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CELLULAR |
1 Muscle Contraction Group, Department of Physiology, School of Medical Sciences, University of Bristol, Bristol BS8 1TD, UK
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Abstract |
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2 mm) with a resting sarcomere length of 2.5 µm at 20°C. Tension rose to a peak during stretch and decayed afterwards to a level which was higher than the prestretch tetanic tension. This residual force enhancement was insensitive to velocity. The tension rise during stretch showed an early transition (often appearing as an inflection) at 1 ms. Both the stretch (L1) and the tension rise at this transition increased in proportion to velocity. A second transition, marked by a reduction in slope, occurred at a stretch of 18 nm per half-sarcomere; the rise in tension at this transition increased with velocity towards a plateau. Based on analyses of the velocity dependence of the tension and modelling, we propose that the initial steep increase in tension arises from increasing strain of all attached crossbridges and that the first transition reflects the tension loss due to the original post-stroke heads executing a reverse power stroke. Modelling indicates that the reduction in slope at the second transition occurs when the last of the heads that were attached at the start of the ramp become detached. Thereafter, the crossbridge cycle is largely truncated, with prepower stroke crossbridges rapidly detaching at high strain and attaching at low strain, the tension being borne mainly by the prestroke heads. Analysis of the tension decay after the ramp and the velocity dependence of the peak tension suggest that a non-crossbridge component increasingly develops tension throughout the stretch; this decays only slowly, reaching at 500 ms after the ramp 20% of its peak value. This is supported by the finding that, in the presence of 10 µM
N-benzyl-p-toluene sulphonamide (a myosin inhibitor), while isometric tension is reduced to 15%, and the crossbridge contribution to stretch-induced tension rise is reduced to 30–40%, the peak non-crossbridge contribution and the residual force enhancement remain high. We propose that the residual force enhancement is due to changes upon activation in parallel elastic elements, specifically that titin stiffens and C-protein–actin interactions may be recruited.
(Received 25 November 2005;
accepted after revision 14 April 2006;
first published online 20 April 2006)
Corresponding author K. W. Ranatunga: Muscle Contraction Group, Department of Physiology, School of Medical Sciences, University of Bristol, Bristol BS8 1TD, UK Email: k.w.ranatunga{at}bristol.ac.uk
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Introduction |
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TopAbstractIntroductionMethodsResultsDiscussionReferences |
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At low stretch velocities the force increment during a ramp stretch occurs in two phases, an initial steep rise followed usually by a slower rise. The transition between the two phases occurs at a stretch of 8–20 nm per half-sarcomere (hs) (Edman et al. 1978, 1981; Flitney & Hirst, 1978; Getz et al. 1998; Lombardi & Piazzesi, 1990; Månsson, 1994; Piazzesi et al. 1992; Stienen et al. 1992). The stretch amplitude required to reach this transition is independent of the velocity of stretch and of sarcomere length (Edman et al. 1981). The force attained at the transition increases with ramp stretch velocity and reaches a plateau at low velocities of 0.1–1 L0 s–1 (where L0 is muscle fibre length) (Flitney & Hirst, 1978; Getz et al. 1998; Piazzesi et al. 2003). The generally accepted mechanism underlying this velocity-dependent increase in tension is that steady lengthening of an active muscle leads to a new steady-state mechanical cycle, characterized by fast detachment of crossbridges when positively strained beyond a critical limit and their rapid reattachment at low strain (Flitney & Hirst, 1978; Lombardi & Piazzesi, 1990). However, there are differences between the models that have been proposed (Getz et al. 1998 and references therein; Månsson, 1994; Piazzesi et al. 1992) and a model consistent with all the findings has yet to emerge.
Flitney & Hirst (1978) observed the presence of an earlier change in slope (termed S1) on the tension rise at a stretch of only 0.2%
L0 (2 nm hs–1) and compared it to the short-range elasticity reported in other studies (e.g. Rack & Westbury, 1974). However, since only low velocities (< 2 L0 s–1) have been used in most other studies, the occurrence of this early tension transition has received little attention.
The velocity-independent increase of force that persists after a ramp stretch (termed ‘residual force enhancement’) is more prominent at longer sarcomere lengths (Edman et al. 1978; Sugi & Tsuchiya, 1988) and increases in magnitude with the amplitude of stretch (Edman et al. 1978, 1982). It has been suggested that it is caused by the development of sarcomere length non-uniformity in the fibres (Abbott & Aubert, 1952; Julian & Morgan, 1979). Herzog & Leonard (2002) and Rassier et al. (2003b), however, have questioned whether such a mechanism alone could account for residual force enhancement observed at optimal and shorter-than-optimal sarcomere lengths.
