Structurally motivated models to explain the muscle's force-length relationship

The force-length relationship is a fundamental property of muscle contraction. Recently, progress has been made in modeling and understanding the relationship between the fiber microstructure and its force-generating capacity (1Williams C.D. Salcedo M.K. Daniel T.L. et al.The length–tension curve in muscle depends on lattice spacing.Proc. Biol. Sci. 2013; 28020130697Google Scholar,2Rode C. Siebert T. Blickhan R. et al.Myosin filament sliding through the Z-disc relates striated muscle fibre structure to function.Proc. Biol. Sci. 2016; 28320153030Google Scholar,3Rockenfeller R. Günther M. Hooper S.L. Muscle active force-length curve explained by an electrophysical model of interfilament spacing.Biophys. J. 2022; 121: 1823-1855Abstract Full Text Full Text PDF PubMed Google Scholar). Rockenfeller et al. (3Rockenfeller R. Günther M. Hooper S.L. Muscle active force-length curve explained by an electrophysical model of interfilament spacing.Biophys. J. 2022; 121: 1823-1855Abstract Full Text Full Text PDF PubMed Google Scholar) describe the muscle fiber’s force-length-activity relationship (FLAR) by an electrophysical model of interfilament spacing. This model may yield substantial progress in our understanding of muscle structure and contraction. However, model inconsistencies might hamper its predictive power concerning the ascending limb of the fiber’s force-length relationship (FLR) at full stimulation. Further, based on misunderstanding, Rockenfeller et al.’s introduction refutes an existing model (2Rode C. Siebert T. Blickhan R. et al.Myosin filament sliding through the Z-disc relates striated muscle fibre structure to function.Proc. Biol. Sci. 2016; 28320153030Google Scholar) capable of explaining the FLR’s ascending limb. Notably, both the Rockenfeller (3Rockenfeller R. Günther M. Hooper S.L. Muscle active force-length curve explained by an electrophysical model of interfilament spacing.Biophys. J. 2022; 121: 1823-1855Abstract Full Text Full Text PDF PubMed Google Scholar) and the Rode (2Rode C. Siebert T. Blickhan R. et al.Myosin filament sliding through the Z-disc relates striated muscle fibre structure to function.Proc. Biol. Sci. 2016; 28320153030Google Scholar) models are based on the novel hypotheses that the thick (myosin-containing) and thin (actin-containing) filaments maintain their lengths and slide through the M-lines and Z-discs when the fiber contracts to short lengths (2Rode C. Siebert T. Blickhan R. et al.Myosin filament sliding through the Z-disc relates striated muscle fibre structure to function.Proc. Biol. Sci. 2016; 28320153030Google Scholar) and, less important, that thick filaments form swiveled cross-bridges (swXBs; falseXBs in (3Rockenfeller R. Günther M. Hooper S.L. Muscle active force-length curve explained by an electrophysical model of interfilament spacing.Biophys. J. 2022; 121: 1823-1855Abstract Full Text Full Text PDF PubMed Google Scholar)) with thin filaments of opposite polarity. We would like to clarify some claims made in (3Rockenfeller R. Günther M. Hooper S.L. Muscle active force-length curve explained by an electrophysical model of interfilament spacing.Biophys. J. 2022; 121: 1823-1855Abstract Full Text Full Text PDF PubMed Google Scholar) concerning the Rode model and inconsistencies in the Rockenfeller model. The main Rockenfeller et al. argument to dismiss the Rode model is the wrong assertion that the Rode model describes the FLR’s ascending limb based on pushing or compressive forces in thin filaments. The thin filaments would be too soft to transmit relevant pushing forces. However, in the Rode model, thin filaments are always pulled away from the Z-disk. Furthermore, Rockenfeller et al. wrongly assert that swXBs generate pushing forces on thin filaments in the Rode model and that the contribution of swXBs would play a decisive role in the Rode model describing the FLR. Like regular cross-bridges (regXBs), swXBs always produce pulling forces on thin filaments relative to the Z-disk. They, however, tend to lengthen the half-sarcomere when the tips of the thick filaments enter the neighboring half-sarcomere. Mechanically inconsistent, the Rockenfeller model assumes that swXBs always tend to shorten the half-sarcomere (see the Rockenfeller model's effective overlap function is mechanically inconsistent on the FLR's ascending limb). Moreover, the amount of swXB force only gradually affects the FLR prediction of the Rode model. swXB force was even zero when explaining the isometric forces generated at very short mammalian fiber lengths (4Tomalka A. Heim M. Siebert T. et al.Ultrastructural and kinetic evidence support that thick filaments slide through the Z-disc.J. R. Soc. Interface. 