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Department of Physiology, University of Pennsylvania School of Medicine, Philadelphia, PA 19104

Correspondence to S.M. Baylor: baylor{at}mail.med.upenn.edu

TOP ABSTRACT INTRODUCTION MATERIALS AND METHODS RESULTS DISCUSSION REFERENCES

 
Ca²⁺ release from the sarcoplasmic reticulum (SR) of skeletal muscle takes place at the triadic junctions; following release, Ca²⁺ spreads within the sarcomere by diffusion. Here, we report multicompartment simulations of changes in sarcomeric Ca²⁺ evoked by action potentials (APs) in fast-twitch fibers of adult mice. The simulations include Ca²⁺ complexation reactions with ATP, troponin, parvalbumin, and the SR Ca²⁺ pump, as well as Ca²⁺ transport by the pump. Results are compared with spatially averaged Ca²⁺ transients measured in mouse fibers with furaptra, a low-affinity, rapidly responding Ca²⁺ indicator. The furaptra f_CaD signal (change in the fraction of the indicator in the Ca²⁺-bound form) evoked by one AP is well simulated under the assumption that SR Ca²⁺ release has a peak of 200–225 µM/ms and a FDHM of 1.6 ms (16°C). f_CaD elicited by a five-shock, 67-Hz train of APs is well simulated under the assumption that in response to APs 2–5, Ca²⁺ release decreases progressively from 0.25 to 0.15 times that elicited by the first AP, a reduction likely due to Ca²⁺ inactivation of Ca²⁺ release. Recovery from inactivation was studied with a two-AP protocol; the amplitude of the second release recovered to >0.9 times that of the first with a rate constant of 7 s^–1. An obvious feature of f_CaD during a five-shock train is a progressive decline in the rate of decay from the individual peaks of f_CaD. According to the simulations, this decline is due to a reduction in available Ca²⁺ binding sites on troponin and parvalbumin. The effects of sarcomere length, the location of the triadic junctions, resting [Ca²⁺], the parvalbumin concentration, and possible uptake of Ca²⁺ by mitochondria were also investigated. Overall, the simulations indicate that this reaction-diffusion model, which was originally developed for Ca²⁺ sparks in frog fibers, works well when adapted to mouse fast-twitch fibers stimulated by APs.

	INTRODUCTION

TOP ABSTRACT INTRODUCTION MATERIALS AND METHODS RESULTS DISCUSSION REFERENCES

 
During normal excitation–contraction (EC) coupling of vertebrate twitch fibers, an action potential (AP) elicits a large and rapid release of Ca²⁺ from the SR. As a result, myoplasmic free [Ca²⁺] rises quickly to a high level near the triadic junctions, the location of the SR Ca²⁺ release channels. Ca²⁺ then spreads by diffusion throughout the sarcomere while binding to sites on myoplasmic Ca²⁺ buffers, including ATP, troponin, parvalbumin, and the SR Ca²⁺ pump. On a longer time scale, the released Ca²⁺ is resequestered within the SR by ATP-dependent transport by the SR Ca²⁺ pump.

Kinetic models have been used to describe some, but not all, of these events in the setting of a functioning muscle cell. The earliest models started with an assumed (Robertson et al., 1981) or measured (Baylor et al., 1983) waveform of the change in (spatially averaged) free [Ca²⁺] ([Ca²⁺]) or an assumed waveform of SR Ca²⁺ release (Gillis et al., 1982); the diffusion of Ca²⁺ and its mobile buffers was ignored. Also ignored were effects due to binding of Ca²⁺ by the SR Ca²⁺ pump. The model of Cannell and Allen (1984) was the first to consider the diffusive aspects of the problem. The myoplasm of a half sarcomere of a myofibril of a frog twitch fiber was subdivided into a large number of compartments of equal volume; radial symmetry was assumed. AP-evoked activity was initiated by SR Ca²⁺ release into one compartment, and changes in Ca²⁺ binding and diffusion were calculated for each compartment by integration of a large set of simultaneous first-order differential equations; transport, but not binding, of Ca²⁺ by the SR Ca²⁺ pump was included. To link the model to experiments, the spatially averaged luminescence change from aequorin was simulated. An acceptable level of agreement was observed between the simulated and measured aequorin signals, which supported the basic concepts of the model. An important conclusion was that large gradients in [Ca²⁺] exist within the sarcomere during, and for a few tens of milliseconds after, SR Ca²⁺ release (20°C).

A similar multicompartment model was developed by Baylor and Hollingworth (1998) to simulate AP-evoked Ca²⁺ movements in frog twitch fibers (16°C). This model incorporated more up-to-date information about the time course of SR Ca²⁺ release and included ATP as a myoplasmic Ca²⁺ buffer. The experimental comparison for these simulations was with [Ca²⁺] estimated with furaptra (Raju et al., 1989), a rapidly responding fluorescent Ca²⁺ indicator with 1:1 stoichiometry (Ca²⁺:indicator). These simulations confirmed the existence of large gradients in [Ca²⁺] within the sarcomere during and shortly after SR Ca²⁺ release. A novel finding was that Ca²⁺ binding by ATP and the diffusion of CaATP have a strong influence on the amplitude and time course with which Ca²⁺ binds to troponin at different locations along the thin filament.

