Zebrafish airinemes optimize their shape between ballistic and diffusive search

  1. Sohyeon Park
  2. Hyunjoong Kim
  3. Yi Wang
  4. Dae Seok Eom  Is a corresponding author
  5. Jun Allard  Is a corresponding author
  1. Center for Complex Biological Systems, University of California, Irvine, United States
  2. Center for Mathematical Biology, Department of Mathematics, University of Pennsylvania, United States
  3. Department of Developmental & Cell Biology, University of California, Irvine, United States
  4. Department of Physics and Astronomy, University of California, Irvine, United States
  5. Department of Mathematics, University of California, Irvine, United States

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 live-cell 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 trade-off between search optimality and directional information. This provides a framework to characterize the shape, and performance objectives, of non-canonical cellular protrusions in general.

Editor's evaluation

This article studies statistical aspects of the role of long-range 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. Beyond the specialized readers interested in modeling and airineme biology, this article will also be of interest to cell biologists and biophysicists interested in intracellular communication.

https://doi.org/10.7554/eLife.75690.sa0

Introduction

The question of optimal search — given a spatiotemporal process, what parameters allow a searcher to find its target with greatest success? — arises in many biological contexts for a variety of spatiotemporal processes. Examples of relevant processes include searchers moving by diffusion or random walks (Lawley et al., 2020; Berg and Purcell, 1977), Levy walks (Fricke et al., 2016), and ballistic motion (straight trajectories [Bressloff, 2020], which, e.g., arises in chromosome search by microtubules [Holy et al., 1994; Paul et al., 2009]), and combinations of these (Berg, 1993). Another type of motion is the persistent random walk (PRW), which has intermediate properties between diffusion and ballistic motion. PRWs have been studied in continuous space (Schakenraad et al., 2020; Großmann et al., 2016; Khatami et al., 2016) and on a lattice (Tejedor et al., 2012), have been used with variants to model cell migration (Jones et al., 2015; Weavers et al., 2016; Harrison and Baker, 2018), and are mathematically equivalent to worm-like chains, which have been used to study the search by a polymer for a binding partner (Mogre et al., 2020). For all the above processes, optimality depends on parameters of the searcher (e.g., whether searchers operate individually or many in parallel; Schuss et al., 2019; Lawley and Madrid, 2020), the target(s), and the environment (De Bruyne et al., 2020; Bressloff, 2020).

One example of a biological search process arises during organismal development, when cells must establish long-range communication. Some of this communication occurs by diffusing molecules (Hu et al., 2010; Govern and ten Wolde, 2012; Bialek and Setayeshgar, 2008; Endres and Wingreen, 2009) like morphogens. However, recently, an alternative cell–cell communication mechanism has been revealed to be long, thin cellular protrusions extending tens to hundreds of micrometers (Eom, 2020; Yamashita et al., 2018; Caviglia and Ober, 2018; Sanders et al., 2013; Bressloff and Kim, 2019; Inaba et al., 2015). These include cytonemes (Kornberg and Roy, 2014), tunneling nanotubes (Zurzolo, 2021), tenocyte projections (Subramanian et al., 2018), and airinemes in zebrafish (Volkening and Sandstede, 2018; Volkening, 2020; Eom and Parichy, 2017; Eom et al., 2015), shown in Figure 1A. One of the difficulties delaying their discovery and characterization is their thin, suboptical width, and the fact that they only form at specific stages of development (Eom, 2020; Yamashita et al., 2018; Caviglia and Ober, 2018).

Airineme-mediated signaling between xanthoblast and target melanophore.

(A) Multiple airinemes extend from xanthoblast (undifferentiated yellow pigment cell, green). Airineme makes successful contact (arrowhead) with melanophore cell (pigment cell, purple). Asterisks indicate airinemes from other sources. Scale bar: 50 µm. (B) Model schematic. A single airineme extends from the source (right, green circle) and searches for the target cell (left, purple circle). Target cell has radius rtarg and has distance dtarg away from the origin. The airineme’s contour length at time t is l(t).

Airinemes are produced by xanthoblasts (undifferentiated yellow pigment cells) and play a role in the spatial organization of pigment cells that produce the patterns on zebrafish skin (Eom et al., 2015; Eom and Parichy, 2017; Eom, 2020; Volkening and Sandstede, 2018). Macrophages recognize a signal on xanthoblasts and begin dragging a protruding airineme from the xanthoblast as they migrate around the tissue, with the airineme trailing behind them. Airineme lengths have a maximum, regardless of whether they reach their target. If the tip complex reaches a target before this length, it recognizes target cells (melanophores) and the macrophage and airineme tip disconnect. The airineme tip contains the Delta-C ligand, which activates Notch signaling in the target cell. Due to experimental limitations on spatial and temporal resolution, the mechanism by which the airineme tip complex (which might include the entire macrophage) recognizes the target is still mysterious, as is the mechanism by which the macrophage hands off the airineme tip. It is also not 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 retracts. In the unrelated context of wound-healing, macrophages are recruited to the site of injury by detecting chemokines released by damaged cells or other immune cells. In contrast, macrophages pulling airinemes during development are not triggered by tissue damage or infections in zebrafish skin (Eom et al., 2015), and there is no experimental evidence that the airineme search process responds to any directional cues.

For diffusing cell–cell signals, dynamics are characterized by a diffusion coefficient. In contrast, cellular protrusions require more parameters to describe, for example, a velocity and angular diffusion, or equivalently a curvature persistence length. Here, we ask, what are these parameters, and what determines their values? We focus on airinemes where quantitative details have been measured (Eom and Parichy, 2017; Eom et al., 2015). We find that airineme shape is most consistent with a finite-length PRW model, and that the parameters of this model exhibit an optimum for minimizing the probability of finding a target (or, equivalently, the mean number of attempts). We compare this with another performance objective, the ability for the airineme to provide a directional cue to the target cell, and find that there is a theoretical trade-off between these two objectives. This work provides an example where a readily observable optimum appears to be obtained by a biological system.

Results

Airinemes are consistent with a finite PRW model

We examined time-lapse live-cell image data as described in Eom and Parichy, 2017 and Eom et al., 2015. We confirmed that the time series and the final state are similar (Figure 2—figure supplement 1), meaning that the shape of the part of the airineme existing at time t does not significantly change after time t, as the tip of the airineme continues to extend. This allows us to consider only the fully extended airineme and infer the dynamics, assuming airinemes extend with constant velocity v = 4.5 μm/min (Eom et al., 2015). This removes artifacts like microscope stage drift and drastically simplifies the analysis. We manually identified and discretized 70 airinemes into 5596 position vectors r(t), and from these, computed the mean squared displacement (MSD) r2. Random walks satisfy r2=4Dt. However, the observed MSD, shown in Figure 2A, does not appear linear in t. We fit it to r2=γtα and found the best-fit exponent α of 1.55 (90% CI in [1.50,1.61]), and indeed it appears that a single exponent is not appropriate across orders of magnitude. We therefore reject the simple random walk description.

Figure 2 with 1 supplement see all
Airinemes are not consistent with simple random walk or Levy-type models.

