiments have generally been done under different conditions, although the data of Harada and colleagues (1990) acquired at 30~ has been helpful in allowing comparison with Toyoshima and colleagues (1990) under similar, but not identical, conditions (H. E. Huxley, 1990). If the data of Harada and colleagues are used to calculate D by Equations 1 and 3 and n = 1, then D = 5I nm (cf. discussion of next paragraph). The same calculation by Toyoshima and colleagues (1990) gives D = 7.8 nm. All of the quantities used to calculate D contribute to this discrepancy. The velocity obtained by Toyoshima and colleagues was 4.6 ~tm s -~, about 40% that of Harada and colleagues, although this value was revised upward to about 7 pm s at the same HMM density in a subsequent work (Uyeda et al., 1990, their Fig. 3, Tables 1 and 2). A comparison of the ATPase activities (per unit length filament) can be made if corrected for differences in head density (see Table 1). The nearest neighbour distance of myosin in the study of Harada and colleagues (1990) was half that of Toyoshima and colleagues (1990) and hence the ratio of the ATPase activities of Toyoshima and colleagues (4 ATP nm -~ s ~) to those of Harada and colleagues (5.5 ATP nm -~ s ~) is corrected to 2(4.0/5.5) = 1.45. An estimate of t~ (Equation 3) can be made by assuming that the minimum length filament (MLF) observed to move is powered by n = 1 head. Although the MLF observed by Harada and colleagues was smaller than that of Toyoshima and colleagues, the head density was twice as high and therefore the corrected MLF ratio would be (1/2)(150nm/ 40 nm)= 1.88 (Table 1, experiments at 30~ The corrected ATPase activity per filament (Uf) of Toyoshima and colleagues was thus (1.45)(i.88)=2.7 times that of Harada and colleagues. The total divergence in D was therefore over sixfold (2.7/0.4; Equations 1 and 3 with n = 1 assumed).

Myosin step size Possible contributions to this discrepancy have been discussed in a recent review (H. E. Huxley, 1990). One of the most significant of these is the possibility that the leading end of a sliding filament first interacts with myosin heads which are 'cocked' in the myosin- ADP- P~ state and which rapidly attach and release Pi into solution. Such behaviour would have at least two effects. First, the apparent rate of attachment into force-generating states would probably be much higher than in steady cycling during which the time for attachment also includes an earlier rate-limiting step. This would result in fewer heads being required to hold a filament to the surface and hence reduce the length of the MLF for a given density of heads. In the presence of a high density of heads, such as used in the Yanagida studies, the end of the filament could always be in the vicinity of several heads and this 'end effect' would be especially significant. By Equation 3, Uf = (ATPase/unit filament length)(length of MLF), and therefore a reduction in MLF increases D calculated by Equations 1 and 3. A control experiment would be to measure the dependence of MLF length on density of myosin heads on the surface. If the MLF length were to fall more than in proportion to a decrease in the nearest-neighbour distance of heads, this might account for the discrepancy in MLF length discussed in the previous paragraph. Second, as pointed out by H. E. Huxley (1990), the ATPase activity per unit length of very short filaments may be higher than that of long filaments. This could result if cocked heads encountered by filament ends rapidly attach and release Pi into solution, thus executing a 'partial cycle' (since it is Pi which is usually measured in the ATPase assay). Short filaments could pass by a given head before additional cycles are completed and hence only the rapid partial cycles would produce significant phosphate release. An error could be introduced if measurements of ATPase activity averaged over all moving filaments, where the longer filaments dominate, are used with the MLF in calculating D via Equation 3. As with MLF length discussed above, this error would be exacerbated under conditions of high head density (Harada et al., 1990). There is some precedent for this effect in studies of acto-S1 and acto-HMM ATPase where it was observed that the ATPase per acto-myosin complex in the presence of a large excess of $1 or HMM was about five times that in the presence of a large excess of actin (Eisenberg & Kielley, 1973). A useful control would therefore be to measure the ATPase of a homogeneous population of short filaments. If the value was higher than for long filaments, D calculated using Equation 3 would decrease proportionately. It should be noted, however, that Harada and colleagues (1990) did not use Equation 3 in calculating D. Rather, by assuming that myosin on the surface is predominately cocked in the myosin. ADP 'Pi state (Yanagida et al., 1991), they attempted to estimate the rate of attachment to actin (koN) from the velocity and length of their MLF, as discussed

