A three filament mechanistic model of musculotendon force and impedance
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This is a valuable study that develops a new model of the way muscle responds to perturbations, synthesizing models of how it responds to small and large perturbations, both of which are used to predict how muscles function for stability but also how they can be injured, and which tend to be predicted poorly by classic Hill-type models. The evidence presented to support the model is solid, since it outperforms Hill-type models in a variety of conditions. Although the combination of phenomenological and mechanistic aspects of the model may sometimes make it challenging to interpret the output, the work will be of interest to those developing realistic models of the stability and control of movement in humans or other animals.
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Abstract
The force developed by actively lengthened muscle depends on different structures across different scales of lengthening. For small perturbations, the active response of muscle is well captured by a linear-time-invariant (LTI) system: a stiff spring in parallel with a light damper. The force response of muscle to longer stretches is better represented by a compliant spring that can fix its end when activated. Experimental work has shown that the stiffness and damping (impedance) of muscle in response to small perturbations is of fundamental importance to motor learning and mechanical stability, while the huge forces developed during long active stretches are critical for simulating and predicting injury. Outside of motor learning and injury, muscle is actively lengthened as a part of nearly all terrestrial locomotion. Despite the functional importance of impedance and active lengthening, no single muscle model has all of these mechanical properties. In this work, we present the viscoelastic-crossbridge active-titin (VEXAT) model that can replicate the response of muscle to length changes great and small. To evaluate the VEXAT model, we compare its response to biological muscle by simulating experiments that measure the impedance of muscle, and the forces developed during long active stretches. In addition, we have also compared the responses of the VEXAT model to a popular Hill-type muscle model. The VEXAT model more accurately captures the impedance of biological muscle and its responses to long active stretches than a Hill-type model and can still reproduce the force-velocity and force-length relations of muscle. While the comparison between the VEXAT model and biological muscle is favorable, there are some phenomena that can be improved: the low frequency phase response of the model, and a mechanism to support passive force enhancement.
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Author response:
The following is the authors’ response to the previous reviews.
The reviewers thoughtful comments have helped us make the manuscript both more comprehensive and clearer. Thank you for your time and effort. We know that this is a long and technical paper. In our responses we refer to three documents:
Original: the first original submission
Revision: the revised document (02 MillardFranklinHerzog2023 v2.pdf)
Difference: a document that shows the changes made to text (but not figures or tables) from the original to revision (03 MillardFranklinHerzog2023 diff.pdf).
Reviewer #1 (Recommendations For The Authors):
(1) In general, the paper is well written and addresses important questions of muscle mechanics and muscle modeling. In the current version, the model limitations are briefly summarized in the abstract. However, the …
Author response:
The following is the authors’ response to the previous reviews.
The reviewers thoughtful comments have helped us make the manuscript both more comprehensive and clearer. Thank you for your time and effort. We know that this is a long and technical paper. In our responses we refer to three documents:
Original: the first original submission
Revision: the revised document (02 MillardFranklinHerzog2023 v2.pdf)
Difference: a document that shows the changes made to text (but not figures or tables) from the original to revision (03 MillardFranklinHerzog2023 diff.pdf).
Reviewer #1 (Recommendations For The Authors):
(1) In general, the paper is well written and addresses important questions of muscle mechanics and muscle modeling. In the current version, the model limitations are briefly summarized in the abstract. However, the discussion needs a more complete description of limitations as well as a discussion of types of data (in vivo, ex vivo, single fiber, wholes muscle, MTU, etc.) that can be modeled using this approach.
Please see the response to comment 23 for more details of the limitations that have been added to the revised document.
(2) The choice of a model with several tendon parameters for simulating single muscle fiber experiments is not well justified.
A rigid-tendon model with a slack length of zero was, in fact, used for these simulations for both the VEXAT and Hill models. In case this is still not clear: a rigid-tendon model of zero length is equivalent to no tendon at all. The text that first mentions the tendon model has now been modified to make it clearer that the parameters of the model were set to be consistent with no tendon at all:
Please see the following text:
Original:
page 17, column 1, line 28 ”... rigid tendon of zero length,”
page 17, column 1, line 51 ”... rigid tendon of zero length.”
Revision:
page 19, column 1, line 19 ”... we used a rigid-tendon of zero length (equivalent to ignoring the tendon)”
page 19, column 1, line 38 ”... coupled with a rigid-tendon of zero-length.”
Difference:
page 21, column 1, line 19 ”... we used a rigid-tendon... ”
page 21, column 1, line 45 ”... rigid-tendon of zero length ...”
(3) A table that clarifies how all model parameters were estimated needs to be included in the main part of the manuscript.
Two tables have been added to the manuscript that detail the parameters of the elastic-tendon cat soleus model (in the main body of the text) and the rabbit psoas fibril model (in an appendix). Each table includes:
A plain language parameter name
The mathematical symbol for the parameter
The value and unit of the parameter
A coded reference to the data source that indicates both the experimental animal and how the data was used to evaluate the parameter.
Please see the following text:
Revision:
page 11
page 42
Difference:
page 11
page 46
(4) The supplemental information is not properly referenced in the main text. There are a number of smaller issues that also need to be addressed.
Thank for your attention to detail. The following problems related to Appendix referencing have been fixed:
Appendices are now parenthetically referenced at the end of a sentence. However, a few references to figures (that are contained within anAppendix) still appear in the body of the sentence since moving these figure references makes the text difficult to understand.
