Diffusion vs. direct transport in the precision of morphogen readout.

Morphogen profiles allow cells to determine their position within a developing organism, but not all morphogen profiles form by the same mechanism. Here, we derive fundamental limits to the precision of morphogen concentration sensing for two canonical mechanisms: the diffusion of morphogen through extracellular space and the direct transport of morphogen from source cell to target cell, for example, via cytonemes. We find that direct transport establishes a morphogen profile without adding noise in the process. Despite this advantage, we find that for sufficiently large values of profile length, the diffusion mechanism is many times more precise due to a higher refresh rate of morphogen molecules. We predict a profile lengthscale below which direct transport is more precise, and above which diffusion is more precise. This prediction is supported by data from a wide variety of morphogens in developing Drosophila and zebrafish.

Introduction

Within developing organisms, morphogen profiles provide cells with information about their position relative to other cells. Cells use this information to determine their position with extremely high precision (Dubuis et al., 2013; Erdmann et al., 2009; Gregor et al., 2007a; Houchmandzadeh et al., 2002; de Lachapelle and Bergmann, 2010). However, not all morphogen profiles are formed via the same mechanism and for some profiles the mechanism is still not well understood. One well-known mechanism is the synthesis-diffusion-clearance (SDC) model in which morphogen molecules are produced by localized source cells and diffuse through extracellular space before degrading or being internalized by target cells (Akiyama and Gibson, 2015; Gierer and Meinhardt, 1972; Lander et al., 2002; Müller et al., 2013; Rogers and Schier, 2011; Wilcockson et al., 2017). Alternatively, a direct transport (DT) model has been proposed where morphogen molecules travel through protrusions called cytonemes directly from the source cells to the target cells (Akiyama and Gibson, 2015; Bressloff and Kim, 2018; Kornberg and Roy, 2014; Müller et al., 2013; Wilcockson et al., 2017). The presence of these two alternative theories raises the question of whether there exists a difference in the performance capabilities between cells utilizing one or the other.

Experiments have shown that morphogen profiles display many characteristics consistent with the SDC model. The concentration of morphogen as a function of distance from the source cells has been observed to follow an exponential distribution for a variety of different morphogens (Driever and Nüsslein-Volhard, 1988; Houchmandzadeh et al., 2002). The accumulation times for several morphogens in Drosophila have been measured and found to match the predictions made by the SDC model (Berezhkovskii et al., 2011). In zebrafish, the molecular dynamics of the morphogen Fgf8 have been measured and found to be consistent with Brownian diffusion through extracellular space (Yu et al., 2009). Despite these consistencies, recent experiments have lent support to the theory that morphogen molecules are transported through cytonemes rather than extracellular space. The establishment of the Hedgehog morphogen gradient in Drosophila is highly correlated in both space and time with the formation of cytonemes (Bischoff et al., 2013), while Wnt morphogens have been found to be highly localized around cell protrusions such as cytonemes (Huang and Kornberg, 2015; Stanganello and Scholpp, 2016). Theoretical studies of both the SDC and DT models have examined these measurable effects (Berezhkovskii et al., 2011; Bressloff and Kim, 2018; Shvartsman and Baker, 2012; Teimouri and Kolomeisky, 2016a), but direct comparisons between the two models have thus far been poorly explored. In particular, it remains unknown whether one model allows for a cell to sense its local morphogen concentration more precisely than the other given biological parameters such as the number of cells or the characteristic lengthscale of the profile.