593 earlier. The time for this step was then summed with the time for earlier non-stroking steps (taken to be the reciprocal of the sum of the forward and reverse rate constants for ATP cleavage measured in solution). The time in stroking states was then calculated by subtracting the total time in non-stroking states from the complete cycle time, which was that measured with long actin filaments. There are of course several uncertainties in this approach, including the assumption that nearly all the heads are in the myosin. ADP 9Pi state. It seems though that their approach would not be compromised by the rapid attachment of cocked heads or an elevated ATPase associated with short filaments. Harada and colleagues (1990) suggested that the lower velocities observed by Spudich and colleagues might be the result of an internal load applied by heat-labile myosin heads. It might seem peculiar that the ATPase should be higher for filaments moving more slowly in the presence of fewer properly functioning heads. However, ATPase has been shown to rise as velocity decreases to about half the maximum in muscle (Kushmerick & Davies, 1969; Homsher et al., 1984) as well as in a motility assay system (Harada et al., 1990, their Fig. 13). Although the increase in ATPase over this range is small in the motility assay result (about 20%) it is larger in shortening muscle, about 50-300%, and this would be consistent with the dis], crepancy in ATPase and velocity. A significant technical advance was achieved by Uyeda and colleagues (1990) with the use of a viscous polymer to reduce lateral diffusion so that filaments remain associated with the surface even with low numbers of attached heads. They monitored very slowly moving filaments and provided evidence for the velocity resulting from a single HMM (Uyeda et al., 1991). These data have resulted in small values of D (Table 1). Although the velocity of long filaments was higher than in the work of Toyoshima and colleagues (1990) ( ~ 6.8 versus 4.6 I.tm s -1 at the same HMM density), there remains a discrepancy between their results and those of Harada and colleagues (1990), especially for short filaments at high head density. In the latter study, velocity measured for short filaments (40 nm) was about 90% of that for long filaments (0.9(11.2 I.tm s-l) = 10 ~m s-l). A comparison can be made to the data of Uyeda and colleagues (1990; their Fig. 3, velocity versus filament length) by correcting the value of MLF = 40 nm for differences in HMM and myosin density (4000 myosin per ~m 2 versus 2400 HMM per I.tm2; MLF' = (x/4000/2400 (40 nm) -- 67 nm). The velocity of a 67 nm filament is predicted to be about 1 lain s -1 from the data of Uyeda and colleagues (1990), which is about 10% of the short filament velocity of Harada and colleagues (1990) at 30~ On the other hand, Harada and colleagues estimated the length of the shortest moving filaments indirectly from the intensity of fluorescence of the labelled actin. It would be useful for such estimates to be verified by direct comparison with images of these filaments in the electron microscope.

594 Some of the uncertainty in quantities used to calculate step size can also be illustrated by comparing results obtained by the same workers in separate reports. For example, at an HMM density of 1000 ~ m -2, the actinactivated M g . ATPase obtained by Toyoshima and colleagues (1990) in the motility assay was 4 s -~. nm -~ of actin filament. The number of heads available to interact with an actin filament using the 'band model' of Uyeda and colleagues (1990; their Table 2) was about 60 heads per I,tm i at the same H M M density. These results give the ATPase per head as 4(1000/60) = 66 s 1. head-~. This is five-fold higher than the value of I3 s -~-head -~ estimated by Uyeda and colleagues (1990) and twice the value obtained in solution. The calculation of a small step size is not invalidated as the high value of ATPase would decrease the estimate of D proportionately. In a subsequent paper, Spudich and colleagues (Uyeda et al., 199i; Table 2) discuss several reasons why their estimate of 5.3 nm for the step size might be low. For example, it was shown in their earlier work (Uyeda et al., 1990) that the velocity obtainable with long filaments falls by a factor of four (7.4 to ~ 1.8 I.tm-s ~) as the HMM density is reduced from 2400 to 40 HMM per btm2. Assuming the surface density of H M M was the same in the two studies for a given loading concentration (40 HMM per ~ m 2 at 1-2 ~g ml -~ (Uyeda et al., 1990) versus 1.5 t,tg ml 1 (Uyeda et al., I991)), the unitary velocity could be corrected to 4(0.33 btm s -*) = 1.3 ~m s -~ for H M M or 0.65 t,tm s 1 for a single myosin head. Furthermore, the ATPase per head could be that estimated from measurements in the motility assay of their earlier work (13 s ~; Uyeda et al., 1990) rather than that from acto-S1 in solution (31 s 2). These two values give D = 650/13 = 50 nm (Equation 4). If the two heads of H M M do not operate independently, the velocity and hence D would have to be revised upward accordingly. Although Yanagida and colleagues regularly obtain estimates of D in excess of 100 nm, the estimates o f f on which these are based vary widely: 1.3% for crab myofibrils, 20% for thick plus thin filaments, and about 60% for myosin plus (actin or thin) filaments (Tables I and 2). Based on measurements of force fluctuations during rapid filament sliding, Ishijima and colleagues (1991) estimate f to be about 90%. Yanagida's values of f given in Tables 1 and 2 are minimum estimates and using 0.9 would obviously increase the estimates of D substantially. It is possible, as discussed above and in publications from the laboratories of Spudich and Yanagida, to have estimates of the step size from the two groups overlap. However, fundamental differences in the basic measurements of velocity and ATPase activity remain. This situation will probably only be resolved if the effects of differences in conditions are assessed. For example, the two laboratories use different protein preparations (HMM stored in 50% glycerol solution versus myosin prepared freshly and used within a few days) and materials for attachment to the surface (nitrocellulose versus silicone).

