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. 2012 Sep 15;590(18):4603-22.
doi: 10.1113/jphysiol.2012.228965. Epub 2012 May 8.

Myocardial twitch duration and the dependence of oxygen consumption on pressure-volume area: experiments and modelling

J-C Han  1 , K Tran, A J Taberner, D P Nickerson, R S Kirton, P M F Nielsen, M-L Ward, M P Nash, E J Crampin, D S Loiselle
Affiliations

Myocardial twitch duration and the dependence of oxygen consumption on pressure-volume area: experiments and modelling

J-C Han et al. J Physiol. .

Abstract

We tested the proposition that linear length dependence of twitch duration underlies the well-characterised linear dependence of oxygen consumption (V(O(2)) ) on pressure–volume area (PVA) in the heart. By way of experimental simplification, we reduced the problem from three dimensions to one by substituting cardiac trabeculae for the classically investigated whole-heart. This allowed adoption of stress–length area (SLA) as a surrogate for PVA, and heat as a proxy for V(O(2)) . Heat and stress (force per cross-sectional area), at a range of muscle lengths and at both 1 mM and 2 mM [Ca(2+)](o), were recorded from continuously superfused rat right-ventricular trabeculae undergoing fixed-end contractions. The heat–SLA relations of trabeculae (reported here, for the first time) are linear. Twitch duration increases monotonically (but not strictly linearly) with muscle length. We probed the cellular mechanisms of this phenomenon by determining: (i) the length dependence of the duration of the Ca(2+) transient, (ii) the length dependence of the rate of force redevelopment following a length impulse (an index of Ca(2+) binding to troponin-C), (iii) the effect on the simulated time course of the twitch of progressive deletion of length and Ca(2+)-dependent mechanisms of crossbridge cooperativity, using a detailed mathematical model of the crossbridge cycle, and (iv) the conditions required to achieve these multiple length dependencies, using a greatly simplified model of twitch mechano-energetics. From the results of these four independent investigations, we infer that the linearity of the heat–SLA relation (and, by analogy, the V(O(2))–PVA relation) is remarkably robust in the face of departures from linearity of length-dependent twitch duration.

