Author response:
The following is the authors’ response to the original reviews.
eLife Assessment
This is a theoretical analysis that gives compelling evidence that length control of bundles of actin filaments undergoing assembly and disassembly emerges even in the absence of a length control mechanism at the individual filament level. Furthermore, the length distribution should exhibit a variance that grows quadratically with the average bundle length. The experimental data are compatible with these fundamental theoretical findings, but further investigations are necessary to make the work conclusive concerning the validity of the inferences for filamentous actin structures in cells.
We think this is an excellent assessment of the article. We suggest adding a sentence after the first one: “The distribution of bundle lengths is not Gaussian but Gumbel, since the bundle length is the length of the longest filament in the bundle.”
Public Reviews:
Reviewer #1 (Public Review):
Actin filaments and their kinetics have been the subject of extensive research, with several models for filament length control already existing in the literature. The work by Rosario et al. focuses instead on bundle length dynamics and how their fluctuations can inform us of the underlying kinetics. Surprisingly, the authors show that irrespective of the details, typical "balance point" models for filament kinetics give the wrong scaling of bundle length variance with mean length compared to experiments. Instead, the authors show that if one considers a bundle made of several individual filaments, length control for the bundle naturally emerges even in the absence of such a mechanism at the individual filament level. Furthermore, the authors show that the fluctuations of the bundle length display the same scaling with respect to the average as experimental measurements from different systems. This work constitutes a simple yet nuanced and powerful theoretical result that challenges our current understanding of actin filament kinetics and helps relate accessible experimental measurements such as actin bundle length fluctuations to their underlying kinetics. Finally, I found the manuscript to be very well written, with a particularly clear structure and development which made it very accessible.
We are grateful to Reviewer #1 for this very favorable assessment.
Reviewer #2 (Public Review):
Summary:
The authors present a theoretical study of the length dynamics of bundles of actin filaments. They first show a "balance point model" in which the bundle is described as an effective polymer. The corresponding assembly and disassembly rates can depend on bundle length. This model generates a steady-state bundle-length distribution with a variance that is proportional to the average bundle length. Numerical simulations confirm this analytic result. The authors then present an analysis of previously published length distributions of actin bundles in various contexts and argue that these distributions have variances that depend quadratically with the average length. They then consider a bundle of N-independent filaments that each grow in an unregulated way. Defining the bundle length to be that of the longest filament, the resulting length distribution has a variance that scales quadratically with the average bundle length.
Strengths:
The manuscript is very well written, and the computations are nicely presented. The work gives fundamental insights into the length distribution of filamentous actin structures. The universal dependence of the variance on the mean length is of particular interest. It will be interesting to see in the future, how many universality classes there are, and which features of a growth process determine to which class it belongs.
Weaknesses:
(1) You present the data in Fig. 3 as arguments against the balance point model. Although I agree that the data is compatible with your description of a bundle of filaments, I think that the range of mean lengths you can explore is too limited to conclusively argue against the balance point model. In most cases, your data extend over half an order of magnitude only. Could you provide a measure to quantify how much your model of independent filaments fits better than the balance point model?
Indeed, we agree that the experimental data we present, each on their own, provide inconclusive evidence of the scaling predicted by our model. However, in aggregate, as presented in Fig. 3E, the data make for compelling evidence of scaling of the variance with the average length squared, as quantified by the power-law fit. Also, we think that Fig. 3E argues strongly against the Balance Point Model, because the data do not conform with simple linear scaling (indicated by the dashed line in Fig. 3E). Regardless, we agree with the referee that better data is needed to make a more convincing case, and we see this paper as a call to arms to collect such data in the future. The published data we used (other than our own data from experiments on yeast actin cables) is from experiments that were not designed with this question in mind, i.e., how do length fluctuations scale with the mean?
(2) Concerning your bundled-filament model, why do you consider the polymerizing ends to be all aligned? Similarly to the opposite end, fluctuations should be present. Furthermore, it is not clear to me, where the presence of crosslinking proteins enters your description. Finally, linked to my first remark on this model, why is the longest filament determining the length of the bundle in all the biological examples you cite? I am thinking in particular about the actin cables in yeast.
