Determinants of force rise time during isometric contraction of frog muscle fibres

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1 J Physiol (2007) pp Determinants of force rise time during isometric contraction of frog muscle fibres K. A. P. Edman and R. K. Josephson Department of Experimental Medical Science, Biomedical Centre, F11, University of Lund, S Lund, Sweden Force velocity (F V ) relationships were determined for single frog muscle fibres during the rise of tetanic contraction. F V curves obtained using isotonic shortening early in a tetanic contraction were different from those obtained at equivalent times with isovelocity shortening, apparently because changing activation early in the contraction leads, in isovelocity experiments, to changing force and changing series elastic extension. F V curves obtained with isotonic and with isovelocity shortening are similar if the shortening velocity in the isovelocity trials is corrected for series elastic extension. There is a progressive shift in the scaling of force velocity curves along the force axis during the course of the tetanic rise, reflecting increasing fibre activation. The time taken for F V curves to reach the steady-state position was quite variable, ranging from about 50 ms after the onset of contraction (1 3 C) to well over 100 ms in different fibres. The muscle force at a fixed, moderately high shortening velocity relative to the force at this velocity during the tetanic plateau was taken as a measure of muscle activation. The reference velocity used was 60% of the maximum shortening velocity (V max ) at the tetanic plateau. The estimated value of the fractional activation at 40 ms after the onset of contraction was used as a measure of the rate of activation. The rate of rise of isometric tension in different fibres was correlated with the rate of fibre activation and with V max during the plateau of the tetanus. Together differences in rate of activation and in V max accounted for 60 80% of the fibre-to-fibre variability in the rate of rise of isometric tension, depending on the measure of the force rise time used. There was not a significant correlation between the rate of fibre activation and V max. The steady-state F V characteristics and the rate at which these characteristics are achieved early in contraction are seemingly independent. A simulation study based on F V properties and series compliance in frog muscle fibres indicates that if muscle activation were instantaneous, the time taken for force to rise to 50% of the plateau value would be about 60% shorter than that actually measured from living fibres. Thus about 60% of the force rise time is a consequence of the time course of activation processes and about 40% represents time taken to stretch series compliance by activated contractile material. (Received 25 August 2006; accepted after revision 12 February 2007; first published online 15 February 2007) Corresponding author R. K. Josephson: Department of Experimental Medical Science, Biomedical Centre, F11, University of Lund, S Lund, Sweden. rkjoseph@uci.edu A number of physiological processes are involved in determining the time course of rising muscle force during an isometric tetanus. Activation of the contractile machinery begins with the triggered release of calcium from internal stores within the muscle fibre, followed by diffusion of the released calcium to activation binding sites on the thin filaments, binding of calcium to these activation sites, and resulting configurational changes in the molecules of the thin filaments that allow attachment of cross bridges to the thin filaments from adjacent thick filaments. It is anticipated that there will be a delay between the exposure of cross bridge binding sites and the early rise of tension because of the finite rate constant for the formation of attached cross bridges between thick and thin filaments (the f parameter in the model of Huxley (1957)). This delay between calcium binding and force production is likely to be appreciable. A simulation study, using Huxley s estimates of rate constants for cross bridge attachment and detachment in frog muscle at 0 C, indicated that even if all cross bridge binding sites were instantaneously available, and there was no series compliance to delay force rise, it would still take about 20 ms for muscle force to rise to 50% of its maximum tetanic value because of the kinetics of cross-bridge attachment and detachment (Josephson & Edman, 1998). For convenience in discussion, the delay attributable to DOI: /jphysiol

2 1008 K. A. P. Edman and R. K. Josephson J Physiol the time course of initial cross-bridge attachment will be considered to be part of the activation process. Later in contraction, as activation reaches its maximal level, the rise of tension is dominated by the force velocity (F V ) properties of the contractile component of the muscle and the stiffness of the series elastic element (SEE) against which the contractile component shortens, the rate of force rise being the product of the shortening velocity of the contractile component and the SEE stiffness (Hill, 1938). Early measurements by A. V. Hill and others suggested that muscle activation is abrupt at the onset of contraction. This conclusion was based on the large force that a muscle could maintain early in a contraction after a stretch which appropriately elongated series elastic elements (Hill, 1949), and on the rapid increase at the end of the latent period in heat production (Hill, 1950a), resistance to stretch (Hill, 1950b), and the velocity of shortening under a light load (Fenn & Marsh, 1935; Abbott & Ritchie, 1951; Hill, 1951). Edman (1970) and Edman & Kiessling (1971) used quick release during a series of partially fused, isometric twitches to map the rise time of that force which the contractile component of the muscle could just maintain without shortening or being lengthened. This force, traditionally termed the active state force, rose rapidly after a 12 ms latent period following stimulation, the middle 40% of the rise time being completed in 3 4 ms (1 2 C). If the rate of muscle activation is rapid compared to the rise time of tetanic tension, as these results suggest, than the rise of tension must be determined largely by the F V characteristics of the activated muscle and the series compliance against which the contractile components operate. In contrast to the reports indicating that there is an abrupt transition between rest and activity in muscle, several studies have emphasized that the time taken for muscle activation is a substantial fraction of the isometric twitch or tetanic rise time. Jewell & Wilkie (1958) and Close (1962) compared the rate of force increase during the rise of an isometric contraction of frog muscle with the time course of force redevelopment following a quick release imposed during the plateau of an isometric tetanus when the muscle is fully activated. The initial rate of force increase was found to be less than that during force redevelopment at an equivalent load, indicating incomplete activation, until ms (0 C) after the initial force onset. Haugen (1987) reports that the shortening velocity under a light load, and the maximum power output of frog muscle fibres, do not reach a maximum until several tens of milliseconds into a twitch contraction (4 6 C). The time course of isometric tension in frog muscle fibres, measured under conditions in which sarcomere length was held fixed and therefore in which compliant elements external to the sarcomere were of little influence, was only slightly faster than that of a fixed-end tetanic contraction, indicating that the rise time in a usual isometric contraction is not substantially delayed because of time taken to stretch external compliant components (Haugen & Sten-Knudsen, 1987). Finally, a series of studies in which isovelocity shortening was used to establish force velocity (F V ) relationships early in tetanic contractions of frog muscle fibres demonstrated that the force at any shortening velocity less than the maximum velocity was lower early in a tetanic contraction than during the plateau, and that the time taken for full fibre activation, measured by the shifting position of the F V curve, was quite variable, ranging from 40 ms to over 100 ms (2 5 C) in different fibres (Cecchi et al. 1978, 1979, 1981; Ambrogi-Lorenzini et al. 1983; Lombardi & Manchetti, 1984; Josephson & Edman, 1998). F V curves obtained early in the redevelopment of tension following quick release were indistinguishable from those during the plateau of contraction, indicating that the long time taken for initial activation is not a consequence of slow cross-bridge formation (Cecchi et al. 1979). The experiments described below were begun to better clarify the early time course of muscle activation in frog muscle fibres, and the relationship between the time course of muscle activation and the rise of tetanic tension. Muscle activation was monitored by the changing position of F V curves along the force axis early in tetanic contractions. In most measurements the F V relationships were determined using isotonic contraction rather than isovelocity contraction as has been employed in most of the previous studies of developing activation. Using isotonic contraction avoids the potential problem, inherent in isovelocity measurements, of changing muscle force and therefore changing elongation of series elastic elements early in a contraction as the level of activation increases. Methods Measurements were made from single muscle fibres from the frog Rana temporaria. The experiments were performed according to procedures approved by the Animal Ethics Committee of the University of Lund. The frogs were killed by decapitation followed by destruction of the spinal cord. Individual fibres were dissected from the tibialis anterior muscle. Slips of tendon were left on each end of a fibre as attachment points. The fibres were normally dissected the afternoon before the experiment and kept in Ringer solution at +4 C overnight. For an experiment a fibre was mounted horizontally between a semiconductor strain gauge (AE 801, Aksjeselskapet Mikroelektronikk, Horton, Norway) and an arm which extended from the moving coil of an electromagnetic puller. Small clips of aluminium foil, wrapped around the slips of tendon at the end of the fibre, were used to attach the fibre to the strain gauge and to the puller. The fibre was suspended in a temperature-controlled, Perspex

3 J Physiol Determinants of force rise time 1009 chamber filled with saline. The saline had the following composition (mm): NaCl, 115.5; KCl, 2.0; CaCl 2, 1.8; Na 2 HPO 4 + NaH 2 PO 4 (total concentration), 2.0; ph 7.0. Details of the muscle dissection, instrumentation, and methods used for measuring fibre length, sarcomere length and fibre cross-sectional area are given in Edman & Reggiani (1984). The saline temperature in different experiments ranged from 1 to 3 C, but in any one experiment the temperature range was less than 0.5 C. The fibre length was adjusted so that the resting sarcomere length, as measured with laser diffraction, was 2.25 μm. A pair of platinum plates placed one on each side of the fibre and approximately 2 mm from it served as stimulating electrodes. The stimuli were 0.2 ms voltage pulses approximately 20% above threshold. During tetanic stimulation the stimulus pulses were in 1 2 s trains at a frequency just sufficient to give full mechanical fusion (14 22 Hz). Tetanic bursts of stimuli were separated by 2 min intervals, and a 20 min period of regularly paced, tetanic stimulation at 2 min intervals preceded data collection in each experiment. The electromagnetic puller was used to rapidly change the fibre length, or to dynamically control the fibre length so as to keep constant the fibre shortening velocity (velocity clamp; isovelocity mode) or the fibre force (force clamp; isotonic mode). Generally the fibre was released from isometric contraction to isotonic or isovelocity shortening at a chosen time after the onset of stimulation. The velocity clamp or force clamp measurements were used to construct F V curves for the rising phase and the plateau of tetanic contractions. In velocity clamp experiments a rapid (less than 0.5 ms) shortening step of varied amplitude preceded the period of constant velocity shortening. This shortening step, which partially unloaded stretched elastic elements, quickly reduced the muscle force. The size of the step was adjusted so as to minimize the difference between the tension at the end of the step and the steady-state tension during isovelocity shortening. There are force or velocity transients immediately following release from isometric contraction to isotonic or isovelocity contraction (e.g. Podolsky, 1960; Huxley & Simmons, 1971; Sugi & Tsuchiya, 1981; Edman & Curtin, 2001). These transient responses are particularly large and long lasting if the force is high following the release. For contractions in which muscle force following release was less than about 0.5F max (F max = maximum isometric tension), measurements for F V curvesweremade15ms after the release to isotonic or isovelocity contraction, by which time the responses had reached an approximate steady state. With contractions in which the force following release was greater