Sarcomere dynamic instability and stochastic heterogeneity drive robust cardiomyocyte contraction

Reviewer #1 (Public review):

Summary:

In this manuscript, the authors present comprehensive experimental observations and a theoretical framework to explain the heterogeneous behaviour of sarcomeres in cardiomyocytes. They show that a stochastic component exists in their contractile activity, which may act as a feedback mechanism regulating physiological function.

Strengths:

Experiments and data analysis are robust and valid. The rigorous statistical analysis and unbiased methods enable the authors to draw well-supported conclusions that go beyond the existing literature. Their outcomes inform about cellular activity at the individual level and the authors explain how the transient dynamics of single sarcomeres are governed by a force-velocity relationship and lead to the complex contractile patterns. The similarity of the results to the study cited in [24] demonstrates the validity of the in vitro setup for answering these questions and the feasibility of such in-vitro systems to extend our knowledge of out-of-equilibrium dynamics in cardiac cells.

Very interesting the suggestion that the interplay between intrinsic fluctuations and the dynamic instability are part of a feedback mechanism for maintaining structural and functional homeostasis.

The addition of the theoretical model and the new text of the manuscript improves the clarity of the study.

Reviewer #2 (Public review):

Summary:

Sarcomeres, the contractile units of skeletal and cardiac muscle, contract in a concerted fashion to power myofibril and thus muscle fiber contraction.

Muscle fiber contraction depends on the stiffness of the elastic substrate of the cell, yet it is not known how this dependence emerges from the collective dynamics of sarcomeres. Here, the authors analyze contraction time series of individual sarcomeres using live imaging of fluorescently labeled cardiomyocytes cultured on elastic substrates of different stiffness. They find that a reduced collective contractility of muscle fibers on unphysiologically stiff substrates is partially explained by a lack of synchronization in the contraction of individual sarcomeres.

This lack of synchronization is at least partially stochastic, consistent with the notion of a tug-of-war between sarcomeres on stiff sarcomeres. A particular irregularity of sarcomere contraction cycles is 'popping', the extension of sarcomers beyond their rest length. The statistics of 'popping' suggest that this is a purely random process.

Strengths:

This study thus marks an important shift of perspective from whole-cell analysis towards an understanding the collective dynamics of coupled, stochastic sarcomeres.

Reviewer #3 (Public review):

The manuscript of Haertter and coworkers studied the variation of the length of a single sarcomere and the response of microfibrils made by sarcomeres of cardiomyocytes on soft gel substrates of varying stiffness.

The measurements at the level of a single sarcomere are an important new result of this manuscript. They are done by combining the labeling of the sarcomeres z line using genetic manipulation and a sophisticated tracking program using machine learning. This single sarcomere analysis shows strong heterogeneities of the sarcomeres that can show fast oscillations not synchronized with the average behavior of the cell and what the authors call popping eveents which are large amplitude oscillations. Another important result is the fact that cardiomyocyte contractility decreases with the substrate stiffness, although the properties of single sarcomeres do not seem to depend on substrate stiffness.

The authors suggest that the cardiomyocyte cell behavior is dominated by sarcomere heterogeneity. They show that the heterogeneity between sarcomere is stochastic and that the contribution of static heterogeneity (such as composition differences between sarcomeres) is small.

Strengths:

All the results are, to my knowledge, new and original. The authors also made a theoretical model where each sarcomere is described by a Langevin equation based on a non-linear coupling between force and velocity of the sarcomeres. This model accounts well for the experimental results including the observation of what the authors call popping events.

Author response:

The following is the authors’ response to the original reviews.

eLife Assessment

This study provides a valuable characterization of individual sarcomere's contractility and synchrony in spontaneously beating cardiomyocytes as a function of substrate stiffness. The authors, however, provide an incomplete explanation for the observed heterogeneous and stochastic dynamics, so that the work remains mainly descriptive. The work will be of interest to scientists working on muscle biophysics, nonlinear dynamics, and synchronization phenomena in biological systems.

We appreciate the reviewer’s insightful comments. A detailed explanation of the described phenomena in the form of a theoretical model and simulations was not included in our manuscript, because we believed it would be most impactful to present a detailed quantitative statistical description of the experiments in one manuscript and then introduce the model, which we already had in preparation, in a separate manuscript to avoid diluting the overall message.

However, following the reviewers’ advice, we have now included a comprehensive model into the revised manuscript. This model qualitatively and quantitatively explains the experimentally observed phenomena and introduces a novel class of coupled relaxation oscillators based on a non-monotonic force-velocity relationship of individual sarcomeres. We believe that this addition significantly strengthens the manuscript.

