Article Information

• PDF (219 kb)
• Full Text with Large Figures
• Cited by in Scopus (2)

PubMed

• Articles by Kenneth S. Campbell

Related Articles

• Relaxation Kinetics Following Sudden Ca Reduction in Single ...

  Relaxation Kinetics Following Sudden Ca Reduction in Single Myofibrils from Skeletal Muscle Biophysical Journal, Volume 83, Issue 4, 1 October 2002, Pages 2142-2151 Chiara Tesi, Nicoletta Piroddi, Francesco Colomo and Corrado Poggesi Abstract To investigate the roles of cross-bridge dissociation and cross-bridge-induced thin filament activation in the time course of muscle relaxation, we initiated force relaxation in single myofibrils from skeletal muscles by rapidly (∼10ms) switching from high to low [Ca] solutions. Full force decay from maximal activation occurs in two phases: a slow one followed by a rapid one. The latter is initiated by sarcomere “give” and dominated by inter-sarcomere dynamics (see the companion paper, Stehle, R., M. Krueger, and G. Pfitzer. 2002. . . 83:2152–2161), while the former occurs under nearly isometric conditions and is sensitive to mechanical perturbations. Decreasing the Ca-activated force preceding the start of relaxation does not increase the rate of the slow isometric phase, suggesting that force-generating cross-bridges do not significantly sustain activation during relaxation. This conclusion is strengthened by the finding that the rate of isometric relaxation from maximum force to any given Ca-activated force level is similar to that of Ca-activation from rest to that given force. It is likely, therefore, that the slow rate of force decay in full relaxation simply reflects the rate at which cross-bridges leave force-generating states. Because increasing [P] accelerates relaxation while increasing [MgADP] slows relaxation, both forward and backward transitions of cross-bridges from force-generating to non-force-generating states contribute to muscle relaxation. Abstract | Full Text | PDF (157 kb)

• Filament Compliance Effects Can Explain Tension Overshoots d...

  Filament Compliance Effects Can Explain Tension Overshoots during Force Development Biophysical Journal, Volume 91, Issue 11, 1 December 2006, Pages 4102-4109 Kenneth S. Campbell Abstract Spatially explicit stochastic simulations of myosin S1 heads attaching to a single actin filament were used to investigate the process of force development in contracting muscle. Filament compliance effects were incorporated by adjusting the spacing between adjacent actin binding sites and adjacent myosin heads in response to cross-bridge attachment/detachment events. Appropriate model parameters were determined by multi-dimensional optimization and used to simulate force development records corresponding to different levels of Ca activation. Simulations in which the spacing between both adjacent actin binding sites and adjacent myosin S1 heads changed by ∼0.06nm after cross-bridge attachment/detachment events 1), exhibited tension overshoots with a Ca dependence similar to that measured experimentally and 2), mimicked the observed -relative tension relationship without invoking a Ca-dependent increase in the rate of cross-bridge state transitions. Tension did not overshoot its steady-state value in control simulations modeling rigid thick and thin filaments with otherwise identical parameters. These results underline the importance of filament geometry and actin binding site availability in quantitative theories of muscle contraction. Abstract | Full Text | PDF (297 kb)

• The Effect of Inorganic Phosphate on Force Generation in Sin...

  The Effect of Inorganic Phosphate on Force Generation in Single Myofibrils from Rabbit Skeletal Muscle Biophysical Journal, Volume 78, Issue 6, 1 June 2000, Pages 3081-3092 C. Tesi, F. Colomo, S. Nencini, N. Piroddi and C. Poggesi Abstract In striated muscle, force generation and phosphate (P) release are closely related. Alterations in the [P] bathing skinned fibers have been used to probe key transitions of the mechanochemical coupling. Accuracy in this kind of studies is reduced, however, by diffusional barriers. A new perfusion technique is used to study the effect of [P] in single or very thin bundles (1–3M in diameter; 5°C) of rabbit psoas myofibrils. With this technique, it is possible to rapidly jump [P] during contraction and observe the transient and steady-state effects on force of both an increase and a decrease in [P]. Steady-state isometric force decreases linearly with an increase in log[P] in the range 500M to 10mM (slope −0.4/decade). Between 5 and 200M P, the slope of the relation is smaller (∼ −0.07/decade). The rate constant of force development () increases with an increase in [P] over the same concentration range. After rapid jumps in [P], the kinetics of both the force decrease with an increase in [P] () and the force increase with a decrease in [P] () were measured. As observed in skinned fibers with caged P, is about three to four times higher than , strongly dependent on final [P], and scarcely modulated by the activation level. Unexpectedly, the kinetics of force increase after jumps from high to low [P] is slower: is indistinguishable from measured at the same [P] and has the same calcium sensitivity. Abstract | Full Text | PDF (163 kb)