(A) Mean squared displacement (MSD) of airinemes. Airineme experimental MSDs have exponent 2 (corresponding to slope in log–log plot) up to 15 min, arguing against a simple random walk model. N=70 airinemes. (B) For airinemes with growth periods longer than 15 min, the distribution of final angles, that is, the angle between the tangent vector at the airineme’s origin and the tangent vector at its tip. This is consistent with isotropy, that is, a uniform distribution in (-π,π), shown as a dashed line (Kolmogorov–Smirnov test, N=26, p-value 0.37). The results of this test are insensitive to the cutoff length (see Figure 2—source data 1). (C) Step length complementary cumulative distribution function (CCDF). Levy models in 2D are characterized by CCDF tails with an exponent between 1 and 3. However, for two different time sampling intervals (10 and 20 min, from N=56 airinemes), the tails of the distribution fit to an exponent greater than 3, and a continued downward curvature instead of a power-law, arguing against Levy models. Inset shows same data, same time sampling intervals, and same axes, but on semi-log plot.

Next, we consider Levy-type models such as those that have been used to describe animal optimal foraging (Viswanathan et al., 2011) and T cell migration (Fricke et al., 2016). These processes have a step size distribution whose tail has exponent between 1 and 3 in 2D (Fricke et al., 2016; Viswanathan et al., 2011), where step size is the displacement during a specified time interval. We revisit the time-series data (i.e., here we do not use the final state approximation) and compute a step length complementary cumulative distribution function (CCDF). For two time interval choices, shown in Figure 2B, the best-fit CCDF exponents are greater than 3 (for 10 min, exponent is 3.81 with 90% CI in [3.68,3.95]; for 20 min, exponent is 4.40 with 90% CI in [4.18,4.61]). Indeed, the CCDFs at two different time sampling intervals have continued downward curvature, indicating that a power-law description is inappropriate. We thus conclude that the process is not consistent with Levy-type models.

Finally, we consider a finite-length PRW. In this model, the tip of the airineme moves at constant speed v, while the direction undergoes random changes with parameter Dθ, the angular diffusion coefficient. This parameter has units inverse minutes and roughly corresponds to the ‘curviness’ of the path. The dynamics are governed by Equations 2–4. A key observation from time-lapse imaging is that airinemes have a maximum length, after which the search process terminates if unsuccessful. Thus, our PRW model is finite-length, meaning that we assume the airinemes extend only up to lmax=250μm. This assumption yields a final length distribution (Figure 3, Figure 3—figure supplement 1) consistent with the observed distribution (Eom et al., 2015).

Figure 3 with 2 supplements see all
Probability to contact a target cell is maximized by a balance between ballistic search and diffusion-like search.

(A, B) Simulated airineme target search for various Dθ. The values of Dθ that leads to the highest search success probability are shown as black open circles. Theoretical values for ballistic limit from Equation 6 are shown as red circles. (A) Varying distances between the target and the source cell, while fixing the target cell radius at 25 µm. We validate simulation results with survival probability PDE at high Dθ from Equation 7, shown in blue circles. (B) Varying target cell radii while fixing the distance between the target and the source. For the biologically relevant parameter (dtarg50μm), optimal angular diffusion is around Dθ0.18min-1. (C) Qualitative behavior of airineme target search depends on Dθ. (D) Optimal Dθ for a larger range of cell-to-cell distances and target cell radii. Rectangular region shows biologically relevant parameters.

The airineme MSD fits the prediction of the PRW model, in Equation 5, up to time point around 15 min. Above this time, the PRW model is consistent with the data, although the low number of long airinemes in our data precludes a strong conclusion from MSD alone. We therefore took all airinemes whose growth time was greater than 15 min and plotted their final angle, that is, the angle between the tangent vector at their point of emergence from the source cell and the tangent vector at their tip. The PRW model predicts that, for long times >1/Dθ, the angular distribution should become isotropic. In Figure 2B, we find that the angular distribution is uniform, that is, isotropic (Kolmogorov–Smirnov test p-value 0.37, N=26). Since there are relatively few data points, we repeated this analysis under various airineme selection criteria, which includes up to N=49 airinemes, and in all cases found the final angular distribution to be consistent with uniformity (Figure 2—source data 1). (In Figure 4, we also check the autocorrelation function and further confirm consistency with the PRW model.) Taken together, the data favor the PRW model, which we use in the following analysis.

Figure 4 with 1 supplement see all
Experimental airineme curvature agrees with the optimal curvature.

(A) Orientation autocorrelation function. We measure tangent angles cos(θ) at 5596 points along 70 airinemes, and then compute the likelihood function (B, bottom) of Dθ fit to Equation 1. Best-fit curve is shown as red with a 90% confidence interval shown in black. Blue dots and heatmap were generated using a moving average with a window of 10 nearest data points and the heatscatter MATLAB function. (B) Bottom: we find that the best-fit airineme curvature from maximum likelihood estimation is Dθ=0.1838min-1. We find that this value is similar to the Dθ that optimizes contact probability for the biologically relevant target cell distance dtarg=50μm (top). The experimentally observed probability of contact per airineme, center estimate (horizontal dashed line), and 90% confidence interval (gray area) are also shown.

We also assume that airinemes operate independently as there is no evidence of airinemes communicating with each other during the search process. Furthermore, airinemes are generated at approximately 0.15 airinemes per cell per hour. Thus, the mean time between airineme initiations is 400min, much larger than the time each airineme extends, which is 56 min. Note that many airinemes emanating from the same source cell may exist simultaneously, but most of the time only one airinemes is extending. Also, while the tissue surface is crowded, the airineme tips (which are transported by macrophages; Eom et al., 2015) appear unrestricted in their motion on the 2D surface, passing over or under other cells unimpeded (Eom and Parichy, 2017). We therefore do not consider obstacles in our model (although these have been studied in other PRW contexts; Schakenraad et al., 2020; Khatami et al., 2016; Hassan et al., 2019). This includes the source cell, that is, we allow the search process to overlie the source cell.

The target cell is modeled as a circle of radius rtarg=1525μm(Eom et al., 2015), separated from the source of the airineme by a distance dtarg50μm, as shown in Figure 1B. Including the position and size of the target, the model has five parameters, all of which have been measured (see Table 1 and Eom et al., 2015; Ryu et al., 2016) except for Dθ.

Table 1
Parameter values.
SymbolMeaningEstimate and value usedSource
lmaxMaximum length of airineme250 μmFigure 3—figure supplement 1
vVelocity of airineme growth4.5 μm (1.4–12 μm, N = 929)Eom et al., 2015
dtargDistance to target cell51 μm (33–84 μm, N = 70)Eom et al., 2015
rtargTarget cell radius15–25 μmRyu et al., 2016
DθAngular diffusion0.1838rad2/minEstimated here

Contact probability is maximized for a balance between ballistic and diffusive search

We performed simulations of the PRW model, testing different angular diffusion values for different values of cell-to-cell distance and target cell radius. For each parameter set, we measured the proportion of simulations that contacted the target. Specifically, contact is defined as the event in which the tip of the growing airineme intersects with the target. In simulations, we assume contact has occurred when the airineme tip reaches within a distance rtarg of the center of the target cell. We refer to rtarg as the target cell radius. However, as discussed above, the mechanism by which contact is detected is unknown, and it could be that the airineme tip has a large effective spatial extent that includes some or all of the macrophage. Note again the search process is finite-length (otherwise in two dimensions would always eventually find the target). These contact probabilities are shown in Figure 3A and B. Equivalently, we plot the inverse, the mean number of attempts, in Figure 3—figure supplement 2.