Similar multicompartment simulations have not yet been reported for mammalian fibers. Mammalian Ca²⁺ measurements have been the focus of much recent work, including investigations into alterations in Ca²⁺ homeostasis by drugs and diseases (e.g., Dirksen and Avila, 2004; Woods et al., 2004; Capote et al., 2005; Yeung et al., 2005; Brown et al., 2007; Pouvreau et al., 2007). Thus, it is important to establish valid modeling methods for interpretation of Ca²⁺ measurements in mammalian muscle. Although it is expected that many of the conclusions from the amphibian studies may apply to mammalian fibers, there are structural, biochemical, and physiological differences between mammalian and amphibian fibers that will influence the results and, possibly, the interpretations. For example, in mammals, the triadic junctions are offset 0.5 µm from the z line (Smith, 1966; Eisenberg, 1983; Brown et al., 1998), whereas, in amphibians, the junctions are located at the z line. The functional properties of the dihydropyridine receptors, the voltage sensors of EC coupling, are also different, as indicated by differences in the amount and voltage dependence of muscle charge movement (e.g., Hollingworth and Marshall, 1981). Differences are also found in the organization of the ryanodine receptors (RyRs), the Ca²⁺ release channels of the SR, at the triadic junction. In fibers from mammalian limb muscles (which are used in the experiments in our laboratory), the RyRs are found in junctional arrays only, whereas, in amphibian fibers, both junctional and parajunctional arrays are present (Felder and Franzini-Armstrong, 2002). Moreover, adult mammalian limb fibers have only the RyR1 isoform (Conti et al., 1996; Flucher et al., 1999), whereas amphibian fibers have a nearly equal mixture of the RyR1 and RyR3 isoforms (sometimes referred to as RyR and RyRß) (Olivares et al., 1991; Lai et al., 1992; Murayama and Ogawa, 1992; O'Brien et al., 1993). Finally, the time course of [Ca²⁺] evoked by an AP is significantly different in mammalian and amphibian fibers. At 16°C, the FDHM (full duration at half maximum) of [Ca²⁺] in mouse fast-twitch fibers is 5 ms, whereas, in frog twitch fibers of similar diameter, it is 8–9 ms (Hollingworth et al., 1996).

In this article, we report multicompartment simulations of sarcomeric Ca²⁺ movements evoked by APs in mouse fast-twitch fibers. The results are compared with spatially averaged Ca²⁺ transients measured in these fibers with furaptra and with simulations based on a single-compartment model. This article also examines how sarcomere length, the location of the triadic junctions, the resting level of [Ca²⁺] ([Ca²⁺]_R), the presence of parvalbumin, and the possible uptake of Ca²⁺ by mitochondria are expected to affect the results.

A preliminary version of the results has appeared in abstract form (Baylor, S.M., and S. Hollingworth. 2007. Biophys. J. 92:311a).

	MATERIALS AND METHODS

TOP ABSTRACT INTRODUCTION MATERIALS AND METHODS RESULTS DISCUSSION REFERENCES

 
Simulations
A multicompartment model was used to simulate myoplasmic Ca²⁺ binding and diffusion as well as SR Ca²⁺ transport in mouse EDL (extensor digitorum longus) fibers activated by APs (16°C). Simulations were performed for a half sarcomere of a myofibril, which is divided into 18 equal volume units (six longitudinal by three radial; Fig. 1) at two sarcomere lengths, 4.0 (Fig. 1 A) and 2.4 µm (Fig. 1 B). The myofibrillar radii were 0.375 and 0.484 µm, respectively, and the computational water volume was 0.884 fL in both cases. The calculations were performed with MLAB (Civilized Software). An 18-compartment model appears to be sufficient for the simulations of this article; in a study of Ca²⁺ binding and diffusion evoked by an AP in frog fibers, simulations of the fluo-3 Ca²⁺ signal performed with an 18- and a 100-compartment model yielded similar results (Hollingworth et al., 2000).

View larger version (14K): [in this window] [in a new window] [Download PPT slide]   Figure 1. Cut-away view of a half sarcomere of one myofibril showing the arrangement of the 18 equal-volume compartments (six longitudinal x three radial) in the simulations at a sarcomere length of 4 µm (A) and 2.4 µm (B). SR Ca2+ release enters the myoplasm near the middle of the thin filament in the outer compartment row (large downward arrow); Ca2+ pump activity occurs within all compartments in the outer row (small upward arrows). In both arrangements, troponin is restricted to the compartments located within 1 µm of the z-line (the region containing thin filaments, average length 1 µm). Because the buffer concentrations in Table I are averages over the entire myoplasmic volume, the actual compartment concentration of troponin is 2.0 (A) or 1.2 (B) times the value listed in Table I, and the actual compartment concentration of Ca2+ pump molecules is three times (both A and B) the value listed in Table I. In both parts, the vertical and horizontal calibrations are different and not to scale.

View larger version (8K): [in this window] [in a new window] [Download PPT slide]   Figure 2. Reaction schemes for ATP (A), parvalbumin (B), troponin (C), the SR Ca2+ pump (D), and furaptra (=Dye; E). Table I gives the total reactant concentrations and Table II the reaction rate constants. With the resting values of free [Ca2+], free [Mg2+], and pH in Table I, the fractional amounts of the various reactants at rest are (A) ATP (1.000), CaATP (0.000); (B) Parv (0.062), CaParv (0.258), MgParv (0.680); (C) Trop (0.993), CaTrop (0.006), Ca2Trop (0.001); (D) E (0.006), CaE (0.006), Ca2E (0.004), MgE (0.123), Mg2E (0.123), HE (0.062), H2E (0.615), H3E (0.062), H4E (0.001); (E) Dye (0.451), CaDye (0.000), PrDye (0.548), CaPrDye (0.000). The resting fraction of furaptra in the protein-bound form, 0.548, is approximately consistent with that estimated from furaptra's apparent myoplasmic diffusion coefficient, 0.68 x 10–6 cm2s–1, measured in frog twitch fibers at 16°C (Konishi et al., 1991). From this diffusion coefficient, Zhao et al. (1996) estimated that the fraction of furaptra in the protein-bound form in resting frog fibers is 0.58.