• …more

Copyright © 2006 The Biophysical Society. All rights reserved.
Biophysical Journal, Volume 90, Issue 4, 1288-1294, 15 February 2006

doi:10.1529/biophysj.105.067504

Muscle and Contractility

Tension Recovery in Permeabilized Rat Soleus Muscle Fibers after Rapid Shortening and Restretch

Kenneth S. Campbell^,  

Department of Physiology, University of Kentucky, Lexington, Kentucky

Address reprint requests to Kenneth S. Campbell, Dept. of Physiology, MS-508 Chandler Medical Center, 800 Rose St., Lexington, KY 40536-0298. Tel.: 859-323-8157; Fax: 859-323-1070;

How quickly do cross-bridges generate force? One way of answering this important question is to measure the rate constants for each kinetic step in the cross-bridge cycle using solution chemistry techniques (reviewed by Howard 1). Such experiments have provided much useful information over the years but cannot assess the importance of mechanical stress or geometrical constraints imposed by the filament lattice.

An alternative strategy is to measure the rate of force generation in an isolated muscle fiber. The obvious technique is to measure the rate of force development at the beginning of an isometric contraction, but this method cannot distinguish between the rate at which the cross-bridges generate force and the rate at which the contractile apparatus is activated. Experiments using permeabilized fibers are further complicated because of uncertainty in the time required for the myofibrilar free Ca²⁺ concentration to reach steady state 2.

Brenner 3 showed that it was possible to circumvent these issues by measuring the kinetics of force generation in chemically permeabilized fibers during sustained Ca²⁺-activated contractures. He argued that if the muscle was allowed to shorten and then rapidly restretched, the rate constant of force redevelopment k_redev should equal the sum of the apparent rate constants f_app and g_app for cross-bridge attachment and detachment, respectively.

Brenner's analytical method has proved exceptionally useful (see review by Gordon et al. 4), but in reality many experimental records deviate from the single exponential form predicted by his two-state model. Tension recovery in rabbit psoas fibers for instance is often more closely approximated by the sum of two exponential components than by a single exponential function 5,6. The recent work of Burton et al. 7 includes detailed examples.

Another type of deviation occurs when tension temporarily exceeds the steady-state value during recovery before declining back to the original isometric level. Such tension “overshoots” appear to be a general feature of tension recovery measurements. They are evident in published records from more than one research group 7,8,9 and have been observed using 1), permeabilized slow skeletal preparations (rat soleus fibers) 10, 2), permeabilized fast skeletal preparations (rabbit psoas fibers) 6,7,8,9, and 3), permeabilized cardiac preparations from rats, dogs, and pigs (K. S. Campbell, unpublished observations). The published records of Fitzsimons et al. 11 show that tension can also overshoot its steady-state value after photo-release of caged Ca²⁺ at fixed muscle length. This is an important finding because it shows that tension overshoots can occur during isometric force development not preceded by stretch.

Although tension overshoots were described in abstract form in 1993 12 they do not appear to have been systematically examined before now. One reason they have not received further attention could be that most published figures illustrating tension recovery records show only a small portion of the return to steady state (see for example McDonald et al. 8). This style of presentation minimizes the visual impact of the overshoot because the decline back to steady-state force is not apparent.

This work presents an analysis of tension overshoots observed in experiments utilizing permeabilized rat soleus fibers. The measurements demonstrate that the temporary increase in muscle force is accompanied by a comparable increase in fiber stiffness. Two additional novel findings are that 1), tension first reaches its steady-state value more quickly at submaximal levels of Ca²⁺ activation than in saturating Ca²⁺ solutions, and that 2), the rate of tension recovery correlates with the relative residual force prevailing immediately after restretch.