All Appendices are now referenced in the main body of the text.
(5) Abstract, line 6: While it is commonly assumed that the short range stiffness of muscle is due to cross bridges, Rack & Westbury (1974) noted that it occurs over a distance of 25-35 nm, and that many cross-bridges must be stretched even farther than this distance (their p. 348 middle). It seems unlikely that cross-bridges alone can actually account for the short-range stiffness.
There are three parts to our response to this comment:
(a) Rack & Westbury’s definition of short-range-stiffness and unrealistic cross-bridge stretches
(b) Rack & Westbury’s definition of short-range-stiffness vs. linear-timeinvariant system theory
(c) Updates to the paper
a. Rack & Westbury’s definition of short-range-stiffness and unrealistic cross-bridge stretches.
As you note, on page 348, Rack and Westbury write that ”If the short range stiffness is to be explained in terms of extension of cross-bridges, then many of them must be extended further than the 25-35 nm mentioned above.” Having re-read the paper, its not clear how these three factors are being treated in the 25−35 nm estimate:
the elasticity of the tendon and aponeurosis,
the elasticity of actin and myosin filaments,
and the cycling rate of the cross-bridges.
Obviously the elasticity of the tendon, aponeurosis, actin, and myosin filaments will reduce the estimated amount of crossbridge strain during Rack and Westbury’s experiments. A potentially larger factor is the cycling rate of each cross-bridge. If each crossbridge cycles faster than 11 Hz (the maximum frequency Rack and Westbury used), then no single crossbridge would stretch by 25-35 nm. So why didn’t Rack and Westbury consider the cycling rate of crossbridges?
Rack and Westbury’s reasoned that a perfectly elastic work loop would necessarily mean that all crossbridges stayed attached: as soon as a crossbridge cycles it would release its stored elastic energy and the work loop would no longer be elastic. Since Rack and Westbury measured some nearly perfect elastic work loops (the smallest loops in Fig. 2,3, and 4), I guess they assumed crossbridges remained attached during the 25-35 nm crossbridge stretch estimate. However, even Rack and Westbury note that none of the work loops they measured were perfectly elastic and so there is room to entertain the idea that crossbridges are cycling.
Fortunately, for this discussion, crossbridge cycling rates have been measured.
In-vitro measurements by Uyeda et al. show that crossbridges are cycling at 30 Hz when moving at 0.5-1.2 length/s. At this rate, there would be enough time for a single crossbridge to cycle nearly 2.72 times for every cycle of the 11 Hz sinusoidal perturbations, reducing its expected strain from 25-35 nm down to 9.2−12.9µm. This effect becomes even more pronounced if crossbridge cycling rate is used to explain the difference in sliding velocity between Uyeda et al.’s in-vitro data (0.5-1.2 length/s) and the maximum contraction velocity of an in-situ cat soleus (4.65 lengths/s, Scott et al.).
b. Rack & Westbury’s definition of short-range-stiffness vs. linear-time-invariant system theory
Rack and Westbury defined short-range-stiffness to describe a specific kind of force response of the muscle to cyclical length changes:
muscle force is linear with length change,
and independent of velocity.
Rack and Westbury’s definition therefore fails when viscous forces become noticeable, because viscous forces are velocity dependent.
On line 6 of the abstract the term ‘short-range-stiffness’ is not used because Rack and Westbury’s definition is too narrow for our purposes. Instead we are using the more general approach of approximating muscle as a linear-timeinvariant (LTI) system, where it is assumed that
the response of the system is linear
and time invariant.
To unpack that a little, a muscle is considered in the ‘short-range’ in our work if it meets the criteria of a linear time-invariant (LTI) system:
the force response of muscle can be accurately described as a linear function of its length and velocity (its state)
and its response is not a function of time (which means constant stimulation, and no fatigue).
In contrast to Rack and Westbury’s definition, the ‘short-range’ in linear systems theory is general enough to accommodate both elastic and viscous forces. In physical terms, small for an LTI approximation of muscle is larger than the short-range defined by Rack and Westbury: an LTI system can include velocity dependence, while short-range-stiffness ends when velocity dependence begins.
c. Updates to the paper
To make the differences between Rack and Westbury’s ‘short-range-stiffness’ and LTI system theory clearer:
We have removed all occurrences of ‘short-range’ that were associated with Kirsch et al. and have replaced this phrase with ‘small’.
On the first mention of Kirsch’s work we have made the wording more specific
Revision:
page 1, column 1, lines 4,5
page 1, column 2, lines 14-21 ”Under constant activation ...”
Difference: page 1, column 2, line 19-26
page 1, column 1, lines 4,5
page 1, column 2, lines 20-27 ”Under constant activation ...”
A footnote has been added to contrast the definition of ‘small’ in the context of an linear time invariant system to ‘short-range’ in the context of Rack and Westbury’s definition of short-range-stiffness.
Revision: page 1, column 2, bottom
Difference: page 1, column 2, bottom
- In addition, we have added a brief overview of LTI system theory to make the analysis and results more easily understood:
Revision: Figure 4 paragraph beginning on page 10, column 2, line 15 ”As long as ...”
Difference: Figure 4 paragraph beginning on page 12, column 1, line 46 ”As long as ...”