Here, we derive fundamental limits to the precision of morphogen concentration sensing for both the SDC and DT models. We investigate the hypothesis that sensory precision plays a major role in the selection of a gradient formation mechanism during evolution, and we test this hypothesis by quantitatively comparing our theory to morphogen data. Intuitively one might expect the DT model to have less noise due to the fact that molecules are directly deposited at their target. Indeed, we find below that the noise arises only from molecular production and degradation, with no additional noise from molecular transport. However, we also find below that for sufficiently large morphogen profile lengthscales, the SDC model produces less noise than the DT model due to it being able achieve a higher effective unique molecule count. By elucidating the competing effects of profile amplitude, steepness, and noise, we ultimately conclude that there should exist a profile lengthscale below which the DT model is more precise and above which the SDC mechanism is more precise. We find that this prediction is quantitatively supported by data from a wide variety of morphogens, suggesting that readout precision plays an important role in determining the mechanisms of morphogen profile establishment.

Results

Several past studies have focused on the formation dynamics of morphogen profiles (Berezhkovskii et al., 2011; Bressloff and Kim, 2018; Shvartsman and Baker, 2012; Teimouri and Kolomeisky, 2016a). Here, we model profiles in the steady state regime, as most of the experimental measurements to which we will later compare our results were taken during stages when the steady state approximation is valid (Grimm et al., 2010; Gregor et al., 2007b; Kicheva et al., 2007; Yu et al., 2009; Kanodia et al., 2009). Precision depends not only on stochastic fluctuations in the morphogen concentration, but also on the shape of the mean morphogen profile, as the shape determines concentration differences between adjacent cells that may adopt different fates. Therefore, as in past studies (Gregor et al., 2007a; Tostevin et al., 2007), we define the precision as , where is the standard deviation of the number of morphogen molecules arriving at cell j, and is the difference between the molecule number in that cell and the adjacent cell. As is typical in studies of both the DT (Teimouri and Kolomeisky, 2016a; Bressloff and Kim, 2018) and SDC (Berezhkovskii et al., 2011; Shvartsman and Baker, 2012; Teimouri and Kolomeisky, 2016b) mechanisms, we focus on a one-dimensional line of target cells. However, we derive analogous results for 2D and 3D systems, and we generally find that the dimensionality does not qualitatively change our results, as we discuss later. In 1D, cells extend in both directions from the source cell, with N cells on each side (Figure 1). We note that in the case of the Bicoid morphogen in the Drosophila embryo, target cells extend only on one side of the source. This will introduce a factor of 2 in the means of both the DT and SDC models and a factor slightly greater than 2 in the variance of the SDC model. This will not affect the agreement of the Bicoid data with our theory in Figure 4B.

Figure 1.

Source cell (green) produces morphogen which is delivered to N target cells (blue) on either side via (A) direct transport (DT) or (B) synthesis-diffusion-clearance (SDC).

A common method for determining the statistical properties of stochastic systems is to express the dynamics of their probability distributions in the form of a master equation. The first moment of the master equation then dictates the dynamics of the mean and becomes the rate equation. Higher order moments can be similarly used to calculate other statistical properties such as the variance. As we will show, in the case of the DT mechanism only the rate equation will be needed as the relevant statistical properties are identical to that of a simple birth-death system which is fully characterized by its mean. Conversely, the SDC mechanism has more complicated noise properties, which we will calculate via the first and second moments. Specifically, we will use a Langevin description which is expressed as a rate equation with noise terms Gardiner, 2004.

Direct transport model

We first consider the DT case, where morphogen molecules are transported via cytonemes that connect a single source cell to multiple target cells (Figure 1A). Cytonemes are tubular protrusions that are hundreds of nanometers thick and between several and hundreds of microns long (Kornberg and Roy, 2014; Kornberg, 2014). They are supported by actin filaments, and it is thought that morphogen molecules are actively transported along the filaments via molecular motors (Kornberg and Roy, 2014; Kornberg, 2014; Sanders et al., 2013; Huang and Kornberg, 2015). It was recently shown that a DT model that includes forward and backward transport of molecules within cytonemes reproduces experimentally measured accumulation times (Teimouri and Kolomeisky, 2016a; Bressloff and Kim, 2018), although the noise properties of this model were not considered. Here, we review the steady state properties of this model and derive its noise properties.