We find that there exists an optimal angular diffusion coefficient that maximizes the chance to contact the target cell. The optimal value balances between ballistic and diffusion-like search. This has been previously shown for infinite, on-lattice PRWs (Tejedor et al., 2012) and worm-like chain models searching for binding partners (Mogre et al., 2020). We heuristically understand it as follows. When Dθ is small (Figure 3C, left), airinemes are straight and therefore move outward a large distance, which is favorable for finding distant targets. However, straight airinemes easily miss targets. On the other hand, for Dθ large (Figure 3C, right), the airineme executes a random walk. Random walks are locally thorough, so do not miss nearby targets, but the search rarely travels far. Thus, if the target cell is small or close, a diffusion-like search process is favored, but if the target cell is far or large, then a ballistic search is favored. We confirm this in Figure 3D, where we plot optimal Dθ over a large range of target cell radii and cell-to-cell distances. For the biologically relevant parameters (rectangular region in Figure 3D), a balance between ballistic and diffusion-like is optimal.

Experimental airineme curvature is approximately optimal

In order to estimate the missing parameter Dθ, we use the angular autocorrelation function

(1) cosθ(s)=exp(-Dθs/v)

shown in Figure 4. We performed manual image analysis and maximum likelihood estimation to fit Equation 1, along with model convolution (Figure 4—figure supplement 1, Gardner et al., 2010, Materials and methods) to estimate uncertainty.

We find the maximum likelihood estimated persistence length of 12.24μm, corresponding to an angular diffusion Dθ=0.184min1. Surprisingly, as shown in Figure 4B, this value matches our simulated optimal angular diffusion value for the biologically relevant parameter values. Moreover, in the experimental data, we find that the proportion making successful contact with target cells is Pcontact=0.15 (horizontal dashed line), from N=49 airinemes with a 90% confidence interval in [0.06,0.24] (gray box in Figure 4B). This also agrees surprisingly well with the model prediction pcontact0.185.

Directional information at the target cell

In some models of zebrafish pattern formation, the target cells receive directional information from source cells (Eom, 2020; Volkening, 2020), that is, the target cell must determine where the source cell is, relative to the target’s current position. We explore the hypothesis that airineme contact itself could provide directional information since the location on the target cell at which the airineme contacts, θcontact as shown in Figure 5A, is correlated with the direction of the source cell. Analogous directional sensing is possible by diffusive signals, where physical limits have been computed in a variety of situations (Lawley et al., 2020; Berg and Purcell, 1977).

Figure 5 with 2 supplements see all
Trade-off between airineme directional sensing information and the probability of contacting the target cell.

(A) Given that the source cell is located at θ=0, the distribution of angles at which the airineme contacts the target cell. (B) Given a source cell is located at θ=0, the distribution of angles at which the airineme contacts the target cell. Angle distributions with higher variance indicate poorer directional sensing. Three angular diffusion values (near ballistic limit, experimentally observed, and near diffusion limit) are shown. (C) Directional sensing accuracy, quantified using the modified Fisher information (FI) Equation 9, for different source-to-target distances and different ranges of angular diffusion values are tested, while fixing target cell radius at 25 µm. Black rectangular region shows the FI values for the observed airineme curvature. (D) Relationship between the contact probability Pcontact and FI for a range of Dθ (increasing with direction of arrow).

We examined the contact angle distribution on the target cell. In Figure 5B, we show this distribution for three values of Dθ: low (ballistic), high (diffusion-like), and the observed value we found above. The source cell is placed at θorigin=0 without loss of generality (its initial direction is still chosen uniformly randomly). We show the distribution of contact angles on the target cell p(θcontact|θorigin=0) as both a radial histogram (top) and cumulative distribution (bottom). Interestingly, we find that the observed airineme parameters lead to a wide distribution of contact angles compared to both ballistic or diffusion-like airinemes.

To quantify the ability of the target cell to sense the direction of the source by arrival angle of a single airineme, we use Fisher information (Fisher, 1997), modified to take into account the probabilistic number of airinemes that a target cell receives, using Equation 9. Using this measure, we observe a minimum at intermediate Dθ, shown in Figure 5C. We understand this intuitively as follows. For very straight airinemes, the allowed contact locations are restricted to a narrow range (a straight airineme can never hit the target’s far side), resulting in high directional information. For high Dθ, we initially expected low and decreasing directional information since there is more randomness. However, these are finite-length searches, and the spatial extent of the search process shrinks as Dθ increases. This leads to a situation where the tip barely reaches the target, and only at closest points (near θcontact=0), resulting again in high directional information.

To compare with experimental observations, we attempted to measure the contact angle distribution of airinemes in contact with target cells. This is complicated by the highly noncircular shape of these cells, so we approximate the angle by connecting three points: the point on the source cell from which the airineme begins, the center of the nucleus of the target, and the point on the surface of the target where the airineme makes contact, as shown in Figure 5—figure supplement 1. We find a modified Fisher information of 5.7×10-5, slightly smaller but similar in magnitude to the angle distribution predicted by the simulation.

Trade-off between directional sensing and contact probability

By inspecting both Pcontact and directional information shown in Figure 5D, we find that there is a trade-off between the searcher’s contact success and the target cell’s directional sensing. Heuristically, this is because the two objectives prefer opposite variances. To maximize contact probability, variance should be maximal, taking full advantage of the surface of the target. On the other hand, to maximize directional information, the variance of contact angle should be minimized.

Interestingly, the experimentally observed Dθ is at a point where either increasing or decreasing its value would suffer one or the other objectives, a property known as Pareto optimality (Alon, 2009; Barton and Sontag, 2013). Note that this is also the Dθ value that maximizes search success, so the data is consistent with either conclusion that the curvature is optimized for search or it is optimized to balance search and directional information. In other words, in the case of zebrafish airinemes, there is no evidence that the shape of these protrusions sacrifices the goal of optimal search in order to achieve increased directional signaling. We wondered whether this is a general feature of search by PRW. The parametric curve in Figure 5D has a peculiar loop, the concave-down region giving rise to the Pareto optimum. In Figure 5—figure supplement 2, we show the parametric curve for a range of distances to the target dtarg and target sizes rtarg. Note that we do not explicitly explore lmax, but since these plots have not been nondimensionalized, the parametric curve for a different lmax can be obtained by rescaling the results shown. At low dtarg (top row of Figure 5—figure supplement 2), the trade-off is amplified, and the parametric curve resembles bull’s horns with two tips representing the smallest and largest Dθ in our explored range, pointing outward so the shape is concave-up. Intuitively, we understand this as follows: since the target is fairly close (relative to lmax), contact is easy. But the only way to get directional specification is by increasing Dθ to be very large, effectively shrinking the search range so it only reaches (with significant probability) the target at the near side at θcontact=0. The parametric curve is concave-up, and there is no Pareto optimum. At high dtarg (bottom row of Figure 5—figure supplement 2), the searcher either barely reaches, and does so at θcontact=0, therefore providing high directional information, or Dθ is high, and the searcher fails to reach, and therefore also fails to provide directional information. So, there is no trade-off. At intermediate dtarg, the curve transitions from concave-up bull’s horn to the no-trade-off diagonal line. Interestingly, it does so by bending forward, forming a loop, and closing the loop as the low-Dθ tip moves toward the origin. At these intermediate dtarg values, the loop offers a concave-down region with a Pareto optimum.