BURTON Another significant difference is the density of heads on the surface; the relatively high value in the studies of Yanagida and colleagues may introduce difficulties as discussed above. The reliability would be greatly enhanced if a method were developed for quantitating the number of heads which can interact with the actin filaments as well as the average number attached. This has been approached in the experiments of Spudich and colleagues (Uyeda ei al., 1991). There are several general problems about motility assays which need to be addressed. (1) What are the lability and integrity of myosin and its proteolytic fragments attached to the substrate? (2) The motors are randomly oriented on the surface with respect to a filament and this may affect the velocity and ATPase. For example, Sellers and Kachar (i990) have shown that the velocity of an actin filament depends on its orientation with respect to myosin in thick filaments. This concern would presumably apply to the results from both the Yanagida and Spudich laboratories. Any heterogeneity in the density of heads on the surface would also contribute to variability in results. (3) How are the molecules oriented with respect to the substrate to which they attach and which portion is involved in attachment? (4) Is the composition of the substrate critical? At present, there is no concensus on the value of the myosin step size based on the results of motility assays. It is therefore appropriate to look for supportive evidence from other motile systems. The next section describes data acquired from shortening striated muscle with emphasis on the relation of the amount of filament sliding to ATPase as well as other properties. This will include data from studies which pre-date the current interest in the coupling of ATP hydrolysis to movement, but also several which have partly been motivated by it.

Shortening skeletal muscle Investigations of biochemical, energetic, structural, and mechanical properties of shortening muscle have addressed questions not only about the size of a myosin step, but also its mechanism. (1) What is the step size calculated from rapidly shortening muscle7 (2) Is there mechanical or structural evidence that > 10-20 nm of filament sliding is required for crossbridges to detach from force or motionproducing states7 (3) On the question of the mechanism of stepping during steady filament sliding, Yanagida and colleagues have suggested that for each ATP split, myosin heads undergo many cycles of detachment and reattachment to motion-producing states

Myosin step size (Harada eta]., 1990; Ishijima et al., 1991; Yanagida et al., 1991). What evidence exists for such rapid cycles of productive interaction? A discussion of step size and properties relevant to its mechanism requires certain concepts to be defined. As stated at the outset, the term 'step size' is defined operationally as the amount of filament sliding that one myosin head produces during one ATPase cycle. An important question is what type of mechanical interaction between myosin and actin is equivalent to the calculated step size; three possibilities will be discussed (Fig. 2). (1) The change in acto-myosin which produces positive force (i.e. causing a sarcomere to shorten) while a head is attached to a given actin site is referred to as a 'working stroke'. (2) The interaction of a head with one site might include a period during which it drags and resists sliding. The sum of movements during the pulling and dragging periods is termed an 'elementary step' here. (3) One or more elementary steps could occur before a head enters non-force-producing states and hydrolyzes an ATP; the sum of these steps will be referred to as an 'interaction distance'. This section first considers step size calculations from muscle and myofibrils and this is followed by discussion of experimental evidence relevant to mechanism. Some theoretical proposals which bear on possible mechanisms will then be reviewed. Distinction between models used to describe movement in motility assays and muscle physiology Calculations of the step size using data from motility assays (Table 1) and shortening muscle (Table 2) are based on a simplified model (Equations 1-3). A myosin head, whether working alone or in parallel with others, attaches to an actin site and moves the filament at velocity V~ for a distance D, and then detaches. Although the description serves the purpose, it differs from that used as a basis for most current models describing muscle function (A. F. Huxley, 1957). In this model, attachment is made with a rate constant f and detachment at a rate g. Attachment to a thin filament site is via an elastic domain and is only allowed if there is positive strain (force in the shortening direction). For a single myosin head driving a filament, the velocity is given as:

595 Working Stroke

~ / / / / / / / / / / / / / / / / / / / A

Elementary Step

~ / / I / l l l f l I / / l I / / I / / A

Interaction Distance

Fig. 2. Possible modes of crossbridge action. Schematic representation of three possible mechanisms by which actin filaments are moved by a myosin head with the hydrolysis of one ATP. Any of the three could thus fulfil the definition of 'step size'; see explanation given in the text. An elastic link is shown to connect the head to the substrate, although the location of this link is not known and the relevance to myosin fragments is uncertain. A curved arrow at the head indicates motion generated by the head itself, a straight arrow refers to motion generated by other attached heads before detachment of the given head. The top and middle panels show a filament at the beginning and end of a given type of motion while the head is attached to a single actin site. The bottom panel shows two sequential movements resulting from the head binding to two sites, the sum of which is the interaction distance; the positions of the filament are offset for clarity.