than about 0.5F max, it was often necessary to increase the delay to the measurement of force and velocity until ms after the release to make the measurements during an approximate steady state. Such high forces, and the associated additional delay to the measurement of force and velocity, occurred only in releases late in the rise or during the plateau of the tetanic contraction. The following equation (Hill, 1938) was fitted to force velocity data using the three dimensional regression method of Wohlfart & Edman (1994): (F + a)(v + b) = a(v max + b) where V max is the maximum shortening velocity (that at F = 0), and a and b are constants. The F V curves for isotonic contraction are based solely on shortening under forces less than or equal to the isometric force at the transition to the force-clamp contraction. Therefore F V curves from early in a contraction do not include a high-force, low-velocity segment. To have obtained the segment of the F V curve with forces higher than those that existed at release would have required initially stretching the muscle. The decay of the mechanical transients following a rapid length change of a frog fibre is much slower after stretch than after release (Huxley & Simmons, 1971), and the duration of the transient response after a stretch is long in proportion to the rise time of an isometric contraction. Because of the difficulties in disentangling increasing activation from declining transient responses, measurements were not made from contractions involving muscle stretch. Muscle force and shortening velocity were measured from photographed oscilloscope traces, analysed with a Nikon profile projector, or from digitized force and velocity records. In all cases time was measured from first detectable force during a contraction. Thus time values do not include the latent period between stimulation and force onset. Results Isotonic and isovelocity force velocity curves F V curves measured during the plateau of tetanic contractions from whole muscles or muscle fibres have been found to be essentially identical when obtained with isotonic or with isovelocity contraction (Bressler & Clinch, 1974; Julian et al. 1986a). We too obtained similar F V curves during the plateau of a tetanic contraction when either force or velocity was the independent variable. Early in tetanic contractions, however, there was a consistent difference between F V curves determined with force clamp and with velocity clamp contractions. In the region of high force and low velocity, curves determined from force clamp contraction had higher shortening velocity at any given force than did those from velocity clamp contraction (Fig. 1A). This result was obtained in each of five preparations in which F V curves were obtained with both force clamp and velocity clamp contraction at equivalent times following the onset of tetanic stimulation and in which the measurements were made sufficiently early that the muscle tension had risen to less than 30%

4 1010 K. A. P. Edman and R. K. Josephson J Physiol of the final isometric level at the release from isometric contraction to the force or velocity clamp condition. The difference between F V curves determined from isotonic and isovelocity contraction is a consequence of changing force and associated change in the length of series elastic components during isovelocity contraction. Early in a tetanus the force rises during isovelocity shortening because of increasing muscle activation (Fig. 2B). This increase in force, which is appreciable only early in the contraction when the state of muscle activation is rising rapidly, is most apparent when the controlled velocity is low and the fibre force high. The increase in force during isovelocity contractions early in a tetanus can be expected to stretch compliant components that are in series with the force generators (the SEE). During an isovelocity contraction in which there is increasing fibre force, there should also be increasing stretch of the SEE. The constant shortening velocity imposed on the fibre is the shortening velocity of the fibre as a whole. If the SEE is of constant length, as it is during isotonic contraction, the shortening velocity of the fibre is also the shortening velocity of the contractile component (CC). If, on the other hand, force is rising and the SEE is becoming elongated, the shortening velocity of the CC is greater than the shortening velocity of the fibre as a whole, greater by the velocity of SEE lengthening. Thus whole fibre shortening velocity underestimates CC shortening velocity when there is increasing fibre force. If the observed difference between isotonic and isovelocity F V curves is a consequence of SEE elongation, then correcting the observed fibre shortening velocity so as to compensate for the SEE s lengthening velocity should reduce or abolish the difference. This correction requires information on the stiffness of the SEE. To measure SEE stiffness, a small ( 0.5%), rapid (< 1 ms) shortening was imposed on the muscle fibre, and the resulting drop Figure 1. Force velocity relationships during trials with force clamp and velocity clamp and the method used to measure muscle stiffness A, force velocity (F V) relationships during trials with force clamp (open symbols) and velocity clamp (filled symbols). The release to force clamp or velocity clamp occurred 32 ms after force onset for the force clamp values, 33 ms after force onset for the velocity clamp values. The force at release was approximately 0.25F max for both force clamp and velocity clamp trials. Measurements of force and velocity were made 15 ms after the release. The continuous curve is the F V relationship during the plateau of the contraction measured with force clamp (see Fig. 4). B, the method used to measure muscle stiffness. A small release (0.04 mm = 0.47% fibre length) was imposed on the muscle at different times during the rising tension of an isometric contraction. L 1 and L 2 are muscle length, F 1 and F 2 muscle force. The figure shows one trial on a slow (L 1, F 1 ) and on a faster (L 2, F 2 ) time base. The change in force, F, was measured as the difference between the force at the onset of release and the minimum force reached at the end of release. The ratio of change in force to change in length ( F/ L) was taken as a measure of muscle stiffness. C, stiffness as a function of muscle force. The force of the abscissa is the average of the force at the onset and at the end of the length change. D, F V relationships as in A, but with the values from velocity clamp trials corrected for changing length of series elastic elements resulting from changing force levels (see text).

