Multiscale modeling of twitch contractions in cardiac trabeculae

Figure 1.. Origin of trabecula series elasticity is complex and might include muscle fiber alignment and connective tissue serial elasticity. (A) A rat right ventricular trabecula mounted between a force transducer and a motor (scale bar, 50 µm). (B) The magnified region displays alignment of muscle fibers along trabeculae. (C) Higher magnification shows stratifications along muscle fibers as well as imperfect alignments showing gaps between and along muscle fibers (scale bar, 10 µm). (D) The simplest method is to include trabecula elasticity (SE) in series with the CE consisting of multiplesarcomere muscle fibers. The change of SE element length uSE is equal to change in multisarcomere fiber length.

Figure 2.. Model of the cross-bridge cycle. (A) Five-state model of the actomyosin cycle includes biochemical states consistent with observed structural states: a detached state 1, M.D.Pi; weakly bound state 2, A.M.D.Pi; strongly bound post–power stroke state 3, A.M.D; rigor-like state 4, A.M; and detached state 5, M.T. The strain-dependent state transition rates are also associated with conformational changes defining the structural conformations of myosin in each state (i.e., power stroke, d, associated with Pi release, and stroke, δ, associated with the ADP release, ATP binding and cross-bridge detachment, and hydrolysis and reverse stroke). (B) Addition of a state representing interaction of myosin heads with the thick filament backbone, the so-called “parked state”, PS, denoted as state 6, into a five-state model from A. The PS is a partial M.D.Pi state with structural conformation associated with thick-filament backbone, reducing the population of the M.D.Pi state capable of binding to actin and therefore fluxes from M.D.Pi state to A.M.D.Pi state (myosin binding to actin) or the reverse hydrolysis to M.T state. The transition rate from PS to M.D.Pi, k+PS, is assumed to be strongly…

Figure S1.. Interaction between myosin heads and actin filaments in 3-D is defined by the triple-helical arrangement of myosin molecules along the myosin filament and the double-helical arrangement of monomers (myosin-binding sites) along the actin filaments. The 3-D geometry of myosin head binding domains and binding sites on actin in a sarcomere requires both longitudinal position matching and angular matching in the azimuthal plane. (A) Each myosin head m can move from its undeformed position XMm along actin filament due to thermal agitation and can reach a few neighboring binding sites on actin. The myosin-binding domain is shown as a yellow oval at the tip of the myosin head, and binding sites on actin are shown as bright blue circles. The range of axial movement is shown as a pale red bar. The relative axial position of a myosin head (cross-bridge) XMmt and adjacent actin sites XAn+lt, where superscripts n+l and l denote the index of an adjacent site on actin, and l=0, 1, 2,…La−1 is the index of accessible sites on actin in neighborhood of m, respectively. The maximum number of adjacent sites on actin reachable by a myosin head m is denoted as La. To bind the site +l, the cross-bridge,…

Figure S2.. Effect of the five-state cross-bridge model rate k−Ao on relaxation of twitch tension in isometric half-sarcomeres (IoSarc.). The simulation, with parameters obtained from fits of isometric half-sarcomere tension at 27.2°C (Fig. 3), shows slow relaxation (dark green line). An additional increase of k−Ao by ∼17-fold significantly increases the rate of relaxation (green line and arrows), but it is still insufficient to reach the observation (dark gray line). The superscript o is omitted from the figure display for clarity.

Figure S3.. SL controlled isometric tension of a half-sarcomere. (A) Observed change in length of trabeculae per half-sarcomere (gray line) and corrected length of trabeculae to truly keep a half-sarcomere length isometric (pink line), which shows a decrease in trabecula length (i.e., motor position) toward the initial length of relaxed muscle. (B) Correction of the tension transient during the twitch due the change in SL (gray line in C). The observed tension (gray line) and the tension after correcting for the PE passive tension due to change of half-sarcomere length (red line) and also after additional correction for titin (passive) forces (pink line). (C) Change in half-sarcomere length (gray line) shows the effect of losing control of HS length that apparently should be constant (pink line). The tension of the trabecula length was fixed (black line in B) is shown for reference. The direction and color of the arrows signify changes in length and tension.