Public Reviews:

Reviewer #1 (Public Review):

Summary:

In this manuscript, the authors experimentally demonstrated the heterogeneous behavior of sarcomeres in cardiomyocytes and that a stochastic component exists in their contractile activity, which cancels out at the level of myofibrils.

Strengths:

The experiments and data analysis are robust and valid. With very good statistics and unbiased methods, they show cellular activity at the individual level and highlight the heterogeneity between biological networks. The similarity of the results to the study cited in [24] demonstrates the validity of the in vitro setup for answering these questions and the feasibility of such in-vitro systems to extend our knowledge of physiology.

Weaknesses:

Compared to the current literature ([24]), the study does not show a high degree of innovation. It mainly confirms what has been established in the past. The authors complemented the published experiments by developing an in vitro setup with stem cells and by changing the stiffness of the substrate to simulate pathological conditions. However, the experiments they performed do not allow them to explain more than the study in [24], and the conclusions of their study are based on interpretation and speculation about the possible mechanism underlying the observations.

We thank the reviewer for contextualizing our work with the literature. We appreciate the comparison to the study by Kobirumaki-Shimozawa et al. which we cite prominently. They observed stochastically varying beating patterns of individual sarcomeres on a beat-to-beat basis. They propose that this arises from a "titin-based mechanism" operating stochastically, which they interpret as being fundamentally linked to sarcomere-length-dependent effects. This interpretation differs from our model. We feel that the inclusion of our comprehensive model in the revised manuscript will emphasize the significance and novelty of our findings. Our work proposes a distinct alternative mechanistic explanation for the observed stochasticity, grounded in the force-velocity relationship and intrinsic stochasticity, and presents additional novel dynamic phenomena (such as popping and high-frequency oscillations) not reported in the literature yet. We outline the key advancements of our study below:

(1) Physiologically Relevant Human Model System: Our study utilizes human induced pluripotent stem cell-derived cardiomyocytes (hiPSC-CMs). Using a human cell model provides direct relevance for understanding human cardiac physiology and pathophysiology, overcoming limitations inherent in translating findings from rodent models. The hiPSC-CMs exhibit key physiological differences from the mouse ventricular myocytes observed in [24], most notably beating at a significantly lower frequency (~1 Hz or 60 bpm) compared to mice (~5-8 Hz or 300-500 bpm). This difference in timescale is critical as it allowed us to resolve complex intra-beat dynamics that may be different and also harder to observe in mouse cardiomyocytes.

(2) Advanced Experimental Methodology and Resolution: We developed a novel assay incorporating our SarcAsM algorithm for high-throughput tracking and analysis of individual sarcomere dynamics. This approach gave us spatial resolution better than 20 nm at significantly higher sampling rates than previous studies, including Kobirumaki-Shimozawa et al. Furthermore, our high-throughput in vitro approach made it possible to analyze vastly larger datasets than, e.g., the study by Kobirumaki-Shimozawa et al. (which reports observations from fewer than 20 myofibrils, encompassing less than 200 sarcomeres in total). While we recognize that in-vivo tissue studies present unique experimental challenges, the substantially greater statistical power of our study is crucial for reliably characterizing the complex, stochastic dynamics we report. The enhanced resolution and statistical robustness are not merely incremental; they enable the detailed identification and analysis of heterogeneous behaviors that were previously inaccessible or could not be characterized with the same level of confidence.

(3) Novel Observed Phenomena: Our high-resolution data reveals specific dynamic behaviors, such as sarcomere "popping" and high-frequency oscillations during contraction, which, to our knowledge, have not been previously reported or characterized in cardiomyocytes. The resolution limitations and the high beating frequency in mouse models may not have permitted the observation of these subtle, but potentially important phenomena.

(4) Distinct Mechanistic Explanation and Model: Kobirumaki-Shimozawa et al. propose a qualitative model where sarcomere motion variability primarily arises from length-dependent activation. This view is essentially a static one, based on a long history of isometric skeletal muscle experiments, where time-dependent forces are not relevant. We argue that in highly dynamic cardiomyocytes this may not be the most useful approach. While we acknowledge length dependence can play a role, our integrated experimental-theoretical work proposes a different primary mechanism. Our model demonstrates that the observed stochastic heterogeneity and beat-to-beat variations, including the oscillatory motion and popping, can be quantitatively explained by dynamic instabilities arising from a non-monotonic force-velocity relationship of individual sarcomeres in conjunction with intrinsic sarcomere-level stochastic fluctuations. The model emphasizes the active, transient nature of force generation rather than solely assuming length dependence. Our model provides an alternative explanation for the observed dynamics, and a quantitative, mechanism-based understanding.