V~ = D / ( 1 / f + l/g) = fgD/(f + g) - f Vs

where f = f / ( f + g ) is the portion of time attached; Vs = gD is the time average stroking velocity for a single head. The actual time to move a distance D is that required to put force on the elastic element, which is on the order of 0.5 ms from the data of Ford and colleagues (1977; O~ from muscle, zero external load), whereas 1/g is several milliseconds (Woledge et aI., 1985; Pate & Cook, 1991). This is to say that under no load the filament is very rapidly moved by a distance D, after which the head remains attached for a time , ~ l / g . This time

average-movement for a single bridge is identical to that for the motility model: V~= f Vs

(5)

The difference between the two models arises when the number of interacting heads is large so that their action overlaps in time. The A. F. Huxley model has the bridges detaching at a rate g, so that under no load a proportion of the bridges are dragged into regions of negative strain, where they resist shortening, before detaching. At zero

596

BURTON Table 2. D estimated from shortening muscle and myofibrils* Source

Preparation

A TPase

f,J

Vo

D

Homsher et al. (1981)

Frog sartorius, ooc

1.84

(0.28)

2.67

203

Brenner (1986, 1988a, b) Higuchi and Goldman (1991)w Yanagida eta]. (1985) Harada eta]. (1990) Ohno and Kodama (1991):1: %

Skinned rabbit psoas fibre, 5~ Skinned rabbit psoas fibre, 20~ Crab myofibrils 5~ Rabbit psoas myofibrils, 22~ Rabbit psoas myofibrils, 20~

8.3% 1

(0.28) 0.12

2.67 1.6

45 192

4.5

0.6

300

40

1"*

> 1/78

4.7

>60

9

(0.2)

7.8

173

0

--

--

--

8.4:I:

(0.2)

20

480

*ATPase as s i. head i, Vo as pm s-1 per hs, unless otherwise noted, f is percentage attached, D in nm. #ATPase during shortening calculated with post-shortening burst of hydrolysis included. w as ATP. head i, Vo as nm. ,]Values in parentheses are estimates used here based on the ratio of isotonic to isometric stiffness in frog and rabbit muscle and assuming that 80% of heads are attached during isometric contraction; this assumption was also applied to the stiffness measurements of Brenner (1988a). **ATPase was given as ATP. s i. (myosin molecule) 1 and therefore D is the sliding imparted by a pair of heads using a total of one ATP. To be consistent with later estimates, the ATPase should be 0.5 s 1. head and D would then be > 120 nm. However, f ~ 1/(78 molecules) was a minimum estimate and this could just as easily have been 1/(156 heads) giving D > 60 nm. :l:%Data taken from shortening initiated by addition of ATP, ~,= 25 raM, s l o = 2.6 ~m.

load there is a balance between bridges under negative and positive strain. The filament velocity is then (Simmons & Jewell, 1974; Woledge et al., 1985; Pate & Cooke, 1991): Vo ~ (1/2)gD = (1/2)Vs

(6)

(note that Vo is also the stroking velocity of each of a large number of heads interacting through the filament). Although an effect of dragging heads might be expected to exist in motility assays, the simple models used to describe these do not have simultaneously attached heads interacting with each other through the actin filament, but simply working in parallel and hence: Vo = Vs.

(7)

If the model used to fit the motility assay data (Uyeda et al., 1990) were correct, it would require quite a different mechanism of myosin action to that described by A. F. Huxley (1957). If Vo is to equal Vs, then a given bridge cannot resist the action of others and this could be accomplished by g being v e r y high at zero strain. However, this would give rise to a velocity of the order of 14 nm per 0.5 ms ~ 30 pm s 1, as opposed to about 2 lain s 1 in flog at 0~ Pate and Cooke (1991) point out an alternative which is to have rate limitation within the working stroke, which would then have to be some process other than the rapid force recovery described by A. F. Huxley and Simmons (1971). The 'thermal ratchet' model of Vale and Oosawa (1990) may offer an alternative, although it is not clear how bridges interact with

