Fig 1.
Representation of the different stages of the force generation mechanism and of the sections where they are discussed in this paper.
Table 1.
List of abbreviations of this paper.
Fig 2.
Representation of the steady-state force-calcium relationship (a), the force-velocity relationship (b) and tension-elongation curves after a fast transient (c).
Fig 3.
Sketch of the proposed CTMC model.
Each RU is described by a 4-state model (top left), whose dynamics is affected by the state nearest-neighboring RUs. An example of nearest-neighbor interactions is shown in the bottom-left box, where the notation for the transition rates kT,i is illustrated by the orange arrows. MHs are described as 2-state elements, whose transition rates are affected by the XB elongation, the sliding velocity between myofilaments and the permissivity state of the RU.
Fig 4.
Representation of the configuration corresponding to the state variable .
The arrows illustrate the meaning of the notation.
Fig 5.
Spatially-explicit model: Each triplet of consecutive RUs can undergo 6 different transitions.
For example, this figure shows the transitions of the configuration associated with the state variable , with the corresponding transition rates. The transition rates computed thanks to Ass. (H1) are highlighted with a colored background.
Fig 6.
Representation of the variables of the distribution-moments equations.
The variable (respectively,
) corresponds to the fraction of permissive (respectively, non-permissive) BSs to which a MH is bound, while the ratio
(respectively,
) corresponds to the average XB distortion within the permissive (respectively, non-permissive) attached BSs.
Fig 7.
Representation of the configuration corresponding to the state variable .
The arrows illustrate the meaning of the notation.
Fig 8.
Mean-field model: Representation of the 4 forward and backward transitions of the configuration associated with the state variable , with the corresponding transition rates.
The transition rates computed thanks to Ass. (H4) are highlighted with a colored background.
Fig 9.
Sketch of the sarcomere model.
The thick filament (MF) is represented in red and two thin filaments (AF) are represented in blue (top). The origin of the frame of reference is the left side of the reference AF. The functions χSF and χM are also represented (bottom).
Table 2.
List of the models proposed in this paper.
For future reference, we assign a name to each model (SE stands for spatially-explicit,MF stands for mean-field). In the second column we report the number of ODEs and PDEs included in each model as a function of NA and NM and we specify the resulting values in the case NA = 32, NM = 18. In the “Assumptions” column,m.f. stands for mean-field assumption.
Fig 10.
The proposed four states Markov model describing each RU.
The terms depending on the intracellular calcium concentration [Ca2+]i are highlighted in red; terms depending on the state of neighbouring RUs (i.e. depending on n) are highlighted in green; terms depending on the position of the RU and the current sarcomere elongation are highlighted in blue.
Table 3.
Parameters of the SE-ODE and MF-ODE models calibrated for room-temperature rat and body-temperature human cells.
Table 4.
List of geometrical constants.
Fig 11.
Steady-state force-calcium curves obtained with the SE-ODE (left) and MF-ODE (right) models with the optimal parameters values reported in Table 3 for SL = 1.85 μm and SL = 2.15 μM, compared with experimental data from intact rat cardiac cells at room temperature, from [90].
Fig 12.
Steady-state force-calcium relationship at different SL obtained with the SE-ODE (left) and the MF-ODE (right) models for intact, body-temperature human cardiomyocytes.
Fig 13.
Dependence of the Hill coefficient nH and of the half-activating calcium concentration EC50 on the sarcomere length SL, for the SE-ODE (blue lines) and MF-ODE (red lines) model, calibrated for intact, room-temperature rat cardiomyocytes (dashed lines) and for intact, body-temperature human cardiomyocytes (solid lines).
Fig 14.
Steady-state force-length relationship at different [Ca2+]i obtained with the SE-ODE (left) and the MF-ODE (right) models for intact, body-temperature human cardiomyocytes.
Fig 15.
Force transients (top line) and phase-loops (bottom line) in isometric twitches, for different values of SL, predicted by the SE-ODE model (left column) and MF-ODE model (right column), in comparison with the experimental measurements from intact rat cardiac cells taken from [94].
Fig 16.
Tension transients during isometric twitches at different SL obtained with the SE-ODE (left) and the MF-ODE (right) models for intact, body-temperature human cardiomyocytes.
Fig 17.
Tension peak, normalized with respect to the value at 2.2 μm (left), and metrics of activation and relaxation kinetics (right) as function of SL during isometric twitches obtained with the SE-ODE and the MF-ODE models for intact, body-temperature human cardiomyocytes.
When possible, model results are compared with experimental data from [95].
Fig 18.
Normalized force-velocity relationships for different combinations of [Ca2+]i and SL obtained with the SE-ODE (left) and the MF-ODE (right) models for intact, room-temperature rat cardiomyocytes in comparison with experimental measurements from [70].
Fig 19.
Normalized force after the application of a fast length step ΔL for intact, room-temperature rat cardiomyocytes.
Solid line: model result; circles: T2-L2 experimental data from [70].
Fig 20.
LV multiscale EM: Deformed geometry and magnitude of displacement d at different times.
Top row: full geometry. Middle row: half domain (top view). Bottom row: half domain (frontal view).
Fig 21.
Time evolution of [Ca2+]i, SL, v and Ta (minimum, maximum and average over the computational domain) and of left ventricle pressure and volume.
Fig 22.
Pressure-volume loops obtained with different preloads (reported in legend).