In the case of the yeast actin cables (which grow from the bud neck into the mother cell), we know that the formins that polymerize the actin filaments are spatially aligned at the bud neck. In the cases of stereocilia and microvilli, again the polymerizing ends of the actin filaments are well-aligned at the growing tips of these bundled actin structures, as indicated by classic EM studies from Lew Tilney and others. The alignment of polymerizing actin filament ends is more difficult to assess at the leading edge of lamellipodia, because of undulating shape of the polymerization (membrane) surface. In fact, this could be the reason why data from the lamellipodia experiments deviate from the line in Fig. 3E, in contrast to the data from the other three structures (this is discussed in some detail in the Supplement). Regarding the actin crosslinkers, the only role they play in our model is keeping the filaments connected in the bundle. As far as the question of why the longest filament in the actin cable is the one that specifies the length of the cable, this is addressed in more detail in our McInally et al., 2024 (PNAS) paper, where we measured cable length by segmenting the fluorescence signal of the cable. Therefore, the filaments in the bundle that extend the furthest define the reported length. Also, given the function of the cables for transporting vesicles, the furthest reach of the filaments in the bundle defines the area from which the vesicles are collected.
Recommendations for the authors:
Reviewer #1 (Recommendations For The Authors):
An important result of the model proposed by the authors is that the relationship between bundle mean length and variance should also inform the number of filaments in the bundle (Equation 13). In the SI the authors thus predict from fitting experimental results that bundles should be made of around 173 filaments, which is larger than most values proposed in the literature (and quoted in this work), except for stereocilia. Can the authors comment on this?
This is an interesting point that we have been thinking about. Indeed, the model does relate the number of filaments to the variance of the length, but this dependence is logarithmic and therefore insensitive to changes in the number of filaments. Consequently, the number 173 comes with very large error bars and should be thought of more like a few hundred filaments in terms of the precision with which we can extract this number from data. We make this point more clearly in the revised SI, where we now say that based on the data the best we can do is say that the number of filaments is between 80 and 400.
Along the same lines, in their derivation of Equations 12 and 13 (a key result of the manuscript) the authors make some approximations that are only valid for large N (number of filaments in the bundle). Is this approximation valid for actin cables or filopodia, estimated to comprise only around 10 filaments?
Indeed, even for N=10 filaments the approximate formulas have errors that are well below what can be measured. We consider the details of the approximation in deriving Equations 12 and 13 from the exact distribution (Equation 11) in the Supplemental section “Distribution of bundle lengths when individual filament lengths are exponentially distributed”. For example, the exact result involves the harmonic number which for N=10 is 2.88, while the approximate formula ln(N) + gamma we use yields 2.92, a fractional error that is < 2%.
A key assumption of the model is that the bundle length corresponds to the maximum individual filament length inside the bundle. Couldn't bundles comprise several filaments one after another, head-to-tail? What do the authors expect then?
Excellent point. Indeed, this is precisely the geometry of the yeast actin cable. In our previously published McInally et al., 2024 (PNAS) paper we worked out the math in that case and found that the main result about the variance holds. In this paper we presented a simpler, model that retains the same features of the one presented in the PNAS paper to better accentuate the origins of the scaling of the variance with the mean length, which is simply the result of bundling and identifying the length of the bundle with the length of the longest filament (or, more precisely, furthest extending filament) in the bundle.
The model also allows us to relate the bundle length fluctuations and average to the individual filament characteristic length (Equations 12 and 13 again). Can the authors comment on the values of 〈l〉 they would obtain for experimental data?
It is hard to give a precise number, as we would need to know also the number of filaments in the bundle, and for that we would need better electron microscopy data (which has proven difficult for the field to obtain). Still with typical numbers in the 10s to 100s the expected average filament lengths are roughly, ln(10) – ln(100), or 2-5 times smaller than the average bundle length.
I find the Methods section a bit underwhelming. In particular, can the authors give more details on their treatment of experimental data? Bootstrapping sampling is mentioned but there is no information on the size of the original data sets, which could affect the validity of such a method.
Thanks for the criticism. We have added details regarding the sizes of the data sets used in the analysis in the Methods section.
Along the same lines, is the graph in Figure 1E the result of a simulation like the ones the authors used to obtain their result or is it just a schematic? If the first, I would suggest replacing it with an actual simulated length trajectory. In general, I think this work would benefit from more detailed explanations and examples of how stochastic trajectories were computed and analysed.
This is also a good point. We still prefer to keep the schematic in this figure since our goal here is to define the question before we commence with computations and data analysis. The stochastic trajectories were generated using the standard Gillespi algorithm and the statistics of length were gathered once the dynamics of length reach steady state. We explain this in the Methods section and give more details in the Supplement.
Finally, while I find the writing in this manuscript to be excellent, I think the figures require some work. The schematics and drawings, which are very low resolution, the font size for the axes, and the choice of colours all make it more cumbersome than necessary to understand what is being shown.