each other through the actin filament in their description. Tawada and Sekimoto (1991) have proposed a model for filament movement in vitro which has a proportion of heads exerting a frictional drag on stroking heads to limit sliding velocity. Filaments sliding at low head density in vitro may buckle, an effect described by a 'transmission efficiency' in Uyeda and colleagues (1990), and this may ameliorate the cooperativity between attached heads. Although the two models predict that Vo is different for a given ~, the data of velocity versus number of heads of Uyeda and colleagues (1990) can be fit by either model. In either case the velocity slows with a reduction in the number of available heads as a result of 'dead time' when there are no heads attached to the filament. Also, in both models, the velocity is proportional to the number of available heads as long as none attach simultaneously. In practice, the estimate of D is probably the same for both models. If Equation 6 is correct, then the estimate of V~ (V~) obtained from Vo is in error by a factor of 2: V ~ - Vo = Vs/2. However, the estimate of f (f') is in error by the same factor: f ' -- V~/V~ = 2 V J V s = 2f by Equations 5 and 6. Therefore, by Equations 1 and 2, D = t c f ' V ~ = t c f V s . However, the concept of attached heads under positive and negative strain may be of importance as discussed under Theoretical Considerations. Calculation of step size A consideration of ATPase rates during shortening at a given velocity indicates that the proportion of heads

Myosin step size which hydrolyse an ATP for each 10-20 nm of sliding is surprisingly small compared with estimates of the fraction attached obtained from stiffness measurements, so that the amount of sliding during the attached part of the cycle is 4 0 - 2 0 0 n m (Bagshaw, 1982). Homsher and colleagues (1981) calculated that only 2% of the myosin heads could be executing 15 nm working strokes at the rate of ATP splitting during shortening at maximum velocity (Vo). Examples of this sort of calculation, using Equations 1 and 2, are shown in Table 2. It is clear that the data obtained from intact frog muscle and skinned rabbit fibres give a large value for the step size. This is also true for myofibrils if f is that used for muscle. There are two values for ATP splitting given in Table 2 from the work of Homsher and colleagues (1981). The first is that measured during shortening and the second includes a burst of hydrolysis which occurs afterwards, but is nevertheless a direct result of the shortening. Some caution is required in using the latter value as it has not been shown that the sum of ATP splitting from the two periods is proportional to the distance shortened. However, such a burst would probably be included in measurements from motility assays as myosin heads would go through the hydrolysis step in the cycle after an actin filament has passed by. The value of 40 nm from Higuchi and Goldman (1991) in Table 2 is a minimum estimate because rigor bridges formed during shortening from a progressive reduction in ATP concentration, thus reducing the velocity. Their approach did however allow a very precise measure of the amount of ATP used, in contrast to the measurements o n whole muscle (Homsher et al., 1981; Table 2), where the standard error was 50% of the mean. Although whole muscle provides a relatively large sample for the measurement of myosin ATPase activity, the time for shortening is limited by the sarcomere length (allowing 0.3 s in the work of Homsher and colleagues (1981)). The motility assay system offers an advantage by allowing steady sliding for extended periods (10-30 min; Harada eta]., 1990; Uyeda et al., 1990). An approach to this problem in shortening muscle may be the method developed by Brenner (1983) where single skinned muscle fibres are repetitively cycled through periods of shortening followed by rapid ( < 1 ms) extension to the original length (Brenner, 1988a). An active rabbit fibre can be continuously cycled in this way for over an hour at 5~ How reliable are the measurements used to calculate step size in shortening muscle and myofibrils? It might be suggested that the velocity of a thin filament in the sarcomere is influenced by cooperativity with others through the Z-disk. This does not seem to be significant as the sliding velocity is similar to that in a motility assay (Harada et al., 1990) and detachment of thin filaments from the Z-disk does not alter the estimate of a large step size ( > 60 nm; Yanagida et aI. (1985)). The estimates of f used to calculate D in shortening muscle and myofibrils have been obtained from stiffness measurements. There