2022; 1920220642Crossref PubMed Scopus (0) Google Scholar). Hence, swXB forces are not essential to explain significant features of the FLRs ascending limb like the change in slope from its shallow to its steep part (4Tomalka A. Heim M. Siebert T. et al.Ultrastructural and kinetic evidence support that thick filaments slide through the Z-disc.J. R. Soc. Interface. 2022; 1920220642Crossref PubMed Scopus (0) Google Scholar, 5Gordon A.M. Huxley A.F. Julian F.J. The variation in isometric tension with sarcomere length in vertebrate muscle fibres.J. Physiol. 1966; 184: 170-192Crossref PubMed Google Scholar) and the force hump at very short lengths (6Ramsey R.W. Street S.F. The Isometric Length-Tension Diagram of isolated skeletal muscle fibers of the frog.J. Cell. Physiol. 1940; 15: 11-34Crossref Google Scholar, 7Knappeis G.G. Carlsen F. The ultrastructure of the Z disc in skeletal muscle.J. Cell Biol. 1962; 13: 323-335Crossref PubMed Google Scholar). The Rockenfeller model assumes a hexagonal filament lattice as in optimal overlap down to 0.4-μm half-sarcomere length (with about 1 μm being the optimal length) and that the lattice distances progressively increase with shortening due to volume constancy. These assumptions ignore several geometrical changes in the half-sarcomere that occur during shortening and hamper the predictive power of the model in the range of the FLR’s ascending limb. For example, the Rockenfeller model predicts no hump in the FLR at very short lengths found in experiments (4Tomalka A. Heim M. Siebert T. et al.Ultrastructural and kinetic evidence support that thick filaments slide through the Z-disc.J. R. Soc. Interface. 2022; 1920220642Crossref PubMed Scopus (0) Google Scholar, 6Ramsey R.W. Street S.F. The Isometric Length-Tension Diagram of isolated skeletal muscle fibers of the frog.J. Cell. Physiol. 1940; 15: 11-34Crossref Google Scholar) and explained with the Rode model (2Rode C. Siebert T. Blickhan R. et al.Myosin filament sliding through the Z-disc relates striated muscle fibre structure to function.Proc. Biol. Sci. 2016; 28320153030Google Scholar, 7Knappeis G.G. Carlsen F. The ultrastructure of the Z disc in skeletal muscle.J. Cell Biol. 1962; 13: 323-335Crossref PubMed Google Scholar). Due to the hypothesis of filament sliding through Z-disk and M-line, the number of filaments in a half-sarcomere increases when the fiber contracts to lengths below the FLR plateau. This reduces interfilament spacing. Accounting simultaneously for volume constancy, Rode et al. estimated that the interfilament spacing remains rather constant on the ascending limb (2Rode C. Siebert T. Blickhan R. et al.Myosin filament sliding through the Z-disc relates striated muscle fibre structure to function.Proc. Biol. Sci. 2016; 28320153030Google Scholar). Moreover, the Z-disk connects thin filaments in a tetragonal grid (7Knappeis G.G. Carlsen F. The ultrastructure of the Z disc in skeletal muscle.J. Cell Biol. 1962; 13: 323-335Crossref PubMed Google Scholar), which hampers a hexagonal lattice when thick filaments approach the Z-disk on the one hand and prohibits a hexagonal lattice below half-sarcomere lengths of 0.8 μm when thick filament tips slide through the meshed Z-disk on the other hand. Thus, neither the filament numbers nor the Z-disk structure allow a hexagonal lattice at short fiber lengths. In addition, thick filaments interact with different numbers of thin filaments in developing filament overlaps, probably affecting the number of XBs that can be formed. The Rockenfeller model neglects these geometrical changes, shedding further doubt on its FLR’s ascending limb predictions. Double thin-filament overlap starts to develop when thin filaments slide through the M-line near optimal half-sarcomere length (left-hand side of the FLR plateau). The Rockenfeller model assumes that XBs are formed in the region of double thin-filament overlap. This assumption leads to effective force in this region and cannot explain Trombitás and Tigyi-Sebes’ experiments (8TrombitÁs K. Tigyi-Sebes A. Cross-bridge interaction, with oppositely polarized actin filaments in double-overlap zones of insect flight muscle.Nature. 