5 J Physiol Determinants of force rise time 1011 in tension was determined (Fig. 1B). The ratio of force change to length change ( F/ L) is a measure of muscle stiffness (Edman & Nilsson, 1968; Joyce & Rack, 1969). Stiffness determinations were made at a number of times during either the rising phase of tetanic contractions or during the tension redevelopment following a large, rapid shortening so as to determine fibre stiffness as a function of muscle force. The relationships between stiffness and force were found to be similar for the initial development of tension and for the redevelopment of tension following a quick release. The stiffness of the frog muscle fibres, like that of rabbit heart muscle (Edman & Nilsson, 1968), increased linearly with muscle force (Fig. 1C). The stiffness determined in a similar manner for cat soleus muscle increased monotonically but not quite linearly with muscle force (Joyce & Rack, 1969). Jewell & Wilkie (1958) characterized the stress strain relationship in frog sartorious muscles using the initial changes in force resulting from rapid releases of varied distance during the plateau of isometric contractions. More recently Curtin et al. (1998) used a similar approach to investigate stress strain relationships in dogfish muscles. In both these studies the muscle stiffness (= the slope of the stress strain curve) increased with increasing muscle force at low forces but became relatively constant at high force (see also Lichtwark & Wilson, 2005). Early in a tetanic contraction when the force is low, the series stiffness is also relatively low. Therefore the rate of force increase for a given shortening velocity of the contractile component is expected to be lower early in an isometric contraction when the force and stiffness are low than later in the contraction when force and SEE stiffness are greater. The following procedure was used to correct the measured shortening velocity for the lengthening velocity of series elastic elements during isovelocity fibre shortening. The instantaneous fibre force and the rate of change of force (df/dt) were measured 15 ms after the transition from isometric to isovelocity shortening. The rate of change of force was measured as the slope of a tangent drawn to the force trace at the appropriate time. The muscle stiffness (df/dx) at the existing force was estimated from the curve relating muscle force and stiffness (e.g. Fig. 1C) constructed with data from the same fibre. The expected lengthening velocity of elastic components was calculated from the ratio of (1) the rate of force change as a function of time and (2) the muscle stiffness: SEE lengthening velocity(dx/dt) = (df/dt)/(df/dx). The lengthening velocity of elastic elements was added to the shortening velocity of the whole fibre to determine the shortening velocity of contractile elements alone. When force was plotted against corrected shortening velocity, F V curves from isovelocity contractions were quite similar to those from isotonic contraction (Fig. 1D), confirming the hypothesis that it is lengthening of compliant elements associated with changing fibre force that produces the differences between isotonic data and uncorrected isovelocity data. Because it is a more direct measure, requiring no correction for series compliance, isotonic contraction was used in analysing the early activation of fibres. Changes in F V relationship early in a tetanus Examples of muscle force and shortening during length clamp contractions are shown in Fig. 3. F V curves obtained from such contractions changed with time in an orderly way following the onset of tetanic stimulation. At other than very light loads the shortening velocity at a given load became progressively greater with increasing time (Fig. 4). In contrast, the maximum shortening velocity, that is velocity at very light load, was greatest at the onset of contraction and then declined over the next ms to reach a value that was maintained through the remaining rise time and the plateau of the contraction. The early decline in maximum shortening velocity is considered in more detail in Josephson & Edman (1998). Figure 2. Force and fibre length during force clamp (A) and velocity clamp (B) contraction The release to force clamp or velocity clamp contraction occurred when the force reached 0.16F max. Note the increasing force during the isovelocity shortening in B.