(6) Page 3, lines 6-8: It also seems unlikely that 25% of cross-bridges are attached at one time (Howard, 1997) even for supramaximal isometric stimulation. The number should be less than 20%. What would the ratio of load path stiffness be for low force movements such as changing the direction of a frictionless manipulandum or slow walking? The range of relative stiffnesses is of more interest than the upper limit.
We have made the following updates to address this comment:
A 20% duty cycle now defines the upper bound stiffness of the actinmyosin load path.
We have also evaluated the lower bound actin-myosin stiffness when a single crossbridge is attached.
The stiffness of titin from Kellermayer et al. has been digitized at a length of 2 µm and 4 µm to more accurately capture the length dependence of titin’s stiffness.
We have added a new figure (Figure 14) to make it easier to compare the range of actin-myosin stiffness to titin-actin stiffness.
The text in the main body of the paper and the Appendix has been updated.
The script ’main ActinMyosinAndTitinStiffness.m’ used to perform the calculations and generate the figure is now a part of the code repository.
Please see the following text:
Revision
The paragraph beginning at page 2, column 2, line 45 ”The addition of a titin element ...”
Appendix A
Figure 14 (in Appendix A)
Difference
The paragraph beginning at page 3, column 1, line 6: ”The addition of a titin element ...”
Appendix A
Figure 14 (in Appendix A)
(7) Page 5, line 12: A word seems to be missing here, ”...together to further...”.
Thank you for your attention to detail. The sentence has been corrected.
Please see the following text:
Revision: page 4, column 2, line 40 ”... into a single ...”
Difference: page 5, column 1, line 18
(8) Page 5, line 24-27: These ”theories” are not mutually exclusive, and it is misleading to suggest they are. There is evidence for binding of titin to actin at multiple locations and there is no reason why evidence supporting one binding location must detract from the evidence supporting other binding locations.
The text has been modified to make it clear to readers that the different titinactin binding locations are not mutually exclusive. Please see the following text:
Revision: page 5, column 1, lines 17-19, the sentence beginning ”As previously mentioned, ...”
Difference: page 5, column 1, lines 41-44
(9) Page 5, lines 48-51: Should cite Kellermayer and Granzier (1996) not Kellermayer et al. (1997).
The reference to ‘Kellermayer et al.’ has been changed to ‘Kellermayer and Granzier’. The comment that the year of the reference should be changed from (1997) to (1996) is confusing: the 1996 paper is being referenced.
For further details please see:
Revision: page 5, column 1, 39-40
Difference: page 5, column 2, line 19-22
(10) Also, Dutta et al. (2018) should be cited as further showing that N2A titin by itself slows actin motility on myosin.
Thank you for the suggestion. The sentence has been modified to include Dutta et al.:
For further details please see:
Revision: page 5, column 1, 40
Difference: page 5, column 2, line 19-22
(11) Figure 2 legend and elsewhere: it is odd to say that experiments used ”a cat soleus” when more than one cat coleus was used. Change to ”cat coleus”. See also page 15, line 15.
Thank you for your attention to detail. All occurrences of ‘a cat soleus’ have been changed, with some sentence revision, to ‘cat soleus’.
(12) Page 6, line 10: It is not clear why an MTU was used to simulate single muscle fiber experiments. What is the justification for choosing this particular model? Also, the choice of model might explain why the version with stiff tendon performs better than the version with an elastic tendon, but this is never mentioned. Why not use a muscle model with no tendon (e.g., Wakeling et al., 2021 J. Biomech.)?
Please see the response to comment 2.
(13) Millard et al.’s activation dynamics model also fails to capture the lengthdependence of activation dynamics (Shue and Crago, 1998; Sandercock and Heckman, 1997), which should be noted in the discussion along with other limitations.
An additional limitations paragraph is in the revised manuscript that addresses this comment specifically. However, we have used Stephenson and Wendt as a reference for the shift in peak isometric force that comes with submaximal activation. In addition, we also reference Chow and Darling for the property that the maximum shortening velocity is reduced with submaximal activations.
Revision: page 22, column 1, line 41 ”Finally, the VEXAT model ...”
Difference: page 24, column 2, line 12 ”Finally, the VEXAT model ...”
In addition, please see the response to comment 23.
(14) Page 6, line 22: ”An underbar...”.
Thank you for your attention to detail, this correction has been made.
(14) Page 7, lines 27-32: This and other issues should be described in the Discussion under a heading of model limitations.
Please see the response to comment 23.
(15) Page 7, lines 43-44: Numerous papers from the last author’s laboratory contradict the claim that there is no force enhancement on the ascending limb by demonstrating that force enhancement does occur on the ascending limb (see e.g., Leonard & Herzog 2002, Peterson et al., 2004 and several papers from the Rassier laboratory).
Thank you for your attention to detail. This statement is in error and has been removed. To improve this section of the paper, a paragraph has been added to briefly mention the experimental observations of residual force enhancement before proceeding to explain how this phenomena is represented by the model.
Please see the following text:
Revision:
the paragraph starting on page 7, column 2, line 43 ”When active muscle is lengthened, ...”
and the following paragraph starting on page 8, column 1, line 3 “To develop RFE, ”
Difference:
the paragraph starting on page 8, column 2, line 15
and the following paragraph starting on page 9, column 1, line 6
(17) Figure 3 legend and elsewhere: The authors use Prado et al. (2005) to determine several titin parameters, however the simulations seem to focus on cat soleus, but Prado et al.’s paper is on rabbits. More clarity is needed about which specific results from which species and muscles were used to parameterize the model.