Clearly our understanding of the molecular processes underlying the stretch-induced tension increase in active muscle remains incomplete. Firstly, there has been no systematic investigation of these components over a wide range of velocities. Secondly, the early change in slope on the tension rise seen at higher velocities, its mechanism and its relation to the later change in slope remain unclear. Thirdly, the non-crossbridge contribution to the tension rise during stretch and to the residual force enhancement and its structural basis deserve further examination. Finally, most of the above studies have been on intact frog muscle/fibres. Although there have been a few studies on skinned fibres (Stienen et al. 1992; Getz et al. 1998; Campbell & Moss, 2000) and on whole muscle (e.g. in situ cat muscle, Joyce & Rack, 1969; Joyce et al. 1969; Morgan et al. 2000; Herzog & Leonard, 2002), comparative data from intact mammalian muscle fibres are not available despite the advantage in using intact mammalian muscle of the availability of biochemical kinetic data. The aim of the present study was to investigate the molecular processes underlying the stretch-induced force increase in an attempt to address these issues. Preliminary results of this study have been reported previously in abstract form (Pinniger et al. 2003; Pinniger & Ranatunga, 2004).
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Methods |
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90% type 2) fibres (Coupland & Ranatunga, 2003). Bundles of 5–10 intact excitable fibres were dissected from the mid-belly of the muscle under dark-field illumination and aluminium foil T-clips were attached to the tendons at either end of the preparation within 0.2 mm of the fibre ends. The preparation was mounted horizontally between a force transducer (AE 801 element; AME, Horten, Norway) and a servomotor in a flow-through stainless steel chamber (volume 2 ml). The force transducer had a natural resonant frequency > 5 kHz (Mutungi & Ranatunga, 1996) and the servomotor was capable of completing a 5%
L0 length step in < 1 ms. The chamber was perfused (0.5 ml min–1) with a physiological saline solution containing (mM): NaCl, 109; KCl, 5; MgCl2, 1; CaCl2, 4; NaHCO3, 24; NaH2PO4, 1; sodium pyruvate, 10; and 200 mg l –1of bovine fetal serum; the solution was continuously bubbled with 95% O2 and 5% CO2. Unless otherwise stated, experiments were conducted at 20°C; temperature control (± 1°C) of the solution was achieved by a Peltier device fitted underneath the chamber and the temperature was monitored with a thermocouple inside the chamber. Experimental protocol
The preparation was set to an initial sarcomere length of 2.5 µm using He–Ne laser diffraction. The length–tension relationship was examined in four experiments (data not shown) which showed that the optimal sarcomere length for maximum tetanic tension was between 2.40 and 2.65 µm. This is slightly higher, but comparable to, the values of 2.14–2.50 µm observed by Elmubarak & Ranatunga (1984) and 2.26–2.43 µm by ter Keurs et al. (1984) in rat muscle fibres.
The mean (±
S.E.M.) fibre length (L0) and width of the fibre bundles was 2.30 ± 0.04 mm and 0.18 ± 0.02 mm, respectively (n
= 14). Following an equilibration period of at least 30 min, control isometric tetanic contractions were recorded at L0 and at a 5% extended muscle length (i.e. the length reached at the end of a ramp stretch of 5%
L0). Using platinum plate electrodes the fibre bundles were stimulated at 70 Hz to give a steady tetanic tension and ramp stretches of 5%
L0 were applied on the tension plateau (see Fig. 1) at velocities ranging from 0.1 to 10 L0 s–1. Tetanic stimulation was maintained at the new stretched length for at least 500 ms after the end of the ramp and the muscle remained at the stretched length for the remainder of the tetanus duration. In some preparations a marker (80 µm wide by 0.2 mm long) was placed on the surface of the muscle fibre bundle to monitor the change in muscle segment length (see Mutungi & Ranatunga, 2001). The fibre segment containing the marker was illuminated and the image of the marker was focused through a high-power microscope to obtain a sharp image. Movement of the marker was monitored using a one-dimensional position-sensitive detector (S3932, Hamamatsu Photonics, Welwyn, Garden City, Hertfordshire, UK).
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Data recording and analysis
The length signal from the motor and the tension transducer signal were collected via a CED 1401 laboratory interface using Signal 2 software (Cambridge Electronic Design Ltd, Cambridge, UK) and stored on a PC. Tension and time course measurements were performed offline using Signal 2 software and additional analysis and nonlinear curve fitting were performed using Fig.P software (Biosoft, Great Shelford, Cambridgeshire, UK).
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Results |
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4% of the tension increase obtained by the same stretch on a tetanized fibre bundle. Figure 1 also shows that the isometric tetanic contractions elicited at the initial length (L0) and at a length 5% larger (labelled L0
+ 5%) developed very similar plateau tensions; the tetanic tension at the extended length was 1.01 ± 0.01 times that at L0 (n
= 8) and the difference was not significant (P > 0.05; Student's t test). On application of a stretch on the plateau of a tetanic contraction at the initial length, the tension rose throughout the ramp to a peak and decayed afterwards to an apparent steady level at the maintained stretched length. As can be seen from Fig. 1, the tension at the stretched length remained higher than the plateau tension of the tetanic contractions recorded at both the initial length and the longer length; as an index of this force enhancement the tension increment above the prestretch tetanic tension was measured at 500 ms after the end of the ramp (and is referred to as P3). Examination of the rising phase of the tension response to a ramp stretch showed the occurrence of two transitions (changes in the slope of tension rise) referred to as P1 and P2. The characteristics of these transitions are described below.