6 1012 K. A. P. Edman and R. K. Josephson J Physiol Figure 3. Examples of load clamp recordings at different force levels during tetanus of a single muscle fibre Arrows indicate the time of release from isometric contraction. A, force clamps began 32 ms after force onset when force had reached 25% F max. B, release to force clamp during the plateau of contraction. a is the time interval surrounding the release; b is the force for the complete contraction at the middle force level in a. The thickened portion of the base line in b marks the duration of the tetanic stimulation. Vertical dotted lines mark the time, 15 ms after the onset of load clamp, at which force and muscle shortening velocity were measured to construct F V curves. Cecchi et al. (1978, 1979, 1981) quantified the changing position of F V curves early in a tetanus by extrapolating the curves to the force axis. The intercept of an extrapolated curve with the force axis was taken as an estimate of the force that would be generated at zero shortening velocity. The distance of extrapolation is very large for curves early in a tetanus when the initial force is low (Fig. 4). Therefore Figure 4. Force velocity relationships for force clamp measurements made early in tetanic contractions and during the plateau of contractions Data are from the same fibre as in Figs 1 and 3. B is an expanded version of the upper left portion of A. The release at 0.25F max occurred 32 ms after force onset; that at 0.13F max was 20 ms after force onset. Velocity was measured 15 ms after release except for the highest force points of the plateau curve, which were measured ms after release. The horizontal, dashed line in B is the reference velocity, 0.6V max, at which force was measured to quantify the changing level of fibre activation. the estimated force at zero velocity has a large uncertainty and the value of the estimated intercept with the force axis is likely to be strongly dependent on the particular model assumed to describe the underlying F V relationship. In order to avoid the uncertainties arising from extrapolation well beyond the range of collected data, we quantified the changing position of F V curves by measuring the expected force at a common shortening velocity. The shortening velocity taken was 0.6 times the maximum shortening velocity (V max ) recorded during the plateau of the tetanic contraction. The force at this shortening velocity will be indicated as F (0.6Vmax ). The velocity, 0.6V max, was taken as a reference because it occurs at forces low enough to be obtained without stretch, even quite early in a contraction, but at forces high enough to be adequately removed from the region in which there is crossover of F V curves from early and late in a contraction and therefore the region for which there is little change in the force at the selected velocity during the contraction (Fig. 4). The expected force at 0.6V max was obtained by fitting a Hill equation to the points and solving the equation to obtain V max and the expected force at 0.6V max. Fibres proved to be quite variable in the rate at which their F V properties approached the steady state of the tetanic plateau. In some fibres F (0.6Vmax ) measured ms after the onset of contraction was nearly as great as that at the tetanic plateau, in other fibres F (0.6Vmax ) was less than half the plateau value at ms after force onset (Fig. 5). Fibres were similarly variable whether F (0.6Vmax ) was measured at equivalent times during the onset of the contraction or when measured following release from isometric to isotonic contraction at similar initial isometric force (Fig. 6). The average value of F (0.6Vmax ) relative to F max, the force at the tetanic plateau, was 72% (s.d. = 12%, n = 17) for fibres released to isotonic

7 J Physiol Determinants of force rise time 1013 Figure 5. Force at a selected shortening velocity (0.6V max ) calculated from force velocity curves obtained at different times after force onset in tetanic contractions Time on the abscissa is the delay between force onset and the release to isotonic contraction plus the delay (15 ms) between release and the measurement of shortening velocity. Lines join points from the same fibre. Different symbols code the amplitude of the force at the release to isotonic contraction. The vertical line at 40 ms indicates the time at which the distribution of force at 0.6V max (designated F (0.6Vmax)) was evaluated for the fibres of the population. Estimates of F (0.6Vmax) at 40 ms were obtained by linear interpolation between values obtained before and after this time (effectively the intersection of the vertical dashed line with the line joining points from an individual fibre). contraction when force was about 25% F max (actual range of force at release = 23 29% F max ) and 90% (s.d. = 9%, n = 13) when the force at release was about 50% F max (actual range 47 53%). Thus activation is nearly complete by the time that isometric force has risen to half its final value. The determinants of force rise time during an isometric tetanus As a first approximation the force trajectory during the rise of an isometric tetanus is determined by two, potentially independent, sets of physiological processes and properties. The first set is composed of activation processes the release, diffusion and binding of calcium and the changes in filament organization allowing cross-bridge attachment. We have included, among the activation events, the initial rate of cross-bridge binding to thin filaments. The second set of determinants is made up of the force velocity properties of the contractile apparatus when activated and the series compliance against which the contractile apparatus shortens during an isometric contraction the more rapid the shortening velocity at any force, and the stiffer the compliance, the more rapid is the force rise. The question to be addressed is the extent to which each of these sets contributes to the time course of force increase during a contraction. We have chosen, as a quantitative measure of activation rate, the expected value of F (0.6Vmax ) reached 40 ms after force onset. This measure, which will be termed the activation rate index (ARI), was obtained by linear interpolation between values of F (0.6Vmax ) obtained at times shorter than and longer than 40 ms after the onset of the tetanic contraction (Fig. 5). The delay of 40 ms was chosen because there is large dispersion among the values of F (0.6Vmax ) at this time, and because it maximized the number of estimates of activation time available from the data set shown in Fig. 5. ARI and V max, the latter a measure of the F V properties of a fully activated fibre, varied considerably from fibre to fibre. For the 20 fibres of Figs 6 and 7 the coefficient of variation (=s.d./mean) was 19% for ARI and 6.6% for V max. The maximum isometric stress for these fibres had a coefficient of variation of 18% (range = kn m 2 ). Two measures were used to characterize the rise time of force in a tetanic contraction: (1) the time taken for the force to reach 50% of the plateau value; and (2) the force, relative to the plateau value, reached 40 ms after force onset. The 40 ms delay for the latter measurement was chosen to be congruent with the time for which ARI was estimated. There was a statistically significant correlation (P < 0.05) between each of the measures of force rise time and both ARI and V max (Fig. 7, Table 1), Figure 6. Variability in the extent of fibre activation at a common time after force onset (left) or at a common isometric force level at release to isotonic shortening (right) The left histogram was obtained as described in the legend of Fig. 5. The right histogram includes all fibres for which the force at release to isotonic contraction was in the range 23 29% F max.