Discussion

As long-range cellular projections like airinemes continue to be discovered in multicellular systems, their mathematical characterization will become increasingly valuable, mirroring the mathematical characterization of diffusion-mediated cell–cell signals. We have measured the in situ shape of airinemes, and find agreeable fit to a finite, unobstructed PRW model, rather than Levy or diffusion-like motion. The mean square curvature, or equivalently the directional persistent length, is close to that which allows optimal search efficiency for target cells. Since airineme tip motion is driven by macrophages, our results have implications for macrophage cell motility, which is relevant in other macrophage-dependent processes like wound healing and infection (Sun et al., 2019; Achouri et al., 2015).

The growing catalog of non-canonical cellular protrusions (Eom, 2020; Yamashita et al., 2018; Caviglia and Ober, 2018; Sanders et al., 2013; Bressloff and Kim, 2019; Inaba et al., 2015; Kornberg and Roy, 2014; Parker et al., 2017; Subramanian et al., 2018; Wang and Gerdes, 2015) includes strikingly different shapes. For example, some tunneling nanotubes in cancer cells (Parker et al., 2017) are straight compared to airinemes. They also have different functions. For example, nanotubes in PC12 cells serve as conduits for organelles (Wang and Gerdes, 2015). This raises an intriguing possibility that different protrusions have a shape optimized for different functions. Besides search success probability and directional information, one obvious candidate for optimization is the efficient transport of signaling molecules after contact has been established (Bressloff and Kim, 2018; Kim and Bressloff, 2018). This might prefer shorter protrusion length, and therefore favor straight morphologies. In the future, it would be intriguing to compare all known non-canonical protrusions in light of the three performance objectives, and others.

Since the airineme tip’s motion is linked to macrophage motion, these results also inform cell migration patterns. Variants of the PRW model have been found to describe cell migration (Harrison and Baker, 2018; Weavers et al., 2016). Specifically, a related model was found to accurately describe macrophages in zebrafish (Jones et al., 2015). Two observations from Jones et al., 2015 are particularly relevant to our work: that macrophages in zebrafish demonstrate a mix of directional persistence and randomness, and that their migration patterns adapt to circumstance (specifically, in their case, distance to wound and time since wounding).

Mathematically, the finite PRW process is equivalent to the worm-like chain model, for which exact formula have been derived for the tip location (Spakowitz and Wang, 2005; Mehraeen et al., 2008). The contact probability corresponds to a survival probability in the presence of an absorbing disk representing the target cell, and therefore an integral of the formulae in Spakowitz and Wang, 2005; Mehraeen et al., 2008. There is an opportunity to find analytic (asymptotic or exact) expressions for the contact probabilities, which would obviate the need for stochastic simulation.

The cell–cell interactions mediated by airinemes contribute to large-scale pattern formation in zebrafish, a subject of previous mathematical modeling (Volkening and Sandstede, 2015; Volkening and Sandstede, 2018; Nakamasu et al., 2009). Our results provide a contact probability per airineme, setting an upper bound on the ability of cells to communicate via this modality, which is itself a function of cell density (related to dtarg in our notation). Thus, our results may inform future pattern formation models. In the reciprocal direction, these models may provide information about the distribution of target cells, which may significantly affect search efficiencies.

Materials and methods

Key resources table
Reagent type (species) or resourceDesignationSource or referenceIdentifiersAdditional information
Genetic reagent (Danio rerio)Tg(tyrp1b: PALM-mCherry)Eom et al., 2015RRID:ZDB-TGCONSTRCT-141218-3
Recombinant DNA reagentaox5: palmEGFP (plasmid)Eom et al., 2015RRID:ZDB-TGCONSTRCT-160414-1
SoftwareImageJhttp://imagej.nih.gov/ij/RRID:SCR_003070
SoftwareMATLAB R2021bhttp://www.mathworks.com/

Zebrafish husbandry and maintenance

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Adult zebrafish were maintained at 28.5°C on a 16 hr:8 hr light:dark cycle. Fish stocks of Tg(tyrp1b:palmmCherry)wp.rt11 (McMenamin et al., 2014) were used. Embryos were collected in E3 medium (5.0 mM NaCl, 0.17 mM KCl, 0.33 mM CaCl2, 0.33 mM MgCl2·6H2O, adjusted to pH 7.2–7.4) in Petri dishes by in vitro fertilization as described in Westerfield with modifications (Westerfield, 2004). Unfertilized and dead embryos were removed 5 hr post-fertilization (hpf) and 1-day post-fertilization (dpf). Fertilized embryos were kept in E3 medium at 28.5°C until 5 dpf, at which time they were introduced to the main system until they were ready for downstream procedures. All animal work in this study was conducted with the approval of the University of California Irvine Institutional Animal Care and Use Committee (protocol #AUP-19-043) in accordance with institutional and federal guidelines for the ethical use of animals.

Time-lapse and static imaging

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The transgenic embryos, Tg(tyrp1b:palmmCherry), were injected with the construct drive membrane-bound EGFP under the aox5 promoter to visualize airinemes in xanthophore lineages and melanophores (Eom et al., 2015). Zebrafish larvae of 7.5 SSL were staged following Parichy et al., 2009 prior to explant preparation for ex vivo imaging of pigment cells in their native tissue environment as described by Budi et al., 2011 and Eom et al., 2012. Time-lapse images, acquired at 5 min intervals for 12 hr, and static images were taken at ×40 (water-emulsion objective) on a Leica SP8 confocal microscope with resonant scanner.

Model definitions, simulation, and analysis

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In the finite-length PRW model, the position of the airineme tip at time t is given by

(2) dxdt=vcos(θ(t))
(3) dydt=vsin(θ(t))
(4) dθ=2DθdWt

where Wt is a Wiener process, and Dθ is related to the directional persistence length lp in 2D by Dθ=v/2lp.

The MSD for PRWs is (Wu et al., 2014; Sadjadi et al., 2020)

(5) r2=4lpvt(1-2lpvt(1-exp(-vt2lp))).

To simulate this model, we use an Euler–Maruyama scheme with timestep Δt1/Dθ, implemented in MATLAB (The MathWorks). To validate these simulations, at two limits of Dθ, search contact probabilities can be solved analytically (Figure 3A and B, filled circles). First, the straight limit Dθ0. Suppose an airineme searches for the target cell centered at (0,0) with radius rtarg, and the airineme emanates from a source at (rtarg+dtarg,0). Let ϕ be the angle between the hitting point on the target cell and the center line. Then,

(6) ϕ=π2-arccos(rtargrtarg+dtarg)

and Pcontact=ϕ/π. At the other limit, lPdtarg, the PRW is approximately equivalent to diffusion with coefficient D=v2/2Dθ. For a finite time 0<t<lmax/v diffusive search process, the probability of hitting the target cell is Pcontact=1-S(r,t), where S(r,t) denotes the survival probability, which evolves according to

(7) S/t=D2S

with S(r,t)=0 on the surface of the target cell. We solve this PDE and display results in Figure 3A and B, blue circles. For these validations, the Dθ values were chosen to fit the blue circles onto the plot.

Image analysis and model fitting

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In order to estimate uncertainty in our analysis method, we used model convolution (Figure 4—figure supplement 1, Gardner et al., 2010). Specifically, we first measured the experimental signal-to-noise ratio and point spread function. We then simulated airinemes with a ground-truth curvature value and convoluted the simulated images with a Gaussian kernel with the signal-to-noise ratio and point spread function measured from experimental data. Since there is a manual step in this analysis pipeline, independent analyses by five people were performed on both simulated data and experimental data. For simulated data, the difference between simulated and estimated Dθ was less than 7% in all cases and usually 2%.