Female Sprague-Dawley rats (150g) were anesthetized by intraperitoneal injection of Pentobarbital (50mg kg⁻¹ body weight) and subsequently killed by surgical excision of the heart. The soleus muscles were isolated, and bundles of ∼20 fibers were chemically permeabilized and stored as described 10. Animal use was approved by the Institutional Animal Care and Use Committee at the University of Kentucky.

Segments of individual muscle fibers (length (l₀) 1000±30μm, sarcomere length 2.59±0.05μm, cross sectional area (estimated assuming a circular profile) 6550±3940μm², measurements performed in pCa (= −log₁₀[Ca²⁺]) 9.0 solution, n=24) were attached between a force transducer and a motor (312B, Aurora Scientific, Aurora, Ontario, Canada, step time 0.6ms) as illustrated in Fig. 1 of Campbell and Moss 10. The experiments illustrated in Figure 1 and Figure 3 and Figure 4 and Figure 5 and Figure 6 were performed using a commercially available force transducer (403, Aurora, resonant frequency 600Hz). The measurements reported in Figure 7 and Figure 8 and Figure 9 required a force transducer with a higher frequency response and utilized a silicon strain-gauge sensor (AE801, SensorOne Technologies, Sausalito, CA, resonant frequency 6.5kHz). Additional mechanical damping (provided by a drop of light machine oil applied between the back of the sensor element and an adjacent end-stop) minimized inappropriate beam oscillation after rapid changes in muscle length.

Display large version of this figure Display high quality version of this figure

Figure 1

Sarcomere length control. Force, sarcomere length, and muscle length records from a representative experiment. pCa 6.0. Steady-state force (P_ss) was 0.48 of maximally Ca²⁺-activated tension (P₀). Measured sarcomere length was held constant after restretch (SL control) in the second trial (shaded traces) using SLControl software (13). The left-hand panels are plotted using a conventional linear time axis. The right-hand panels are plotted using a logarithmic axis to clarify the muscle's behavior immediately after restretch. The horizontal dashed lines in the top panels indicate P_ss.

Real-time measurement of sarcomere length was accomplished by projecting a HeNe laser beam through the central 0.5mm segment of each fiber preparation and monitoring the position of a first-order diffraction line incident on a lateral effects photodiode detector (Model 1239 detector, 301-DIV amplifier, bandwidth 5kHz, UDT Instruments, Baltimore, MD). Signals representing force, sarcomere length, and motor position (indicative of muscle length) were sampled at fixed rates of between 2 and 25kHz depending on the experimental protocol and saved to computer files using SLControl software (http://www.slcontrol.com). Sarcomere length control was implemented in the experiments illustrated in Fig. 1 by updating the motor command voltage at 0.5ms intervals in such a way as to minimize measured changes in sarcomere length. Full details of the feedback algorithm have been published 13.

Subsequent analysis utilized custom-written MATLAB routines (The MathWorks, Natick, MA). Results are reported as mean±SD. The steady-state isometric tension measured in pCa 9.0 solution was subtracted from each P_resid value (defined as in Fig. 2) to correct for passive elastic properties.

Display large version of this figure Display high quality version of this figure

Figure 2

Definitions. Schematic traces of force and muscle length plotted using linear (left-hand panels) and logarithmic (right-hand panels) time axes. Annotations define steady-state isometric force (P_ss), the residual force prevailing immediately after restretch (P_resid), and the maximum force attained in the 5s period after the length perturbation (P_max). t_ct (the “crossing time”) is a measure of how long the muscle takes to regenerate P_ss. In records where there is no clear tension overshoot, force approaches P_ss asymptotically and the exact time at which force reaches P_ss cannot be precisely determined. To overcome this difficulty, t_ct is defined in this work as the time required for force to rise from P_resid to 0.97 P_ss. k_tr values were calculated as −ln(½)/(t_½), where t_½ is the time required for force to rise from P_resid to ½ (P_max+P_resid). Similar experimental trends were observed in a comparable rate constant calculated by fitting a single exponential function to each force trace during the t_max interval. These values were, however, ∼0.1s⁻¹ slower than the k_tr values calculated as above.

Solutions (pH 7.0) with pCa values ranging from 9.0 (1 nM free Ca²⁺) to 4.5 (32μM free Ca²⁺) were prepared as described 10. All experiments were performed at 15°C. Steady-state isometric force in pCa 4.5 solution (P₀) was 79±29kNm⁻². pCa 9.0 steady-state tension averaged 0.006±0.004 of P₀.