Reviewer #2 (Public Review):

Summary:

Sarcomeres, the contractile units of skeletal and cardiac muscle, contract in a concerted fashion to power myofibril and thus muscle fiber contraction.

Muscle fiber contraction depends on the stiffness of the elastic substrate of the cell, yet it is not known how this dependence emerges from the collective dynamics of sarcomeres. Here, the authors analyze the contraction time series of individual sarcomeres using live imaging of fluorescently labeled cardiomyocytes cultured on elastic substrates of different stiffness. They find that reduced collective contractility of muscle fibers on unphysiologically stiff substrates is partially explained by a lack of synchronization in the contraction of individual sarcomeres.

This lack of synchronization is at least partially stochastic, consistent with the notion of a tug-of-war between sarcomeres on stiff sarcomeres. A particular irregularity of sarcomere contraction cycles is 'popping', the extension of sarcomeres beyond their rest length. The statistics of 'popping' suggest that this is a purely random process.

Strengths:

This study thus marks an important shift of perspective from whole-cell analysis towards an understanding of the collective dynamics of coupled, stochastic sarcomeres.

Weaknesses:

Further insight into mechanisms could be provided by additional analyses and/or comparisons to mathematical models.

We thank the reviewer for the feedback. We have enhanced the manuscript by a comprehensive dynamic model, that we also contrast with previously proposed models.

Reviewer #3 (Public Review):

Summary:

The manuscript of Haertter and coworkers studied the variation of length of a single sarcomere and the response of microfibrils made by sarcomeres of cardiomyocytes on soft gel substrates of varying stiffnesses.

The measurements at the level of a single sarcomere are an important new result of this manuscript. They are done by combining the labeling of the sarcomeres z line using genetic manipulation and a sophisticated tracking program using machine learning. This single sarcomere analysis shows strong heterogeneities of the sarcomeres that can show fast oscillations not synchronized with the average behavior of the cell and what the authors call popping events which are large amplitude oscillations. Another important result is the fact that cardiomyocyte contractility decreases with the substrate stiffness although the properties of single sarcomeres do not seem to depend on substrate stiffness.

The authors suggest that the cardiomyocyte cell behavior is dominated by sarcomere heterogeneity. They show that the heterogeneity between sarcomeres is stochastic and that the contribution of static heterogeneity (such as composition differences between sarcomeres) is small.

Strengths:

All the results are to my knowledge new and original and deserve attention.

Weaknesses:

However, I find the manuscript a bit frustrating because the authors only give very qualitative explanations of the phenomena that they observe. They mention that popping could be explained by a nonlinear force-velocity relation of the sarcomere leading to a rapid detachment of all motors. However, they do not explicitly provide a theoretical description. How would the popping depend on the parameters and in particular on the substrate stiffness? Would the popping statistics be affected by the stiffness? It is also not clear to me how the dependence on the soft gel stiffness of the cardiomyocyte cell can be explained by the stochasticity of the sarcomere properties. Can any of the results found by the authors be explained by existing theories of cardiomyocytes? The only one I know is that of Safran and coworkers.

I also found the paper very difficult to read. The authors should perhaps reorganize the structure of the presentation in order to highlight what the new and important results are.

We are grateful for this detailed and critical feedback. The observed phenomena (stochastic heterogeneity, popping, high-frequency oscillatory motion) can indeed be explained by a nonmonotonic force-velocity relation along with stochastic fluctuations of individual sarcomeres. At the time of initial submission of this manuscript, we already had a theoretical model in preparation, which both qualitatively and quantitatively explains the observed phenomena. As a result, we included certain interpretations preemptively, which caused some lack of clarity in the absence of the full model. We have now added the model to this manuscript, providing a mechanistic interpretation of our findings. The model is different from prior models in that it emphasizes time-dependent forces, typically disregarded in models built to understand isometric skeletal muscle experiments.

We have shortened, streamlined and restructured our manuscript to improve the readability and accessibility of our study.

Recommendations for the authors:

There is a consensus among reviewers that the link between the stiffness dependence of the observed stochastic dynamics and the proposed tug-of-war mechanism is unclear. More quantitative support and discussion is required, possibly using theoretical modeling.

We are grateful for the insightful and comprehensive feedback by both editor and reviewers. As suggested, we have now added a comprehensive model explaining the observed phenomena and presenting a new conceptual view on cardiac muscle dynamics.