The new parameter table includes coded entries to indicate the literature source for experimental data, the animal it came from, and how the data was used. For example, the ‘ECM fraction’ has a source of ‘R[57]’ to show that the data came from rabbits from reference 57. For further details, please see the response to comment #3
Please see the following text:
Revision: page 11, column 2, table section H: ‘ECM fraction’.
Difference: page 11, column 2, table section H: ‘ECM fraction’.
To address this comment in a little more detail, we have had to use Prado et al. (2005) to give us estimates for only one parameter: P, the fraction of the passive force-length relation that is due to titin. Prado et al.’s measurements relating to P are unique to our knowledge: these are the only measurements we have to estimate P in any muscle, cat soleus or otherwise. Here we use the average of the values for P across the 5 muscles measured by Prado et al. as a plausible default value for all of our simulations.
(18) Figure 4 seems unnecessary.
Figure 4 has been removed.
(19) Page 10, lines 17-18: provide the abbreviation (VAF) here with the definition (variance accounted for).
Thank you for your attention to detail. The abbreviation has been added.
Please see these parts of the manuscripts for details:
Revision: page 12, column 2, line 13
Difference: page 13, column 2, line 32
(20) Page 11, lines 2-3: Here and elsewhere, it is clear that some model parameters have been optimized to fit the model. The main paper should include a table that lists all model parameters and how they were chosen or optimized, including but not limited to the information in Table 1 of the supplemental information section.
See response to comment 3.
(20) Page 17, lines 45 -49: Again, a substantial number of ad hoc adjustments to the model appear to be required. These should be described in the Discussion under limitations, and accounted for in the parameters table. See also legends to Fig. 12 and 13, page 19, lines 23-26.
Please see the response to comment #3: a coded entry now appears to indicate the data source, the animal used in the experiment, and the method used to process the data. This includes entries for parameters which were estimated
‘E’ so that the model produced acceptable results in the simulations presented. In addition, the new discussion paragraph includes a number of sentences that use the adjustment to the active-titin-damping coefficient as an opening to discuss the limitations of the VEXAT’s titin-actin bond model and the circumstances under which the model’s parameters would need to be adjusted.
Please see responses to comments 3 and 23 for additional details. In addition, please see the specific discussion text mentioning the change to βoPEVK:
Revision: page 22, column 1, line 30 ”In Sec. 3.3 we had ...”
Difference: page 24, column 1, line 49
(22) Page 20, lines 50-11: It should be noted here that Tahir et al.’s (2018) model has both series and parallel elastic elements, provided by superposition of rotation (series) and translation (parallel) of a pulley.
While it is true that Tahir et al.’s (2018) model has series and parallel elements, as do the other models mentioned, these models do not have the correct structure to yield a gain and phase response that mimics biological muscle. The text that I originally wrote attempted to explain this without going into the details. As you note, this explanation leaves something to be desired. The original text commenting on the models of Forcinito et al, Tahir et al, Haeufle et al., and Gunther et al. has been updated to be more specific.¨ Please see the parts of the following manuscripts for details:
Revision: page 22, column 2, line 20, the paragraph beginning ”The models of Forcinito ...”
Difference: page 24, column 2, line 44
(23) Discussion: This section should include a description of model limitations, including the relatively large number of ad hoc modifications and how many parameters must be found by optimization in practice. The authors should discuss what types of data are most compatible for use with the model (ex vivo, in vivo, single fiber, whole muscle, MTU), requirements for applying the model to different types of data, and impediments to using the model on different types of data.
An additional limitations paragraph has been added to the discussion.
Please see the following text:
Revision: the paragraph beginning on page 22, column 1, line 11 ”Both the viscoelastic ...”
Difference: the paragraph beginning on page 24, column 1, line 27.
Reviewer #2 (Recommendations For The Authors):
(1) If it is possible to compare the output of this model to other more contemporary models which incorporate titin but are also simple enough to implement in whole-body simulation (such as the winding filament model), this would seem to greatly strengthen the paper.
That’s an excellent idea, though beyond the scope of this already lengthy paper. Even though the Hill model we evaluated is a bit old it is widely used, and so, many readers will be interested in seeing the benchmark results. As benchmarking work is both difficult to fund and undertake, we do hope that others will evaluate their own models using the code and data we have provided.
(2) I’m a little unclear on the basis for the transition between short- and midrange length changes, both in reality and in the model. And also about the range of strains that qualify as ”short”. It seems like there is potential for short range stiffness, although I would have thought more in the range of 1-2% strains than >3%, to be due to currently attached crossbridges. There is clear evidence that active titin is responsible for the low stiffness at very large strains that exceed actin-myosin overlap. But I am not clear on how a transitional stiffness on the descending limb of the force-length relationship is implemented in the model, and what aspect of physiology this is replicating. It may be helpful to clarify this further and indicate where in the model this stiffness arises.
This question has several parts to it which I will paraphrase here:
A Short-range stiffness acts over smaller strains than 3.8%. How is shortrange defined?
B Where is the transition made between short-range and mid-range force response, both in reality and in the model. Also how does this change on the descending limb?
C What components in the model contribute to the stiffness of the CE?
A. Short-range stiffness acts over smaller strains than 3.8%. How is shortrange defined?
The response to Reviewer 1’s comment # 5 directly addresses this question.