The extracted data and analysis routines are available openly at: https://github.com/sohyeonparkgithub/Airineme-optimal-target-search, (copy archived at swh:1:rev:366ad6e1e3e5061c0cf395c8e3be784872903922; Park, 2021).

Directional information

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To measure the directional information that, stochastically, an airineme provides its target cell, we use the Fisher information,

(8) I1(θorigin)=E(2θorigin2logp(θtarget|θorigin)).

This quantity can be intuitively understood by noting that, for Gaussian distributions, Fisher information is the inverse of the variance. So, high variance implies low information and low variance implies high information. If multiple independent and identically distributed airinemes provide information to the target cell, then the probability densities of each airineme multiply, and the Fisher information is I(θorigin)=nhitI1(θorigin), where nhit is the number of successful attempts, which is proportional to Pcontact. Therefore, we define the modified Fisher information as

(9) Modifed FI=I(origin)Pcontact.

With this modification, a target cell that receives almost no airinemes will score low in directional information.

Experimental measurement of directional information

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Images capturing incidences of airinemes with membrane-bound vesicles extended from xanthoblasts, stabilizing on melanophores were captured and imported into ImageJ for angle analysis between (1) originating point of airinemes on xanthoblasts, (2) center of target cells (i.e., melanophores), and (3) docking site of airineme vesicles on target cells. Two intersecting lines were drawn as follows: (1) connect the originating point of an airineme on a xanthoblast with the center of the target melanophore to draw the first line, and (2) connect the center of the target melanophore to the docking site of the airineme vesicle on the target melanophore to draw the second line. The angle between the three points connected by the two intersecting lines was then generated automatically with the angle tool in ImageJ. Coordinates of each point and the corresponding angle were recorded with ImageJ and exported to an Excel worksheet for further analysis. Each angle was assigned a ± sign in the 180° system based on the relative location of the three points at the time of the airineme incident. The 0° line was defined as the line passing through the center of the target melanophore. Thus, a positive angle was assigned when the originating point of an airineme on a xanthoblast lies on the 0° line with the docking site of airineme vesicles on the target melanophore lies above the 0° line, and vice versa.

The extracted data and analysis routines are available openly at https://github.com/sohyeonparkgithub/Airineme-optimal-target-search.

Data availability

Data and computational scripts are available in a repository mentioned in the manuscript (on GitHub) https://github.com/sohyeonparkgithub/Airineme-optimal-target-search, (copy archived at swh:1:rev:366ad6e1e3e5061c0cf395c8e3be784872903922).

The following data sets were generated

References

  1. Book
    1. Berg HC
    (1993)
    Random Walks in Biology
    New Jersey, United States: Princeton University Press.
    1. Fisher RA
    (1997) On the mathematical foundations of theoretical statistics
    Philosophical Transactions of the Royal Society of London. Series A, Containing Papers of A Mathematical or Physical Character 222:309–368.
    https://doi.org/10.1098/rsta.1922.0009
  2. Book
    1. Westerfield M
    (2004)
    A Guide for the Laboratory Use of Zebrafish
    Eugene, OR: University of Oregon Press.

Decision letter

  1. Pierre Sens
    Reviewing Editor; Institut Curie, CNRS UMR168, Paris, France
  2. Naama Barkai
    Senior Editor; Weizmann Institute of Science, Rehovot, Israel
  3. Elena F Koslover
    Reviewer; University of California, San Diego, United States

Our editorial process produces two outputs: (i) public reviews designed to be posted alongside the preprint for the benefit of readers; (ii) feedback on the manuscript for the authors, including requests for revisions, shown below. We also include an acceptance summary that explains what the editors found interesting or important about the work.

Decision letter after peer review:

Thank you for submitting your article "Zebrafish airineme shape is optimized between ballistic search and diffusive search" for consideration by eLife. Your article has been reviewed by 4 peer reviewers, one of whom is a member of our Board of Reviewing Editors, and the evaluation has been overseen by Naama Barkai as the Senior Editor. The following individual involved in review of your submission has agreed to reveal their identity: Elena F Koslover (Reviewer #1).

The reviewers have discussed their reviews with one another, and the Reviewing Editor has drafted this to help you prepare a revised submission.

Essential revisions:

1) (Reviewer#3 and reviewer#2 – point 2) Some aspects of the biological situation under study must be explained better, as early as possible in the main text:

– 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 is fundamentally different from a self-guided cellular protrusion since it is driven by a macrophage. Therefore, it is essential to focus on the "search-and-find" walk of the macrophage and not the passively dragged airineme. In the light of this discussion, it is not clear 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. Furthermore, since the protrusion tip is directed by a macrophage, contact mean that the driving macrophage must contact the target cell and attached the airineme to it. So the airineme tip has a large spatial extent (the macrophage size), which will certainly affect the contact probability. The consequence of this for the probability of establishing contact must be discussed.

– In the current version of the paper, one must go to the material method section to understand that there is a maximal length for airinemes. For clarity it should be mention in the main text, because it is an important point of the discussion of Section 2.2 and Figure 3A. Indeed it is very well known that a 2D a diffusive walker will always find any target, which makes very surprising the Figure 3A until one understands that there is a maximal length in the model.

2) (Reviewer # 1, Reviewer #3 – point 1, Reviewer #4 ) One possibly surprising results is the fact 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. This necessitates more explanation. Is this true in general or does this rely on the particular range of parameter explored? Is it applicable to other systems involving a semiflexible structure reaching for a target or a moving agent executing a PRW?

3) Provide a point-by-point response to the reviewers' comments appended below.

Reviewer #1 (Recommendations for the authors):

I found this to be an interesting and well-written manuscript. Most of my recommendations are along the lines of suggestions for clarifications and further discussion placing this work into context.

1) Figure 2A is not very compelling in terms of the long-time scaling. Are there any other metrics that could be shown to bolster the case for approximately diffusive behavior at long times. Velocity correlation functions perhaps? Or step size distributions over long time intervals?

2) I was persistently confused when reading the paper (until I finally found it in the methods) about the definition of contact probability. It should be made clearer in the main text that this is the probability for a fixed length of airineme that somewhere along the length (or is it just at the tip?) it will intersect a circular target.

3) Some background context is provided in the intro and discussion linking the models here to previously explored stochastic processes that are described as persistent random walks. However, as I understand it, the persistent random walk is also mathematically equivalent to a wormlike chain in the polymer physics field. Given the authors are mostly exploring fixed structures of a mechanical object rather than particles moving through time, this is an analogy that could use further highlighting. There is extensive literature available on the distribution properties of wormlike chains. For example, I believe the distribution of angular source positions (used in calculating directional information) could be computed analytically using known wormlike chain distribution functions (such as in Spakowitz and Wang, 2005). In Mogre et al., Biophys J, 2020 very similar problems are discussed in the context of a wormlike chain polymer needing to contact a target with its tip, and the trade-off between a stiff, narrow path and a more meandering one depending on the polymer flexibility. The numerical calculations done in this paper are sufficient and reasonable for the problems addressed, but drawing connections to similar past work in the discussion may be helpful.

4) It would also be helpful to the reader to provide further context on the biological function and regulation of airinemes. In particular, the PRW model here necessarily assumes that the airineme tips grow in an unguided manner (as opposed to following potential signals that indicate target location). Is there any evidence that this is indeed the case? What is the functional role of the airinemes -- what is it they transport and how? Are there diffusing molecules that move through them? Motor-carried particles? Signaling waves? Do they exert mechanical forces on the target? I realize that incorporating transport processes along the airinemes is outside the scope of the calculations in this paper, but further discussion of these issues would be helpful to place the work in context.

