Zebrafish airinemes optimize their shape between ballistic and diffusive search
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Evaluation Summary:
This paper studies statistical aspects of the role of longrange cellular protrusions called airinemes as means of intracellular communication. The authors use published data showing how airinemes approach a target cell and describe these movements with a mathematical model for an unobstructed persistent random walk. The impact of this study will be on the specialised reader interested in modelling and airineme biology.
(This preprint has been reviewed by eLife. We include the public reviews from the reviewers here; the authors also receive private feedback with suggested changes to the manuscript. Reviewer #1 agreed to share their name with the authors.)
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Abstract
In addition to diffusive signals, cells in tissue also communicate via long, thin cellular protrusions, such as airinemes in zebrafish. Before establishing communication, cellular protrusions must find their target cell. Here, we demonstrate that the shapes of airinemes in zebrafish are consistent with a finite persistent random walk model. The probability of contacting the target cell is maximized for a balance between ballistic search (straight) and diffusive search (highly curved, random). We find that the curvature of airinemes in zebrafish, extracted from livecell microscopy, is approximately the same value as the optimum in the simple persistent random walk model. We also explore the ability of the target cell to infer direction of the airineme’s source, finding that there is a theoretical tradeoff between search optimality and directional information. This provides a framework to characterize the shape, and performance objectives, of noncanonical cellular protrusions in general.
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Author Response:
Reviewer #1 (Public Review):
This article focuses on a quantitative description of airineme morphology and its consequences for contact and communication between cells via these long narrow projections. The primary conclusions are
 Airineme shapes are consistent with a persistent random walk model (analogous to a wormlike polymer chain), unhindered by the presence of other cells.
The authors convincingly demonstrate, using analysis of the meansquareddisplacement along the airineme contour, that the structures cannot be described by a diffusive growth process (ie: a Gaussian chain) as would be expected if there were no directional correlations between consecutive steps. Furthermore, by observing the airineme growth and looking at the distribution of stepsizes, they show that these steps do not exhibit the …
Author Response:
Reviewer #1 (Public Review):
This article focuses on a quantitative description of airineme morphology and its consequences for contact and communication between cells via these long narrow projections. The primary conclusions are
 Airineme shapes are consistent with a persistent random walk model (analogous to a wormlike polymer chain), unhindered by the presence of other cells.
The authors convincingly demonstrate, using analysis of the meansquareddisplacement along the airineme contour, that the structures cannot be described by a diffusive growth process (ie: a Gaussian chain) as would be expected if there were no directional correlations between consecutive steps. Furthermore, by observing the airineme growth and looking at the distribution of stepsizes, they show that these steps do not exhibit the expected longtail distributions that would imply a Levywalk behavior. The persistent random walk (PRW) is presented as an alternative that is not inconsistent with the data. However, given the high level of noise due to low sampling, the claimed scaling behavior of the MSD at long lengths is not fully convincing. Nevertheless, the PRW provides a plausible potential description of the airineme shapes.
To reiterate the comment: the MSD analysis allows us to reject the simple random walk model, and it is consistent but alone is not strongly supportive of the PRW model, especially at high time of around 15 minutes (long lengths of around 65 microns). As the Reviewer points out, this is due to low numbers of long airinemes.
This prompted us to investigate the longlength data using multiple analysis approaches. In the new manuscript, new Fig 2B, we took all airinemes whose growth time was greater than 15 min, and plotted their final angle, i.e., the angle between the tangent vector at their point of emergence from the source cell and the tangent vector at their tip. At long times (>1/D_theta), the PRW model predicts that the angular distribution should become isotropic.
In new 2B, we find that the angular distribution is uniform, i.e., isotropic, using a KolmogorovSmirnov test (pvalue 0.37, N=26).
Since there are relatively few data points, we repeated this analysis under various airineme selection criteria, and in all cases found the final angular distribution to be consistent with uniformity (new Supplemental Data Figure 1). For example, if we set the threshold at 10min, which includes N=49 airinemes, the KolmogorovSmirnov test against a uniform angular distribution gives a pvalue of 0.32.
We here add a few additional notes
● Note that there is significantly less data used in this test than in the MSD analysis or the autocorrelation function maximum likelihood analysis. In order to perform a hypothesis test, we wanted to be sure that the data points are independent, so we take only one from each airineme (unlike MSD and autocorrelation analyses, for which we take every interval of a particular length, whether in the same airineme or not.)
● Finally, although the >10min KS test has more data than the >15min KS test (N=49 compared to N=26), we have chosen to present the >15min KS test in the Main Text. As we mentioned above, the conclusion is unchanged for >10min (see Supporting Data). The reason is that >15min is the first test we ran to check angular distribution against a uniform (pi,pi) distribution, and we did not want to bias our testing.