1984; 309: 168-170Crossref PubMed Scopus (13) Google Scholar). Trombitás and Tigyi-Sebes stretched rigor muscle fibers leading to the detachment of thin filaments from the Z-disk. Then, upon activation, symmetrically (at both ends of the sarcomere) detached thin filaments centered around the M-line, completely overlapping. They concluded from this and other results that no effective force is produced in the range of double thin-filament overlap. Trombitás and Tigyi-Sebes explain their conclusion suggesting that “actin filaments slide into (double overlap) regions where they are of the wrong polarity to form cross-bridges, and where they inhibit the existing cross-bridges,” i.e., no XBs form in the range of double thin-filament overlap. Rockenfeller et al. refute the first part of this explanation based on literature and judge that this is “throwing the (28) conclusions into question”. However, they only discard a part of one possible explanation. This does not invalidate Trombitás and Tigyi-Sebes’ conclusion that no effective force is produced in the range of double thin-filament overlap. In their model, Rockenfeller et al. assume that swXBs and regXBs form in the range of double thin-filament overlap. This assumption leads to an effective force in this region. Hence, detached thin filaments would always move in the direction determined by their polarity and not come to a halt in an overlapping configuration. This is in contrast with the results of Trombitás and Tigyi-Sebes. It remains to be shown whether XBs form in the range of double thin-filament overlap and, if yes, how the results of Trombitás and Tigyi-Sebes can still be explained. One basis of the Rockenfeller model’s force calculation is the effective filament overlap, the theoretical capacity to produce isometric force depending on half-sarcomere length without effects of interfilament spacing. Surprisingly, the effective filament overlap (their Figure 1 in (3Rockenfeller R. Günther M. Hooper S.L. Muscle active force-length curve explained by an electrophysical model of interfilament spacing.Biophys. J. 2022; 121: 1823-1855Abstract Full Text Full Text PDF PubMed Google Scholar)) is straight at about 0.8-μm half-sarcomere length. First, when thick-filament tips slide through the Z-disk at this length, the myosin heads meet thin filaments of opposite polarity and form swXBs in their model. In addition to the further developing double thin-filament overlap, this process would increase the proportion of swXBs relative to regXBs in the half-sarcomere. Hence, a kink should occur in Rockenfeller et al.’s effective filament overlap function at 0.8-μm half-sarcomere length. Second, Rockenfeller et al. assume that swXBs always contribute to half-sarcomere shortening. Although this might theoretically be true in the range of double thin-filament overlap that develops when thin filaments slide through the M-line (but would be in contrast with experiments (8TrombitÁs K. Tigyi-Sebes A. Cross-bridge interaction, with oppositely polarized actin filaments in double-overlap zones of insect flight muscle.Nature. 1984; 309: 168-170Crossref PubMed Scopus (13) Google Scholar)), the swXBs formed when thick-filament tips slide through the Z-disk tend to lengthen the half-sarcomere, pushing the thick filaments back. Thus, the force-producing capacity of the half-sarcomere decreases even more below lengths of 0.8 μm, exacerbating the expected kink at 0.8-μm half-sarcomere length in Rockenfeller et al.’s effective filament overlap function. This influences Rockenfeller et al.’s parameter estimation. Concluding, it might be worthwhile to combine the Rockenfeller and Rode models to describe the complete FLAR. C.R. provided the first draft of the manuscript. C.R., T.S., A.T., and R.B. contributed to the manuscript and to the revision. This research was funded by the Deutsche Forschungsgemeinschaft (DFG) under Grants SI841/17-1 and RO5811/1-1. This research was funded by the Deutsche Forschungsgemeinschaft (DFG) under grant 405834662 (SI841/17-1 and RO5811/1-1) as well as partially funded by the DFG as part of the German Excellence Strategy – EXC 2075–390740016. The authors declare no competing interests.