Reviewer #1 (Recommendations For The Authors):

The authors addressed an interesting question related to the dynamics of cardiac cells and their multiscale dynamics. They did a good job in terms of experimental design and data analysis. However, I fear that they do not contribute enough new information to the topic.

The authors should refer to the study in [24] and explain better the difference between these two studies. Although the different approaches are quite obvious, it is not clear to me what additional insights they add to the problem. They conducted their experiments with different stiffnesses. However, the conclusions they draw from the study are based on speculation (e.g. about the behavior of myosin heads in relation to shortening and relaxation), while their data mainly confirm previous studies. They need to address more explicitly the novelty of their study.

Novelty and Comparison with Previous Studies: We understand the concern about distinguishing our contribution from prior work, specifically Kobirumaki-Shimozawa et al., 2021.

As detailed in our public response, these are the key advances:

Use of a medically relevant human iPSC-CM model vs. mouse cardiomyocytes.

Superior spatial and temporal resolution via our SarcAsM algorithm, revealing novel phenomena like popping and high-frequency oscillations not previously reported.

Significantly greater statistical power due to our high-throughput in vitro assay.

We added a distinct mechanistic explanation based on the dynamic force-velocity relationship and sarcomere-level stochasticity, contrasting with the static, deterministic titin/length-dependence focus of previous studies.

Interpretation and Speculation: We acknowledge that without the explicit model, some interpretations in the initial submission appeared speculative. As noted in our public response, we had already started to develop a theoretical model explaining our observations at the time of submission, targeting a second follow-up publication. Including interpretations based on this unpublished model prematurely clearly caused confusion. We now include the full model in the revised manuscript.

Integration of the Theoretical Model: We have now fully integrated the model into the revised manuscript. The model explicitly demonstrates how the non-monotonic force-velocity relationship of individual sarcomeres leads to dynamic instabilities around a critical force threshold. This instability along with stochasticity drives a 'tug-of-war' between coupled sarcomeres, generating complex emergent behaviors.

Mechanistic Explanation Beyond Length-Dependence: Our model quantitatively reproduces all key experimental findings (stochastic heterogeneity, popping, oscillations) without relying on length-dependent activation effects. This strongly supports our conclusion that the active, transient dynamics of individual sarcomeres governed by the force-velocity relationship are fundamental drivers of these complex contractile patterns. We believe this provides a significant conceptual advance, highlighting a potentially underappreciated aspect of sarcomere dynamics. Previous models focused mostly on length-dependence, historically based on skeletal muscle fiber experiments that were often done under static, isometric conditions. We feel that the new model represents a substantial paradigm shift in understanding highly dynamic muscles such as heart muscle.

We are confident that the inclusion of the model addresses the majority of the reviewer's concerns.

Additional comments:

The authors write of a tug-of-war competition between the sarcomeres, and I'm not sure what they mean by that. I would spend more words explaining this point, especially because it seems to be an important point to describe their results. Similarly, they talked about an all-or-nothing phenomenon when they described the elongation of sarcomeres. What do they mean by this?

We have revised the manuscript where clarification was needed and now define the terms mentioned more explicitly.

(1) "Tug-of-War": We used this term metaphorically to describe the mechanical competition between linearly coupled sarcomeres within a myofibril, especially when contracting against rigid external boundary conditions. While it is not a perfect analogy, the metaphor intuitively captures the inherent instability of this interaction: similar to how a team in a real tug-of-war might suddenly yield when one person tires and the rest of team gets overloaded, rather than steadily losing ground, the dynamic instability arising from the non-monotonic force-velocity relationship (detailed in our model, lines 300ff) can cause individual sarcomeres to abruptly change state (e.g., shorten or rapidly lengthen) while under tension from their neighbors. We have removed the term from the title and now use it more sparingly within the manuscript to better reflect its role as an illustrative analogy.

(2) "All-or-Nothing" Elongation (Popping): The term "popping" describes our experimental observation of sudden, rapid, and extensive elongation of individual sarcomeres. This typically occurs late in the contraction cycle during early relaxation, when overall force may be declining, but individual sarcomeres can still experience significant tension from their neighbors. We described this specific type of rapid elongation in the original manuscript as an "all-or-nothing" phenomenon because, typically, sarcomeres in these events yield rapidly and strongly overshoot their resting length without recovering in a given activation cycle. The speed of popping events is substantially higher than the speed of coordinated gradual shortening observed during systoles that is driven by bound myosin heads. This observation strongly suggests an instability-driven, avalanche-like unbinding of myosin heads from the actin filaments during these events.