Taken together, the data are even more strongly supportive of the PRW model. We are grateful for the Reviewer in encouraging us to further explore the hightime data.
 The flexibility (ie: persistence length) of the airineme shapes is one that maximizes the probability of a given airineme (of fixed length) contacting the target cell.
This optimum arises due to the balance between straightline paths that reach far from the source but cover a narrow region of space and diffusive paths that compactly explore space but do not reach far from the starting point. Such optimization has previously been noted in unrelated contexts both for search processes of moving particles and for semiflexible chains that need to contact a target. The authors present a compelling case (Fig 4B) that the measured angular diffusion of the airinemes falls close to the predicted optimum. Furthermore, the measured probability of hitting the target cell also lies close to the model prediction, providing a strong test of the applicability of their model.
 Airineme flexibility engenders a tradeoff between contact probability and directional information (ie: the extent to which the target cell can determine the position of the source).
This calculation proposes an alternative utility metric for communication via airinemes. The observed flexiblity is shown to be at a Pareto optimum, where changes in either direction would decrease either the probability of contact or the directional information. Again the absolute value of the metric (Fisher information for angular distribution) is within the predicted order of magnitude from the model. Thus, while the importance of maximizing this metric remains speculative, its quantitative value provides an additional test for the applicability of the PRW model.
Overall, this paper provides an interesting exploration of optimization problems for communication by long thin projections. A particular strength is the quantitative match to experimental data  indicating not just that the experimental parameters fall along a putative optimum but also that the metrics being optimized are wellpredicted by the model. Defining an optimization problem and showing that some parameter sits at the optimum is a common approach to generating insight in biophysical modeling, albeit invariably suffering from the fact that it is difficult to know which optimization criteria actually matter in a particular cellular system. The authors do an excellent job of exploring multiple optimization criteria, quantifying the balance between them, and pointing out inherent limitations in knowing which is most relevant.
A minor weakness of the manuscript is its focus on a very narrowly defined cellular system, with the general applicability of the results not being highlighted for clarity. For example, the fact that the same flexiblity optimizes contact probability and the balance between contact and directional information is an interesting conclusion of the paper. Is this true in general? Is it applicable to other systems involving a semiflexible structure reaching for a target or a moving agent executing a PRW?
The Reviewer’s question is an excellent question: Is the tradeoff between contact and directional information a general property of searchers that obey persistent random walks? To address this question, we now include the analysis previously contained in Figure 5D, but for a full parameter space exploration. This is done in new Figure 5 Supplemental Figure 1. In doing so, we found fascinating behavior that sheds some light on the loop in Fig 5D.
At low d_targ, the tradeoff is amplified, and the parametric curve resembles bull's horns with two tips representing the smallest and largest D_theta in our explored range, pointing outward so the shape is concaveup. Intuitively, we understand this as follows: since the target is fairly close (relative to l_max), contact is easy. The only way to get directional specification is by increasing D_theta to be very large, effectively shrinking the search range so it only reaches (with significant probability) the target at the near side (“3oclock'' in Fig. 5A). At low d_targ, the parametric curve is concaveup, and there is no Pareto optimum.
At high d_targ, the searcher either barely reaches (when D_theta is high), and does so at 3oclock, therefore providing high directional information, or D_theta is low, and the searcher fails to reach, and therefore also fails to provide directional information. So, at high d_targ, there is no tradeoff.
At intermediate d_targ, the curve transitions from concaveup bull's horn to the notradeoff line. To our surprise, it does so by bending forward, forming a loop, and closing the loop as the lowD_theta tip moves towards the origin. At these intermediate d_targ values, the loop offers a concavedown region with a Pareto optimum.
So, to answer the specific question of the Reviewers: No, the Pareto optimum is not a general feature of persistent random walk searchers. It only exists in a particular parameter regime, sandwiched between a regime where there is a strict tradeoff with no Pareto optimum and a regime in which there is no tradeoff.
All of these results are now discussed in the main text.
(Note that although we do not explicitly explore lmax, since these plots have not been nondimensionalized, the parametric curve for a different lmax can be obtained by rescaling the results).
Reviewer #2 (Public Review):
Signalling filopodia are essential in disseminating chemical signals in development and tissue homeostasis. These signalling filopodia can be defined as nanotubes, cytonemes, or the recently discovered airinemes. Airinemes are protrusions established between pigment cells due to the help of macrophages. Macrophages take up a small vesicle from one pigment cell and carry it over to the neighbouring pigment cell to induce signalling. However, the vesicle maintains contact with the source cell due to a thin protrusion  the airineme. In support of these data, the authors find that the extension progress of the airinemes fits an "unobstructed persistent random walk model" as described for other macrophages or neutrophils.