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Figures

Figure 1
Figure 1. Schematic representation of variables appropriate to the 3-D intact heart (A, C and E) and the isolated cardiac trabecula (B, D and F)
A, PTI: pressure-time integral. B, t50% and t95%: duration of twitch at 50% and 95% of relaxation from the peak; ‘active’ or ‘developed’ stress is given by the difference between ‘Total’ and ‘Passive’ stress; C, PVA: pressure–volume area, ESPVR and EDPVR: end-systolic and end-diastolic pressure–volume relations, respectively; D, SLA: force–length–area; E, formula image: oxygen consumption per beat; F, putative form of the heat–SLA relation for cardiac trabeculae, to be determined.
Figure 2
Figure 2. Dependence of peak stress and SLA on muscle length
Average (n= 10) and representative (insets) peak active stress (A) and stress–length area (SLA, B) as functions of relative length (L/Lo), at 2 Hz stimulation and at two values of [Ca2+]o, 1 mm (open symbols and thin lines) and 2 mm (filled symbols and continuous lines) SLA (B) calculated by integrating the stress–length relations shown in A. For the averaged curves in A: 1 mm Ca2+: R2= 0.998, sy.x= 0.728 kPa; 2 mm Ca2+: R2= 0.997, sy.x= 0.966 kPa. For the averaged curves in B: 1 mm Ca2+: R2= 0.999, sy.x= 0.055 kPa; 2 mm Ca2+: R2= 0.999, sy.x= 0.045 kPa.
Figure 3
Figure 3. Linear regression of heat on stress–length area (SLA) of rat isolated cardiac trabeculae undergoing fixed-end contractions at steady state
Results are from representative trabeculae (A and C) and from the averages of 10 trabeculae (B and D). Thin lines and open symbols: 1 mm[Ca2+]o; thick lines and filled symbols: 2 mm[Ca2+]o. A and B, 0.2 Hz; C and D, 2 Hz. B, average regression lines for 0.2 Hz: H= (2.92 ± 0.5) + (3.2 ± 0.3) × SLA (R2= 0.973, sy.x= 0.725 kJ m−3); and H = (5.80 ± 0.5) + (3.4 ± 0.3) × SLA (R2= 0.959, sy.x= 1.335 kJ m−3); for 1 mm and 2 mm[Ca2+]o, respectively. D, average regression lines for 2.0 Hz: H= (3.97 ± 0.5) + (2.6 ± 0.3) × SLA (R2= 0.959, sy.x= 0.734 kJ m−3), and H= (6.59 ± 0.5) + (2.3 ± 0.3) × SLA (R2= 0.934, sy.x= 1.045 kJ m−3) for 1 mm and 2 mm[Ca2+]o, respectively.
Figure 7
Figure 7. Time-course of Ca2+-activation derived from length impulses
A–C,ΔT, an index of the time course of Ca2+ activation of contraction (see legend of Fig. 6 for details), plotted as functions of relative muscle length (A), twitch duration (B) and peak active stress (C); circles: ΔT50, triangles: ΔT95. D, twitch duration as a function of peak active stress; circles: t50%, triangles: t95%. In panels A, C and D, data for ‘50%’ fitted by linear regression; data for ‘95%’ fitted by quadratic regression. Data in panel B fitted by linear regression. Same trabecula as in Fig. 6.
Figure 6
Figure 6. Experimental length impulses
A, superimposed stress response (thin lines) upon application of length impulses of amplitude ∼0.06–0.1Lo (60–100 μm) and half-duration 10–18 ms at different times (20–220 ms) from stimulus pulse during the course of twitches at three different muscle lengths less than Lo, at 2 mm[Ca2+]o. The thick lines represent steady state twitch stress in the absence of ‘length-impulse’ perturbation. The profiles of stress redevelopment after application of length impulses were fitted by 5th order polynomials. The initial rates of stress redevelopment (B), and their normalised equivalents (C), were computed from the derivatives of the fitted polynomials, and plotted as a function of time (relative to the stimulus pulse) at which the impulses ended, for five different muscle lengths (the filled arrow indicates increasing muscle length). These ‘derivative’ data, plotted as a function of time, were fitted by 3rd order polynomials (C). The double-headed arrows (ΔT50 and ΔT95) provide indices of the time course of Ca2+ activation of contraction. The diagonal arrows indicate observations at increasing muscle lengths. The trabecula was 210 μm in diameter and 2.0 mm long.
Figure 4
Figure 4. Dependence of stress and twitch duration on muscle length