We agree that the term "all-or-nothing" is not precise, and we have removed it, as it is not essential for describing the observed "popping" dynamics.

The authors claim that the popping frequency increases as a function of stiffness. However, Figure 4E does not really seem to be a common practice in terms of statistical significance. A better description could help to remove this doubt.

We clarified the presentation of popping frequency data and its statistical interpretation.

(1) Popping Frequency vs. Substrate Stiffness (previously Figure 4D, now Figure 3G):

We first corrected that the dependence of popping frequency on substrate stiffness was presented in Figure 4D, not 4E. In the revised, shortened manuscript it can be now found in Fig. 3G. Due to the large number of observations (N) in our dataset, the slight upward trend in popping frequency with increasing substrate stiffness shown in Figure 4D does reach statistical significance using standard tests. For details see Figure captions.

(2) Popping Frequency vs. Sarcomere Resting Length (previously Figure 4E, now Figure 3H):

Figure 4E addresses the relationship between popping frequency and the individual sarcomere's resting length. To generate this plot, we binned sarcomeres based on their measured resting length (in intervals of 0.02 µm) and calculated the mean popping frequency within each bin across all conditions. We have now clarified this in the figure caption.

(3) Interpretation of Length Dependence:

While Figure 3H clearly shows that longer sarcomeres are more prone to popping, we argue this is likely a modulating factor rather than the sole underlying cause. Two key observations support this interpretation:

Even very short sarcomeres (e.g., < 1.65 µm resting length) exhibit a non-zero popping frequency (around 5-10%), indicating that popping is not exclusive to long sarcomeres.

The distribution of resting lengths, now added to the graph, is narrower than the wide range (1.6-2.0 µm) plotted in Figure 3H. Popping still occurs stochastically within a myofibril of sarcomere with relatively similar resting lengths.

Therefore, while length clearly influences the probability of popping, the phenomenon itself appears to be fundamentally stochastic, occurring across a range of lengths. This is consistent with our model in which dynamic instabilities (driven by the non-linear force-velocity relationship) and stochastic fluctuations are the primary triggers, while length affects probability of occurrence.

Changes in Manuscript:

We have revised the text associated with Figures 3G and 3H to clarify the distinction between stiffness and length dependence.

We have added a statement in the Methods section and figure legends (e.g., Legend for Fig 3) explaining our approach to statistical analysis and interpretation for large datasets where standard p-values may be less informative.

We believe these clarifications directly address the reviewer's concerns about the data presentation and interpretation in Figure 3.

Reviewer #2 (Recommendations For The Authors):

This is an interesting study, which however could and should be extended, see below. The current manuscript contains much less information than its length suggests; its figures contain partially redundant data.

Taking into account this critical feedback, we have restructured, streamlined and shortened the manuscript to improve readability and accessibility.

(1) How regular are the cellular contraction cycles?

Have the authors computed a coefficient of variation of cycle durations?

Does this regularity depend on substrate stiffness?

We have substantially improved the detection accuracy of contraction intervals compared to our initial submission (details see SarcAsM, https://www.biorxiv.org/content/10.1101/2025.04.29.650605v1). We calculated the beating rate variability (defined as the standard deviation of cycle durations), and found a low variability of on average less than 0.05 s across the tested conditions. The distribution of this variability is positively skewed, with the majority of values clustering near zero. We have added new panels showing these results to Fig. S2B.

(2) Which experiments could the authors perform to identify the origin of the apparent 3-Hz oscillations?

Would these oscillations persist even if the cardiomyocytes would not beat?

We now address these questions in the revised manuscript.

(1) Active Nature: The ~3 Hz oscillations are clearly linked to active contraction. They are absent in quiescent, non-beating cardiomyocytes observed under identical conditions, confirming that they are not passive fluctuations or baseline cellular tremors.

(2) Signal Fidelity: We are confident these are genuine physiological events, not artifacts. Our high temporal resolution (~15 ms frame time) and tracking accuracy (< 20 nm) allow reliable detection because events are well above system noise. This is now explained in the revised manuscript.

(3) Can the authors augment their study by modeling?

For example, could the experimental data be fitted by a Kuramoto-type model of the form d phi_i / dt = eps*sin( Omega - phi_i ) + lambda*sin( phi_i - phi_i+1 ) + xi_i, combining phase-locking of sarcomere oscillations with phase phi_i to intracellular calcium oscillations with phase Omega, and anti-phase synchronization between neighboring sarcomeres, as well as noise xi?

If yes, how would the coupling strength depend on subtrate stiffness?