Ca²⁺ Release.
In mammalian fibers, the triadic junctions are located at the periphery of a myofibril in a narrow region that is offset 0.5 µm from the z-line, both in slack and stretched fibers (Brown et al., 1998). This Ca²⁺ release location is consistent with the experiments of Gomez et al. (2006) on mouse fast-twitch fibers from flexor digitorum brevis muscle stimulated by APs, in which Ca²⁺ release domains offset 0.5 µm from the z-line were detected with confocal microscopy. In the simulations, the SR Ca²⁺ release flux is assumed to enter the outer compartment that includes the region 0.5 µm from the z-line; for the simulations at sarcomere lengths 4.0 and 2.4 µm, this is the second and third compartment, respectively (large downward arrows in Fig. 1).

The time course of SR Ca²⁺ release in response to an AP was calculated with an empirical function (compare Baylor and Hollingworth, 1998):

(1)

The time-shift parameter T was set to 1.4 ms to simulate the delay from the time of AP generation to the onset of SR Ca²⁺ release. L, 1, and 2 were set to 5, 1.3 ms, and 0.5 ms, respectively, so that the FDHM of the release flux would be 1.6 ms, the same as the mean value estimated from the 11 fibers of this study in a single-compartment analysis of their furaptra Ca²⁺ transients (compare last section of Results). The value of R was adjusted to give good agreement between the amplitudes of the simulated and measured furaptra Ca²⁺ transients (next section).

Experimental Procedures
Spatially averaged myoplasmic Ca²⁺ signals evoked by APs were measured with furaptra (also known as mag-fura-2) in intact EDL fibers from 6–14-wk-old mice. Furaptra is a fluorescent indicator whose dissociation constant for Ca²⁺ is relatively large (49 µM in a standard salt solution with 1 mM Mg²⁺; 16°C) and which responds rapidly to [Ca²⁺] in muscle fibers (Konishi et al., 1991; Rome et al., 1996). Because furaptra's dissociation constant for Mg²⁺ is large (5.3 mM in a standard salt solution at 16°C), interference in the Ca²⁺ measurements from changes in [Mg²⁺] are expected to be small. As argued previously (Baylor and Hollingworth, 2003), furaptra appears to supply the most accurate measurements to date of (spatially averaged) [Ca²⁺] in mammalian muscle fibers stimulated by APs.

Most results from the furaptra EDL experiments were reported previously (Hollingworth et al., 1996; Baylor and Hollingworth, 2003); the results with the two-AP protocol (see Figs. 10 and 11) are new to this study. In each experiment, one fiber within a small, manually dissected bundle of fibers was microinjected with a membrane-impermeant form of furaptra (K₄mag-fura-2; Invitrogen); the average myoplasmic furaptra concentration was 0.1 mM. To minimize movement artifacts in the fluorescence records, the fibers were stretched to long-sarcomere length, 3.8 ± 0.2 µm (mean ± SD). F/F_R, the change in indicator fluorescence divided by its resting fluorescence, was recorded from a 300-µm length of fiber in response to either one external shock, a five-shock train at 67 Hz, or two shocks separated by a delay of 15 to 400 ms. In most experiments, the fluorescence excitation and emission wavelengths were 410 ± 20 nm and >480 nm, respectively.

Comparisons between the Multicompartment Simulations and the Measurements
The signal that can be most directly compared between the simulations and the measurements is the spatially averaged change in the fraction of furaptra bound with Ca²⁺ (denoted f_CaD). The spatially averaged F/F_R measurements were converted to spatially averaged f_CaD as described previously ( Hollingworth et al., 1996; Baylor and Hollingworth, 2003). With 410-nm excitation, f_CaD is calculated from F/F_R with the equation:

(2)

Spatially averaged f_CaD in the simulations was obtained by averaging the simulated f_CaD waveforms in the 18 compartments. These waveforms, which include changes in both protein-free and protein-bound indicator (Fig. 2 E and Table II, E), were calculated as ([CaDye]+[CaPrDye])/[Dye_Total], where [Dye_Total] is the total furaptra concentration (usually 100 µM, Table I). Within each compartment, the time courses of the [CaDye] and [CaPrDye] waveforms were always virtually identical; peak [CaDye], however, was always somewhat larger than peak [CaPrDye] (see dissociation constants in Table II, E).

Single-Compartment Estimation of [Ca²⁺] from f_CaD
For single-compartment modeling (compare Figs. 3 and 4 and the last section of Results), f_CaD was converted to (spatially averaged) [Ca²⁺] with the equation:

(3)

Eq. 3 is based on the idea of a spatially uniform, kinetically rapid, 1:1 binding reaction between furaptra and Ca²⁺. K_D, furaptra's apparent dissociation constant for Ca²⁺ in the myoplasm, is assumed to be 96 µM (Hollingworth et al., 1996; Baylor and Hollingworth, 2003). This K_D was selected previously to make the estimated peak of [Ca²⁺] evoked by an AP in a frog twitch fiber match that estimated with PDAA (purpurate diacetic acid) (Konishi et al., 1991). PDAA is a rapidly responding absorbance indicator whose fractional binding to myoplasmic constituents of large molecular weight is relatively small (0.2–0.5; Hirota et al., 1989; Konishi and Baylor, 1991) and whose K_D for Ca²⁺ (0.9 mM in a standard salt solution) is sufficiently large that nonlinearities in Ca²⁺ binding near the SR Ca²⁺ release sites is expected to be small (Hirota et al., 1989). In contrast, with furaptra, whose K_D is an order of magnitude smaller, substantial nonlinear responses are expected near the release sites, while nearly linear responses are expected in regions removed from the release sites. It is expected that use of Eq. 3 to estimate [Ca²⁺] evoked by APs cannot be accurate at all time points because of the nonlinear Ca²⁺ binding by furaptra near the release sites (Baylor and Hollingworth, 1998; see also Results and legend to Table II).