Characteristics of the P2 transition
Figure 2A shows sample records of the tension rise in response to ramp stretches at three low velocities (0.10, 0.25 and 0.50 L0 s–1). As found in previous studies, the tension rise during lengthening is biphasic with the tension rising faster initially before a transition is reached after which tension rose more slowly. In some cases, particularly at faster stretch velocities, the change in slope was abrupt and the rise in tension (P2) and the corresponding length (L2) at the transition were determined directly from the data records. During slower stretches, however, the transition between the initial fast and later slow rise in tension was more gradual and P2 was determined from the point of intersection of two linear regressions fitted on either side of the transition (see Fig. 2A). Figure 2B is a plot of the mean (±
S.E.M.) P2 against the full range of velocities examined in seven fibre bundles. P2 increased steeply with velocity and reached an approximate plateau of 0.7 P0 at velocities above 2 L0 s–1.
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18 nm hs–1. This value is in the upper range of that obtained in experiments with sarcomere length control (12–20 nm hs–1; see Introduction). Our value (18 nm hs–1) is likely to be an over-estimate because of the effect of end-compliance; however, it is below the range previously reported from intact mammalian (cat soleus) muscle (20–35 nm hs–1, Rack & Westbury, 1974) in which tendon compliance may have been larger. In 95% of cases, the tension continued to increase at a slower rate after the P2 transition and hence the peak tension coincided with the end of the ramp. The increase in tension after the P2 transition (Pk – P2) was 0.20 ± 0.01 P0 (n = 5). In the remaining 5% of cases, the tension either remained at a steady level or decreased after P2. The cause of such differences in these few cases is not known although similar observations have been made on frog muscle/fibres (Flitney & Hirst, 1978; Lombardi & Piazzesi, 1990) and skinned psoas fibres (Getz et al. 1998).
Characteristics of the P1 transition
Figure 3A shows three tension responses to fast stretches (1, 2 and 4 L0 s–1) displayed at different time scales to highlight the features of the rising phase. These records illustrate that the P2 transition is preceded by an earlier transition (labelled P1). Since the rise in tension (P1) at this transition increased as the velocity was increased, it was easier to identify it at higher velocities which may have contributed to the limited observation and description of this transition in previous studies.
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Close inspection of the P1 transition revealed that the change of slope in the tension record did not occur instantaneously but usually consisted of a decrease in slope followed by an increase, producing an inflection. To quantify the inflection, linear regressions were fitted to the data on either side of the P1 transition (Fig. 4B). The points at which the tension record deviated from the two linear regressions were taken as the beginning and the end of the transition and were used to estimate its duration; this was 0.57 ± 0.06 ms (n = 22 from 6 fibres).
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We took the P1 inflection to be caused by a process producing a tension decrease superposed over the underlying tension increase due to the stretch of the compliant elements. The inflection point was taken to be the midpoint of this process. To obtain a value for the tension decrease associated with this process, we measured the vertical displacement between the extrapolated initial linear regression line and the point of inflection or, in the few cases in which the inflection point was not clearly defined, the mid-point of the transition period (vertical arrow in Fig. 4B). From analyses of 26 tension traces from seven fibre bundles, the mean vertical tension offset at the mid-point for velocities > 2 L0 s–1 was 5.97 ± 0.91 kN m–2: hence the total tension decrease associated with this process was 12 kN m–2. There was some indication that the values were greater at higher velocities.
Analysis of the development of residual force enhancement
Figure 5A shows superimposed tension, segment length and motor length records from one preparation at a range of velocities. The segment length recordings mirror the length changes imposed by the motor indicating uniform length changes throughout the fibre. The velocity dependence of Pk, P3 and P1 is shown in Fig. 5B. P1 was proportional to stretch velocity; from the pooled data, P1 was 3.88 ± 0.97 kN m–2 s/L0.Pk increased with velocity towards a plateau.
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To further investigate the nature of residual force enhancement, experiments were conducted with N-benzyl-p-toluene sulphonamide (BTS), a myosin inhibitor that suppresses tension generation by crossbridges (Cheung et al. 2002; Shaw et al. 2003). BTS does not affect the calcium-activation mechanisms and the force depression is accompanied by a corresponding reduction in rapid stiffness (Pinniger et al. 2005). Figure 6A shows the tension response to stretch in the presence and absence of 10 µM BTS. Despite an 70% reduction in tetanic tension in the presence of BTS, the incremental peak tension during stretch (Pk) decreased only slightly and the residual force enhancement after the stretch (P3) was not depressed at all. It is also important to note that, in the presence of BTS, the tension rises nearly linearly throughout the ramp stretch so that the P1 transition is absent (see Fig. 6B) and the P2 transition is not apparent. When measured from the slopes of the tension rise and length change at a range of velocities, this monotonous tension rise corresponded to a stiffness of 10.7 ± 0.98 P0 (control)/L0 (n
= 14 from four preparations); this is similar to the stiffness estimated for the tension rise from P2 to Pk in the absence of BTS from the same preparations (5.9 ± 1.32 P0 (control)/L0). These observations strongly support the idea that P3 is a non-crossbridge tension.
