Fig 1.
Schematic of passive and fully activated stress–stretch relations and corresponding energy—stretch relations.
Top: Consider the case that muscle force is completely described by the passive stress (black solid curve) and total isometric stress (red dashed curve). Then, a fixed-length contraction of two in-series-arranged half-sarcomeres on the descending limb of the total stress—stretch relation (blue dot) is unstable. Small initial half-sarcomere length differences lead to a stronger, shorter half-sarcomere and a weaker, longer half-sarcomere. Hence, the short half-sarcomere will shorten further, stretching the longer half-sarcomere until static force equilibrium is established again on ascending albeit different limbs of the total stress—stretch relation (red dots). Such behaviour is not observed in experiments [8, 9]. Bottom: Passive, convex energy (black curve) and total, non-convex energy (red dashed curve).
Fig 2.
Overview of the multi-scale skeletal muscle model.
Each box indicates a submodel. The coupling between the submodels is illustrated by the arrows in combination with the exchanged variables. (For further details see [19, 30, 32].)
Fig 3.
Schematic representation of the half-sarcomere model.
The half-sarcomere model consists of a phenomenological description of the membrane voltage and the intracellular calcium concentration [41], that is coupled to a simplified Huxley-type model [42] and the ‘sticky—spring’ titin model [31].
Fig 4.
Top: Isometric force responses of the microscopic half-sarcomere model due to different stimulations. Single—twitch (black), submaximal contraction at 20 Hz (red), and tetanic contraction at 100 Hz (blue). Bottom: Force—frequency relation of the half-sarcomere model. Under conditions of not completely fused twitches, the range between the minimum and maximum values of the active force is shown.
Fig 5.
Total stress of the half-sarcomere model with actin—titin interaction versus the half-sarcomere length at which activation started, lhs[0], and the active stretch increment, Δlhs.
The thick black curve at the right edge of the surface corresponds to the static total stress—stretch relation shown in Fig 8 (top).
Fig 6.
Isometric active force—length relations of the different models.
Models and MATI reproduce the force—length relation as proposed in [1] that is included at the half-sarcomere level of all models. Models MFv and
predict different forces on the descending limb. Under steady-state conditions (after 2.5 s) the results of models MFv and
, as well as those of models
and MATI, coincide.
Fig 7.
Distribution of half-sarcomere lengths, lhs, in the cubic muscle specimen for different muscle lengths ranging from L/L0 = 1.1 to 1.7 during fixed-length contractions of models , MFv, MATI, and
.
Results after 2.5 s are displayed (steady-state conditions), where the results of models and MFv (left), as well as those of MATI and
(right) coincide.
Fig 8.
Top: Total, active, and passive force—length relations. Superimposed on the total stress curve are, for a muscle length of L/L0 = 1.5, the half-sarcomere lengths predicted by models and MFv (red crosses), and models MATI and
(blue dot). The value predicted by models MATI and
coincides with the theoretically predicted stress—stretch value. Bottom: Histogram of half-sarcomere lengths in models
and MFv (red), and models MATI and
(blue, scaled for readability). Under conditions of steady state (after 2.5 s), the results of models
and MFv, as well as those of models MATI and
, coincide.
Fig 9.
Coefficient of variation (CoV; standard deviation/mean*100%) of the half-sarcomere lengths during an isometric contraction at muscle length L/L0 = 1.2, for models (blue line), MFv (red line), MATI (black solid line) and
(black dashed line).
While actin—titin interactions can damp the magnitude of half-sarcomere—length heterogeneities, the force-velocity relation damps the their temporal evolution.
Fig 10.
Initial half-sarcomere length inhomogeneity is introduced via variation of passive (left column) and active (right column) material properties (see text). Top: Total stress evolution in models (black dashed line) and MATI (red line) following a protocol of passive stretch from L/L0 = 1.0 to 1.05 (phase I, i.e., computing the passive stretch as shown in Fig 6), full activation at fixed length (phase II), active stretch at a velocity of v/vmax = 13.3% from L/L0 = 1.05 to 1.25 (phase III), and subsequent fixed-length contraction (phase IV). Additionally, the stress resulting from a fixed-length contraction (model
) at the same final length L/L0 = 1.25 is shown (blue dotted line). The kink in the red line during the active stretch marks the transition from the plateau to the descending limb of the force—length relation. Note that we excluded stresses from models MFv and
from these figures, since these models cannot reproduce the expected forces on the descending limb (previously shown, see e. g. Fig 6). Bottom: Coefficient of variation (CoV, standard deviation/mean*100%) of the half-sarcomere lengths corresponding to the above stretch experiments.
Fig 11.
Histogram of half-sarcomere lengths in model with distributed material properties after an active stretch from L/L0 = 1.05 to 1.25 (red) and after a fixed-length contraction at the same final length L/L0 = 1.25 (blue).
This figure compares steady-state results, i. e. transient effects of the force-velocity relation have disappeared. Left: Variable passive properties, right: Variable active properties.