When permeabilized muscle preparations are subjected to a rapid shortening/restretch protocol (0.2 l₀, 20ms duration), tension can temporarily exceed (or overshoot) the steady-state isometric value during the recovery process. Fig. 1 illustrates an experiment designed to test whether the overshoot arises as a result of series compliance in the muscle attachments.

The figure shows recordings of force, sarcomere length, and muscle length for two successive trials imposed during a prolonged contraction in submaximally activating pCa 6.0 solution. In the first trial (black traces), the motor was held at a fixed position after restretch (muscle length (ML) control). In the second trial (shaded traces, sarcomere length (SL) control), negative feedback was imposed 5ms after restretch to minimize changes in measured sarcomere length. The magnitudes of the two tension overshoots are not markedly different.

Similar experiments were performed in pCa 6.0 solution using six additional fibers. P_max/P_ss (Fig. 2) averaged 1.14±0.05 in ML control experiments. The corresponding statistic measured under SL control was 1.17±0.06. These values are not significantly different (paired t-test, p>0.05, n=7), indicating that the tension overshoot is unlikely to reflect potential extension of the muscle fiber near its attachments.

Fig. 3 presents illustrative recordings from a fiber activated in solutions with different pCa values. The magnitude of the tension overshoot was small in pCa 6.5 and pCa 4.5 solutions but relatively large at P_ss/P_0∼ 0.5. Summary statistics from 10 fibers are shown in Fig. 4.

Display large version of this figure Display high quality version of this figure

Figure 3

Records obtained at different levels of Ca²⁺ activation. Superposed force and muscle length records (plotted on linear and logarithmic time axes for clarity) for a fiber subjected to the same rapid shortening/restretch protocol in each of 10 solutions. pCa values ranged from 6.5 (very low Ca²⁺ activation) to 4.5 (maximal activation). pCa 9.0 traces were recorded for each experiment but are omitted from this figure because P_ss under these conditions was <1% of P₀. Force records are normalized to P₀.

Display large version of this figure Display high quality version of this figure

Figure 4

Analysis of collated data. Symbols show mean±SD (n=10 fibers) for each parameter for an individual pCa value. (A) Relative Ca²⁺-activated tension (P_ss/P₀). Solid line is a best fit of the form . (B) k_tr values (defined as in Fig. 2). (C) P_max/P_ss (Fig. 2). The dashed line indicates a ratio of unity, i.e., no tension overshoot. P_max/P_ss was greater (ANOVA, Tukey multiple comparison test, p<0.01) in pCa 6.0 solution (P_ss/P₀=0.55±0.04) than during either pCa 6.5 (P_ss/P₀=0.05±0.02) or pCa 4.5 (P_ss/P₀≡1) activations. (D) Crossing time t_ct (Fig. 2). Stars indicate the predicted t_ct values (−ln(0.03)/k_tr) if tension recovered with an exponential time course and did not overshoot P_ss.

k_tr values increased progressively with the level of Ca²⁺ activation for P_ss/P₀ values >0.2 (Figure 4B). This result might be interpreted as suggesting that soleus fibers generate force more quickly at higher levels of Ca²⁺ activation. However t_ct values (a measure of how long the muscle takes to regenerate P_ss after the length perturbation) are lower (ANOVA, Tukey multiple comparison test, p<0.05) for 5.7≤pCa≤5.9 activations (0.67≤P_ss/P₀≤0.80) than in maximally activating pCa 4.5 solution (Figure 4D).

Fig. 5 reinforces this point with illustrative recordings. The soleus fiber in this example redeveloped isometric force 0.76s earlier when immersed in pCa 5.8 solution than at maximal Ca²⁺ activation. This is despite the fact that k_tr equaled 4.6s⁻¹ in pCa 4.5 solution and only 3.4s⁻¹ during the submaximal activation.

Display large version of this figure Display high quality version of this figure

Figure 5

Recovery times. Superposed force and muscle length records (shown on both linear and logarithmic timescales) for a fiber subjected to the same length perturbation in pCa 5.8 (P_ss/P₀=0.74) and pCa 4.5 solutions. Both force records are normalized to their respective steady-state values. Force reached P_ss 0.77s after the start of the record (0.65s after restretch) in pCa 5.8 solution. This was 0.76s earlier than the corresponding transition point in pCa 4.5 solution.