8 1014 K. A. P. Edman and R. K. Josephson J Physiol Table 1. Linear regression analysis of relationships between V max, Activation Rate Index (ARI), and force rise time Measure of force X 1 X 2 X 1 + X 2 Rise time V max (L s 1) ) ARI (%) Force onset to 50% F max (ms) = X 1 = X 2 = X1 0.50X 2 r 2 = 0.21 r 2 = 0.49 r 2 = 0.66 Force at 40 ms (% F max ) = X 1 = X 2 = X X 2 r 2 = 0.37 r 2 = 0.51 r 2 = 0.83 with the correlations being somewhat stronger for rise time measured as relative force at 40 ms than as the time to reach 50% F max. There was no significant correlation between ARI and V max, the maximum shortening velocity during the tetanic plateau (r 2 < 0.001). The large variability of V max, ARI, and the measures of force rise time, and the independent variability of ARI and V max, make it possible to use correlation analysis to evaluate the extent to which the rise time of force in an isometric contraction is determined by the time course of activation, and to what extent by the F V properties achieved by the activated contractile machinery as indicated by the plateau value of V max. Multiple linear regression analysis indicated that about 66% of the variability in the time taken for the force to rise to 50% F max and 83% of the variability in the force 40 ms after tetanic onset is accounted for by the variability in ARI and V max taken together (r 2 values in Table 1). While both activation time and F V properties are involved in determining the rise time of tension, the first is the stronger determinant. The ratio a/f max is often used as a measure of the curvature of the F V relation; the greater the value of this ratio the straighter is the F V curve and the greater is the shortening velocity at any given force. The mean Figure 7. Relationships between tetanic rise time, V max and activation rate index Relationships between (1) tetanic rise time, measured from onset to 50% F max ; (2) V max, the maximum shortening velocity during the tetanic plateau; and (3) activation rate index (ARI), a measure of the rate of development of the fibre s force velocity properties (= expected force, as percentage F max, if it were to be measured 40 ms after the tetanic onset and at V = 0.6V max ). The correlations in A and B are both statistically significant (P < 0.01). There is no correlation between the ARI and the value of V max at the plateau (plot C, r 2 = 0.005).

9 J Physiol Determinants of force rise time 1015 value of a/f max for the 20 fibres of Fig. 7 was 0.36 with a coefficient of variation of 16.5%. In agreement with Edman et al. (1985), we again found that there was not a statistically significant correlation between a/f max and V max in single muscle fibres (r 2 = 0.1, P > 0.15). Nor was there a significant correlation between a/f max and ARI or either of the two measures used for force rise time. What would be the rise time of isometric force if activation were instantaneous? Available information on series elastic compliance and on the force velocity properties of the contractile component allows prediction of the expected trajectory of force rise during an isometric tetanus were the muscle to become totally activated instantaneously. The equation for the linear relationship between SEE stiffness (df/dx) and force (F)inFig.1C is: df/dx = C 2 F + C 1 (1) where x is the shortening distance of the contractile component from resting length (and also the lengthening distance of the SEE), C 1 is the intercept of the regression line, and C 2 the slope. The length of the fibre in Fig. 1 was 8.5 mm. The values of the slope and intercept constants expressed as strain in the 8.5 mm long fibre, are: C 1 = 1131 kn m 2 (per muscle length) C 2 = 124(a dimensionless number, but more easily thought of as per muscle length ) We will assume the values for slope and intercept in Fig. 1C are not atypical, and will use them in the further development. Integrating eqn (1), and using the boundary condition that F = 0 when x = 0, gives force as a function of shortening distance: Solving eqn (4) for V, and substituting for dx/dt in eqn (3) gives: df/dt = C 1 (exp(c 2 x))(b(f max F)/(F + a)) (5) Equation (5) was solved numerically to obtain force as a function of time. Force values were calculated at sequential time steps, starting from t = 0, using either the Euler method or a fourth order Runga Kutta approach. The duration of the time steps was decreased progressively in a series of trials until the calculated force curves converged satisfactorily on a common trajectory, which occurred with time steps of 0.5 ms or less. The values used for F max and a were 277 kn m 2 and 99 kn m 2, respectively, and were chosen because these were the average values for the 20 fibres of Fig. 7. The value used for b was varied to examine the effect of changing the maximum shortening velocity. Force curves obtained with short time steps were essentially indistinguishable when using either the Euler or the Runga Kutta method. The force rise predicted by the model like that in live muscle fibres during isometric, tetanic contraction is sigmoidal (Fig. 8). The rate of force rise initially increases with time and later decreases as force approaches the tetanic plateau. The shortening velocity of the contractile component is greatest early in the contraction when the force is low, but when the force is low the stiffness is also low and it is only later in the contraction, when the stiffness is greater but the velocity lower, that the rate of force reaches its maximum. The total SEE stretch predicted by the model at the tetanic plateau is 2.8% of the muscle length. For comparison, the minimum quick release required to bring muscle force to 0 in a slack test measurement, which is a measure of SEE stretch, was about 2% during the tetanic plateau in the fibre of Fig. 3 in Edman (1979). As anticipated, the time course of the force rise F = (C 1 /C 2 )((exp(c 2 x)) 1) (2) Differentiating eqn (2) with respect to time, t, gives: df/dt = C 1 (exp(c 2 x))dx/dt (3) dx/dt here is the shortening velocity, V, which we assume to be an inverse function of muscle force as described by the Hill equation: (F + a)(v + b) = (F max + a) b (4) where F max is the maximum isometric force and a and b are constants. Figure 8. Predicted time course of isometric force following instantaneous fibre activation The Hill parameter values (F max = 277 kn m 2, a/f max = 0.36) were chosen to be the average for the fibres of Figure 7. The thicker line shows the predicted force rise for the average value of V max of the fibres in Fig. 7 (1.9 s 1 ).

