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First published online December 28, 2007
Journal of Experimental Biology 211, 180-186 (2008)
Published by The Company of Biologists 2008
doi: 10.1242/jeb.013466
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A spatially explicit model of muscle contraction explains a relationship between activation phase, power and ATP utilization in insect flight

Bertrand C. W. Tanner1,*, Michael Regnier1 and Thomas L. Daniel1,2

1 Department of Biology, University of Washington, Seattle, WA 98195, USA
2 Department of Bioengineering, University of Washington, Seattle, WA 98195, USA


Figure 1
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  		Fig. 1. (A) Kinetics of thin filament regulation and cross-bridge cycling, modeled as coupled three-state cycles. Transition rates (kij) between cross-bridge states (X1–X3) are strain dependent. Transition rates (rij) between thin filament states (T1–T3) explicitly encode spatial information about troponin binding Ca2+ and tropomyosin movement. B) We simulated force production in a network of linear springs, using spring constants for thick filaments (km), thin filaments (ka) and cross-bridges (kxb). Thick and thin filament nodes (white circles between springs) represent modeled points from which cross-bridges extend from the thick filament backbone or actin binding sties along the thin filament where cross-bridges. At each time step, Monte-Carlo methods simulate likelihoods of Ca2+ regulated cross-bridge attachment to thin filaments, then forces balance about each node throughout the filament lattice.

 

Figure 2
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  		Fig. 2. Work loop simulations oscillated muscle length and intracellular Ca2+ concentration as a function of time, while monitoring force and ATP utilization. These panels show the initial half second of a simulation where muscle strain amplitude ({epsilon}, normalized peak to peak) was 0.025 and the Ca2+ transient had a 0.2 duty cycle ({delta}Ca) at a 0.033 phase of activation ({phi}Ca). Force and ATP utilization were calculated at each time step (dt=1 ms). Our standard mechanical parameters apply to these simulations: kxb=5, ka=5229 and km=6060 pN nm–1 (Tanner et al., 2007Go).

 

Figure 3
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  		Fig. 3. Each simulation (10 s long) produced 250 work loops, constructed from the phase portrait of force and muscle strain over any single oscillation period (40 ms). This work loop is the average phase portrait for the simulation shown in Fig. 2. The counter-clockwise direction (arrows) denotes positive work output (548.6±46.6 pN nm, mean ± s.d., N=245 work loops).

 

Figure 4
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  		Fig. 4. Work and ATP utilization vary with respect to the phase of Ca2+ activation ({phi}Ca). Data points (connected with a line) represent average values (mean ± s.d.) from 245 individual work loops at a duty cycle ({delta}Ca) of 0.5. The {phi}Ca values producing maxima and minima varied for work (A) and ATP (B), therefore yielding a unique {phi}Ca that maximized efficiency (ratio of work to ATP; C). Strain amplitude {epsilon} was 0.05 (left) and 0.025 (right), and primarily affected work magnitude. Note the difference in ordinate scales between the left and right panels in A and C when {epsilon}=0.05 versus 0.025. Mechanical parameters for the filaments and cross-bridges are listed in Fig. 2.

 

Figure 5
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  		Fig. 5. Duty cycle and filament stiffness influence how work and ATP vary with respect to phase of activation ({phi}Ca). Similar to Fig. 4, lines represent mean work (A), ATP (B) and efficiency (C) values; however error bars are not shown. Duty faction ({delta}Ca) varied between 0.1 (blue), 0.2 (green) and 0.5 (black). Broken lines represent our standard mechanical parameters for the filaments and cross-bridges, listed in Fig. 2. Therefore, the mean data shown in Fig. 4 correspond to the grey broken lines shown here in Fig. 5. Solid lines represent a more compliant filament lattice, where both ka and km were simultaneous scaled by a factor of 0.1 (kxb did not change). As in Fig. 4, note the difference in ordinate scale between the left and right panels in A and C when {epsilon}=0.05 (left) versus 0.025 (right).

 




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