View larger version (12K): [in this window] [in a new window] [Download PPT slide]   Figure 3. Spatially averaged responses elicited by a single AP (left) and a 67-Hz train of 5 APs (right) initiated at 0 time. (A and B) Experimental fCaD and [Ca2+] responses averaged from four fibers in which fCaD was relatively free of movement artifacts; [Ca2+] was calculated from fCaD with Eq. 3 in Materials and Methods. The mean sarcomere length was 3.8 µm (range, 3.7–3.9 µm) and the mean myoplasmic concentration of furaptra was 89 µM (range, 64–113 µM). Fiber identification numbers: 032596.2, 040596.1, 040896.1, and 040996.3. (C and D) fCaD and [Ca2+] responses averaged over the 18 compartments in the multicompartment simulations. The bottom traces show the release fluxes used to drive the simulations. The peaks of the release fluxes are 205 µM/ms in C and 208, 53, 45, 38, and 32 µM/ms in D (APs 1–5, respectively); the FDHM of all release fluxes is 1.6 ms. The values of [CaTotal] are 349 µM in C and 354, 90, 76, 66, and 55 µM in D.

View larger version (13K): [in this window] [in a new window] [Download PPT slide]   Figure 4. Comparison of spatially averaged waveforms in the experiments and simulations. (A) superposition of the fCaD waveforms in Fig. 3 C and Fig. 3 A (noise-free and slightly noisy traces, respectively). (C) Superposition of the [Ca2+] waveform in Fig. 3 C (noise-free trace) and the [Ca2+] waveform in Fig. 3 A (noisy trace). (E) Superposition of the [Ca2+] waveform in Fig. 3 C (continuous trace) and [Ca2+] calculated with Eq. 3 from the fCaD waveform in Fig. 3 C (dashed trace). (B, D, and F) Similar presentation for the measurements and simulations in Fig. 3 (B and D).

	RESULTS

TOP ABSTRACT INTRODUCTION MATERIALS AND METHODS RESULTS DISCUSSION REFERENCES

Fig. 4 (A and B) shows a comparison of the experimental and simulated f_CaD waveforms in Fig. 3. Good agreement is observed between these waveforms. This is one of the central results of this article and shows that the reaction-diffusion equations of spark Model 3, which was developed for frog fibers (Hollingworth et al., 2006), work well on this time scale when adapted to mouse fast-twitch fibers activated by APs. Columns 2 and 3 of Table IV give additional information about the amplitude and time course of the waveforms in Fig. 3 (A and C). The values of peak amplitude and FDHM of f_CaD are 0.155 and 5.3 ms, respectively, in the measurements and 0.155 and 5.1 ms in the simulations.

The traces in Fig. 4 (E and F) reveal that the difference in the waveforms in Fig. 4 (C and D) is due to an error associated with the single-compartment method for estimating [Ca²⁺]. The continuous traces in Fig. 4 (E and F) are the same as the simulated traces in Fig. 4 (C and D), i.e., are the "true" spatially averaged [Ca²⁺] waveforms as determined in the multicompartment simulations. The dashed traces are the single-compartment estimates of spatially averaged [Ca²⁺] based on the simulated f_CaD waveforms in Fig. 4 (A and B) and Eq. 3. This comparison reveals that the single-compartment method overestimates [Ca²⁺]. The error arises from the assumption, inherent in a single-compartment model, that [Ca²⁺] is homogenous throughout the sarcomere. However, the multicompartment simulations reveal highly nonuniform changes, particularly at times during and immediately after Ca²⁺ release (Figs. 5 and 6, described below). As noted in the last section of Materials and Methods, the large [Ca²⁺] in and near the release compartment produces substantial nonlinearities in furaptra's response that vary with time. As a result, it is not possible to accurately calibrate the entire [Ca²⁺] waveform from f_CaD based on Eq. 3 and a single value of K_D.

View larger version (18K): [in this window] [in a new window] [Download PPT slide]   Figure 5. Ca2+ concentration changes in the 18 compartments in the simulation of Fig. 3 C. [CaDye] includes both protein-free and protein-bound furaptra (Fig. 2 E); [CaTrop] and [CaPump] include the Ca2+ ions that are both singly and doubly bound to these buffers (Fig. 2, C and D). In each panel, peak amplitudes decline with distance from the release compartment; dashed traces denote changes in compartments adjacent to the z-line. In several panels, there is an apparent grouping of the traces in triplets, which corresponds to the three radial compartments at a given longitudinal location; at some longitudinal locations, the changes in the three radial compartments are indistinguishable. To facilitate comparisons among panels, the calibrations of the ordinates are referred to the total concentrations listed in Table I, which are spatially averaged. Because troponin and the Ca2+ pump are not located in all compartments, the simulated amplitudes of [CaTrop] and [CaPump] in their respective compartments are larger than shown in the plots by the ratio of the total site concentrations in these compartments to the spatially averaged concentrations (see Fig. 1, legend).

View larger version (19K): [in this window] [in a new window] [Download PPT slide]   Figure 6. Ca2+ concentration changes in the 18 compartments in the simulation of Fig. 3 D. The presentation is like that in Fig. 5.