P_resid, the residual force prevailing immediately after restretch (Fig. 2), also varied systematically with the level of Ca²⁺ activation. Figure 6A shows P_resid/P_ss as a function of relative steady-state isometric tension; Figure 6B illustrates the relationship between P_resid/P_ss and k_tr. The two parameters are correlated (p<0.001).

Display large version of this figure Display high quality version of this figure

Figure 6

Residual forces. (A) P_resid/P_ss (Fig. 2) plotted as a function of relative tension (P_ss/P₀). (B) k_tr plotted against P_resid/P_ss. The solid line is a robust fit minimizing the absolute deviation between the experimental values and a linear model. (This technique is more appropriate than linear regression for experimental data with uncertainty in two dimensions 35.) k_tr and P_resid/P_ss are correlated.

During tension overshoots at submaximal levels of Ca²⁺ activation, isometric force is elevated above its steady-state value. In principle this “extra” force could reflect 1), a temporary increase (relative to steady-state conditions) in the number of attached cross-bridges, 2), a temporary increase in the mean force per cross-bridge, or 3), some combination of the two mechanisms.

Fig. 7 illustrates an experimental approach designed to distinguish between these possibilities. Single soleus fibers were activated in pCa 6.0 solution and subjected to repeated trials, each consisting of two 1% step length changes (motor step time 0.6ms) interposed by a single shortening/restretch perturbation (0.2 l₀, 20ms duration). Recordings lasted for 15s, after which the fiber was returned to l₀ and held at that length for at least 6s before the next trial was initiated. τ, the time interval from restretch to the second length step, was adjusted between pseudorandomly ordered preset values in successive trials.

Display large version of this figure Display high quality version of this figure

Figure 7

Length step protocol. Force, sarcomere length, and muscle length records from a representative experiment. pCa 6.0. P_ss/P₀=0.50. The first step length change always occurred 60ms before the shortening/restretch perturbation. τ varied between preset values ranging from 30ms to 13s in successive trials. P_prev is the prevailing tension immediately before the second step length change. Insets show force and sarcomere length traces on an expanded timescale. ΔP₁ and ΔP₂ are the force changes resulting from the first and second steps, respectively.

Soleus muscle fibers are ideal preparations for this type of experiment because they remain mechanically stable during prolonged activations (see, for example, Fig. 3 of Campbell and Moss 10). Figure 8A shows force traces recorded during a sustained activation which exceeded 25min in duration. Tension overshoots recorded near the end of the experiment were not noticeably different from those recorded near the beginning of the activation. Calculated values of P_ss, P_prev, ΔP₁, and ΔP₂ (Fig. 7) from each trial are plotted in Figure 8B. The values are remarkably consistent given the prolonged nature of the experiment.

Display large version of this figure Display high quality version of this figure

Figure 8

Stability of the fiber preparation. (A) Slow time base recording of force during a prolonged activation in pCa 6.0 solution. P_ss/P_0∼ 0.55. Insets show force records for the 10th and 75th trials on an expanded (logarithmic) timescale. τ=9s for both trials. (B) Values of P_ss, P_prev (top panel), and ΔP₁, ΔP₂ (bottom panel) calculated for each trial from the experiment shown in panel A. P_ss and ΔP₁ do not vary with the recovery time τ but are plotted aligned with their corresponding P_prev and ΔP₂ values to enhance clarity.

P_prev/P_ss and ΔP₂/ΔP₁ ratios are plotted as functions of τ in Figure 9A. Both ratios reached maximum values ∼2.5s after restretch and declined gradually back to unity thereafter. F-tests showed that functions of the form y=α – β×e^−γ×τ+δ× e^−ɛ×τ, where α, β, γ, δ, and ɛ are all greater than zero fitted both the P_prev/P_ss and the ΔP₂/ΔP₁ parameter plots significantly better than single exponential recoveries (p<0.001). This result indicates that both ratios significantly exceed their steady-state values during the recovery process.