We now added a model. While a Kuramoto-type phase model is powerful for studying synchronization, we determined that a more mechanistic approach was required. Crucially, sarcomeres are mechanically coupled in series within a myofibril, and this direct physical linkage is not well-represented by the abstract, phase-based coupling of a Kuramoto model.

Instead, our model comprises serially coupled sarcomeres, each governed by an underdamped Langevin equation. This framework allowed us to infer the force-velocity relation without any prior assumptions directly from our experimental data, revealing a critical non-monotonic characteristic. As we now emphasize in the revised manuscript, this behavior is mathematically equivalent to a Van-der-Pol relaxation oscillator, which reflects the instability-driven nature of the system.

Furthermore, and in line with the reviewer's suggestion, our model incorporates a stochastic noise term which we found essential for reproducing the observed phenomena. Without this noise term, the characteristic sarcomere dynamics do not emerge (Fig. 5).

(4) What is the maximally extended length of titin, and how does this length correspond to the maximal length of popping sarcomeres?

The force-extension curves of titin have been measured in single-molecule experiments (and the packing density of titin is known) - can the authors use this information to infer the forces acting inside sarcomeres?

We thank the reviewer for this thoughtful question. While sarcomere length during popping can be measured, inferring the corresponding intra-sarcomeric force is not straightforward in a living, contracting cardiomyocyte. The relationship between extension and force is complex and dynamic, involving multiple molecular components.

Our data show elongations up to 0.5 μm during popping events. While this magnitude is plausibly within the extensibility range of titin and other mechanically relevant components (Caporizzo & Prosser, 2021; Loescher & Linke, 2023), directly inferring force from this observation is challenging. In such a multi-component system with both active and passive elements, total force comprises several factors that cannot be disentangled from a simple length measurement alone. First, the system is dominated by active, velocity-dependent force generation of cross-bridges, which our model shows is non-monotonic. Second, titin exhibits a restoring force that is strongly strain-rate dependent (Rief et al., 1997), critical during rapid elongation. Third, viscous drag forces within the sarcomere are also highly strain-rate dependent, contributing significantly during rapid length changes. Fourth, other structural elements such as microtubules and intermediate filaments contribute to viscoelastic properties, particularly at high strains (Caporizzo & Prosser, 2021). This complex interplay makes it impossible to map a given sarcomere length to a unique force value using single-molecule titin data alone.

(5) I urge the authors to make their raw data openly available.

We agree on the importance of data availability. While the complete raw imaging dataset is several hundred gigabytes and thus impractical to deposit, we have uploaded a comprehensive dataset to Zenodo to ensure full reproducibility. This repository includes a representative subset of raw imaging data (50 cells per condition), with corresponding sarcomere motion data provided in a readable JSON format. Crucially, the deposition also contains the complete aggregated data underlying all figures and statistical analyses presented in the manuscript. All provided data can be programmatically accessed and analyzed using our `SarcAsM` Python API. The data can be accessed at: https://doi.org/10.5281/zenodo.17564384.

Minor

(1) How did the authors determine the start and end of contraction cycles when analyzing their data?

The start and end points of each contraction cycle were identified using ContractionNet, a custom convolutional neural network we developed for this purpose. This method, used for all analyses in the revised manuscript, detects contraction intervals with high accuracy directly from sarcomere dynamics time-series data and significantly outperforms the threshold-based approach used previously. The complete methodology, algorithm description, and validation of ContractionNet are detailed in our companion paper on the SarcAsM analysis software

(www.biorxiv.org/content/10.1101/2025.04.29.650605v1, see Fig. S6).

(2) What are the measurement errors in determining Delta_SL?

The measurement error for the Z-band trajectories is approximately 17 nm. This high tracking accuracy is achieved with our deep-learning-based Z-band segmentation approach, which employs a 3D convolutional neural network (3D U-Net) to leverage both spatial and temporal context for robust Z-band segmentation in noisy, high-speed recordings. A full description of this validation is available in our SarcAsM companion paper (see Figure S3 therein).

(3) Does popping occur while other sarcomeres are still contracting?

This is an important point. Yes, popping frequently occurs while other sarcomeres within the same myofibril are still actively shortening. This simultaneity is clearly visualized in the newly added Movie M1, which displays a phase-space plot (velocity vs. length change relative to rest) for all tracked sarcomeres over time. In this visualization, popping events appear as trajectories moving into the top-right quadrant (rapid elongation), while concurrently, other sarcomeres are represented by points in the left quadrants (negative velocity), indicating ongoing shortening. We have included Movie M1 as supplementary material.