The traces in the other panels in Fig. 5 follow a similar pattern. The largest responses are found in the release compartment, and responses in the other compartments decline with distance from the release compartment; the dashed traces denote responses in the compartments next to the z-line. These other panels show changes in the concentration of Ca²⁺ bound to furaptra ([CaDye]; B), troponin ([CaTrop]; C), parvalbumin ([CaParv]; D), and the SR Ca²⁺ pump ([CaPump]; E). Panel F shows the concentration of released Ca²⁺ returned to the SR by the transport step of the Ca²⁺ pump ([CaPumped]). Not shown are the changes in Ca²⁺ binding to ATP ([CaATP]) in the 18 compartments. The time courses of these waveforms are very similar to those of [Ca²⁺] in the corresponding compartment (panel A). The amplitudes of the [CaATP] and [Ca²⁺] waveforms are related by empirical factors that vary slightly with compartment location. For example, with 1 AP, peak [CaATP] in the release compartment is 3.0 times peak [Ca²⁺] (240 and 80 µM, respectively). In the compartment farthest from the release compartment, peak [CaATP] is 3.7 times peak [Ca²⁺] (10.54 and 2.85 µM, respectively). In the other compartments, peak [CaATP] is between 3.4 and 3.8 times peak [Ca²⁺]. Fig. 6 shows analogous spatially resolved information for the 5-AP simulation of Fig. 3 D.

The large gradients in [Ca²⁺], in combination with the differing reaction schemes in Fig. 2 and the differing diffusion constants in Table III (B) generate the variety of waveforms in the panels of Figs. 5 and 6. With troponin, which binds Ca²⁺ relatively quickly, differences among the compartments occur primarily at time <10 ms. With parvalbumin and the SR Ca²⁺ pump, the differences persist for longer times. Of importance for activation of the myofilaments, peak [CaTrop] in response to both 1 and 5 APs is large in all troponin-containing compartments, 210 µM. Thus, peak occupancy is 0.88 times the concentration of Ca²⁺-free troponin sites at rest, 239.3 µM (=240 µM minus the resting occupancy, 0.7 µM; see Table I and Fig. 1, legend).

Progressive Changes during High-Frequency Stimulation
According to the simulations in Fig. 3 D, the first AP in a 67-Hz train of APs releases 350 µM total [Ca²⁺] ([Ca_Total]), whereas APs 2–5 in the train release 0.25–0.15 times the first release. The large reduction in release with subsequent APs in the train is probably due to Ca²⁺ inactivation of Ca²⁺ release (Baylor et al., 1983; Schneider and Simon, 1988; Jong et al., 1995) produced by the rise in myoplasmic [Ca²⁺] caused by the prior release(s).

It is possible that myoplasmic Ca²⁺ buffering due to the presence of 100 µM furaptra increases SR Ca²⁺ release above its normal value by reducing [Ca²⁺] and thus Ca²⁺ inactivation of Ca²⁺ release. This possibility was explored in simulations with 0 and 100 µM furaptra. According to these simulations, the peak of [Ca²⁺] in the release compartment (compare Fig. 1 A), which is likely to contain the receptor for Ca²⁺ inactivation of Ca²⁺ release, is 86 µM without furaptra and 80 µM with 100 µM furaptra; the time course of [Ca²⁺] in this compartment is essentially identical with 0 and 100 µM furaptra. Any perturbation in SR Ca²⁺ release due to the 7% reduction in the peak of [Ca²⁺] is expected to be minor.

An interesting feature of the waveforms in Fig. 3 (B and D) is that the changes in f_CaD and [Ca²⁺] evoked by the individual APs in the train decay progressively more slowly at later times. This feature is quantified in Fig. 7. The circles and X's in panel A show measured and simulated values, respectively, of the rate constant characterizing the early decay of f_CaD plotted as a function of the time of AP stimulation during the train. As expected from the similarity of the measured and simulated f_CaD traces in Fig. 4 B, the circles and X's in Fig. 7 A have similar values at all time points. The largest reduction in the rate constants occurs between the first and second shocks in the train (Hollingworth et al., 1996).

View larger version (11K): [in this window] [in a new window] [Download PPT slide]   Figure 7. Analysis of changes in spatially averaged waveforms during a five-shock, 67-Hz stimulus (A, measurements; A–D, simulations). (A) The fCaD waveforms in Fig. 3 (B and D) were analyzed to estimate the rate constants characterizing the early decay of the changes evoked by the five APs. For each change, a decaying single-exponential function without an offset was fitted to the falling phase of fCaD during a 4–5-ms period that began 1–1.5 ms after peak; the time points that specify the beginning and end of each fitting period are denoted t1 and t2. The fitted rate constants are plotted vs. the times of the corresponding external shocks (open circles, measurements; X's, simulations). (B) Repeat of the analysis in A for the [Ca2+] waveform in Fig. 3 D; for each evoked change, rate constants were again determined between t1 and t2. (C) [Site] denotes the simulated spatially averaged concentration of sites on troponin (circles), parvalbumin (squares), and the Ca2+ pump (triangles) that are not bound with Ca2+, Mg2+, or H+ at time t1; the values were determined from the simulations for each of the five evoked changes. (D) Simulated values of [CaSite](t2) – [CaSite](t1), defined as the change in the spatially averaged concentration of Ca2+ bound to troponin (circles), parvalbumin (squares), and the pump (triangles) between times t1 and t2. As in C, the values were determined from the multicompartment simulations for each of the five evoked changes.

To quantify the decrease in the concentration of buffer sites available to bind Ca²⁺ during the train, the simulated concentration of sites that are free of bound ions shortly after each peak of f_CaD—and thus are immediately available to bind Ca²⁺ during the decay phase of f_CaD—were determined. The spatially averaged concentrations of these sites (denote [Site]) are plotted in Fig. 7 C for troponin, parvalbumin, and the Ca²⁺ pump (the more slowly reacting Ca²⁺ buffers) vs. the time of the corresponding AP stimulus during the train. These concentrations were determined at the time points that marked the beginning of the period during which the simulated rate constants in Fig. 7 (A and B) were estimated (see legend). Fig. 7 C shows that the [Site] values for troponin and parvalbumin but not the Ca²⁺ pump decrease substantially during the train.