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| (1) |
We therefore considered an alternative case in which the non-crossbridge component is considered viscoelastic but with a relaxation rate much lower than the reciprocal of the longest ramp stretch duration used here. Consequently, the non-crossbridge contribution at the end of the ramp (P3*) was greater than the measured P3:
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All this suggests that a non-crossbridge tension develops during the stretch reaching P3* at the end of the ramp, but decays 500 ms after the ramp to P3. From the pooled data (see Table 1), the P3* calculated by curve fitting was 50 kN m–2 (0.37 P0) which is five times higher than P3 (9.7 kN m–2; 0.08 P0). Our analysis is consistent with the conclusion of Edman & Tsuchiya (1996) that the tension contribution of (visco)elastic elements progressively declines after the ramp.
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80 kN m–2 (0.6 P0). The critical velocity (Vc) was 2.3 L0 s–1. Flitney & Hirst (1978) determined a similar index for frog muscle (‘give’
–
Katz, 1939) in the tension–lengthening velocity relation. Their values ranged between 0.14 and 0.27 L0 s–1 for a temperature range of 0–12°C in frog muscle. Piazzesi et al. (2003) give higher values of 0.9–2.4 L0 s–1 between 5 and 11°C from single frog fibres; the differences may be due to differences in preparations, conditions and analyses. Analysis of the effect of velocity on Pk, as described above, was also made in four experiments where 10 µM BTS was included. The BTS/control ratios were 0.15 for P0, 0.33 for PL and 0.83 for P3*; thus, the non-crossbridge contribution remains high (see Fig. 7A).
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Thus, both analyses show that when the isometric force is reduced to 15% using 10 µM BTS, the crossbridge tension rise during stretch is reduced to 30–40% whereas the peak non-crossbridge tension, stiffness and the residual force enhancement remain high at 80% of the control. All this suggests that the residual force enhancement is provided by non-crossbridge elements.
The tension response to stretch in slow muscle fibres
Further insight into the non-crossbridge contribution to the tension response to stretch has been obtained from experiments on a different muscle fibre type. Figure 7B shows a preliminary experiment of the tension response to a ramp stretch (5% L0) of soleus (slow) muscle fibres at 5 L0 s–1 and the corresponding isometric tension records at the initial and extended length. As reported from skinned fibre experiments, slow fibres develop a higher P2/P0 ratio during stretch than fast fibres (Linari et al. 2004). Note that after the end of the ramp, the tension decays to the same level as the isometric tension at the extended length (P3 = 0), i.e. there is no residual force enhancement in this muscle. This absence of residual force enhancement is accompanied by a lack of tension rise after the P2 transition (see inset). The tension records shown by Stienen et al. 1992) from skinned fibres at 15°C (their Figs 4 and 5) show a similar difference between fast and slow fibres. This is consistent with both the residual tension after stretch (P3) and the tension rise after the P2 transition arising from a non-crossbridge mechanism.
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Discussion |
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18 nm hs–1 which was independent of velocity; this corresponds to the critical stretch described in previous studies. An additional earlier transition, P1, was observed (at 0.9 ms) particularly at higher velocities; the stretch (L1) at which this took place increased with velocity, but even at the highest velocity we used, it remained the earlier event. An early ‘breakpoint’ has been reported before in frog muscle (Flitney & Hirst, 1978), cat soleus muscle (Rack & Westbury, 1974) and single frog fibres (Bagni et al. 1998; Cecchi et al. 1997; Månsson, 1994), but its underlying mechanism has been unclear. The modelling we now describe indicates that the P1 and P2 transitions arise mainly from crossbridge characteristics, whereas the residual tension after stretch arises entirely from non-crossbridge elements. Crossbridge model simulations of the tension response
The mechano-kinetic model we used was based on the Lymn-Taylor kinetic scheme for actomyosin in solution. This is shown in Scheme 1 with two attached states (AM.ADP.Pi in a prestroke conformation and AM.ADP in a poststroke conformation) and two detached states (M.ATP and M.ADP.Pi). Step 1 is a reversible attachment, step 2 is the power stroke, step 3 is a sequence including a slow release of ADP and completion of the cycle by binding of ATP and detachment of the crossbridge, and step 4 is ATP hydrolysis on the detached crossbridge that is associated with its repriming. The strain dependence of the rate constants of the steps was defined by transition-state theory. The rate constants and the parameters defining their dependence on strain were refined against the shortening and lengthening limbs of the force–velocity curve.
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To understand the events occurring during lengthening, it is necessary to define the effect of strain on the rate constants of the crossbridge cycle. If the strain in a prestroke head is x, then the energy stored in the elastic element is hx2/2, where h is the stiffness of the elastic element. If the head executes a power stroke of l, the strain will increase to x
+
l and the energy stored will increase to h(x
+
l)2/2. Consequently, compared with an untethered head in solution, the additional energy required to execute a power stroke is hl(2x
+
l)/2. Hence the equilibrium constant for the power stroke for a prestroke strain of x is given by Kpowerexp[–hl(2x
+
l)/2 kBT] where Kpower is the equilibrium constant in solution, kB is the Boltzmann constant and T the absolute temperature (Smith & Geeves, 1995a). Hence the equilibrium constant of the powerstroke decreases steeply and exponentially with x. Given Kpower
= 103 (Smith & Geeves, 1995a), a powerstroke distance of 11 nm and a stiffness of 0.56 pN nm–1 used in our model, the equilibrium constant for the power stroke equals 103 when x
=
–l/2 =
–5.5 nm but reduces to 1 when x
–1 nm. Hence the power stroke is energetically unfavourable unless the strain in the prestroke heads is appreciably negative (Smith & Geeves, 1995a; Getz et al. 1998; Duke, 1999). A key issue is how the work term is distributed between the forward and reverse power stroke. We wanted our model to explain not only steady state behaviour but the tension transients after a step release or stretch; the initial rate of tension recovery should increase rapidly with the magnitude of release but be slow for step stretches (Huxley & Simmons, 1971). This can be achieved if the transition state is reached when the lever arm has swung more than half of its total traverse (in the current model by 65%).