10 1016 K. A. P. Edman and R. K. Josephson J Physiol is faster the greater the maximum shortening velocity of the muscle (Fig. 8). The average value of V max for the 20 fibres of Fig. 7 was 1.9 s 1 (s.d. = 0.1 s 1 ), and the average time from force onset to 50% F max for these fibres was 52.4 ms (s.d. = 10.6 ms). In the model, with instantaneous activation, the time from force onset to 50% F max with a V max of 1.9 s 1 was 21.2 ms (Fig. 8). Thus it would seem that in real fibres about 40% of the delay to 50% F max can be attributed to the time required to stretch series elastic elements and 60% to activation time. Discussion Determinants of isometric rise time There was substantial variability among the muscle fibres of this study in maximum isometric stress, F max (coefficient of variation = 18%), maximum shortening velocity, V max (c.v. = 6.6%), the force rise time (c.v. for time to 50% F max = 20%) and in ARI, a measure of fibre activation rate based on the changing position in the force velocity curve early in contraction (c.v. = 19%). The fibres were taken each from a different frog anterior tibialis muscle with no obvious selection. The variability in maximum isometric stress was quite similar to that reported by Elzinga et al. (1989; c.v. = 19%); and in our study, as in that one, maximum shortening velocity was relatively less variable than was F max. V max and ARI apparently varied independently of one another with no statistically significant correlation between the two. But there are significant correlations between force rise time and both V max and ARI, with ARI being a somewhat better predictor of rise time than is V max (Table 1). Variation in ARI and V max together account for 66% or 83% of the variation in force rise time depending on the measure of rise time selected. The force rise time clearly depends on both the rate at which the contractile machinery becomes turned on, and the force velocity properties achieved when it is activated. It is interesting that there was no significant correlation between V max and ARI. Hill (1970, p. 57) remarked that it nearly always happens in biology that structures or cells which are quicker in some respects are quicker also in others; one would be astonished to find a muscle fibre possessing a higher intrinsic speed which did not also have a shorter latent period, or develop its active state more quickly, or begin to relax earlier, or need a higher frequency of stimulation to give a smooth contraction. But intrinsic speed (=V max ) and the rapidity of development of activation depend largely on independent factors the former reflecting the kinetics of cross-bridge attachment, translation and detachment; the latter the time course of calcium release and diffusion, and the kinetics of calcium binding to regulatory sites. It should not be astonishing to find, as is the case (Fig. 7), that the fibre activation time and intrinsic speed are not correlated parameters, at least not within the limited range offered by fibres from a single type of frog muscle. A simulation study was used as a supplement to the correlation analysis in order to estimate the relative contributions to the trajectory of rising force of activation time and of the force velocity properties of the muscle when activated. The model was based on assumed force velocity properties of fully activated frog muscle fibres and on measured series compliance in fibres as a function of fibre stress. Muscle activation was assumed to be instantaneous. The predicted force rise time in the model, to 50% F max, was about 60% shorter than that actually measured from living fibres, from which we conclude that about 60% of the force rise time in living fibres is a consequence of the time course of activation processes. It should be emphasized that this conclusion applies to frog muscle fibres that have been mechanically isolated and in which most of the external connective tissue and series tendon has been removed. The series elastic compliance may be substantially greater in some intact muscles than in isolated fibres, especially those muscles with long tendons. The greater the series compliance the greater is the required shortening of contractile elements to reach the force at the tetanic plateau and the more important are the force velocity properties of the fully activated fibre in determining the force trajectory. Measuring muscle activation The force velocity curve of a muscle or a muscle fibre defines its mechanical capacity the maximum shortening velocity (V max ), the maximum stress that can be generated (F max ), and the expected power output against any load less than F max or at any shortening velocity between 0 and V max. Therefore it is appropriate to use parameters of the F V relationship in quantifying mechanical activation of muscle. Some reports indicate that V max increases with increasing muscle activation (e.g. Julian & Moss, 1981; Julian et al. 1986b). From most accounts, however, it appears that the principal effect of changing muscle activation during the course of a twitch or tetanus, or of experimental manipulations that are expected to alter the level of muscle activation, or of changes in muscle length and therefore myofilament overlap, is simply to alter the scaling of the F V curve along the force axis with V max and the curvature of the relationship (= a/f max ) remaining more-or-less constant (Podolsky & Teichholz, 1970; Cecchi et al. 1978; Edman, 1979; Podolin & Ford, 1986; Ford, 1991; Malamud & Josephson, 1991). In our experiments V max was highest at the onset of contraction and then declined over the first few tens of milliseconds after the latent period (Josephson & Edman, 1998), but otherwise, for most of the early tetanus from onset to plateau, the most obvious change in the F V curve was a progressive shift in the scaling of the F V curve along the force axis.