Fig. 7 D shows the associated changes in Ca²⁺ binding by these buffers. The changes in the simulated values of [CaTrop], [CaParv], and [CaPump] were determined between the starting and ending times (denoted t₁ and t₂, respectively) during which the simulated decay rate constants in A and B were estimated. These values (denoted [CaSite](t₂) – [CaSite](t₁)) are plotted in Fig. 7 D vs. the time of AP stimulation during the train. As expected from Fig. 7 C, the concentrations of Ca²⁺ bound by troponin and parvalbumin but not the pump decrease substantially during the train. (Note that ATP and furaptra respond rapidly to [Ca²⁺], and the values of [CaSite](t₁) – [CaSite](t₂) for ATP and furaptra [not depicted] are negative rather than positive at all time points, i.e., ATP and furaptra release rather than bind Ca²⁺ between t₁ and t₂. Thus, decreased Ca²⁺ binding by ATP and/or furaptra does not underlie the decrease in the simulated rate constants in Fig. 7 (A and B). Also, the simulated concentration of Ca²⁺ transported by the Ca²⁺ pump progressively increases rather than decreases during the train (Fig. 6 F); thus, decreased transport of Ca²⁺ by the pump does not underlie the decrease in the simulated rate constants.)

Fig. 7 (C and D) indicates that a reduction in Ca²⁺ binding by troponin and parvalbumin but not the pump is primarily responsible for the decrease in the simulated rate constants in Fig. 7 (A and B) and presumably also in the measured rate constants in Fig. 7A.

The Effect of Parvalbumin on the FDHM of [Ca²⁺], f_CaD, and [CaTrop]
Parvalbumin is thought to facilitate muscle relaxation in many types of fast-twitch muscle fibers (e.g., Gillis et al., 1982; Baylor et al., 1983; Rome et al., 1996). In twitch fibers of mice, the concentration of parvalbumin is strongly correlated with the fiber type (Heizmann et al., 1982; Ecob-Prince and Leberer, 1989). Slow-twitch fibers (type I) and fast glycolytic fibers (type IIa) have negligible and small-to-negligible concentrations, respectively; in contrast, fast oxidative-glycolytic fibers (type IIb) generally have large parvalbumin concentrations, comparable to that found in frog twitch fibers (Table I). In mouse EDL muscle, whose fiber-type composition is reported to be 60% type IIb and 40% type IIa (Ecob-Prince and Leberer, 1989), the average parvalbumin concentration is 4.4 g/kg wet weight (Heizmann et al., 1982). This corresponds to a myoplasmic concentration of Ca²⁺/Mg²⁺ sites of 1.26 mM, as calculated from (a) parvalbumin's molecular weight (12,000), (b) the presence of two Ca²⁺/Mg²⁺ sites per parvalbumin molecule, and (c) the factor 1/0.58 to convert a kilogram of muscle wet weight to a liter of myoplasmic water (Baylor et al., 1983). The value 1.26 mM is close to the standard parvalbumin site concentration in our simulations, 1.5 mM (Table I). If parvalbumin is present exclusively in the type IIb fibers of EDL muscle, the concentration of parvalbumin sites in these fibers could be 2 mM. Because the parvalbumin concentration is reported to vary widely between type IIa and type IIb fibers, it was of interest to compare simulation results with different concentrations of parvalbumin sites (denoted [Parv_Total]).

In these simulations, [Parv_Total] was varied between 0 and 2,000 µM. At each concentration, the amplitude of the release flux (set by the value of R in Eq. 1) was adjusted so that peak f_CaD was 0.155, the same as in Fig. 3 (A and C); the values of the total concentration of released Ca²⁺ varied from 298 µM (for [Parv_Total] = 0) to 365 µM (for [Parv_Total] = 2,000 µM). Fig. 8 shows the relations between [Parv_Total] and the values of FDHM of spatially averaged [Ca²⁺] (squares), f_CaD (circles), and [CaTrop] (X's). An increase in [Parv_Total] from 0 to 2,000 µM produces a large reduction in all three FDHMs; by 61%, 74%, and 83%, respectively. As expected from the constraint of a constant peak f_CaD, much smaller changes were observed for peak [Ca²⁺] and peak [CaTrop], +9% and –2%, respectively. These results supply a multicompartment confirmation that Ca²⁺ binding by parvalbumin is expected to shorten the duration of the twitch by abbreviating the time that [Ca²⁺] is elevated above the contractile threshold (Gillis et al., 1982) and hence abbreviating the time of thin filament activation, which is roughly proportional to the FDHM of [CaTrop].

View larger version (9K): [in this window] [in a new window] [Download PPT slide]   Figure 8. Effect of [ParvTotal] (the total concentration of the Ca2+/Mg2+ sites on parvalbumin) on the values of FDHM of spatially averaged fCaD and [Ca2+] (circles and squares, respectively, calibrated on the lefthand ordinate) and of spatially averaged [CaTrop] (X points, calibrated on the righthand ordinate) in the multicompartment simulations. At each value of [ParvTotal], the value of R in Eq. 1 was adjusted to give peak fCaD = 0.155, the value in column 2 of Table IV, A; no changes were made to other simulation variables.