597 are several assumptions in doing this. The first is that stiffness during shortening, relative to some reference condition (such as rigor), is proportional to the numbers of heads attached. An error is introduced if an elastic element is common to the two heads of a crossbridge (e.g. within the $2 link) so that stiffness is the same when either one or both heads of a molecule are bound. There is some evidence for this from EPR studies of muscle in the presence of AMPPNP (Fajer et al., 1988). At the extreme, f would have to be reduced by a factor of two. A second uncertainty is whether the stiffness of attached states during shortening is the same as under conditions where the number is known (e.g. rigor). One way the measured stiffness per head could change is if the detachment rate varied by state and was rapid on the timescale of the length change used to measure stiffness (Schoenberg, 1985). On the other hand, Ford and colleagues (1985) have shown that changes in stiffness on going from isometric contraction to steady shortening can be described as a reduction in the number of crossbridges--at constant stiffness for each--with a reduction in their average strain. A third difficulty arises because f should, by Equations 1 and 2, refer to heads which are producing force in the shortening direction and contributing to sliding. The stiffness measurements sample all attached heads which can resist the imposed movement, whether or not they contribute to shortening. In this sense the elementary step may be longer than the working stroke. Estimates of ts (Equation 1) have also been made on the basis of biochemical kinetic analyses. Shortening myofibrils were studied by Taylor (1989). At 20~ ts was no more than 0.01 s, giving D = 0.01 Vo. Thus D = 20 nm for Vo = 2 I,tm s -1, or 65 nm for Vo = 6.5 ~m s ~ (this latter Vo is from 7.8 btm s -I at 22~ and a Q~0 of 2.4; Harada eta]. (1990)). White and colleagues (1992) used the second order rate constants obtained in solution for the dissociation of actomyosin-S1 by several nucleotides and the shortening velocity in skinned fbres to calculate a value of 4 nm for the average distance over which a myosin head can remain attached to an actin site during unloaded shortening. Measures of an interaction distance There are properties other than ATP hydrolysis which might correspond to an interaction distance in shortening muscle. An approach to this question is to monitor the timecourse (and length change) during shortening required for crossbridges to redistribute from the attached states characteristic of an isometric fibre to detached states during steady shortening. The most relevant properties would be those associated with the working stroke, but those indicating attachment will be discussed below. This type of experiment is essentially one of monitoring relaxation kinetics after a perturbation of steady-state conditions. It is somewhat different in that length constants are of interest as well as time constants. If the

598

working stroke size of 10-20 nm is a paradigm, then the properties indicating detachment should be complete within this amount of filament sliding. A structural measure of an association between crossbridges and thin filaments, if not attachment per se, is the intensity of the 1,0 and 1,1 equatorial reflections of the X-ray diffraction pattern. A decrease in I~,0and an increase in Ii,~ are consistent with a movement of mass from the thick filament towards the thin filament (Haselgrove & Huxley, 1973). It has recently become possible with synchrotron X-ray sources to monitor the timecourse of changes in these and other reflections when an isometric muscle is allowed to shorten (H. E. Huxley et al., 1988, I989). At high velocities of shortening, these are known to partially return towards their values in relaxed muscle (H. E. Huxley et al., 1988; Matsubara & Yagi, 1985), indicating a net detachment. The surprising finding was that the fall in the I~,~ did not occur rapidly but required about 35 ms or roughly 50-60 nm of filament sliding to complete. This slow timecourse was not limited to the equatorial reflections but was also observed in the change in spacing of the 14.5 nm meridional reflection, which arises from the axial repeat of crossbridges along the thick filament. Furthermore, Lowy and Poulsen (I990) have shown that the diffuse scatter in the X-ray pattern, which they attribute to disordered crossbridges, changes during shortening with a similar timecourse. It is known that stiffness falls considerably in rapidly shortening muscle (Julian & Sollins, 1975; Tsuchiya et al., 1982; Ford et al., 1986). As stiffness is a measure of crossbridge attachment, the timecourse of this change is of interest. Using single skinned rabbit fibres, stiffness was measured by imposing length steps during isovelocity shortening (Burton & Simmons, 1991). If the imposed steps were small ( < 10 nm per hs) such that crossbridges probably remain attached during the length change, then the stiffness fell most of the way (90%) towards its steady value within 20 nm of filament sliding, which is within conventional crossbridge theory. However, if the imposed length steps were large (> 20 nm per hs), or if a rapid stretch was applied equal in magnitude to the previous shortening (Brenner, 1983), then the force response to the stretch required about 50-60 nm of sliding to approach completion. This was true for loads of 5-30% of the isometric force so that the time required (75 ms at low load) varied in inverse proportion to the velocity of shortening. An extended timecourse for the fall in stiffness was also reported by Tsuchiya and colleagues (1982), who found that stiffness measured by small (0.6%) steps continued to fall even after shortening 6% of the initial length. Another mechanical property which requires time to develop is the 'pullout' phenomenon where a load is suddenly applied during shortening (A. F. Huxley, 1971). The magnitude and velocity of the resultant lengthening continue to increase for several tens of milliseconds at intermediate loads (40-75% Po) (Sugi & Tsuchiya, 1981).