5) It would also help to highlight which of the results encountered are generalizable to PRWs in many different systems, not just in airinemes. In particular, the fact that the optimal flexibility both maximizes contact probability and the trade-off between contact and directional information -- is this very specific to the particular length parameter or target size picked? Or is it a general feature of PRWs? If the former, what are the parameter criteria for which this relationship holds? Exploration of this would help future researchers looking to apply these results to biologically unrelated processes that show similar PRW behaviors.

Reviewer #2 (Recommendations for the authors):

The modelling suggests that the shape of the long-range projections can be established by macrophages and thus fits their random walk model. Indeed, such a mechanism would fit very nicely to previously published data describing the chemotaxis movement of macrophages in zebrafish wound healing (Phoebe et al., 2015; Inference of random walk models to describe leukocyte migration). The authors could explore this more in detail and propose a comparative analysis of macrophage movements in different contexts.

Airinemes seem to be protrusions transferring signals to a distant cell. This would be a similar aspect as for nanotubes and cytonemes, defined as signalling filopodia. There is now a good amount of literature on nanotubes from PC12 cells (e.g. structural components) and cytonemes in zebrafish (e.g. dynamics), which deliver the signal directly to a neighbouring cell. I believe the "search-and-find mode" could also be applied to these protrusions? The authors could use their model in the context of these actively extending signalling protrusions.

The authors mention that the "shape of an airineme does not change throughout extension". However, this is an unclear expression because the shape certainly refers also to the length.

Reviewer #3 (Recommendations for the authors):

My recommendation, following the list made in the public review are

1) Discuss the robustness of the conclusion regarding optimisation. hoe can the system be optimised both with respect to optimal contact and to the balance between optimal contact and optimal directionality information.

2) enhance the discussion regarding the biology of the system. What are you really modelling? The motion of a cellular protrusion whose velocity and persistence is related to its molecular constituent (cytoskeleton) or the motion of an entire cell (the macrophage quiding the protrusion).

3) Discuss the data in more detail, in particular how well they really agree with the model.

4) Discuss the assumption of the model more precisely, in particular regarding directional information.

Reviewer #4 (Recommendations for the authors):

I have only a few remarks that could be taken into account to improve clarity of the manuscript.

In the current version of the paper, one must go to the material method section to understand that there is a maximal length for airinemes. For clarity it should probably be better to mention it in the main text, because it is an important point of the discussion of Section 2.2 and Fig 3A. Indeed it is very well known that a 2D a diffusive walker will always find any target, which makes very surprising the Figure 3A until one understands that there is a maximal length in the model.

- Again for clarity it could be useful to present Fig1B also in semi-log scales since this type of curved lines in log-log scales may simply be exponentials. Identifying an exponential law for the step size distribution would certainly lead to a rejection of Levy type walks.

Please also define clearly what is the "best fit exponent" (Section 2.1, first paragraph) : which exponent is it (I guess it is the exponent in the MSD) ? Also what is the step size shown on Fig 2B (is it the distance travelled during a specific time ?)

- To avoid any confusion, it would be useful to draw Fig5A with an airineme that is not perpendicular to the cell surface, so that there is no confusion between the angle that the airineme's tip makes with the cell surface, and the contact angle.

https://doi.org/10.7554/eLife.75690.sa1

Author response

Essential revisions:

1) (Reviewer#3 and reviewer#2 – point 2) Some aspects of the biological situation under study must be explained better, as early as possible in the main text:

– 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 is fundamentally different from a self-guided cellular protrusion since it is driven by a macrophage. Therefore, it is essential to focus on the "search-and-find" walk of the macrophage and not the passively dragged airineme. In the light of this discussion, it is not clear 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. Furthermore, since the protrusion tip is directed by a macrophage, contact mean that the driving macrophage must contact the target cell and attached the airineme to it. So the airineme tip has a large spatial extent (the macrophage size), which will certainly affect the contact probability. The consequence of this for the probability of establishing contact must be discussed.

We have added a new paragraph in the Introduction emphasizing the role of the macrophage, and we have changed the language throughout the text. 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 neutral phrase “search process”.

We have also clarified that, in the biological system, the details of contact are unclear (e.g., what mechanism in the macrophage-airineme-vesicle 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.

– In the current version of the paper, one must go to the material method section to understand that there is a maximal length for airinemes. For clarity it should be mention in the main text, because it is an important point of the discussion of Section 2.2 and Figure 3A. Indeed it is very well known that a 2D a diffusive walker will always find any target, which makes very surprising the Figure 3A until one understands that there is a maximal length in the model.

The finite length (after which the search process terminates if unsuccessful) is now discussed in the Introduction, and again in the first Results section, referring to supplemental Figure S4. The Reviewers’ statement, that two-dimensional diffusion is recurrent so always successful, is very important. So, in this version, we state this fact about 2d diffusion explicitly, and repeat the finite-length property upon first mention of search success probability.

2) (Reviewer # 1, Reviewer #3 – point 1, Reviewer #4 ) One possibly surprising results is the fact 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. This necessitates more explanation. Is this true in general or does this rely on the particular range of parameter explored? 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 trade-off 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 Figure 5D.

At low d_targ, the trade-off 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 concave-up. 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 (“3-o-clock'' in Figure 5A). At low d_targ, the parametric curve is concave-up, and there is no Pareto optimum.

At high d_targ, the searcher either barely reaches (when D_theta is high), and does so at 3-o-clock, 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 trade-off.

At intermediate d_targ, the curve transitions from concave-up bull's horn to the no-tradeoff line. To our surprise, it does so by bending forward, forming a loop, and closing the loop as the low-D_theta tip moves towards the origin. At these intermediate d_targ values, the loop offers a concave-down 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 trade-off with no Pareto optimum and a regime in which there is no trade-off.

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 #1 (Recommendations for the authors):

I found this to be an interesting and well-written manuscript. Most of my recommendations are along the lines of suggestions for clarifications and further discussion placing this work into context.

1) Figure 2A is not very compelling in terms of the long-time scaling. Are there any other metrics that could be shown to bolster the case for approximately diffusive behavior at long times. Velocity correlation functions perhaps? Or step size distributions over long time intervals?

See above discussion of long-time behavior, repeated here for convenience:

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 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.

This prompted us to investigate the long-length data using multiple analysis approaches. In the new manuscript, new Figure 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 Kolmogorov-Smirnov test (p-value 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 Kolmogorov-Smirnov test against a uniform angular distribution gives a p-value 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 high-time data.

2) I was persistently confused when reading the paper (until I finally found it in the methods) about the definition of contact probability. It should be made clearer in the main text that this is the probability for a fixed length of airineme that somewhere along the length (or is it just at the tip?) it will intersect a circular target.

We have clarified the definition of contact when it is first discussed in Results. Since any already-created section of airineme does not move significantly after its creation (see Figure 1 Supplemental Figure 1), contact can only occur at the tip. (We assume in the model that, if contact had occurred somewhere behind the tip, then the airineme would have stopped growing).