View larger version (8K): [in this window] [in a new window] [Download PPT slide]   Figure 9. Relation between the peak amplitude and FDHM of spatially averaged fCaD in the measurements and simulations. The large symbols show results from 11 rested EDL fibers stimulated by a single AP; circles, squares, and triangles give the values from the four fibers of Fig. 3 (A and B), the three fibers of Fig. 10 A, and the remaining fibers of this study, respectively. The curves through the small filled symbols show the relations determined in the multicompartment simulations with many values of R in Eq. 1 at three values of [Ca2+]R: 150 nM (top curve), 50 nM (the standard value in the simulations; middle curve), and 0 nM (bottom curve). For the middle curve, the range in release fluxes that corresponds to the range in the experimental amplitudes of fCaD is 191–233 µM/ms.

[Ca²⁺]_R and the Relation between the Peak and FDHM of f_CaD

The relation in Fig. 9 was also simulated at [Ca²⁺]_R = 50 nM with values of [Parv_Total] of 1,000 and 2,000 µM (not depicted). Compared with the continuous curve in Fig. 9, these relations are shifted to higher and lower values on the ordinate, respectively. These effects of [Ca²⁺]_R and [Parv_Total] on the simulated relation are reasonable because increases/decreases in [Ca²⁺]_R will decrease/increase the concentration of metal-free sites available on parvalbumin, which will increase/decrease FDHM of f_CaD (Figs. 7 and 8).

The explanation of the failure of the data points in Fig. 9 to lie on a single curve includes one or more of the following: fiber-to-fiber variation in [Ca²⁺]_R, fiber-to-fiber variation in [Parv_Total], and noise in the experimental determination of peak and FDHM of f_CaD. Overall, the data are generally consistent with [Ca²⁺]_R = 50 nM and [Parv_Total] = 1,500 µM, as used in the standard simulation conditions (continuous curve).

Recovery of SR Ca²⁺ Release after a Conditioning Stimulus
Changes in the amount of SR Ca²⁺ release and changes in the decay phase of f_CaD subsequent to a first stimulus were described above in connection with Fig. 3 (B and D) and Fig. 7. This section describes measurements with a two-pulse protocol that was used to study the ability of the fiber to recover from the effects of a first stimulus. In this protocol, a first AP elicits a first Ca²⁺ release; then, after a variable waiting period, a second AP elicits a second release. Analysis of the amplitudes of the associated release fluxes is expected to give information about recovery of the release system from Ca²⁺ inactivation of Ca²⁺ release (e.g., Jong et al., 1995). The analysis is also expected to give information (described in the next section) about the time required for the slowly reacting myoplasmic Ca²⁺ buffers to recover their ability to bind Ca²⁺ and thus speed the decay of f_CaD.

The two-AP protocol was performed in three fibers, all of which were at a sarcomere length of 4.0 µm. Five common waiting periods between pulses were used in these experiments: 40, 80, 160, 320, and 400 ms; in addition, in two of the three experiments, a 15-ms waiting period was used. The top superimposed traces in Fig. 10 A show the averaged f_CaD signals recorded with the five common waiting periods. The f_CaD amplitudes evoked by the second stimulus are smaller than that evoked by the conditioning stimulus, and the FDHM values are larger; with time, these values recover toward those of the conditioning stimulus. In some of the records, a small movement artifact is apparent during the later falling phase of the f_CaD evoked by the conditioning stimulus, as indicated by a slight undershoot of the baseline (time of peak undershoot 90 ms). The time course of the movement artifact appeared to be similar to the time course of twitch tension (unpublished data), as both the tension transient and the undershoot of f_CaD returned to baseline with very similar time courses. The tension transient was therefore used to correct for the movement artifact in f_CaD. To do so, the tension transient was inverted and scaled so that its time course during the period 90–250 ms matched that of the negative phase of f_CaD; the scaled tension transient was then subtracted from each of the f_CaD waveforms. The corrected waveforms are shown as the bottom superimposed traces in Fig. 10 A.

View larger version (16K): [in this window] [in a new window] [Download PPT slide]   Figure 10. Spatially averaged fCaD responses with a two-AP protocol with waiting periods of 40, 80, 160, 320, and 400 ms between APs. (A) fCaD responses averaged from three EDL fibers (top) and these responses after correction for a small movement artifact by the method described in the text (bottom). A rest period of 1 min was used between repetitions of the paired stimuli. Fiber identification numbers: 021097.1, 042597.1, and 042597.2. (B) Top: simulated multicompartment fCaD responses to mimic the lower traces in A; bottom: the Ca2+ release waveforms used to drive the simulations. The release flux amplitudes are 227 µM/ms for the conditioning stimulus and 0.358, 0.498, 0.703, 0.871, and 0.917 times that of the conditioning pulse (waiting periods of 40, 80, 160, 320, and 400 ms, respectively).

View larger version (12K): [in this window] [in a new window] [Download PPT slide]   Figure 11. Analysis of measured and simulated responses with the two-AP protocol. (A) The large circles show the amplitudes of the measured fCaD signals in Fig. 10 A (bottom traces) normalized by 0.178, the mean amplitude evoked by the conditioning stimulus (filled circle, t = 0 ms); each fCaD amplitude was determined as the peak of fCaD minus its starting value on the fCaD trace evoked by the conditioning stimulus. The small circle at t = 15 ms shows the normalized amplitude averaged from the responses of the two fibers of Fig. 10 A that had measurements with a 15-ms waiting period between APs. The X's show similar information for fCaD in the simulations of Fig. 10 B, as well as for a simulation with a 15-ms waiting period. The triangles show the amplitudes of the release fluxes in the simulations normalized by the release flux of the conditioning stimulus (227 µM/ms). The curve through the triangles is a least-squares fit of the points at t 40 ms with the function f(t) = R0 + (Rmax – R0) · (1 – exp(–r · t)). The fitted values of R0, Rmax, and r are 0.164, 0.965, and 6.87 s–1, respectively. (B) Decay rate constants of fCaD in the measurements (circles) and simulations (X's), determined as in Fig. 7 (A and B). (C and D) Values of [Site] and [CaSite](t2) – [CaSite](t1), determined as in Fig. 7 (C and D).