In an isometric contraction heads attach at strains centred around zero, but only those prestroke heads having more negative strains undertake the power stroke. This results in the prestroke heads having a small positive (1.5 nm) average strain and therefore contributing appreciably to positive tension, and the poststroke heads having an average x value of about –2.4 nm and hence a strain (x
+ l) =
+8.6 nm. In our model about 60% of the attached heads are prestroke and only 40% poststroke. Hence the average strain is 4.3 nm for all attached heads. The poststroke heads, although in the minority, contribute a disproportionately high fraction (f) (about 79%) of the isometric tension.
Initial slope. In the simulation shown in Fig. 8A where the muscle has been stretched at a velocity of 1 L0 s–1 (= 1 nm ms–1 per hs), the tension rises initially steeply and linearly due to stretching of the compliant element of all attached heads. In our experiments we found that the slope of the initial tension rise in ramp stretches was similar to the T1 stiffness. Since T1 stiffness is a measure of the stiffness of all attached heads, this confirms that at the very beginning of the stretch there is no time for any steps in the crossbridge cycle to occur to an appreciable extent and the initial tension rise is indeed due to the extension of all attached heads.
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1.8 ms) by an inflection, simulating the P1 transition, after which the tension rises less steeply. This inflection may also be seen in the simulations of Månsson (1994). Figure 8C shows that this inflection coincides with a sharp drop in the occupancy of the poststroke heads and a concurrent rise in that of the prestroke heads due to many of the poststroke heads executing a reverse power stroke. During the rest of the stretch the occupancies of the post- and prestroke heads remain nearly constant. That a ramp stretch should induce the reversal of the power stroke is only what would be expected from the analysis by Huxley & Simmons (1971) of their step stretch experiments. By modelling we can distinguish the fate of the heads (‘original’) that were already attached at the beginning of the stretch from that of the heads (‘new’) that attach only after the beginning of the ramp. As with the total poststroke heads, there is a sharp fall in the occupancy of the original poststroke heads coincident with the tension inflection and an increase in the occupancy of the original prestroke heads (Fig. 8D). At this velocity 88% of the original poststroke heads execute a reversal of the power stroke (step 2) causing the tension inflection at P1, while the remaining 12% complete the cycle by step 3. At higher velocities an even larger fraction executes the reverse power stroke.
To understand why the reverse power stroke occurs, consider first the attached heads just before the ramp. For heads that are attached to actin subunits with appreciably negative x values, the rate constant for the reverse power stroke is negligible and their only fate is to undertake the power stroke and then complete the cycle by step 3. For heads that are attached with less negative x (say
–1 to –2 nm) the rate constant of the reverse power stroke is no longer negligible and after executing the power stroke, they may either complete the cycle via step 3 or execute the reverse power stroke, perhaps more than once. For those heads that are attached with x > –1 nm, the rate constant of the reverse power stroke exceeds that of the forward power stroke and the execution of the power stroke will occur more rarely. As the muscle is extended the x-value of all attached heads will increase. This will increasingly inhibit the forward power stroke and the irreversible step but increasingly accelerate the reverse power stroke. In the simulation of Fig. 8, when the half-sarcomere length (hsl) has increased by 2 nm the filaments have slid by 1.9 nm and the x-values of all the original attached heads will have increased by that amount. This will bring the poststroke heads to an average x value of 0.5 nm at which they are destabilized and rapidly undergo a reverse power stroke.
If at P1 the only event was that all the original poststroke heads undertook the reverse power stroke, the slope of the tension-extension record would fall from its initial value before rising again to the same value; the decrease in tension between these parallel lines would be fcP0 where f is the fraction of isometric tension due to poststroke heads and c is the fraction of the sarcomere compliance due to crossbridges. Assuming f
= 0.79, c
= 0.25 and P0
= 150 kN m–2, the decrease in tension would be 30 kN m–2. However, the fall in tension due to the poststroke heads undertaking the reverse power stroke is partially offset by newly attached heads executing the forward power stroke. Correcting for this and the fact that a minority of the poststroke heads complete the cycle, the expected fall in tension at P1 is only 20 kN m–2, to be compared with our estimated tension offset of 12 kN m–2.
The near constancy of the time at which the P1 transition occurs over a range of lengthening velocities (see Fig. 3C) is understandable if the rate constant for the recovery process (k+ + k– in the Huxley-Simmons terminology) is relatively insensitive to strain. That is consistent with the recovery half-time for step stretches being insensitive to their magnitude (Huxley & Simmons, 1971; Ford et al. 1977; Ranatunga et al. 2002).