B. Where is the transition made between short-range and mid-range forceresponse, both in reality and in the model. Also how does this change on the descending limb? We are going to rephrase the question because of changes in terminology that we have made in response to Reviewer 1’s comment #5.
(i) What is the basis for the transition between the muscle behaving like an LTI system? Both in reality, and in the model. (ii) What happens outside the LTI range? (iii) Also how does this change on the descending limb?
We will address this question one part at a time:
(i) What is the basis for the transition between the muscle behaving like an LTI system? Both in reality, and in the model.
A system’s response can be approximated as a linear-time-invariant (LTI) system as long as it is time-invariant, and its output can be expressed as a linear function of its input. In the context of Kirsch et al.’s experiment, the ‘system’ is the muscle, the ‘input’ is the time series of length data, and the ‘output’ is the time series of force data. Due to the requirement for timeinvariance, two experimental conditions must be met to approximate muscle as an LTI system:
• the nominal length of the muscle stays constant over long periods of time,
• and the nominal activation of the muscle stays constant.
These conditions were met by default in Kirch et al.’s experiment, and also in our simulations of this experiment. The one remaining condition to assess is whether or not the muscle’s response is linear.
To evaluate whether the muscle’s force is a linear function of the length change, Kirch et al. evaluated (Cxy)2 the coherence squared between the length and force time-series data. Even though the mathematical underpinnings of (Cxy)2 are complicated, the interpretation of (Cxy)2 is simple: muscle can be accurately approximated as a linear system if (Cxy)2 is close to 1, but the accuracy of this approximation becomes poor as (Cxy)2 approaches 0. Kirsch et al. used (Cxy)2 to identify a bandwidth in which the response of the muscle to the 1−3.8%ℓoM length changes was sufficiently linear for analysis: a lower bound of 4 Hz was identified using (Cxy)2 and the bandwidth of the input signal (15 Hz, 35 Hz, or 90 Hz) set the upper bound. In Fig. 3 of Kirsch et al. the (Cxy)2 at 4 Hz has a value of at least 0.67 for the 15 Hz and 90 Hz signals. To minimize error in our analysis and yet be consistent with Kirsch et al., we analyze the bandwidth common to both (Cxy)2 ≥ 0.67 and Kirsch et al.’s defined range. Though the bandwidth defined by the criteria (Cxy)2 ≥ 0.67 is usually larger than the one defined by Kirsch et al., there are some exceptions where the lower frequency bound of the models is higher than 4 Hz (now reported in Tables 4D and 5D).
(ii) What happens outside the LTI range?
When a muscle’s output cannot be considered a LTI it means that either that its length or activation is time-varying, or the relationship between length and force is no longer linear. In short, that the muscle is behaving as one would normally expect: time-varying and non-linearly. The wonderful part of Kirsch et al.’s work is that they found a surprisingly large region in the frequency domain where muscle behaves linearly and can be analyzed using the powerful tools of linear systems and signals.
(iii) Also how does this change on the descending limb?
Since nominal length of Kirsch et al.’s experiments is ℓoM it is not clear how the results of the perturbation experiments will change if the nominal length is moved firmly to the descending limb. However, we can see how the stiffness and damping values will change by examining Figure 9C and 9D which shows the calculated stiffness and damping of the VEXAT and Hill models as ℓM is lengthened from ℓoM down the descending limb: the stiffness and damping of the VEXAT model does not change much, while the Hill model’s stiffness changes sign and the damping coefficient changes a lot. What cannot be seen from Figure 9C and 9D is how the bandwidth over which the models are considered linear changes.
We have made a number of updates to the text to more clearly communicate these details of our response to part (i):
• Text has been edited so that it is clear that the terms ’short-range stiffness’ and ’small’ from Rack and Westbury’s work is not confused with ’stiffness’ and ’small’ from the LTI system’s analysis. Please see our response to comment # 5 for details.
• We have added text to the main body of the paper to explain how the coherence squared metric was used to select a bandwidth in which the response of the system is approximately linear:
- Revision: the paragraph that starts on page 11, column 1, line 3 ”Kirsch et al. used system identification ...”
– Difference: page 13, column 2, line 1
– Coherence is defined in Appendix D
– Coherence is now also included in the example script ‘main SystemIdentificationExample.m’
• The bandwidth over which model output can be considered linear (coherence squared > 0.67) has been added to Tables 4 and 5
– Revision: see Table 4D, and Table 5D in Appendix E
– Difference: see Table 4D, and Table 5D in Appendix E
• Figures 6 and Figures 16 are annotated now if the plotted signal does not meet the linearity requirement of Cxy > 0.67.
C. What components in the model contribute to the stiffness of the CE?
There are three components that contribute to the stiffness of the CE which are pictured in Figure 1, appear in Eqn. 15, and are listed explicitly in Eqn. 76:
(a) The XE, as represented by the afL(ℓ˜S+L˜M)k˜oX term in Eqn. 15.
(b) The elasticity of the distal segment of titin, f2(ℓ˜2). Only f2(ℓ˜2) appears in Eqn. 15 because ℓ˜1 is a model state.
(c) The extracellular matrix, as represented by the fECM(ℓ˜ECM)
There is also a compressive element fKE, but it plays no role in the simulations presented in this work because it only begins to produce force at extremely short CE lengths (ℓ˜M < 0.1ℓoM).