The authors describe the characteristics of an airineme as it would be a signalling filopodia, e.g. a nanotube or a cytoneme, which sends out to target a cell. An airineme, however, is fundamentally different. Here, a macrophage approaches a pigment cell binds to the airineme vesicle. Then, the macrophage approaches a target pigment cell and hands over the airineme vesicle. During this process, the airineme vesicle maintains a connection to the source pigment cell by a thin protrusion. Then, the macrophage leaves the target cell, but the airineme vesicle, including the protrusion, is stabilized at the surface and activates signalling. Indeed nearly all airinemes observed have been associated with macrophages (Eom et al., 2017).
Therefore, it is essential to focus on the "searchandfind" walk of the macrophage and not the passively dragged airineme. In the light of this discussion, I am not sure if statements like "allow the airineme to hit the target cell" are helpful as it would point towards an actively expanding protrusion like a filopodium.
We have added a new paragraph in the Introduction emphasizing the role of the macrophage, and we have changed the language. In particular, we want to remove agency from the airineme, since it is indeed moving with the macrophage. In the mathematical sections, we opt for the phrase “search process”.
We have also clarified that, in the biological system, the details of contact are unclear (e.g., what mechanism in the macrophageairinemevesicle is responsible for distinguishing the target cell). Therefore, in the model, we have clarified that contact is declared when the airineme tip arrives at a distance r_targ from the center of the target cell, and this critical distance might be larger than the size of the target cell, since it might include part or all of the macrophage.
Reviewer #3 (Public Review):
This paper studies statistical aspects of the role of longrange cellular protrusions called airinemes as means of intracellular communication. The mean square distance of an airineme tip is found to follow a persistent random walk with a given velocity and angular diffusion. It is argues that this distribution with these parameters is the one that optimise the probability of contact with the target cell. The authors then evaluate the directional information (where in space did the airineme come from) and found that, again, the measure diffusion coefficient optimise the tradeoff between high directional information (small diffusion) and large encounter probability.
I found this paper well written and clear, and addressing an interesting problem (longrange intracellular communication) using rigorous quantitative tools. This is a very useful approach, which appears to have been appropriately done, that in itself makes this paper worthy of interest.
 The main conclusion of this paper is that the airineme properties optimises something that has to do with their function. Although rather appealing, I find this kind of conclusion often questionable considering the large uncertainty surrounding many parameters.
We agree that conclusions about optimality need to be expressed carefully, to avoid teleological statements and overstating our knowledge about the constraints and variability faced by the living system. In the revised manuscript, we strive to use language to point out that the parameter extracted from data (an average) and the parameter predicted to be optimal (on average) are approximately equal, and avoid speculation about the evolutionary process that may have led to these parameters.
Here, optimality is shown from a practical perspective, using measure parameters. For instance, the optimal diffusion coefficient for hitting the target varies by 2 orders of magnitude when the distance between cells is varied (Fig.3A). The measured coefficient is optimal for cells about 25 µm distant. Does this reflect anything about the physiological situation in which these airinemes operate?
To find the physiological regime in which the airinemes operate, we extracted distancetotarget measurements from imaging data, and found an average distance of 51 microns (note possible typo in referee comment), with a range of 33𝜇m − 84𝜇m, 𝑁 = 70. We report this in updated Table 1). The optima we find is in the average number of attempts before success (so, a single instance of an airineme may either succeed or fail, stochastically), when the distance to the target is 50 microns. These are both averages, across an entire fish epithelium (which contains ~10^5 source cells). So, for a particular cell generating airinemes, there may be different optimal parameters given a priori knowledge of its environment, but, across the whole fish epithelium, we assume the overall success corresponds to the average singlecell success we simulate.
Another rather puzzling claim is that the diffusion coefficient is optimised both for finding the target, AND for finding the best compromised between finding the target and providing directional information, while the latter must necessarily require weaker diffusion. Hence the last paragraph of p.6 ("the data is consistent with either conclusion that the curvature is optimized for search, or it is optimized to balance search and directional information"), although quite honest, gives the feeling that the conclusions are not very robust. I would welcome a discussion of these points.
We have clarified the result about directional information in the new manuscript.
First, it is not optimized for maximal directional information, in the sense that there are other parameters that would give more directional information – we apologize for the lack of clarity. Rather, the parameters observed are such that changing them would either reduce search success or directional information. In the study of multiple optimization, this property is called “Pareto optimality”.