Display large version of this figure Display high quality version of this figure

Figure 9

Relative stiffness and relative tension. (A) P_prev/P_ss and ΔP₂/ΔP₁ values plotted as a function of the recovery time τ on linear (left-hand) and logarithmic (right-hand) axes. Each symbol indicates the mean±SD of the appropriate ratio calculated from seven separate experiments using different muscle fibers. (B) ΔP₂/ΔP₁ values plotted as a function of P_prev/P_ss. In this plot, each symbol represents the mean value observed at a single recovery time for a single fiber (n=7 fibers). The solid line is a regression line constrained to pass through the origin.

Figure 9B shows that the ratios are also correlated (p<0.001). Since step sarcomere length changes did not vary with τ (ANOVA, p>0.05), this result demonstrates that the temporary increase in force observed during the tension overshoot is accompanied by a comparable increase in fiber stiffness.

Although tension overshoots have been observed in a wide range of different permeabilized preparations 6,7,8,9,10,11, they have not been systematically examined before now, to the best of my knowledge. There are at least three reasons why they deserve careful attention.

First, tension overshoots share many similarities with stretch activation responses observed after much smaller length changes 14,15,16,17 and may arise from a similar mechanism. Second, during an overshoot muscle's force generating capacity is temporarily increased above its steady-state value. Discovering how this happens increases our understanding of how muscles work. Third, the fact that tension overshoots occur at all has significant implications for measurements of k_tr, an important parameter in most models of contractile regulation. This discussion focuses on the second and third points above. Stretch activation is the subject of a recent review by Moore 18.

One possibility before these experiments were carried out was that the temporary elevation in muscle force characteristic of a tension overshoot reflected the development of sarcomere length inhomogeneities during tension recovery 19. Three separate arguments suggest that this is unlikely to be the case: 1), P_max/P_ss ratios during pCa 6.0 activations were unaffected when sarcomere length was held constant after restretch (Fig. 1). 2), Sarcomere length heterogeneity should be greatest at the highest levels of Ca²⁺ activation, whereas the relative size of the tension overshoot drops at Ca²⁺ concentrations greater than pCa₅₀ (Figure 4C). 3), Since sarcomere length heterogeneities are by definition unstable, they seem unlikely to be capable of producing consistent mechanical behavior during activations sustained in excess of 25min (Fig. 8).

Neither are overshoots likely to reflect viscoelastic mechanisms due to structural elements such as titin 20,21. Such effects would be greatest at low levels of Ca²⁺ activation (where passive components form the greatest proportion of measured tension), whereas P_max/P_ss ratios peak at about pCa₅₀. Potential Ca²⁺-dependent changes in titin stiffness 22 can probably be discounted as well because the tension overshoots occur in fibers immersed in solutions with fixed free Ca²⁺ concentrations. The most likely alternatives seem to involve some sort of cross-bridge mechanism.

Potential cross-bridge explanations for tension overshoots can be subdivided into three broad categories: those which involve 1), a temporary increase (relative to steady-state conditions) in the number of attached cross-bridges, 2), a temporary increase in the mean force per cross-bridge, and 3), some combination of 1 and 2. The experimental results presented in Fig. 9 suggest that the first explanation, a temporary surfeit of attached cross-bridges ∼2.5s after rapid shortening and restretch, is the most likely.

This conclusion follows from the fact that ΔP₂/ΔP₁ (a measure of the muscle's relative stiffness) correlated with P_prev/P_ss (the muscle's relative tension) during the recovery process. If instead tension overshoots were to have resulted from a temporary increase in the mean force per cross-bridge, ΔP₂/ΔP₁ should not have exceeded unity.

Two points require further consideration. The ΔP₁ and ΔP₂ values (Fig. 7) measured in this work are underestimates of T₁ forces 23 which could potentially have been measured using faster instrumentation (length steps complete in ∼0.1ms as opposed to the 0.6ms steps used in these experiments). This is a limitation of these experiments, but it is unlikely to have affected the current interpretation. ΔP₁ and ΔP₂ were not regarded in this study as separate measures of the preparation's “instantaneous” stiffness but instead used to evaluate the muscle's relative stiffness at different time points in the recovery process. Expressing ΔP₂ relative to the corresponding ΔP₁ value obviates most potential concerns relating to the finite frequency response of the experimental apparatus.