We have made the following changes to make these components clearer
Figure 1A has been updated:
– The symbols for a spring and a damper are now defined in Figure 1A
– The ECM now has a spring symbol. Now all springs and dampers have the correct symbol in Figure 1A.
– The caption now explicitly lists the rigid, viscoelastic, and elastic elements in the model
The equations for the VEXAT’s CE stiffness and damping are now compared and contrasted to the the Hill model’s stiffness and damping in Sec. 3.1.
– Revision: starting at page 14, column 2, line 1: Eqn. 28 and Eqn. 29 and surrounding text
– Difference: page 17, column 1, line 22
(3) This model appears to be an amalgamation of a phenomenological (forcelength and force-velocity relationships) and a mechanistic (crossbridge and titin stiffness and damping) model. While this may improve predictions, and so potentially be useful, it also seems like it limits the interpretation of physiological underpinnings of any findings. It may be helpful to explore in greater detail the implications of this approach.
We have added a limitations paragraph to the discussion which addresses this comment and can be found in:
• Revision: the paragraph beginning on page 22, column 1, line 11 ”Both the viscoelastic ...”
• Difference: the paragraph beginning on page 24, column 1, line 27
(4)As a biologist, I found the interpretation of phase and gain a little difficult and it may help the reader to show in greater detail the time series data and model predictions to highlight conditions under which the models do not accurately capture the magnitude and timing of force production.
It is important that the ideas of phase and gain are understood, especially because little information can be gleaned from the time series data directly. There is some time series data in the paper already that compares each model’s response to its spring-damper of best fit: plots of the force response of each model and its spring damper of best fit can be found in Figures 6A, 6D, 6G, 6J, 16A, 16D, 16G, and 16J in the revised manuscript. While it is clear that models with a higher VAF more closely match the spring-damper of best fit, there is not much more that can be taken from time series data: the systematic differences, particularly in phase, are just not visually apparent in the time-domain but are clear in gain and phase plots in the frequency-domain.
To make the meaning of phase and gain plots clearer, Figure 4 (Figure 5 in the first submission) has been completely re-made and includes plots that illustrate the entire process of going from two length and force timedomain signals to gain and phase plots in the frequency-domain. Included in this figure is a visual representation of transforming a signal from the time to the frequency domain (Fig. 4B and 4C), and also an illustration of the terms gain and phase (Fig. 4D). In addition, a small example file ’main SystemIdentificationExample.m’ has been added to the matlab code repository in the elife2023 branch to accompany Appendix D, which goes through the mathematics used to transform input and output time domain signals into gain and phase plots of the input-output relation. Small updates have been made to Figure 6 and 16 in the revised paper (Figures 7 and 18 in the first submission) to make the time domain signals from the spring-damper of best fit and the model output clearer. Finally, I have re-calculated the gain and phase profiles using a more advanced numerical method that trades off some resolution in frequency for more accuracy in the magnitude. This has allowed me to make Figures 6 and 16 easier to follow because the gain and phase responses are now lines rather than a scattering of points. We hope that these additions make the interpretation of gain and phase clearer.
Please see
Revision:
– Figure 4 and caption on page 12
– The opening 2 paragraphs of Sec 3.1 starting on page 10, column 2, line 4 ”In Kirsch et al.’s ...”
– Figure 6 & 16: spring damper and model annotation added, plotted the gain and phase as lines
– Appendix D: Updated to include coherence and the more advanced method used to evaluate the system transfer function, gain, and phase.
Difference:
– Figure 4 and caption on page 12
– The opening 2 paragraphs of Sec 3.1 starting on page 12, column 1, line 34 and ending on page 13, column 2, line 29
– Figure 6 & 16: spring damper and model annotation added
– Appendix D
(5) The actin-myosin and actin-titin load pathways are depicted as distinct in the model. However, given titin’s position in the center of myosin and the crossbridge connections between actin and myosin, this would seem to be an oversimplification. It seems worth considering whether the separation of these pathways is justified if it has any effect on the conclusions or interpretation.
We have reworked one of the discussion paragraphs to focus on how our simulations would be affected by two mechanisms (Nishikawa et al.’s winding filament theory and DuVall et al.’s titin entanglement hypothesis) that make it possible for crossbridges to do mechanical work on titin.
• Revision: the paragraph beginning on page 21, column 2, line 42 “The active titin model ...”
• Difference: the paragraph beginning on page 23, column 2, line 48
References
Nishikawa KC, Monroy JA, Uyeno TE, Yeo SH, Pai DK, Lindstedt SL. Is titin a ‘winding filament’? A new twist on muscle contraction. Proceedings of the royal society B: Biological sciences. 2012 Mar 7;279(1730):981-90.
DuVall M, Jinha A, Schappacher-Tilp G, Leonard T, Herzog W. I-Band Titin Interaction with Myosin in the Muscle Sarcomere during Eccentric Contraction: The Titin Entanglement Hypothesis. Biophysical Journal. 2016 Feb 16;110(3):302a.
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eLife assessment
This is a valuable study that develops a new model of the way muscle responds to perturbations, synthesizing models of how it responds to small and large perturbations, both of which are used to predict how muscles function for stability but also how they can be injured, and which tend to be predicted poorly by classic Hill-type models. The evidence presented to support the model is solid, since it outperforms Hill-type models in a variety of conditions. Although the combination of phenomenological and mechanistic aspects of the model may sometimes make it challenging to interpret the output, the work will be of interest to those developing realistic models of the stability and control of movement in humans or other animals.