Second, the Reviewer’s intuition is that weaker diffusion (straighter airinemes) would provide more directional information. This was indeed our intuition as well, prior to this study. To our surprise, we found that very weak diffusion or very strong diffusion both give local maxima of directional information. The intuitive explanation is that the searchers are finitelength, and high diffusion leads to a smaller search extent which only reaches the target cell at its very nearest region. We provide this intuitive explanation (which was indeed a surprise to us) in the Results section.
Third, the Reviewer asks about the generality of the result about directional information. This is an excellent question. The comment, and similar comments from other Reviewers, prompted us to perform a parameter exploration study. This is contained in a new Supplemental Figure and new paragraphs in the Results section.
The Reviewer’s question is an excellent question: Is the tradeoff between contact and directional information a general property of searchers that obey persistent random walks? To address this question, we now include the analysis previously contained in Figure 5D, but for a full parameter space exploration. This is done in new Figure 5 Supplemental Figure 1. In doing so, we found fascinating behavior that sheds some light on the loop in Fig 5D.
At low d_targ, the tradeoff is amplified, and the parametric curve resembles bull's horns with two tips representing the smallest and largest D_theta in our explored range, pointing outward so the shape is concaveup. Intuitively, we understand this as follows: since the target is fairly close (relative to l_max), contact is easy. The only way to get directional specification is by increasing D_theta to be very large, effectively shrinking the search range so it only reaches (with significant probability) the target at the near side (“3oclock'' in Fig. 5A). At low d_targ, the parametric curve is concaveup, and there is no Pareto optimum.
At high d_targ, the searcher either barely reaches (when D_theta is high), and does so at 3oclock, therefore providing high directional information, or D_theta is low, and the searcher fails to reach, and therefore also fails to provide directional information. So, at high d_targ, there is no tradeoff.
At intermediate d_targ, the curve transitions from concaveup bull's horn to the notradeoff line. To our surprise, it does so by bending forward, forming a loop, and closing the loop as the lowD_theta tip moves towards the origin. At these intermediate d_targ values, the loop offers a concavedown region with a Pareto optimum.
So, to answer the specific question of the Reviewers: No, the Pareto optimum is not a general feature of persistent random walk searchers. It only exists in a particular parameter regime, sandwiched between a regime where there is a strict tradeoff with no Pareto optimum and a regime in which there is no tradeoff.
All of these results are now discussed in the main text.
(Note that although we do not explicitly explore lmax, since these plots have not been nondimensionalized, the parametric curve for a different lmax can be obtained by rescaling the results).
 on p.4: "the airineme tips (which are transported by macrophages [30]) appear unrestricted in their motion". I don't understand what it means that the airineme tips are transported by macrophage, and I missed the explanation in the cited article. Is airineme dynamics internally generated (i.e. by actin/microtubule polymerisation) or does it reflect to motility of cells dragging the airineme along? This is discussed in passing in the Discussion, but I think that this should be explainde in more detail right from the start. Aslo, if a cell is indeed directing the tip, what does contact mean? Does it mean that the driving macrophage must contact the target cell and somehow attached the airineme to it? IF yes, that means that the airineme tip has a large spatial extent, which will certainly affect the contact probability.
These are very good questions. Airinemes have been characterized in a few studies since their discovery in 2015. We are saddened (and excited) to say that: the answers to all of these questions are currently unknown. To paraphrase the Reviewer, the questions are: First, what is the force generation mechanism that leads to airineme extension (additionally, if there are multiple coordinated force generators, e.g., the airineme’s internal cytoskeleton and the macrophage, how are these forces coordinated)? And second, what are the molecular details of airineme tip contact establishment upon arrival at a target cell?
We present an extended biological background discussion addressing these questions, including what is known and what remains unknown. We have incorporated a shortened version of this as a new paragraph in the introduction.
Airinemes are produced by xanthophore cells (also called yellow pigment cells) and play a role in the spatial organization of pigment cells that produce the patterns on zebrafish skin. Xanthophores have bleblike structures at their membrane, and those blebs are the origin of the airineme vesicles at the tip. Those blebs express phosphatidylserine (PtdSer), an evolutionarily conserved ‘eatme’ signal for macrophages. Macrophages recognize the blebs, ‘nibble,’ and ‘drag’ as they migrate around the tissue and the filaments trailing and extending behind. Airineme lengths have a maximum, regardless of whether they reach their target. If the airineme reaches a target before this length, the airineme tip complex recognizes target cells (melanophores) and the macrophage and airineme tip disconnect.