BURTON

An important property of contracting muscle is the energy released in the form of heat and work, of which a large portion is derived from actomyosin ATP hydrolysis, and the rate of which increases when filaments are allowed to slide (Fenn, 1923; Hill, 1938). Measurements of heat and energy produced as a function of the amount of filament sliding in whole muscle have shown that there is an initial 'burst' of heat production during which the rate is high, but this slows to reach a steady state with further shortening (Irving & Woledge, 1981). The amount of sarcomere shortening required is about 0.1-0.I2 gm, or about 50-60 nm of filament sliding. It is striking that this appears to be a 'length constant', as it seems to hold over a range of velocities from 0.25 Vo to Vo (Homsher & Yamada, 1988). Therefore, the time required to reach a steady state varies by a factor of four in those experiments. The extra heat liberated during shortening has been shown to be proportional to the amount of overlap of thick and thin filaments and it can therefore reasonably be attributed to steps in the crossbridge cycle. It could have been suggested that these steps occur in the detached part of the cycle and that the extended timecourse bears no relation to the amount of filament sliding. However, the direct dependence of the rate of the heat burst on shortening velocity and the similarity of its length-dependence to the structural and mechanical properties discussed above suggest that it is determined by some step(s) in the attached part of the cycle (Kodama & Yamada, 1979). These studies seem consistent with the idea that crossbridges can retain several properties of attached bridges during considerable filament sliding. If this is true, then what sort of attachment might such myosin heads maintain? The simplest model would be one in which each head remains attached to the same actin site for > 50 nm of filament sliding. This would be an elementary step (including periods under positive and negative strain) and might correspond, for example, to a 4-60 ~ 'tilt' of the $1 + 'short' $2 portion (length ~ 6 0 nm (Walzth6ny et al., 1986); thin-thick filament spacing ~ 2 6 n m (centre-centre)). Views of myosin heads 'peeled' away from isolated thick filaments have been obtained by electron microscopy of shadowed preparations (Trinick & Elliot, 1982; bridges project out 40-50 nm on hydrophilic carbon substrates) and it would be interesting if such distorted crossbridges were present in EM views of shortening muscle using recently developed rapid freezing technology (Tsukita & Yano, 1985; Padr6n et al., 1988; Bennett & Elliott, 1989). An obvious alternative to this would be for a myosin head to interact with several binding sites before making the transition responsible for a change in a given property. This type of process has been suggested as a possible explanation for the slow change in the equatorial reflections (H. E. Huxley el al., 1988) as well as from considerations arising from the burst of heat production and low ATPase rate during shortening (Homsher, 1987).

Myosin step size If crossbridges were forced to detach after some elementary step size (e.g. 10-20 nm) and this detachment is not apparent from measurements of several properties of the system, then it is reasonable to suppose that such bridges reattach rapidly. For example, if detachment occurs once every 10 nm and filaments are sliding at 1 lam s 1 (rabbit muscle at 5~ then the detachment rate is 100 s x. If these bridges were to remain attached > 90% of the time, then the reattachment rate would be at least on the order of a 1000 s 1. This is two orders of magnitude higher than attachment rates inferred from the recovery of force after a period of shortening (Edman, 1980; Brenner & Eisenberg, 1986, ~ 3 s -1 at 5~ in skinned rabbit fibres). The next section considers mechanical and structural evidence bearing on this question.

Rapid detachment and reattachment of crossbridges during contraction The amount of lengthening necessary to force crossbridge detachment has been taken to be 10-20 nm, based on results from a large number of studies (Flitney & Hirst, 1978; Griffiths et al., 1980; Edman et al., 1981; Sugi & Tsuchiya, 1988; Lombardi & Piazzesi, 1990). The evidence is that force rises nearly linearly for 10-20 nm of lengthening after which there is a 'give' followed by little additional increase. This has been taken to indicate a detachment or 'slipping' of bridges from their sites of attachment. It could be argued that this non-linearity is within each crossbridge. However, in this case a release imposed immediately after the 'give' during lengthening should result in a reversal of the non-linear force response. It has been shown that this is not the case (Griffiths et al., 1980; Burton, unpublished data) and the force response is linear early in the release, over the same force range as the 'give' during lengthening. The rate of reattachment of bridges forcibly detached in this way has been assessed by measuring stiffness during lengthening after the 'give' in force. Stiffness (taken as a measure of the number of attached crossbridges) does not fall during rapid (Griffiths et al., 1980; Burton, 1992) or slow (Colomo et al., 1989; Lombardi & Piazzesi, 1990) lengthening beyond about 20 nm per hs. In the study of Colomo and colleagues (1989), rapid reattachment after forced detachment was also implied from the high rate of recovery to a steady tension after shortening steps applied during lengthening. In the case of rapid stretches of 100 nm per hs applied during steady shortening, the 'give' is reached in less than 0.1ms; a shortening step applied 0.6ms later shows that the stiffness is not lower than that before the lengthening. Assuming first order kinetics for reattachment, the rate has to be at least 5000 s -*. Rates of several thousand per second were estimated by Griffiths and colleagues (1980) and Lombardi and Piazzesi (1990). An approach to measurement of the attachment and detachment rates without forcing detachment via large length changes has been described by Schoenberg (1985). This method is to