Our studies with BTS provide supporting evidence for the hypothesis that the P1 transition is due to the reverse power stroke. When the tetanic tension was depressed to 15% by BTS, the peak incremental tension produced by a stretch was not depressed, whereas the P1 transition was not observed (see Fig. 6B; Pinniger & Ranatunga, 2004). The suppression of force generation by BTS has been attributed, in part, to the inhibition of Pi release (Shaw et al. 2003), which would greatly reduce the occupancy of AM.ADP and hence the reduction in tension due to stretch-induced reversal of the power-stroke. Additional evidence supporting this hypothesis is that when two identical triangular stretch-releases were applied to skinned fibres, the early transition (P1) was present in the first response but absent in the second if it occurred within 1 s after the first (Campbell & Moss, 2000); the peak tension to stretch, however, remained similar. Tension traces from similar experiments on cat soleus muscle (Rack & Westbury, 1974; Fig. 2) also show an early transition on the first lengthening but not in the ones that follow soon afterwards.
The question arises as to what would happen at very low Pi concentrations where the slow Pi binding to AM.ADP would presumably limit the rate of the reverse power stroke on AM.ADP.Pi. The possibility should be considered that the force-induced reversal of the power stroke can occur in the AM.ADP state without requiring Pi binding and that the more weakly bound prestroke head produced then rapidly detaches to form M.ADP.
P2 transition. In the simulation shown in Fig. 8, the rate of tension rise decreases rather rapidly at about 14 ms corresponding to an hsl increase of 14 nm. The tension then plateaus simulating the experimental P2 transition. Lombardi & Piazzesi (1990) proposed that the extension at the P2 transition (L2) might be ‘the amount of filament sliding necessary to switch from the force response of crossbridges already attached before the lengthening starts to the force response of crossbridges interacting afresh during lengthening’. We have tested this notion by distinguishing the fate of the original and the newly attached heads. Although the occupancy of the total poststroke and total prestroke heads remains nearly constant after the P1 transition, the proportion of original heads declines, while the proportion of newly attached heads increases. New heads attach at low strain to actin subunits as they pass by and become dragged to higher strain, gradually building up a broad distribution of strains that increasingly resembles that in steady state lengthening (Matsubara & Yagi, 1985; Yagi et al. 1995; Linari et al. 2000). As the extension continues, the original heads are also dragged to increasingly higher strains. Although diminishing in numbers due to their ever-increasing rate constant of detachment, these original heads continue to have a narrow distribution of high strain and therefore exert a disproportionately high tension. Figure 8D shows that the P2 transition occurs when the occupancy of the original heads is dropping rapidly toward zero. Within a short time the last of them rapidly detach leaving only the newly attached heads and hence a near steady state. So the modelling vindicates Lombardi & Piazzesi's proposal with the proviso that the P2 transition occurs not when the switchover from original to new heads is half-way but when it is essentially complete. Further stretch will cause little further change in the distribution of strain in attached heads and therefore little change in tension. Any further increase in tension after the steady state has been attained, i.e. from P2 to Pk, is attributed to the non-crossbridge component. After the end of the ramp, the tension drops rapidly, ultimately to reach the same level as the original isometric tension. Concomitantly, the occupancies of the pre- and poststroke attached heads return to their isometric levels.
Unlike the P1 transition, the time at which the P2 transition occurs is sensitive to lengthening velocity such that the extension L2 at which the transition occurs is constant. This is understandable if the rate constant of detachment of prestroke heads is relatively low for small strains but rises steeply at higher strains.
Comparison with previous models
In our mechano-kinetic model, during lengthening, heads in the M.ADP.Pi state attach to actin at low strains, get carried to higher strains where they exert increasingly higher forces and then detach. Our model is simpler than previous models (Lombardi & Piazzesi, 1990; Piazzesi et al. 1992; Månsson, 1994; Smith & Geeves, 1995b; Getz et al. 1998) in being most closely based on the Lymn-Taylor scheme. Nevertheless, it shares features with these previous models, especially with that of Getz et al. (1998). In common with previous models, as the lengthening velocity is increased, the crossbridge cycle becomes progressively truncated, the power-stroke occurring with decreasing frequency. This explains the low turnover rate of ATP in lengthening muscle (Abbott & Aubert, 1951; Hill & Howarth, 1959; Infante et al. 1964; Curtin & Davies, 1973). At high lengthening velocities in our model there are only three predominant states, AM.ADP.Pi, M.ADP.Pi and M.ATP and at such velocities it and the model of Getz et al. (1998) converge.
Secondly, during steady-state lengthening, as in the models of Lombardi & Piazzesi (1990) and Getz et al. (1998), the tension is borne principally by the prestroke heads (in the AM.ADP.Pi state in our model and that of Getz et al. (1998), and in the first (A1) attached state in the model of Lombardi & Piazzesi (1990)). In our model the unimolecular equilibrium constant for binding of M.ADP.Pi to actin is quite low (1) and the rate constant of detachment is moderate (100 s–1). Such heads cannot be described as strongly bound as their equilibrium constant is several orders of magnitude lower than those for AM and AM.ADP and their rate constant of detachment substantially higher. But the relatively low rate constants for attachment and detachment of M.ADP.Pi in our model also contrast with the rate constants found for crossbridges in relaxed muscle at low ionic strength and temperature (Brenner, 1987). Although such crossbridges are attached, they produce stiffness only at high lengthening velocities. It was concluded that detachment and attachment of such crossbridges was very fast (5 x 104 s–1) and they were described as weakly attached. The prestroke AM.ADP.Pi heads in our model are clearly different from such weakly attached heads. They have a much longer lifetime (several milliseconds) and it is this feature that enables them to hold tension at moderate lengthening velocities. We therefore qualitatively describe the prestroke, AM.ADP.Pi heads in our model as being of intermediate strength to differentiate them from both strongly and weakly attached heads.