The airineme tip contains the receptor DeltaC, which activates Notch signaling in the target cell. The mechanism by which a macrophage hands off the airineme tip is still mysterious, due to temporal and spatial resolution limits. It is also known what other signals, if any, are carried by the airineme. If no target cell is found by the maximum length, the macrophage and airineme disconnect, and the airineme the extension switches to retraction. Thus, macrophages do not keep dragging the airineme vesicles until they find the target melanophores. However, how macrophages determine when to engulf the untargeted airineme vesicles is not understood.
Regarding the Reviewer’s specific question about the implications for the macrophage on how we model contact establishment: This would indeed change the interpretation of the model parameter r_targ. Specifically, contact is declared when the airineme tip arrives at a distance r_targ from the center of the target cell, and this critical distance might be larger than the size of the target cell, since it might include part or all of the macrophage. We have added this to the first part of Results, when the parameter is introduced.
 Fig. 2A shows the airinemes MSD and the fit using the PRW model. I don't find the agreement so good. The power law t^2 seems good almost up to 10 minutes, and the scaling above that, if there is one, is clearly larger than linear. So I would say that the apparent agreement with the PRW model reflects the fact that there is a crossover from a ballistic motion to something else, but that this something else is not a randow walk. The MSD does look quite strange at long time, where it apparently decays. This made me wonder whether there might be a statistical biais in the data, for instance, the longest living airinemes are those who didn't find their target and hence those who travel less far, on average. I tried to get more information on the data from the ref.[29,30], but could not find anything. The authors should discuss these data and possible biais in more detail. For instance, do the data mix successful and unsuccessful airinemes? This is somewhat touched upon in Fig.s$, but I did not gain any useful information from it, except that the authors find the agreement "good" while it does not look so good to me.
To reiterate the comment, which is closely related to comments from other Reviewers: the MSD analysis allows us to reject the simple random walk model, and it is consistent but alone is not strongly supportive of the PRW model, especially at high tau of around 15 minutes (long lengths of around 65 microns). As the Reviewer points out, this is due to low numbers of long airinemes.
We agree, and have performed new analysis. The following is repeated here for convenience:
This prompted us to investigate the longlength data using multiple analysis approaches. In the new manuscript, new Fig 2B, we took all airinemes whose growth time was greater than 15 min, and plotted their final angle, i.e., the angle between the tangent vector at their point of emergence from the source cell and the tangent vector at their tip. At long times, the PRW model predicts that, for long times >1/D_theta, the angular distribution should become isotropic. In new 2B, we find that the angular distribution is uniform, i.e., isotropic, using a KolmogorovSmirnov test (pvalue 0.37, N=26).
Since there are relatively few data points, we repeated this analysis under various airineme selection criteria, and in all cases found the final angular distribution to be consistent with uniformity (new Supplemental Data Figure 1). For example, if we set the threshold at 10min, which includes up to N=49 airinemes, the KolmogorovSmirnov test against a uniform angular distribution gives a pvalue of 0.32.
We here add a few additional notes
● Note that there is significantly less data used in this test than in the MSD analysis or the autocorrelation function maximum likelihood analysis. In order to perform a hypothesis test, we wanted to be sure that the data points are independent, so we take only one from each airineme (unlike MSD and autocorrelation analyses, for which we take every interval of a particular length, whether in the same airineme or not.)
● Finally, although the >10min KS test has more data than the >15min KS test (N=49 compared to N=26), we have chosen to present the >15min KS test in the Main Text. As we mentioned above, the conclusion is unchanged for >10min (see Supporting Data). The reason is that >15min is the first test we ran to check angular distribution against a uniform (pi,pi) distribution, and we did not want to bias our testing.
Taken together, the data are even more strongly supportive of the PRW model. We are grateful for the Reviewer in encouraging us to further explore the hightime data.
 Regarding the directionality discussion, some aspect are a bit vague so that we are left to guess the assumptions made. For instance, the source cell is place at \theta=0 "without loss of generality" (p.6). Apparently (sketch Fig.5A) this also means that the airineme starting point from the source is at \theta=0, which clearly involves loss of generality, since the airineme could start from anywhere, its path could be hindered by the body of the source cell, and its contact angle would then be much less likely to be close to 0. It might be that in practice, only those airineme starting close to theta=0 do in fact make contact, but this should be discussed more thoroughly. Also, why is there to maxima in the Fisher information (Fig.5C) for very high and very low diffusion coefficient at short distance?
The sketch was indeed not clear about generality, so we have edited it so that the angles are no longer perpendicular. We also now also clarify in the Main Text that, in all simulations (both measuring contact probability and directional sensing), the airineme begins at a specified point in an orientation uniformly random in (pi,pi). We apologize that this was not clear in the previous sketch.