599 measure stiffness as a function of velocity of the imposed length change. The measured stiffness is reduced if the step is not fast enough to be complete before some crossbridges detach. Brenner (1991) has found that for small displacements between the thick and thin filaments, detachment of force-generating crossbridges occurs at rates of 50-1000 s -~ while reversal of detachment is at least an order of magnitude faster. Structural evidence from X-ray diffraction studies in support of rapid detachment and reattachment to new actin sites has been described by H. E. Huxley and colleagues (1983). They suggested that this is one way of accounting for the partial recovery of the 14.3 nm meridional reflection following a decrease associated with step shortening, at a time when force had recovered by only about 20%. This response was coupled with a transient appearance of the 21.5 nm reflection present in resting muscle. If a step stretch was applied within a few milliseconds of the shortening, the intensity fall was reversed, suggesting that crossbridges were still bound to the original site. However, if the stretch was applied after the rapid partial recovery of the 14.3 nm reflection, then no reversal was observed, consistent with crossbridges having attached to new sites. They also point out that such behaviour would be consistent with the small reduction in the number of attached crossbridges, obtained from the intensity of the equatorial reflections, which occurs during shortening at one-half isometric tension. It is not obvious which of the myosin nucleotide states thought to account for active tension and stiffness (myosin. ADP or myosin alone) can attach to actin at such high rates. The scheme proposed by Geeves and colleagues (1984) has initial attachment to an 'A' state (Geeves, 1991) occurring rapidly (104-10~ M is ~), but a second step is required to reach the 'R' state suggested to be force-generating. The transition to the 'R' state is reported to be slow (4 s 1) for myosin-ADP and fast (2000 s ~) for myosin without nucleotide. Myosin alone might therefore be a candidate. In the recent study of Taylor (1991), however, the rate of transition to the strong binding states is 150-200 s -1 for both myosin.ADP and myosin. Comparison of reattachment rates measured in muscle to on/off rates measured in solution is difficult for at least two reasons. First, it is necessary to bear in mind that the solution rates refer to heads under zero strain and that these steps may be strain-dependent. Second, the solution experiments start with equilibrium mixtures of actin and myosin, whereas myosin and actin forced to dissociate by filament motion may not return to their equilibrium states before the heads rebind (see ideas of T. L. Hill under Theoretical Considerations).

Force production by bridges which rapidly reattach The suggestion that Yanagida and colleagues have put forward to account for a large apparent step size goes

600

BURTON

beyond myosin heads sequentially attaching to several actin sites for each ATP split, but that this binding leads to an active interaction which further contributes to movement. This problem can be approached in studies of muscle mechanics. Brenner (1988a) has measured the force response to imposed length changes, as discussed earlier, and found evidence for rapid detachment and attachment during activation. The transients of force following a large restretch during isotonic shortening at different loads have been studied, including a transient minimum of force following the restretch and before a slower recovery to steady-state tension (Fig. 3b). He has reported that the transient minimum in force increases with isotonic load and stiffness and that the rate of force redevelopment increases tenfold as load approaches zero. Model simulations of these data suggested that

reattachment leads back to a force-generating configuration. Another approach is to probe bridges which have rapidly reattached separately from those which attach slowly to produce redevelopment of isometric force after shortening. If a large lengthening ( ~ 100 nm per hs) is applied rapidly ( ( 0.5 ms) during shortening, there is a spike of tension during the lengthening step, followed by a fall to a minimum value after I0-20 ms, after which there is a slow force recovery at a rate of 4-5 s ~ (skinned rabbit fibres at 5~ to the isometric value (Fig. 3b). The magnitude of force and stiffness within I0 ms of the lengthening step is about tenfold larger than can be accounted for by the slow force recovery, and has been attributed to crossbridges which have rapidly reattached after being forcibly detached (Brenner, 1988a; Burton, 1989). Shortening steps during this period result in a fall (b) ,,-,, 2 . 2

v

E E

-r

I

2.t

I.-

(3 Z

(a) E

2.2o5

--

2.0

hi

m

nr

ILl

1.9

,.

2.200

Z L/J

-J 2.195

'"

6O

2.190

O~

0 0 n, 2 . t 8 5