A and C, overlaid twitches at Lo (uppermost traces), and at five muscle lengths at, or below, Lo, from representative trabeculae at 0.2 Hz (A) and 2 Hz (C). B and D, same twitches as in A and C, but normalised to peak amplitude. The arrows indicate increasing muscle length. Note difference of abscissae scales. E and G, t50% and t95%, respectively, as functions of relative muscle length (L/Lo) at 0.2 Hz (upper pairs of lines) and 2.0 Hz (lower pairs of lines) at 2 mm[Ca2+]o (filled symbols and thick lines) and 1 mm[Ca2+]o (open symbols and thin lines) for representative trabeculae. F and H, equivalent averaged data for n= 10 trabeculae (same line conventions as in E and G). Statistically significant quadratic trends at 0.2 Hz, for both 1 mm and 2 mm[Ca2+]o at t95% (H). F, 0.2 Hz, 1 mm Ca2+: R2= 0.964, sy.x= 14.6 ms; 2 mm Ca2+: R2= 0.981, sy.x= 8.4 ms; 2.0 Hz, 1 mm Ca2+: R2= 0.989, sy.x= 4.2 ms; 2 mm Ca2+: R2= 0.989, sy.x= 4.1 ms. H: 0.2 Hz, 1 mm Ca2+: R2= 0.973, sy.x= 17.4 ms; 2 mm Ca2+: R2= 0.985, sy.x= 12.8 ms; 2.0 Hz, 1 mm Ca2+: R2= 0.977, sy.x= 9.0 ms; 2 mm Ca2+: R2= 0.988, sy.x= 6.5 ms.
Figure 5
Figure 5. Steady state 340/380 Fura-2 fluorescence ratio (an index of [Ca2+]i) as a function of time at increasing muscle length (as indicated by the arrows)
A and C, original traces at 0.2 Hz and 2.0 Hz, respectively; B and D, data normalised to individual peak fluorescence ratios. Each trace represents the average of 8–9 Ca2+ transients recorded in the presence of 2 mm extracellular Ca2+. Note different scales of abscissae between upper and lower pairs of panels. Trabecula length = 3.4 mm, diameter = 190 μm.
Figure 8
Figure 8. The metabolite-sensitive, thermodynamically constrained crossbridge model of Tran et al. (2010)
Normal cycling proceeds in a clockwise sense. Bold lettering indicates crossbridge states; U: unattached, A1: attached in a low-force configuration; A2 and A3: attached in high-force configurations. Rate constants (specifying the rates of transition between states) indicated by ‘k’; some are sarcomere-length-dependent (SL) and some are sensitive to the strain of the cross-bridge head (φ). In the text, formula image has been abbreviated to formula image. The sarcomere-length dependence of formula image arises explicitly from the sarcomere-length dependence of formula image, since it has been computed using the thermodynamic constraint (Loiselle et al. 2010). Hence, its effect is implicitly included in simulations reported in the Text. Reproduced from: Loiselle et al. (2010).
Figure 9
Figure 9. Results generated by the mathematical model of Tran et al. (2010)
Simulated 10 ms length impulses applied at various times during a twitch in the presence (A and B) and absence (C and D) of length dependence of Ca2+ both on its binding to troponin-C and on the kinetic constant (k−1; see Fig. 11). A and C show simulated results at 0.92 SLrest (2.3 μm).
Figure 10
Figure 10. Output of the mechano-energetics model
A, simulated twitches in response to stepwise increases of sarcomere length (as indicated by the arrow) from 1.7 μm to 2.3 μm. B, twitches normalised to their peak values. C, ATP consumption as a function of FLA (force–length area, calculated from the simulated end-systolic force–SL relation). D, twitch duration, at both 50% and 95% of relaxation from peak force, as a function of sarcomere length.
Figure 11
Figure 11. Simulated mechano-energetic relations under all four combinations of presence and absence of ‘upstream’ and ‘downstream’ effects of Ca2+
A, force as a function of sarcomere length. B, ATP consumption as a function of force–length area. Twitch duration at 50% relaxation (C) and 95% relaxation (D). In each panel, the bold symbols and curves denote the behaviour of the model with all components included.
Figure 12
Figure 12. Simulations with linear length dependence of peak force on twitch duration
A, triangular twitches at various values of SL. B, peak force as a function of SL as given by a(SL) (eqn (A1)) a(SL). C, normalised (triangular) twitches. D, ATP vs. FLA plot, where ATP is calculated from the area of the triangular twitches (eqn (A3)) and FLA is given by the integral of a(SL) (eqn (A2)) a(SL).
Figure 13
Figure 13. Simulations with linear length dependence of both peak force and twitch duration
A, triangular twitches at various values of SL. B, peak force as a function of SL as given by a(SL) (eqn (A1)). C, normalised (triangular) twitches. D, ATP vs. FLA, where ATP is calculated from the area of the triangular twitches (eqn (A5)).
Figure 14
Figure 14. Normalised plots of FLA FLA(SL2)and ATP as functions of SL
The minima of these two functions do not coincide, since they are not proportional to one other.
Figure 15
Figure 15
Simulations with linear length dependence of twitch height (i.e. peak force; a in eqn (A1)) and twitch duration (b in eqn (A4)) proportional to one-another – i.e. both pass through zero at SLmin (eqn (A2))
See this image and copyright information in PMC

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