Regarding hindrance by the source cell: While the tissue surface is crowded, the airineme tips appear unrestricted in their motion on the 2d surface, passing over or under other cells unimpeded (Eom et al., 2015, Eom and Parichy, 2017). We therefore do not consider obstacles in our model. This includes the source cell, i.e., we allow the search process to overlie the source cell. We now state this explicitly in the Main Text.
Regarding two maxima in Figure 5C (which was a surprise to us): We understand it with the following intuitive picture. For low D_theta, i.e., for very straight airinemes, the allowed contact locations are in a narrow range (by analogy, imagine the dayside of the planet Earth, as accessible by straight rays of sunlight), resulting in high directional information. For high D_theta, i.e., for very random airinemes, we initially expected low and decreasing directional information, since there is more randomness. However, these are finitelength searches, and the range of the search process shrinks as D_\theta increases. This results in a situation where the tip barely reaches only the closest point on the target cell, resulting again in high directional information. We have added this intuitive reasoning in the Main Text.

Evaluation Summary:
This paper studies statistical aspects of the role of longrange cellular protrusions called airinemes as means of intracellular communication. The authors use published data showing how airinemes approach a target cell and describe these movements with a mathematical model for an unobstructed persistent random walk. The impact of this study will be on the specialised reader interested in modelling and airineme biology.
(This preprint has been reviewed by eLife. We include the public reviews from the reviewers here; the authors also receive private feedback with suggested changes to the manuscript. Reviewer #1 agreed to share their name with the authors.)

Reviewer #1 (Public Review):
This article focuses on a quantitative description of airineme morphology and its consequences for contact and communication between cells via these long narrow projections. The primary conclusions are
 Airineme shapes are consistent with a persistent random walk model (analogous to a wormlike polymer chain), unhindered by the presence of other cells.
The authors convincingly demonstrate, using analysis of the meansquareddisplacement along the airineme contour, that the structures cannot be described by a diffusive growth process (ie: a Gaussian chain) as would be expected if there were no directional correlations between consecutive steps. Furthermore, by observing the airineme growth and looking at the distribution of stepsizes, they show that these steps do not exhibit the expected longtail …
Reviewer #1 (Public Review):
This article focuses on a quantitative description of airineme morphology and its consequences for contact and communication between cells via these long narrow projections. The primary conclusions are
 Airineme shapes are consistent with a persistent random walk model (analogous to a wormlike polymer chain), unhindered by the presence of other cells.
The authors convincingly demonstrate, using analysis of the meansquareddisplacement along the airineme contour, that the structures cannot be described by a diffusive growth process (ie: a Gaussian chain) as would be expected if there were no directional correlations between consecutive steps. Furthermore, by observing the airineme growth and looking at the distribution of stepsizes, they show that these steps do not exhibit the expected longtail distributions that would imply a Levywalk behavior. The persistent random walk (PRW) is presented as an alternative that is not inconsistent with the data. However, given the high level of noise due to low sampling, the claimed scaling behavior of the MSD at long lengths is not fully convincing. Nevertheless, the PRW provides a plausible potential description of the airineme shapes.
 The flexibility (ie: persistence length) of the airineme shapes is one that maximizes the probability of a given airineme (of fixed length) contacting the target cell.
This optimum arises due to the balance between straightline paths that reach far from the source but cover a narrow region of space and diffusive paths that compactly explore space but do not reach far from the starting point. Such optimization has previously been noted in unrelated contexts both for search processes of moving particles and for semiflexible chains that need to contact a target. The authors present a compelling case (Fig 4B) that the measured angular diffusion of the airinemes falls close to the predicted optimum. Furthermore, the measured probability of hitting the target cell also lies close to the model prediction, providing a strong test of the applicability of their model.
 Airineme flexibility engenders a tradeoff between contact probability and directional information (ie: the extent to which the target cell can determine the position of the source).
This calculation proposes an alternative utility metric for communication via airinemes. The observed flexiblity is shown to be at a Pareto optimum, where changes in either direction would decrease either the probability of contact or the directional information. Again the absolute value of the metric (Fisher information for angular distribution) is within the predicted order of magnitude from the model. Thus, while the importance of maximizing this metric remains speculative, its quantitative value provides an additional test for the applicability of the PRW model.
Overall, this paper provides an interesting exploration of optimization problems for communication by long thin projections. A particular strength is the quantitative match to experimental data  indicating not just that the experimental parameters fall along a putative optimum but also that the metrics being optimized are wellpredicted by the model. Defining an optimization problem and showing that some parameter sits at the optimum is a common approach to generating insight in biophysical modeling, albeit invariably suffering from the fact that it is difficult to know which optimization criteria actually matter in a particular cellular system. The authors do an excellent job of exploring multiple optimization criteria, quantifying the balance between them, and pointing out inherent limitations in knowing which is most relevant.