The triangles in Fig. 11 A show the amplitudes of the release fluxes in the simulations, normalized by that of the conditioning release. The amplitudes of the second releases increase progressively from 0.239 (t = 15 ms) to 0.917 (t = 400 ms) times that of the first release. The triangles at t 40 ms have been fitted with a decaying single-exponential function having three adjustable parameters: an initial value, a final value, and a rate constant (see legend). The points are well fitted by this function (continuous curve in Fig. 11 B); the curve also intersects the point at t = 15 ms, which was not included in the fit. According to the curve, the large reduction in release after the first AP recovers to 0.97 of the initial release with a rate constant of 6.9 s^–1 (time constant of 146 ms). If the receptor site for Ca²⁺ inactivation of Ca²⁺ release has a dissociation constant for Ca²⁺ that is large compared with the values of [Ca²⁺] at times 40 ms, the dissociation rate constant of Ca²⁺ from this site should be –6.9 s^–1. On the other hand, if the K_D of the receptor site is small, the dissociation rate constant may be larger than 6.9 s^–1.

Recovery of the Fast Decay of f_CaD after a Conditioning Stimulus
In Fig. 10 A, the five waveforms evoked by the second AP differ in time course as well as in amplitude. With a short delay between stimuli, the decay of the second f_CaD is markedly slower than that of the conditioning stimulus (compare Fig. 7 A); with longer delays, the decay becomes faster and approaches that of the conditioning stimulus. Similar changes are observed in the simulated waveforms (Fig. 10 B).

The simulations were analyzed to determine what changes in Ca²⁺ binding by the slow Ca²⁺ buffers are associated with recovery of the fast decay of f_CaD. The circles in Fig. 11 B plot the decay rate constants of the individual f_CaD measurements in Fig. 10 A; as in Fig. 11 A, Fig. 11 B also includes the data point at t = 15 ms (small circle). The X's in Fig. 11 B plot the decay rate constants of the simulated f_CaD responses. Both plots reveal that the decay rate constant at t = 15 and 40 ms is substantially smaller than that at t = 0 ms and, as time increases, the rate constant recovers toward the initial value. There are some discrepancies, however, between the simulated and measured rate constants. The simulated value for the conditioning stimulus, 132 s^–1 (t = 0 ms), is not as large as the measured value, 150 s^–1, and the simulated values between 15 and 320 ms are all noticeably larger than the measured values. The precise reason(s) for these differences is not known.

Another difference between the simulated and measured rate constants in Fig. 11 B is that the recovery of the simulated rate constant slows at late times and the measured value overtakes the simulated value. A possible explanation for this discrepancy is that [Ca²⁺] evoked by the conditioning AP returns to baseline less rapidly in the simulations than in the measurements, thus delaying the recovery of parvalbumin; for example, at t = 400 ms, simulated spatially averaged [Ca²⁺] in response to the conditioning AP is 108 nM above resting. A slower return of [Ca²⁺] could occur if the turnover rate of the Ca²⁺ pump in the simulations is smaller than the actual turnover rate, or if some other slow process or buffer not included in the simulations contributes to the actual decline of [Ca²⁺] on this time scale. As demonstrated in an earlier section ([Ca²⁺]_R and the Relation between the FDHM and the Peak of f_CaD), release fluxes that produce f_CaD waveforms with identical peak amplitudes are associated with larger values of FDHM of f_CaD (and hence smaller decay rate constants) if [Ca²⁺]_R is larger. The conclusion that [Ca²⁺] returns to baseline more slowly in the simulations than the measurements is supported by preliminary experiments with fluo-3, which can be used to follow the time course of the decline of [Ca²⁺] with much greater accuracy than is available with furaptra. These experiments indicate that [Ca²⁺]'s return is indeed somewhat slower in the simulations than in the measurements (Baylor, S.M., and S. Hollingworth. 2007. Biophys. J. 92:311a). The reason for this shortcoming in the model merits further investigation.

Fig. 11 (C and D) (which is analogous to Fig. 7, C and D) shows the analysis of the simulated changes on troponin, parvalbumin, and the Ca²⁺ pump (the slowly reacting Ca²⁺ buffers) during the time periods when the decay rate constants in Fig. 11 B were estimated. As expected from Fig. 7 C, Fig. 11 C reveals that the conditioning AP produces a marked decline in the concentration of sites on troponin and parvalbumin, but not on the pump, that are immediately available to bind Ca²⁺ early in the falling phase of f_CaD in response to a release initiated 15 ms after the first release. Between 15 and 320–400 ms, the available site concentrations on troponin and parvalbumin, but not the pump, recover substantially. Fig. 11 D shows that a similar pattern is observed for the simulated concentrations of Ca²⁺ actually bound by troponin, parvalbumin, and the pump during the early decay of f_CaD. Presumably, recovery of the ability of troponin and parvalbumin to bind Ca²⁺ is primarily responsible for the reappearance of the fast decay rate of f_CaD in the measurements (Fig. 11 B, open circles).

Simulations at Sarcomere Length = 2.4 µm
To our knowledge, myoplasmic Ca²⁺ transients have not been measured in the same mammalian fibers at short- and long-sarcomere length. This is difficult to do on a bundle of intact fibers, only one of which contains a Ca²⁺ indicator (as in our experiments). We have therefore used simulations to estimate sarcomeric Ca