(4) The authors argue that their data on popping sarcomeres is consistent with homogeneous popping probabilities.

(5) Can the authors assess in simulations how dispersed the popping probabilities of individual sarcomeres could be before they would notice a statistically significant difference to the homogeneous case?

This question touches on a key challenge in analyzing these complex dynamics. A direct statistical test of popping probability for each individual sarcomere is not feasible, as the number of events per sarcomere over our observation time is too low for robust single-unit analysis. Consequently, our approach relies on testing the cumulative distributions of inter-event spatial distances and temporal gaps across all sarcomeres within a given region (LOI).

In nearly half of the analyzed LOIs, these cumulative distributions were statistically indistinguishable (p > 0.05) from the geometric distribution expected for a single, homogeneous stochastic process. This provides strong support for our primary conclusion that popping is fundamentally a random phenomenon.

For the cases that deviate from the homogeneous model, we argue that this does not refute the underlying stochasticity of the events. Instead, we propose this is the expected statistical signature of pooling data from a population of sarcomeres that have slight, intrinsic variations in their individual popping probabilities due to factors like resting length or structural integrity. Even if each sarcomere's popping is a locally random event, a cumulative test performed on a population with varied baseline probabilities is expected to detect a deviation from a simple, homogeneous model.

Regarding the requested simulation study: While we agree this would be methodologically informative, the sensitivity to detect probability dispersion depends on multiple interacting factors (number of sarcomeres per LOI, observation time, event rates, and the assumed form of heterogeneity). Any single simulation scenario would therefore be highly model-dependent and of limited generality. Rather than introducing additional assumptions, we base our conclusions on the observed agreement with the homogeneous model in approximately half of LOIs and the correlation of deviations with measurable properties (Fig. 4E). A comprehensive statistical analysis would constitute a substantial methodological study beyond the scope of this mechanistically focused manuscript.

(6) Can the authors measure sarcomere rest length and check if this rest length is correlated with the popping probability of individual sarcomeres?

Yes, we performed this analysis. As shown in Figure 3H (previously Fig. 4E), we found a positive correlation between sarcomere resting length and popping frequency, confirming that longer sarcomeres have a higher probability of popping.

Importantly, however, the popping probability remains non-zero even for shorter sarcomeres. As detailed in our response to Reviewer #1 regarding this figure, we interpret resting length as a significant modulating factor that influences popping probability, rather than the sole determinant of the phenomenon.

(7) Several mathematical models of sarcomere contraction exist (e.g., crossbridge models).

(8) Could the authors perform computer simulations of several such stochastic sarcomere models coupled in series?

Alternatively, could the authors discuss this?

As I understand, references 16-18 model myofibril contraction assuming static variability of sarcomeres, but do not account for stochasticity in the contractility of individual sarcomeres.

We thank the reviewer for this excellent suggestion. We have performed such simulations, and the theoretical model is a central component of our revised manuscript (new Figures 4 and 5; manuscript lines 316ff).

As the reviewer points out, previous models (e.g., refs 12 and 14 in our manuscript) have often relied on predefined static variability between sarcomeres to explain heterogeneous behavior. Our work takes a fundamentally different approach. We model the myofibril as a chain of serially coupled sarcomeres, where the dynamics of each unit are governed by an underdamped Langevin equation. This formulation inherently incorporates stochasticity and describes the interplay between a non-monotonic, velocity-dependent active force, a length-dependent passive force, and the mechanical coupling to its neighbors.

Crucially, the model parameters were not assumed, but were instead inferred by fitting the model directly to our experimental data using a gradient-free optimization algorithm. This data-driven stochastic model was sufficient to quantitatively reproduce key observed phenomena, including high-frequency oscillations and popping events. Our central finding is that these complex behaviors emerge naturally from the coupled system, driven by the non-monotonic force-velocity relationship and intrinsic stochastic fluctuations. This demonstrates that predefined static heterogeneity is not required to explain the observed dynamics.

(9) The manuscript could be shortened (e.g., lines 52-56 in the introduction provide little extra value).

We have significantly revised the entire manuscript to improve clarity and readability. We have removed sentences in the introduction as suggested and substantially restructured major sections. One of the main reasons for this was the integration of our theoretical model, which was originally prepared as a separate manuscript. This required us to completely reframe the introduction and reorganize the figures and results.

We are confident that these extensive changes have resulted in a stronger, more concise and impactful paper that now integrates our experimental findings with a theoretical model.

(10) Figure 2 is overloaded with data. Several panels could be moved to the SM without compromising the key message.

Introducing the notation in panels Figures 2A-C does not seem ideal to me; maybe add a cartoon?