A minor weakness of the manuscript is its focus on a very narrowly defined cellular system, with the general applicability of the results not being highlighted for clarity. For example, the fact that the same flexiblity optimizes contact probability and the balance between contact and directional information is an interesting conclusion of the paper. Is this true in general? Is it applicable to other systems involving a semiflexible structure reaching for a target or a moving agent executing a PRW?

Reviewer #2 (Public Review):
Signalling filopodia are essential in disseminating chemical signals in development and tissue homeostasis. These signalling filopodia can be defined as nanotubes, cytonemes, or the recently discovered airinemes. Airinemes are protrusions established between pigment cells due to the help of macrophages. Macrophages take up a small vesicle from one pigment cell and carry it over to the neighbouring pigment cell to induce signalling. However, the vesicle maintains contact with the source cell due to a thin protrusion  the airineme. In support of these data, the authors find that the extension progress of the airinemes fits an "unobstructed persistent random walk model" as described for other macrophages or neutrophils.
The authors describe the characteristics of an airineme as it would be a signalling …
Reviewer #2 (Public Review):
Signalling filopodia are essential in disseminating chemical signals in development and tissue homeostasis. These signalling filopodia can be defined as nanotubes, cytonemes, or the recently discovered airinemes. Airinemes are protrusions established between pigment cells due to the help of macrophages. Macrophages take up a small vesicle from one pigment cell and carry it over to the neighbouring pigment cell to induce signalling. However, the vesicle maintains contact with the source cell due to a thin protrusion  the airineme. In support of these data, the authors find that the extension progress of the airinemes fits an "unobstructed persistent random walk model" as described for other macrophages or neutrophils.
The authors describe the characteristics of an airineme as it would be a signalling filopodia, e.g. a nanotube or a cytoneme, which sends out to target a cell. An airineme, however, is fundamentally different. Here, a macrophage approaches a pigment cell binds to the airineme vesicle. Then, the macrophage approaches a target pigment cell and hands over the airineme vesicle. During this process, the airineme vesicle maintains a connection to the source pigment cell by a thin protrusion. Then, the macrophage leaves the target cell, but the airineme vesicle, including the protrusion, is stabilized at the surface and activates signalling. Indeed nearly all airinemes observed have been associated with macrophages (Eom et al., 2017).
Therefore, it is essential to focus on the "searchandfind" walk of the macrophage and not the passively dragged airineme. In the light of this discussion, I am not sure if statements like "allow the airineme to hit the target cell" are helpful as it would point towards an actively expanding protrusion like a filopodium.

Reviewer #3 (Public Review):
This paper studies statistical aspects of the role of longrange cellular protrusions called airinemes as means of intracellular communication. The mean square distance of an airineme tip is found to follow a persistent random walk with a given velocity and angular diffusion. It is argues that this distribution with these parameters is the one that optimise the probability of contact with the target cell. The authors then evaluate the directional information (where in space did the airineme come from) and found that, again, the measure diffusion coefficient optimise the tradeoff between high directional information (small diffusion) and large encounter probability.
I found this paper well written and clear, and addressing an interesting problem (longrange intracellular communication) using rigorous …
Reviewer #3 (Public Review):
This paper studies statistical aspects of the role of longrange cellular protrusions called airinemes as means of intracellular communication. The mean square distance of an airineme tip is found to follow a persistent random walk with a given velocity and angular diffusion. It is argues that this distribution with these parameters is the one that optimise the probability of contact with the target cell. The authors then evaluate the directional information (where in space did the airineme come from) and found that, again, the measure diffusion coefficient optimise the tradeoff between high directional information (small diffusion) and large encounter probability.
I found this paper well written and clear, and addressing an interesting problem (longrange intracellular communication) using rigorous quantitative tools. This is a very useful approach, which appears to have been appropriately done, that in itself makes this paper worthy of interest.
1. The main conclusion of this paper is that the airineme properties optimises something that has to do with their function. Although rather appealing, I find this kind of conclusion often questionable considering the large uncertainty surrounding many parameters. Here, optimality is shown from a practical perspective, using measure parameters. For instance, the optimal diffusion coefficient for hitting the target varies by 2 orders of magnitude when the distance between cells is varied (Fig.3A). The measured coefficient is optimal for cells about 25 µm distant. Does this reflect anything about the physiological situation in which these airinemes operate? Another rather puzzling claim is that the diffusion coefficient is optimised both for finding the target, AND for finding the best compromised between finding the target and providing directional information, while the latter must necessarily require weaker diffusion. Hence the last paragraph of p.6 ("the data is consistent with either conclusion that the curvature is optimized for search, or it is optimized to balance search and directional information"), although quite honest, gives the feeling that the conclusions are not very robust. I would welcome a discussion of these points.