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Reviewer #1 (Public Review):
Muscle models are important tools in the fields of biomechanics and physiology. Muscle models serve a wide variety of functions, including validating existing theories, testing new hypotheses, and predicting forces produced by humans and animals in health and disease. This paper attempts to provide an alternative to Hill-type muscle models that includes contributions of titin to force enhancement over multiple time scales. Due to the significant limitations of Hill-type models, alternative models are needed and therefore the work is important and timely.
The effort to include a role for titin in muscle models is a major strength of the methods and results. The results clearly demonstrate the weaknesses of Hill models and the advantages of incorporating titin into theoretical treatments of muscle mechanics. …
Reviewer #1 (Public Review):
Muscle models are important tools in the fields of biomechanics and physiology. Muscle models serve a wide variety of functions, including validating existing theories, testing new hypotheses, and predicting forces produced by humans and animals in health and disease. This paper attempts to provide an alternative to Hill-type muscle models that includes contributions of titin to force enhancement over multiple time scales. Due to the significant limitations of Hill-type models, alternative models are needed and therefore the work is important and timely.
The effort to include a role for titin in muscle models is a major strength of the methods and results. The results clearly demonstrate the weaknesses of Hill models and the advantages of incorporating titin into theoretical treatments of muscle mechanics. Another strength is to address muscle mechanics over a large range of time scales.
The authors succeed in demonstrating the need to incorporate titin in muscle models, and further show that the model accurately predicts in situ force of cat soleus (Kirsch et al. 1994; Herzog & Leonard, 2002) and rabbit posts myofibrils (Leonard et al. 2010). However, it remains unclear whether the model will be practical for use with data from different muscles or preparations. Several ad hoc modifications were described in the paper, and the degree to which the model requires parameter optimization for different muscles, preparations and experiment types remains unclear.
I think the authors should state how many parameters require fitting to the data vs the total number of model parameters. It would also be interesting for the authors to discuss challenges associated with modeling ex vivo and in vivo data sets, due to differences in means of stimulation vs. model inputs.
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Reviewer #2 (Public Review):
This model of skeletal muscle includes springs and dampers which aim to capture the effect of crossbridge and titin stiffness during the stretch of active muscle. While both crossbridge and titin stiffness have previously been incorporated, in some form, into models, this model is the first to simultaneously include both. The authors suggest that this will allow for the prediction of muscle force in response to short-, mid- and long-range stretches. All these types of stretch are likely to be experienced by muscle during in vivo perturbations, and are known to elicit different muscle responses. Hence, it is valuable to have a single model which can predict muscle force under all these physiologically relevant conditions. In addition, this model dramatically simplifies sarcomere structure to enable this …
Reviewer #2 (Public Review):
This model of skeletal muscle includes springs and dampers which aim to capture the effect of crossbridge and titin stiffness during the stretch of active muscle. While both crossbridge and titin stiffness have previously been incorporated, in some form, into models, this model is the first to simultaneously include both. The authors suggest that this will allow for the prediction of muscle force in response to short-, mid- and long-range stretches. All these types of stretch are likely to be experienced by muscle during in vivo perturbations, and are known to elicit different muscle responses. Hence, it is valuable to have a single model which can predict muscle force under all these physiologically relevant conditions. In addition, this model dramatically simplifies sarcomere structure to enable this muscle model to be used in multi-muscle simulations of whole-body movement.
In order to test this model, its force predictions are compared to 3 sets of experimental data which focus on short-, mid- and long-range perturbations, and to the predictions of a Hill-type muscle model. The choice of data sets is excellent and provide a robust test of the model's ability to predict forces over a range of length perturbations. However, I find the comparison to a Hill-type muscle model to be somewhat limiting. It is well established that Hill-type models do not have any mechanism by which they can predict the effect of active muscle stretch. Hence, that the model proposed here represents an improvement over such a model is not a surprise. Many other models, some of which are also simple enough to be incorporated into whole-body simulations, have incorporated mechanistic elements which allow for the prediction of force responses to muscle stretch. And it is not clear from the results presented here that this model would outperform such models.
The paper begins by outlining the phenomenological vs mechanistic approaches taken to muscle modelling, historically. It appears, although is not directly specified, that this model combines these approaches. A somewhat mechanistic model of the response of the crossbridges and titin to active stretch is combined with a phenomenological implementation of force-length and force-velocity relationships. This combination of approaches may be useful improving the accuracy of predictions of muscle models and whole-body simulations, which is certainly a worthy goal. However, it also may limit the insight that can be gained. For example, it does not seem that this model could reflect any effect of active titin properties on muscle shortening. In addition, it is not clear to me, either physiologically or in the model, what drives the shift from the high stiffness in short-range perturbations to the somewhat lower stiffness in mid-range perturbations.