Thank you for pointing this out. We have made better versions of the figures.
Reviewer #2 (Recommendations For The Authors):
"In this case, the length distribution of the bundle derived from extreme value statistics, leads to a peaked non-Gaussian distribution, even when filaments within the bundle are unregulated and exponentially distributed."
You mention "extreme value statistics" only once, in the introduction. I would suggest that you come back to this notion and explain how your results connect to extreme value statistics or delete it from the manuscript.
Good point. We added a sentence to draw the reader’s attention to the fact that our result is an extreme value distribution (Equation 11 is the Gumbel distribution) used in statistics of extreme events.
This is a follow-up of one of my major points of criticism: Fig. 3A: why do you fit (if I understand correctly) the blue and orange data points with the same power law? For (A-- D) The data extend over less than an order of magnitude. Why is a power law fit appropriate? Can you quantify how much better your fits are compared to a linear dependence? Bundling the data of all structures yields a common matter curve (with the exception of filopodia). This is quite remarkable, I think, and merits some more discussion than currently given in the manuscript.
Good point. We should have been more clear. In Figures 3A-D we show individual data sets for the different bundle structures and compare the prediction of the Balance Point Model (dashed line) to the data. We also do a fit to a power law to show that the data is consistent with the Bundle model. This comparison is made much more clear in Figure 3E.
Fig 1B, right does not show the addition and removal of subunits - Fig. 1C does. Panel C is not explained in the caption. The second appearance of (D) in the caption could be omitted.
Good points. We fixed these issues in the new version of the Figure and caption.
"For individual actin filaments (...)" I found this and the following paragraph slightly confusing at first reading: as long as you write about single filaments, do you have annealing in mind, where two filaments merge and form a longer filament? In case you consider a bundle, do you consider a filament that is cross-linked to other filaments and thereby added to the bundle? Similarly for removing filament segments (severing or unbundling)? Probably, my confusion is a consequence of you seemingly using filament to describe bundles as well as single actin filaments.
Sorry for the confusion. We tried to be consistent throughout the text and use “filament” to denote a single actin filament and “bundle” a collection of parallel filaments crosslinked together. The assembly and disassembly dynamics of the filaments in the bundle are only relevant to the extent that they affect the length distribution of individual filaments. The main result is largely independent of that (as demonstrated in the Supplement by considering different single filament distributions) once we decide that the length of the bundle is given by the length of the longest filament in the bundle. This is the point of extreme value statistics where a universal, Gumbel distribution for the length of the longest filament in the bundle arises independent of the length distribution of a single filament (this result is akin to the Central Limit Theorem which predicts a Gaussian distribution of the mean of a large number of random numbers irrespective of the distribution they’re drawn from.)
In Figure 4D, the variance of the filopodia lengths" Probably Figure 3D?
Yes. Thank you. We fixed this.
"The filopodia data seemingly has the same slope (...) but with variances higher than what is measured for other actin structures." This finding does not contradict the main statement of a nonlinear scaling of the variance with the mean length, right? I therefore find this discussion slightly peripheral and also confusing. Also, what is the reason to assume that EM might get the actual length of filopodia wrong by a factor of 2 to 3?
The issue with filopodia is that the way the lengths are measured is by the extent to which the structure as a whole protrudes from the cell. This leaves unresolved the lengths of the actual filaments in the structure, and we suspect that they are longer as they extend into the cytoplasm. This would contribute to the shift off the common curve in the direction that is observed (larger variance associated with smaller average length). We have no way to justify that this would lead to a 2-3 factor other than that would be enough to collapse the data onto the common curve. Clearly more careful experiments are needed to resolve the issue. We added some clarifying remarks to this effect into the discussion.
Eq.(14) What is Z?
Thanks for pointing out this omission. Z = L/ and we have added that in the formula where Z appears.
LIST OF CHANGES
Here we summarize the changes we made to the manuscript and the Supplementary material in response to the reviewers.
(1) Fixed typo: Figure 1 legend had two parts labelled D which has been changed into a D and a C. The explanation of panel C has been added.
(2) Fixed typo: The incorrect call to Figure 4D is now corrected to Figure 3D.
(3) In the Supplementary material we made more precise our estimate of the number of filaments. The wording “From this we can estimate the number of filaments. We find, with a confidence interval of…” we have changed to “From this we can estimate the number of filaments to be between 80 and 400 which compares favourably to the typical number of filaments in the different actin structures that were analyzed.”
(3) In the Methods section we added the number of measured filament lengths in the different data sets used in the analysis.
(4) We made better (higher resolution) versions of all the Figures.