The remaining issue relates to the precise relationship between the ΔP₂/ΔP₁ ratio and the relative number of attached cross-bridges. Muscle stiffness used to be regarded as a good measure of the number of attached cross-bridges 24, but later measurements of thick and thin filament compliance 25,26,27,28 brought the relationship into question. Filament compliance effects can of course be included in stiffness calculations 29, but these approaches inevitably incorporate assumptions about which sections of the sarcomere extend the most. In the absence of definitive evidence, ΔP₂/ΔP₁ ratios have been assumed in this work to increase with the relative number of attached cross-bridges. This is undoubtedly a simplification but it is probably not wholly inaccurate, particularly at the lower levels of Ca²⁺ activation at which tension overshoots are most apparent.

One question that has not been addressed in this work is how the temporary increase in attached cross-bridges might occur. There are many possibilities: rapid filament movements might dislodge tropomyosin molecules from their normal positions, cross-bridges bound immediately after restretch could take many seconds to detach from the thin filament, etc. An additional and intriguing possibility is that the overshoot reflects “compliant realignment” of thick and thin filaments 30.

Large shortening/restretch perturbations similar to those used in this work (0.2 l₀, 20ms duration) are commonly used to measure cross-bridge kinetics in permeabilized muscle preparations. The basic technique was developed by Brenner 3 who argued that tension should redevelop after such a perturbation at a rate (k_redev) equal to the sum of the apparent cross-bridge attachment (f_app) and detachment (g_app) rate constants.

g_app can be calculated from ATPase measurements and appears to be insensitive to the prevailing free Ca²⁺ concentration 3. k_redev on the other hand increases with the relative level of Ca²⁺ activation in a wide variety of different muscle preparations 3,5,6,9,31,32,33. The implication is that cross-bridge attachment is Ca²⁺ dependent (i.e., f_app increases with the free Ca²⁺ concentration) though whether this reflects Ca²⁺ sensitivity of one or more myosin state transitions or an increase in actin binding site availability is uncertain. The field has been reviewed by Gordon et al. (4).

This work analyzes the time course of tension recovery using two separate parameters (Fig. 2). k_tr is a rate constant calculated from the time required for force to rise from P_resid to &frac12; (P_max+P_resid). t_ct is a direct measure of the time required for force to rise from P_resid to 97% of P_ss. The two parameters would exhibit one-to-one mapping if tension recovered with an exponential time course and did not exceed P_ss.

Figure 4B shows that k_tr values increased progressively for all activations with P_ss/P₀ values >0.2 31. t_ct values in contrast fall significantly below the mean pCa 4.5 value during pCa 5.9, 5.8, and 5.7 activations. The conclusions must be that 1), the k_tr and t_ct parameters are not measures of the same physical processes, and that 2), the k_tr parameter underestimates the rate at which soleus fibers redevelop P_ss at submaximal levels of Ca²⁺ activation.

One interpretation of these results is that t_ct values are dominated by the rates at which cross-bridges attach to and detach from the thin filament near the beginning of the recovery process, whereas the k_tr parameter incorporates additional recruitment of a new pool of cycling cross-bridges. This additional pool is only temporarily available at submaximal levels of Ca²⁺ activation and manifests as a tension overshoot.

Why then does tension not exceed P_ss in pCa 4.5 solution? The answer might be that cross-bridges once recruited at maximal Ca²⁺ activation are not released and continue to contribute to isometric force.

Although tension overshoots are a common feature of tension recovery measurements performed using permeabilized muscles, they have not yet been reported in intact muscle fibers, to my knowledge. This suggests that the underlying mechanism may be an artifact of the permeabilization process, but an alternative possibility is that overshoots are negligibly small in the intracellular conditions pertaining to a fused tetanus. Tension overshoots are most noticeable in permeabilized soleus fibers at approximately half-maximal Ca²⁺ activation. This experimental condition is difficult to reproduce in an intact muscle fiber.

Figure 6B shows that k_tr correlated with the P_resid/P_ss ratio. If this ratio is indicative of the proportion of cross-bridges attached between the filaments immediately after restretch 34, the linear relationship between k_tr and P_resid/P_ss can be explained by a mechanism in which attached cross-bridges activate adjacent actin binding sites through cooperative mechanisms in the thin filament, leading in turn to additional cross-bridge binding.

The low value of the y-intercept in Figure 6B indicates that this would be the dominant mechanism in rat soleus fibers and thus that the rate of tension recovery is controlled primarily by the availability of actin binding sites and not by Ca²⁺ regulation of a cross-bridge state transition.