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eLife assessment
This is an important study that develops a new model of the way muscle responds to perturbations, synthesizing models of how it responds to small and large perturbations, both of which are important to predict how muscles function for stability but also how they can be injured. The evidence presented to support the model is solid, but the work is incomplete as it is particularly lacking a more detailed analysis of the trade-offs associated with using the model to simulate different types of preparations, especially single muscle fibers compared to whole muscles or whole muscles and tendons. With a clearer discussion of the limitations of the model and the situations in which it is best applied, the work will be of interest to those developing realistic models of the stability and control of movement in humans or other …
eLife assessment
This is an important study that develops a new model of the way muscle responds to perturbations, synthesizing models of how it responds to small and large perturbations, both of which are important to predict how muscles function for stability but also how they can be injured. The evidence presented to support the model is solid, but the work is incomplete as it is particularly lacking a more detailed analysis of the trade-offs associated with using the model to simulate different types of preparations, especially single muscle fibers compared to whole muscles or whole muscles and tendons. With a clearer discussion of the limitations of the model and the situations in which it is best applied, the work will be of interest to those developing realistic models of the stability and control of movement in humans or other animals.
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Reviewer #1 (Public Review):
Muscle models are important tools in the fields of biomechanics and physiology. Muscle models serve a wide variety of functions, including validating existing theories, testing new hypotheses, and predicting forces produced by humans and animals in health and disease. This paper attempts to provide an alternative to Hill-type muscle models that includes contributions of titin to force enhancement over multiple time scales. Due to the significant limitations of Hill-type models, alternative models are needed and therefore the work is important and timely.
The effort to include a role for titin in muscle models is a major strength of the methods and results. The results clearly demonstrate the weaknesses of Hill models and the advantages of incorporating titin into theoretical treatments of muscle mechanics. …
Reviewer #1 (Public Review):
Muscle models are important tools in the fields of biomechanics and physiology. Muscle models serve a wide variety of functions, including validating existing theories, testing new hypotheses, and predicting forces produced by humans and animals in health and disease. This paper attempts to provide an alternative to Hill-type muscle models that includes contributions of titin to force enhancement over multiple time scales. Due to the significant limitations of Hill-type models, alternative models are needed and therefore the work is important and timely.
The effort to include a role for titin in muscle models is a major strength of the methods and results. The results clearly demonstrate the weaknesses of Hill models and the advantages of incorporating titin into theoretical treatments of muscle mechanics. Another strength is to address muscle mechanics over a large range of time scales. Weaknesses include the decision to use a MTU model to simulate experiments from single muscle fibers, and failure to systematically address the limitations of the model, including equations for activation dynamics with no length dependence. It would also be useful for readers if the authors provided a discussion of the types of data that can be simulated using the model, along with potential pitfalls and how to determine model parameters.
The authors succeed in demonstrating the need to incorporate titin in muscle models. However, it remains unclear whether it will be practical for others to use this particular model for different types of data. Several ad hoc modifications were described in the paper, and the degree to which the model requires parameter optimization for different muscles, preparations and experiment types is also unclear.
-
Reviewer #2 (Public Review):
This model of skeletal muscle includes springs and dampers which aim to capture the effect of crossbridge and titin stiffness during the stretch of active muscle. While both crossbridge and titin stiffness have previously been incorporated, in some form, into models, this model is the first to simultaneously include both. The authors suggest that this will allow for the prediction of muscle force in response to short-, mid- and long-range stretches. All these types of stretch are likely to be experienced by muscle during in vivo perturbations, and are known to elicit different muscle responses. Hence, it is valuable to have a single model which can predict muscle force under all these physiologically relevant conditions. In addition, this model dramatically simplifies sarcomere structure to enable this …
Reviewer #2 (Public Review):
This model of skeletal muscle includes springs and dampers which aim to capture the effect of crossbridge and titin stiffness during the stretch of active muscle. While both crossbridge and titin stiffness have previously been incorporated, in some form, into models, this model is the first to simultaneously include both. The authors suggest that this will allow for the prediction of muscle force in response to short-, mid- and long-range stretches. All these types of stretch are likely to be experienced by muscle during in vivo perturbations, and are known to elicit different muscle responses. Hence, it is valuable to have a single model which can predict muscle force under all these physiologically relevant conditions. In addition, this model dramatically simplifies sarcomere structure to enable this muscle model to be used in multi-muscle simulations of whole-body movement.
In order to test this model, its force predictions are compared to 3 sets of experimental data which focus on short-, mid- and long-range perturbations, and to the predictions of a Hill-type muscle model. The choice of data sets is excellent and provide a robust test of the model's ability to predict forces over a range of length perturbations. However, I find the comparison to a Hill-type muscle model to be somewhat limiting. It is well established that Hill-type models do not have any mechanism by which they can predict the effect of active muscle stretch. Hence, that the model proposed here represents an improvement over such a model is not a surprise. Many other models, some of which are also simple enough to be incorporated into whole-body simulations, have incorporated mechanistic elements which allow for the prediction of force responses to muscle stretch. It is not clear from the results presented here that this model would outperform such models.
The paper begins by outlining the phenomenological vs mechanistic approaches taken to muscle modelling, historically. It appears, although is not directly specified, that this model combines these approaches. A somewhat mechanistic model of the response of the crossbridges and titin to active stretch is combined with a phenomenological implementation of force-length and force-velocity relationships. This combination of approaches may be useful in improving the accuracy of predictions of muscle models and whole-body simulations, which is certainly a worthy goal. However, it also may limit the insight that can be gained. For example, it does not seem that this model could reflect any effect of active titin properties on muscle shortening. In addition, it is not clear to me, either physiologically or in the model, what drives the shift from the high stiffness in short-range perturbations to the somewhat lower stiffness in mid-range perturbations.
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