3) Some background context is provided in the intro and discussion linking the models here to previously explored stochastic processes that are described as persistent random walks. However, as I understand it, the persistent random walk is also mathematically equivalent to a wormlike chain in the polymer physics field. Given the authors are mostly exploring fixed structures of a mechanical object rather than particles moving through time, this is an analogy that could use further highlighting. There is extensive literature available on the distribution properties of wormlike chains. For example, I believe the distribution of angular source positions (used in calculating directional information) could be computed analytically using known wormlike chain distribution functions (such as in Spakowitz and Wang, 2005). In Mogre et al., Biophys J, 2020 very similar problems are discussed in the context of a wormlike chain polymer needing to contact a target with its tip, and the trade-off between a stiff, narrow path and a more meandering one depending on the polymer flexibility. The numerical calculations done in this paper are sufficient and reasonable for the problems addressed, but drawing connections to similar past work in the discussion may be helpful.

Thank you for this suggestion. We became aware of the optimum in persistence length reported in Mogre et al., 2020 after submission, and we are happy to now include it in the Intro, and when we report the optimum in Results. We also have added a paragraph in Discussion referring to the analytic solutions found in Spakowitz 2005 and Mehraeen et al., 2007.

4) It would also be helpful to the reader to provide further context on the biological function and regulation of airinemes. In particular, the PRW model here necessarily assumes that the airineme tips grow in an unguided manner (as opposed to following potential signals that indicate target location). Is there any evidence that this is indeed the case? What is the functional role of the airinemes -- what is it they transport and how? Are there diffusing molecules that move through them? Motor-carried particles? Signaling waves? Do they exert mechanical forces on the target? I realize that incorporating transport processes along the airinemes is outside the scope of the calculations in this paper, but further discussion of these issues would be helpful to place the work in context.

The following is some background on airinemes, 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 bleb-like 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 ‘eat-me’ 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.

Regarding the specific question about the mechanism by which airinemes functionally enact cell-cell communication: The airineme tip contains the Δ-C ligand, 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 specific question about chemotactic signals: In a different context, during wound-healing, macrophages are recruited to the site of injury by detecting chemokines released by damaged cells or other immune cells. However, airineme pulling macrophage behaviors are not triggered by tissue damage or infections in zebrafish skin, and there is no experimental evidence that they respond to any directional cues.

Indeed, this last statement (that there is no experimental evidence of directional cue in the airineme search process) is further bolstered by the new Figure 2B, which shows a uniform long-time angular distribution of tip orientation.

5) It would also help to highlight which of the results encountered are generalizable to PRWs in many different systems, not just in airinemes. In particular, the fact that the optimal flexibility both maximizes contact probability and the trade-off between contact and directional information -- is this very specific to the particular length parameter or target size picked? Or is it a general feature of PRWs? If the former, what are the parameter criteria for which this relationship holds? Exploration of this would help future researchers looking to apply these results to biologically unrelated processes that show similar PRW behaviors.

See above discussion of generality of theoretical tradeoff, repeated here for convenience:

The Reviewer’s question is an excellent question: Is the trade-off 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 Figure 5D.

At low d_targ, the trade-off 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 concave-up. 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 (“3-o-clock'' in Figure 5A). At low d_targ, the parametric curve is concave-up, and there is no Pareto optimum.

At high d_targ, the searcher either barely reaches (when D_theta is high), and does so at 3-o-clock, 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 trade-off.

At intermediate d_targ, the curve transitions from concave-up bull's horn to the no-tradeoff line. To our surprise, it does so by bending forward, forming a loop, and closing the loop as the low-D_theta tip moves towards the origin. At these intermediate d_targ values, the loop offers a concave-down 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 trade-off with no Pareto optimum and a regime in which there is no trade-off.

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 (Recommendations for the authors):

The modelling suggests that the shape of the long-range projections can be established by macrophages and thus fits their random walk model. Indeed, such a mechanism would fit very nicely to previously published data describing the chemotaxis movement of macrophages in zebrafish wound healing (Phoebe et al., 2015; Inference of random walk models to describe leukocyte migration). The authors could explore this more in detail and propose a comparative analysis of macrophage movements in different contexts.

We agree completely that this is an important connection to make. We have added a paragraph in Discussion putting this work in the context of several previously published work describing cell migration, some of which use variants of persistent random walks. The data and analysis by Phoebe Jones et al., is particularly relevant because it studies macrophages in zebrafish. Unfortunately, while both models are qualitatively similar, having persistence and randomness, a direct comparison is not possible because of differences in the models (specifically, they report that their parameter “w” is strongly correlated with their parameter “b”, which is the parameter that would be comparable with our l_P or D_theta). Nonetheless, two observations from Jones et al., are intriguing in relation to our work: that macrophages in zebrafish demonstrate persistence, and that their migration patterns adapt to circumstance (specifically in their case, distance to wound and time since wounding).

Airinemes seem to be protrusions transferring signals to a distant cell. This would be a similar aspect as for nanotubes and cytonemes, defined as signalling filopodia. There is now a good amount of literature on nanotubes from PC12 cells (e.g. structural components) and cytonemes in zebrafish (e.g. dynamics), which deliver the signal directly to a neighbouring cell. I believe the "search-and-find mode" could also be applied to these protrusions? The authors could use their model in the context of these actively extending signalling protrusions.

This is an intriguing project idea. And, indeed, we are indeed planning to do a “comparative biology” study of the morphology (shape, length) of all noncanonical protrusions for which we can collect or find data. This raises the appealing possibility that different protrusions have a shape optimized for different functions, besides search success probability and directional information. One obvious candidate functional objective – that might apply especially to PC12 nanotubes – is the efficient transport of signaling molecules after contact has been established (see, e.g., Bressloff and Kim 2018, and Kim and Bressloff 2018). The transport time would decrease monotonically with protrusion length (in other words, it would favor straight protrusions). We are in the process of seeking resources for such a study. We believe it is outside the scope of the current manuscript. However, we have an extended, edited paragraph in Discussion with the exposition of this idea.

The authors mention that the "shape of an airineme does not change throughout extension". However, this is an unclear expression because the shape certainly refers also to the length.

Good point. We now use language, “the shape of the part of the airineme existing at time t does not significantly change after time t, as the tip of the airineme continues to extend.”

Reviewer #3 (Recommendations for the authors):

My recommendation, following the list made in the public review are

1) Discuss the robustness of the conclusion regarding optimisation. hoe can the system be optimised both with respect to optimcal contact and to the balance between optimal contact and optimal directionality information.

This is closely related to the Reviewer’s public comment #1. The text below is largely repeated for convenience.

We have clarified the result about directional information in the new manuscript.

First, it is not optimized, 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 fact is called “Pareto optimality”.

Second, the prior 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 both very weak diffusion and very strong diffusion both give local maxima of directional information. The intuitive explanation is that the searchers are finite-length, 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. See our comments above, repeated here for convenience:

The Reviewer’s question is an excellent question: Is the trade-off 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 Figure 5D.

At low d_targ, the trade-off 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 concave-up. 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 (“3-o-clock'' in Figure 5A). At low d_targ, the parametric curve is concave-up, and there is no Pareto optimum.

At high d_targ, the searcher either barely reaches (when D_theta is high), and does so at 3-o-clock, 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 trade-off.

At intermediate d_targ, the curve transitions from concave-up bull's horn to the no-tradeoff line. To our surprise, it does so by bending forward, forming a loop, and closing the loop as the low-D_theta tip moves towards the origin. At these intermediate d_targ values, the loop offers a concave-down 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 trade-off with no Pareto optimum and a regime in which there is no trade-off.

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).