We agree that the Fig. 2 was dense. We have redesigned panels A-F to improve clarity and better guide the reader. We now use a consistent color-coding scheme to link the extrema in the phase portraits (A-C) to the corresponding distributions of individual sarcomeres (E-G). We have also revised the accompanying text to make the figure's logic more transparent.

We have considered moving panels A-C to the supplementary materials. However, we believe their placement in the main text is crucial for two reasons:

(1) Revealing Core Dynamics: The length-velocity phase portrait is the first visualization that reveals the underlying near-oscillatory dynamics of individual sarcomeres. This was not an assumed behavior but a critical experimental observation that directly motivated our entire theoretical modeling effort. We now also provide animated versions of these plots (Movies X-Y) to further illustrate these complex dynamics.

(2) Enabling Model-Experiment Comparison: A phase portrait is a standard tool for comparing experimental data with theoretical models. Retaining it in the main text allows us to directly compare data and model in our new Figures 4 and 5, providing a clear validation of our model.

(11) Similarly, Figures 4F, G, and H seem dispensable to me.

(I also wonder how clear the analogy of a coin flip is if a biased coin with probabilities p and 1-p needs to be used.)

We agree that the previous Figure 4F, which served a purely illustrative purpose, was dispensable and have removed it. The "coin flip" analogy was potentially confusing and we have removed it.

As part of a broader restructuring of the manuscript, the quantitative analyses from the original Figures 4G and 4H are now presented as Figures 3I and 3J. They provide important supporting evidence for the stochastic nature of the resulting popping events. We believe retaining this quantitative analysis is valuable, and we hope that by streamlining the figure and removing the analogy, we have addressed the reviewer's concerns.

(12) Equation (1) is unnecessarily complicated. The same holds for Equation (2).

It might make sense to separate definitions for serial and mutual correlations.

(This would also simplify the axes labels in Figure 3C.)

(13) The notation used in Equation (1) is not fully clear.

I assume t denotes a unit-less time index and T is the unit-less duration of a contraction cycle, measured in multiples of a fixed time interval?

Regarding comments (12) and (13):

We thank the reviewer for these helpful suggestions. In response to comment (12), we have separated the definitions for the mutual (r_m) and serial (r_s) correlation coefficients, presenting them as distinct calculations rather than as special cases of a single, more complex formula. This makes their definitions more direct and explicit. The calculation for the serial correlation coefficient has also been streamlined into a concise inline definition.

In response to comment (13), we have clarified the notation in Equation (1). In the manuscript text (lines 208f), we now explicitly state that 𝑡 represents the discrete, unitless time index (i.e., the frame number) within a time-series, and 𝑇 is the total number of frames (i.e., the total duration in frames) of a given contraction cycle.

While Equation (1) itself is the standard definition for the uncentered correlation coefficient and cannot be algebraically simplified, we have added text to specify this and justify its use. This metric (equivalent to cosine similarity) is appropriate for our analysis as it assesses the similarity in the shape of motion patterns, independent of their mean values.

Finally, to further streamline the paper, we have removed the velocity correlation analysis and the corresponding parts of Figure 3.

(14) The authors should make clear in all figures what is experiment and what is simulation.

We have now clarified the nature of each graph in the figure captions.

(15) The caption of Figure 3C could be simplified.

We have simplified all figure captions.

(16) I found Figure 3A hard to understand.

We concluded that Figure 3A was confusing and did not add essential information to the manuscript. We have removed it entirely.

Reviewer #3 (Recommendations For The Authors):

In conclusion, l think that the manuscript would gain a lot if some more precise and more quantitative interpretation of the results were given. This might require a collaboration with theorists.

We have integrated a novel theoretical framework into the revised manuscript (new Figures 4 and 5; manuscript lines 300ff as described above.

This new section introduces a data-driven, stochastic dynamical model that simulates the myofibril as a chain of serially coupled sarcomeres. Each sarcomere's motion is governed by an underdamped Langevin equation, a formulation that inherently accounts for stochasticity. Crucially, our model incorporates a non-monotonic force-velocity relationship inferred directly from our experimental data, rather than relying on predefined static variability between sarcomeres a key distinction from previous theoretical work.

This integrated model successfully and quantitatively reproduces all major experimental phenomena described in the paper, including high-frequency oscillations and stochastic "popping" events. It demonstrates that these complex behaviors emerge naturally as dynamic instabilities from the coupled system. This addition elevates the manuscript from a descriptive study to one that provides a predictive, mechanism-driven framework for understanding sarcomere dynamics.