Figure S4.. Significance of the PS in twitch relaxation phase. The simulations of a half-sarcomere isometric twitch with the five-state model showed slow relaxation (dark green line) compared with adjusted observations of Janssen and de Tombe, 1997 (gray line), even when the baseline [Ca2+] was reduced (pink dashed line). The simulations also showed high muscle resting tension at observed baseline [Ca2+] (not shown), similar to that shown by Prodanovic et al. (2020) at comparable low [Ca2+]. In contrast, the simulations with the six-state model, which includes the PS, significantly increased the speed of relaxation (indicated by arrow), but the model predictions state (red line) followed the poorly observed unusual flat top to the tension transient reported by Janssen and de Tombe, 1997. However, the model predictions approximately matched the simulation of tension reported by Niederer et al. (2006) (dark gray dotted line), suggesting possible inconsistency of the observation with other similar data. In the simulations, we used the same model parameters as shown in Table 1, except k+Ao, k−Ao, and k+Do, which are presented in the text here for the five-state model and in Table 2 for the…

Figure S5.. Comparative tension–displacement loops obtained from Janssen and de Tombe and Caremani et al. (red and orange symbols and lines, respectively). Related to Janssen and de Tombe, 1997; Caremani et al., 2016. Both loops indicated viscoelastic behavior showing higher tension during shortening and lower tension during relaxation accompanied with lengthening. In both cases, the peak force per myofilament and maximum shortening were about the same, but Janssen and de Tombe’s loop is much wider and more asymmetric about the mean value than that of Caremani et al. (see also Fig. 4 B). The Janssen and de Tombe loop (red line) also shows a large drop in tension after reaching maximum shortening and, after the drop in tension, relaxes slowly during lengthening, having a higher value of tension than the initial value at displacement <4 nm. This deviation of the descending part of the loop shows some uncertainty in the measurements compared with that of Caremani et al. Thus, for simplicity, we derived a nonlinear serial springlike elasticity from Caremani et al.’s average tension–displacement relationship (thick gray line) that shows more consistent behavior at displacement <40 nm and a nicely…

Figure S6.. Comparison of ATP consumption rate, ATPase, during twitch contractions of isometric half-sarcomere and fixed-length trabeculae. During twitch in an isomeric half-sarcomere, the ATPase rate increases up to 15% due to relative movement between extensible filaments compared with the ATPase with rigid filaments (Mijailovich et al., 1996). Therefore, ATPase shows only a minor increase if the extensibility of thick and thin filaments is taken into account. In contrast, the ATPase in fixed-length trabeculae shows more than an order of magnitude change during tension rise, having a peak value ∼21 s−1. This large increase is associated with a large degree of sarcomere shortening (80 nm) that requires multiple detachments and reattachments of the cross-bridges. The ATPase is calculated as the rate of cross-bridge detachment over the number of cross-bridges participating in an active cross-bridge cycle; that is, the cross-bridge in PS is excluded.

Figure S7.. ATPase is strongly correlated to sarcomere-shortening velocities in fixed-length trabeculae during force development (in SL per second). The nondimensional ratio of the ATPase and sarcomere-shortening velocity slowly increases from ∼25 at 1 SL/s to ∼50 at 0.25 SL/s. The progressive increase of the normalized ATPase suggests that there is a steady-state component of ATPase that becomes more evident at low shortening velocities <0.25 SL/s. Interestingly, about the same behavior is observed for the simulations of twitches and force development in fixed-length trabeculae. The shortening velocities rise sharply up to 3 SL/s and quickly drop to ∼1 SL/s and further decrease as the rate of force rise decreases.

Figure S8.. Changes in twitch tension from Fig. 5 B (black line) by ±20% variation in binding rate, k+Ao. Increase in k+Ao increases the peak of the twitch tension and shows delay in relaxation. The superscript o is omitted from the figure display for clarity.

Figure S9.. Changes in twitch tension from Fig. 5 B (black line) by ±20% variation in detachment rate, k−Ao. Increase in k−Ao increases the rate in tension relaxation. The superscript o is omitted from the figure display for clarity.

Figure S10.. Changes in twitch tension from Fig. 5 B (black line) by ±20% variation in ADP release rate, k+Do. Increase in k+Do only slightly increases the peak of the twitch tension and the relaxation starts at earlier times. The superscript o is omitted from the figure display for clarity.