2. on p.4: "the airineme tips (which are transported by macrophages [30]) appear unrestricted in their motion". I don't understand what it means that the airineme tips are transported by macrophage, and I missed the explanation in the cited article. Is airineme dynamics internally generated (i.e. by actin/microtubule polymerisation) or does it reflect to motility of cells dragging the airineme along? This is discussed in passing in the Discussion, but I think that this should be explainde in more detail right from the start. Aslo, if a cell is indeed directing the tip, what does contact mean? Does it mean that the driving macrophage must contact the target cell and somehow attached the airineme to it? IF yes, that means that the airineme tip has a large spatial extent, which will certainly affect the contact probability.
3. Fig. 2A shows the airinemes MSD and the fit using the PRW model. I don't find the agreement so good. The power law t^2 seems good almost up to 10 minutes, and the scaling above that, if there is one, is clearly larger than linear. So I would say that the apparent agreement with the PRW model reflects the fact that there is a crossover from a ballistic motion to something else, but that this something else is not a randow walk. The MSD does look quite strange at long time, where it apparently decays. This made me wonder whether there might be a statistical biais in the data, for instance, the longest living airinemes are those who didn't find their target and hence those who travel less far, on average. I tried to get more information on the data from the ref.[29,30], but could not find anything. The authors should discuss these data and possible biais in more detail. For instance, do the data mix successful and unsuccessful airinemes? This is somewhat touched upon in Fig.s$, but I did not gain any useful information from it, except that the authors find the agreement "good" while it does not look so good to me.
4. Regarding the directionality discussion, some aspect are a bit vague so that we are left to guess the assumptions made. For instance, the source cell is place at \theta=0 "without loss of generality" (p.6). Apparently (sketch Fig.5A) this also means that the airineme starting point from the source is at \theta=0, which clearly involves loss of generality, since the airineme could start from anywhere, its path could be hindered by the body of the source cell, and its contact angle would then be much less likely to be close to 0. It might be that in practice, only those airineme starting close to theta=0 do in fact make contact, but this should be discussed more thoroughly. Also, why is there to maxima in the Fisher information (Fig.5C) for very high and very low diffusion coefficient at short distance?

Reviewer #4 (Public Review):
In this work, the authors analyze the ability of zebrafish cells to find neighboring cells (playing the role of targets) by extended protrusions (airinemes) that display a "persistent random walk". In this kind of motion, a key parameter is the angular diffusivity, and the main finding is that, for the parameters of the experiments, the probability to find the target is maximized for the experimentally observed value of the angular diffusivity. Basically, a low value of angular diffusivity means that airinemes will be completely straight and miss easily small targets, but large values of angular diffusivities imply that the maximal spatial extension is reduced, disabling the cell to find targets that are to far away (as soon as one assumes that there is a maximal length of airinemes, as seems to be the case …
Reviewer #4 (Public Review):
In this work, the authors analyze the ability of zebrafish cells to find neighboring cells (playing the role of targets) by extended protrusions (airinemes) that display a "persistent random walk". In this kind of motion, a key parameter is the angular diffusivity, and the main finding is that, for the parameters of the experiments, the probability to find the target is maximized for the experimentally observed value of the angular diffusivity. Basically, a low value of angular diffusivity means that airinemes will be completely straight and miss easily small targets, but large values of angular diffusivities imply that the maximal spatial extension is reduced, disabling the cell to find targets that are to far away (as soon as one assumes that there is a maximal length of airinemes, as seems to be the case experimentally). As the result of a tradeoff between these effects, there is an optimal value of angular diffusivity which turn out to be the experimentally observed value, for the experiments analyzed in the paper. In a second part, the authors ask whether the cell can gain directional information on where the target is located ; again the observed value of diffusivity seems to correspond to a tradeoff between directional sensing and the ability to find targets.
I find that these results interesting since the paper brings arguments to understand which kind of stochastic motion are the most suitable to ease communication between cells. The analysis is performed by using simulations of persistent random walks, and image analysis of real cells. The main weakness of the approach consists in the fact that it is difficult, after having read the manuscript, to understand if the observed « optimality » seems is specific to the values of the cell sizes and celltocell distances of the present experiment, or if it would also be approximately the case for other cell sizes, densities, etc. Nevertheless, quantifying such optimality for one experimental situation is already an interesting result.