2) Enhance the discussion regarding the biology of the system. What are you really modelling? The motion of a cellular protrusion whose velocit¥ and persistence is related to its molecular constituent (cytoskeleton) or thge motion of an entire cell (the macrophage quiding the protrusion).

This is closely related to the Reviewer’s public comment #2. The text below is largely repeated for convenience.

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 they 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. See response above, repeated here for convenience. 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 bleb-like 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 ‘eat-me’ 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 Δ-C, 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.

In a different context, during wound-healing, macrophages are recruited to the site of injury by detecting chemokines released by damaged cells or other immune cells. However, airineme pulling macrophage behaviors are not triggered by tissue damage or infections in zebrafish skin, and there is no evidence that they respond to any directional cues.

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.

3) Discuss the data in more detail, in particular how well they really agree with the model.

This is closely related to the Reviewer’s public comment #3. The text below is largely repeated for convenience.

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:

The lack of strong agreement prompted us to investigate the long-length data using multiple analysis approaches. In the new manuscript, new Figure 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 Kolmogorov-Smirnov test (p-value 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 Kolmogorov-Smirnov test against a uniform angular distribution gives a p-value 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 high-time data.

4) Discuss the assumption of the model more precisely, in particular regarding directional information.

This is closely related to the Reviewer’s public comment #4. The text below is largely repeated for convenience.

The sketch in Figure 5 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 (which are transported by macrophages (Eom et al., 2015)) appear unrestricted in their motion on the 2d surface, passing over or under other cells unimpeded (Eom et al., 2015). 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: 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 day-side 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 finite-length 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.

Reviewer #4 (Recommendations for the authors):

I have only a few remarks that could be taken into account to improve clarity of the manuscript.

In the current version of the paper, one must go to the material method section to understand that there is a maximal length for airinemes. For clarity it should probably be better to mention it in the main text, because it is an important point of the discussion of Section 2.2 and Fig 3A. Indeed it is very well known that a 2D a diffusive walker will always find any target, which makes very surprising the Figure 3A until one understands that there is a maximal length in the model.

The finite length (after which the search process terminates if unsuccessful) is now mentioned and justified in the Introduction and again in the first Results section, referring to supplemental Figure S4. The Reviewer also makes an important statement, that two-dimensional diffusion is recurrent so always successful. We feel this is important to point out. So, in this version, we state this explicitly, and repeat the finite-length property upon first mention of search probability.

- Again for clarity it could be useful to present Fig1B also in semi-log scales since this type of curved lines in log-log scales may simply be exponentials. Identifying an exponential law for the step size distribution would certainly lead to a rejection of Levy type walks.

We now include the same data (the complementary cumulative distribution of the step size) on semi-log axes, as an inset (new figure 2C).

Please also define clearly what is the "best fit exponent" (Section 2.1, first paragraph) : which exponent is it (I guess it is the exponent in the MSD) ? Also what is the step size shown on Fig 2B (is it the distance travelled during a specific time ?)

We have added definitions of both best fit exponent and step size. The Reviewer was correct about both of these.

- To avoid any confusion, it would be useful to draw Fig5A with an airineme that is not perpendicular to the cell surface, so that there is no confusion between the angle that the airineme's tip makes with the cell surface, and the contact angle.

Agreed. Done. In the new schematic, the departure point is also not centered on the source cell, to further reduce confusion.

https://doi.org/10.7554/eLife.75690.sa2

Article and author information

Author details

  1. Sohyeon Park

    Center for Complex Biological Systems, University of California, Irvine, Irvine, United States
    Contribution
    Data curation, Formal analysis, Investigation, Software, Validation, Visualization, Writing – original draft, Writing – review and editing
    Contributed equally with
    Hyunjoong Kim
    Competing interests
    No competing interests declared
  2. Hyunjoong Kim

    Center for Mathematical Biology, Department of Mathematics, University of Pennsylvania, Philadelphia, United States
    Contribution
    Conceptualization, Investigation, Mathematical model development, Software, Validation
    Contributed equally with
    Sohyeon Park
    Competing interests
    No competing interests declared
    ORCID icon "This ORCID iD identifies the author of this article:" 0000-0002-3534-2102
  3. Yi Wang

    Department of Developmental & Cell Biology, University of California, Irvine, Irvine, United States
    Contribution
    Formal analysis, Investigation
    Competing interests
    No competing interests declared
    ORCID icon "This ORCID iD identifies the author of this article:" 0000-0001-7409-4335
  4. Dae Seok Eom

    1. Center for Complex Biological Systems, University of California, Irvine, Irvine, United States
    2. Department of Developmental & Cell Biology, University of California, Irvine, Irvine, United States
    Contribution
    Conceptualization, Funding acquisition, Methodology, Project administration, Supervision, Writing – original draft, Writing – review and editing
    For correspondence
    dseom@uci.edu
    Competing interests
    No competing interests declared
    ORCID icon "This ORCID iD identifies the author of this article:" 0000-0002-0617-8788
  5. Jun Allard

    1. Center for Complex Biological Systems, University of California, Irvine, Irvine, United States
    2. Department of Physics and Astronomy, University of California, Irvine, Irvine, United States
    3. Department of Mathematics, University of California, Irvine, Irvine, United States
    Contribution
    Conceptualization, Funding acquisition, Project administration, Supervision, Writing – original draft, Writing – review and editing
    For correspondence
    jun.allard@uci.edu
    Competing interests
    No competing interests declared
    ORCID icon "This ORCID iD identifies the author of this article:" 0000-0002-2758-4515

Funding

National Science Foundation (DMS-1454739)

  • Jun Allard

National Institutes of Health (R35GM142791)

  • Yi Wang
  • Dae Seok Eom

National Science Foundation (DMS 1763272)

  • Sohyeon Park
  • Jun Allard

Simons Foundation (594598)

  • Sohyeon Park

Simons Foundation (Math+X U Penn)

  • Hyunjoong Kim

The funders had no role in study design, data collection and interpretation, or the decision to submit the work for publication.

Acknowledgements

We thank Sean Lawley (University of Utah), Jay Newby (University of Alberta), and Yoichiro Mori (University of Pennsylvania) for valuable discussion. We acknowledge support from NSF CAREER award DMS-1454739 to JA, NIH R35GM142791 to DSE, NSF grant DMS 1763272 and two grants from the Simons Foundation (594598, QN and Math+X grant to the University of Pennsylvania).

Ethics

All animal work in this study was conducted with the approval of the University of California Irvine Institutional Animal Care and Use Committee (Protocol #AUP-19-043) in accordance with institutional and federal guidelines for the ethical use of animals.

Senior Editor

  1. Naama Barkai, Weizmann Institute of Science, Rehovot, Israel

Reviewing Editor

  1. Pierre Sens, Institut Curie, CNRS UMR168, Paris, France

Reviewer

  1. Elena F Koslover, University of California, San Diego, United States

Publication history

  1. Preprint posted: October 26, 2021 (view preprint)
  2. Received: November 19, 2021
  3. Accepted: April 25, 2022
  4. Accepted Manuscript published: April 25, 2022 (version 1)
  5. Version of Record published: May 12, 2022 (version 2)

Copyright

© 2022, Park et al.

This article is distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use and redistribution provided that the original author and source are credited.

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  1. Sohyeon Park
  2. Hyunjoong Kim
  3. Yi Wang
  4. Dae Seok Eom
  5. Jun Allard
(2022)
Zebrafish airinemes optimize their shape between ballistic and diffusive search
eLife 11:e75690.
https://doi.org/10.7554/eLife.75690

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