Thirdly, although in common with previous models we suppose that the high forces generated in lengthening muscle accelerate detachment, we see no reason to suppose that they do so by a new pathway (‘forced detachment’). The application of force to a protein complex is now considered to speed up the spontaneous rate of dissociation, rather than dissociation occurring by a separate mechanism (Evans, 2001). Nor do we suppose that there is a critical strain, below which force has no effect on detachment and at which there is a large step increase in detachment rate. Finally, we do not require poststroke heads in the AM.ADP state to detach to form M.ADP during lengthening, as supposed in some models.
The non-crossbridge contribution to the tension increase during and after stretch
Our modelling suggests that crossbridges can account for the main features of both P1 and P2 transitions but not the P3 component. P3 was insensitive to stretch velocity but scaled with stretch amplitude and clearly is the same as the residual force enhancement after stretch that has been studied in single frog fibres (Edman et al. 1978, 1982; Bagni et al. 2002), whole isolated muscle or fibre bundles from frog and mouse (Morgan et al. 2000; Schachar et al. 2002) and human skeletal muscle in vivo (Lee & Herzog, 2002). Residual force enhancement has been linked to the development of sarcomere inhomogeneity during stretch (Julian & Morgan, 1979; Edman et al. 1978, 1982; Morgan, 1990, 1994; Edman & Tsuchiya, 1996). When an active muscle fibre in the descending limb of the force–length curve is allowed to shorten, non-uniformity may develop with the shorter (and therefore stronger) sarcomeres shortening at the expense of the longer (and therefore weaker) sarcomeres (Julian & Morgan, 1979). If instead the fibre is stretched in the plateau region of the force–length curve, non-uniformity still develops but it is much less marked, all regions lengthening although to varying extents. This concept of sarcomere inhomogeneity was developed into a specific proposal that during stretch, when weaker sarcomeres reach their yield tension, they rapidly extend (‘pop’) to a much longer sarcomere length where passive structural elements bear the tension (Morgan, 1990, 1994). The consequences of some sarcomeres popping during stretch are not easy to predict, so that the subsequent slow changes in sarcomere length may conceivably contribute to some persistent force enhancement. Our data do not completely rule out such a sarcomere-popping mechanism; however, in our view, there is no convincing evidence to support sarcomere popping during muscle stretch.
Firstly, if sarcomeres pop during stretch, most sarcomeres would not be significantly lengthened, the extension primarily occurring in the few sarcomeres that pop. Thus, the spacing of the first-order diffraction from the fibre would be expected to remain unchanged during stretch, whereas it increased (Edman et al. 1982). Secondly, microscopy indicates that not only are the inhomogeneities smaller than expected, but all sarcomeres extend during lengthening (Edman et al. 1982: Brown & Hill, 1991). Thirdly, in our view, the micrographs of Talbot & Morgan (1996) do not convincingly show popping at the predicted frequency. Finally, and tellingly, although single myofibrils might be expected to show greater instability than fibres, when active myofibrils are extended, all sarcomeres lengthen and no popping is observed (Rassier et al. 2003a; Telley et al. 2006). Thus, while there is evidence for some sarcomere inhomogeneity, there appear to be mechanisms resisting the anticipated popping during stretch.
Edman & Tsuchiya (1996) showed that in frog fibres the tension rise from the P2 transition to the peak and the residual force enhancement after stretch are associated with strain building up in viscoelastic elements other than crossbridges. They took this tension rise (i.e. Pk
–
P2) to be a measure of the non-crossbridge tension at the end of the ramp and showed that it decayed to about half after the stretch. In our experiments on rat muscle, similarly, Pk
–
P2 was 0.2 P0 whereas P3 was about half this. However, we consider that the non-crossbridge tension develops from the start of the stretch and that the total contribution of non-crossbridge elements to the peak tension is best measured by P3* (0.365 P0), which is substantially greater than Pk
–
P2. Since the tension remained steady after this time, the non-crossbridge tension may be complex, consisting of a true elastic component as well as a slowly decaying viscoelastic component.