11 J Physiol Determinants of force rise time 1017 If F V curves at different levels of activation differed only in the scaling along the force axis if two F V curves representing different levels of activation could be superimposed by multiplying the force values of one of them by an appropriate constant then any measure of the relative position of the F V curve along the force axis would be an appropriate measure of activation. Hill s active state measured activation by the intercept of the F V curve with the force axis, which is an obvious measure of the position of the curve along the force axis. But determining F max from F V curves obtained early in a contraction may require extrapolation well beyond the limits of collected data (e.g. Fig. 4), with associated uncertainty about the estimated value. Long extrapolation can be avoided by using relative force at a fixed velocity less than V max as a measure of the position of the F V curve. This is the approach that we have used in quantifying muscle activation. Our results do not fully support the assertion that F V curves at different levels of activation differ only in their scaling along the force axis. The estimated values of F max tended to increase and those of a/f max to decrease as F V curves were collected closer and closer to the onset of contraction. The change in a/f max indicates that F V plots have greater curvature early in a tetanus than later. Hints of the increased curvature of early F V plots are seen in Figs 1 and 4. Because of the greater curvature, F V curves collected very early in a contraction tend to extrapolate to an asymptote parallel to but above the zero velocity axis. The high force low velocity points of the early curves, those points that represent isotonic shortening at forces quite close to the isometric level that existed just before the release, lie at a slightly higher velocity for a given force than would be expected by scaling down the plateau F V curve. Why this should be so is not obvious, but the consequences are pronounced. These high force low velocity points are particularly influential in extrapolations of the further trajectory of the F V curve into the high force region, and small changes in their values are amplified into large changes in the extrapolated value of F max. Our measure of fibre activation, the muscle force at a selected shortening velocity, is a valid indicator of the changing capacity of the fibre, but its predictive value about events elsewhere along the F V relation other than at the chosen velocity is diminished if there are, as is the case, changes in V max or curvature of the F V relation during activation as well as changes in the scaling along the force axis. Measuring force during isovelocity contraction would seem, at first sight, to be the most direct way to ascertain the degree of muscle activation if this is to be determined on the basis of force at a given shortening velocity. However, a difficulty with isovelocity contraction is that force may change during the shortening, leading to change in the length of series elastic elements and a discrepancy between the strain rate imposed on the fibre and the actual strain rate of the contractile component itself. An increase in force during isovelocity contractions is to be expected early in a tetanus as activation rises. Cecchi et al. (1978), however, reported that force during isovelocity shortening was constant, even early in tetanic contractions. They suggested that deactivation associated with shortening suppressed the development of activation. We find, in contrast, that force does increase during isovelocity shortening early in a tetanus (Fig. 2). Changing force introduces uncertainties about the actual shortening velocity of the contractile component during isovelocity shortening. Isotonic contraction therefore is preferable to isovelocity shortening as a means to characterize the F V properties of a fibre. However, to determine the expected force at a given velocity during isotonic contraction may require measurements at several loads and interpolation between the F V values obtained. Having pointed out a potential error in using isovelocity contraction to establish F V characteristics, we should also note that our conclusions about the time course of muscle fibre activation, estimated from isotonic contractions, are in general agreement with results obtained earlier from isovelocity contraction (Cecchi et al. 1978, 1981; Cecchi et al. 1979; Ambrogi-Lorenzini et al. 1983; Lombardi & Manchetti, 1984). It is apparent from F V curves obtained early in contraction with either isovelocity or isotonic shortening that fibres are quite variable in the rapidity with which they became active; that activation times are rather long, on the order of 100 ms at low temperature; and that activation is nearly complete by the time that isometric force has risen to half its plateau value. Ford (1991), in reviewing the problem of characterizing muscle activation, proposed using maximum power (P max ) as an index of the degree of muscle activation. Maximum power is certainly a reasonable and defensible measure of the mechanical capacity of muscle, but it is not useful for assessing activity very early in a contraction. Maximum power occurs at an optimum shortening velocity, V opt. If F V curves at different levels of activation differ only in their scaling along the force axis, V opt is independent of the level of activation. For a F V curve described by the Hill equation, V opt is a function of the value of the ratio b/v max (= a/f max, Woledge et al. 1985). This ratio is likely to be about 0.25 in many muscles and muscle fibres, and the associated value of V opt is about 0.31V max. This shortening velocity is too slow to be obtainable early in a contraction with forces equal to or less than existing isometric force. Neither of the two early F V curves in Fig. 4 extends to forces high enough and velocities low enough to reach P max. Early in a contraction an initial stretch would be required to raise the fibre force to that required to reach P max, and the effects of rapid stretch may be profound and long lasting (e.g. Morgan, 1990). The shortening velocity under a light load has also been used as a measure of the degree of activation (Fenn & Marsh, 1935; Abbott & Ritchie, 1951; Hill, 1951; Malamud