The identity of the non-crossbridge elastic elements bearing tension has yet to be established. To contribute to residual force enhancement the elastic element must be part of a continuous chain of structures along a sarcomere (which might include thick or thin filaments) and it should be large enough to allow a strain of up to 30 nm to be developed on stretching (Edman & Tsuchiya, 1996). It is noteworthy that the tension responses to stretch in resting muscle fibres are qualitatively similar to those in active muscle (Mutungi & Ranatunga, 1998; Ranatunga, 2001). In the present study, the tension rise for a 5% stretch in resting fibres was only 3.4 kN m–2 while the tension rise in non-crossbridge elastic elements in active muscle (i.e. P3*) was 50 kN m–2. Thus parallel elastic elements would have the features required to explain residual force enhancement, provided their stiffness increases more than 10-fold on activation. In our view the difficulty in identifying the non-crossbridge structural elements responsible may be caused by adhering to the classical assumption that the parallel elastic elements present in relaxed muscle fibres are unchanged on activation. The presence of residual force enhancement in single myofibrils (Rassier et al. 2003a) indicates that the non-crossbridge component resides within the myofibril. A likely candidate is titin (Bagni et al. 2002) which possesses the characteristics necessary to account for the force increase during stretch of resting muscle (Kellermayer et al. 1997). Residual force enhancement in active muscle may be explained by an increase in titin stiffness upon activation, through calcium-dependent titin–actin interactions (Kellermayer & Granzier, 1996). Upon activation the PEVK region of the titin filament becomes anchored to the thin filament so that the flexible region of titin effectively shortens thereby stiffening the filament (Bagni et al. 2002). A contributory cause of increase in stiffness on activation may be the binding of calcium ions (Labeit et al. 2003). The suggestion that titin is the non-crossbridge component is consistent with our findings that the tension rise above P2 and residual force enhancement are absent in intact slow muscle (soleus) fibres; slow fibres contain larger titin isoforms with their stress–strain relation shifted to longer sarcomere length than fast fibres (Wang et al. 1991). Such structural features of slow muscle fibres seem suited for their role in postural motor control in which they are often exposed to lengthening contractions. However, further detailed work incorporating stiffness measurements are necessary to quantify the fibre type specificity of non-crossbridge contribution to stretch-induced force enhancement.
Another non-crossbridge component that may contribute to residual tension is C-protein. C-protein, although a component of the myosin filament (Offer et al. 1973), binds reversibly to actin in a calcium-dependent manner (Moos, 1981) and may make transient interactions with the activated thin filament (Moos et al. 1978; Squire et al. 2003). Such interactions would be viscoelastic and contribute to the tension resisting stretch in active but not in resting muscle. Hence the C-protein actin interaction appears to have the appropriate characteristics to contribute to the residual tension.
Energy storage in lengthening muscle
A description of the molecular processes underlying lengthening of active muscle should account for the energy balance between the work done on the muscle and the heat produced (Linari et al. 2003). In the early stages of a ramp stretch the increasing extension of the compliant elements of the crossbridges and the thick and thin filaments would require net energy absorption. As the crossbridges get increasingly extended, their rate of detachment would increase. Furthermore, the strain energy in a crossbridge, which is liberated as heat when it detaches, is proportional to the square of the strain. Hence the rate of heat production would increase during the stretch, as observed by Linari et al. despite the reduction in ATPase. When a steady state is reached, the rate of heat production from crossbridges would then be constant, matching the rate of work done on them. The energy absorption in the first part of the stretch (phase 1) plateaued with velocity and may correspond to the strain energy in crossbridges and filaments. Linari et al. (2003) observed, however, that the energy absorption in the later part of the stretch (phase 2), unlike the tension, did not plateau with velocity. Furthermore, two-thirds of the total energy absorption by muscle in a ramp stretch (0.09 P0L0) could not be accounted for by strain energy in cross-bridges and thick and thin filaments. They suggested that highly extended titin filaments in a small number (5%) of popped sarcomeres could contribute as much as 0.015 P0L0 to the total energy absorption. In our 5%
L0 stretches the non-crossbridge component contributed 0.365 P0 to the peak tension, corresponding to an energy absorption of 0.365 x 0.05/2 = 0.009 P0L0. Assuming that in the 10% stretches used by Linari et al. the contribution to peak tension doubles, the energy absorption in the non-crossbridge component would be 0.037 P0L0, which is 40% of the total energy absorption. So the non-crossbridge component rather than the crossbridges (Cavagna et al. 1994) may play a major role in animal locomotion by storing energy during lengthening which in the short term could be used in a subsequent release.
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), P3 (
) and P1 (
) from one fibre bundle. The Pk tension increased with velocity towards a plateau, P3 was relatively insensitive to velocity and P1 increased in proportion to velocity with a slope of 6.97 kN m–2 per L0 s–1. The dashed line represents the curve fit to Pk using eqn (1) in the text. A better fit (continuous line) was obtained using eqn (2), where Vc
= 1.46 L0 s–1, PL
= 100 kN m–2, and P3*
= 73.45 kN m–2. In this example, P3* is approximately fivefold greater than the measured P3 tension. C, velocity dependence of Pk
–
P3 (
) and Pk
–
P3* (
). For Pk
–
P3 data, the dashed curve represents the fit of eqn (1) and the continuous curve of eqn (2), where Vc
= 1.30 L0 s–1, PL
= 111 kN m–2, and P3*
= 44.71 kN m–2. Note that Pk
–
P3* data and the fitted curve (pure viscoelastic equation, eqn (2) with P3*
= 0) characterize the velocity dependence of force in the crossbridge component only. Note the intercept on the Pk axis (
D parameter defining the effect of strain on this rate (