Consider a single source cell that produces morphogen at rate . Morphogen molecules enter each cytoneme at rate . The cytoneme that leads to the jth target cell has length , where a is the cell radius. Once inside a cytoneme, morphogen molecules move forward toward the target cell with velocity or backwards toward the source cell with velocity , and can switch between these states with rates (forward-to-backward) or (backward-to-forward). Once a molecule reaches the forward (backward) end of the cytoneme, it is immediately absorbed into the target (source) cell. Molecules within a target cell spontaneously degrade with rate . An alternative model could involve neglecting this degradation step by counting the arrival of morphogen molecules rather than the concentration. This method can at best reduce the variance by a factor of 2 as the noise from degradation is eliminated but the noise from arrival remains. This would cause negligible change to our results presented in Figure 3 and Figure 4 given the order of magnitude difference in precision seen between our two models.

The dynamics of the mean number of morphogen molecules in the source cell and jth target cell , and the mean density of forward-moving molecules and backward-moving molecules in the jth cytoneme are (Bressloff and Kim, 2018)where the summation in the first line runs from to excluding 0 (the source cell). Additionally, the boundary conditions and are imposed to reflect the rate at which morphogen molecules enter the cytoneme from either end. This creates a steady-state solution of the form

This solution is identical to that found in Bressloff and Kim, 2018 with the exception of the factor of that accounts for target cells existing on both sides of the source cell in our model. Here, is the effective transport rate of morphogen molecules to the jth target cell, and and are defined in terms of the average distance a molecule would move forward or backward within a cytoneme before switching direction. The parameter sets the shape of , and thus of m: when the profile is constant, ; when it is exponential, ; and when it is a power law for large j, . The parameter sets the lengthscale of the profile, defined aswhere we approximate the sum as an integral for . We use this expression to eliminate , writing in Equation 2 entirely in terms of and .

Despite the complexity of the transport process in Equation 1, we find that it adds no noise to m. In fact, here we prove that any system in which molecules can only degrade in the target cells and cannot leave the target cells has the steady-state statistical properties of a simple birth-death process. First, consider the special case of only one target cell. Because each morphogen molecule produced in the source cell acts independently of every other morphogen molecule, we define as the probability density that any given molecule will enter the target cell a time after it is created in the source cell. Next, we define as the probability that a morphogen molecule will enter the target cell between t and . This event requires the molecule to have been produced between and , which occurs with probability ; to arrive at the target cell a time later and to enter the target cell within the window , which occurs with probability ; and we must integrate over all possible times . Therefore,where the last step follows from normalization.

This generalized version of the DT model is visualized in Figure 2. We see that regardless of the form of , the probability of a morphogen molecule entering the target cell in any given small time window is simply . Since the mechanism by which morphogen molecules go from the source cell to the target cell can only affect , this result holds regardless of the specifics of the mechanism so long as the condition that the morphogen cannot leave the target cell other than by degradation is maintained. This result also holds when the system is expanded to have multiple target cells, as then is replaced with , the probability density that the molecule enters the jth target cell a time after being produced. In this case, evaluates to , the total probability the morphogen molecule is ultimately transported to the jth target cell, and is simply replaced with . Combined with the constant degradation rate of morphogen molecules within the target cell, this is precisely a birth-death process with birth rate and death rate . For our system in Equation 2. The conditions that morphogen molecules act independently and can only degrade within the target cell are critical as the former ensures the noise sources are linear and the later will be violated by the SDC mechanism.

Figure 2.

Diagram outlining a generalized version of the DT model.

Between times and a molecule is produced with probability . This molecule then undergoes a transport process over a time with probability density . Finally, the molecule is deposited into the target cell at the end of the transport process. Integrating over all possible values of then yields , the probability of the molecule entering the target cell between times t and .

We now assume that each cell integrates its morphogen molecule count over a time T (Berg and Purcell, 1977; Gregor et al., 2007a). The variance in the time average is simply that of a birth-death process, given by (Fancher and Mugler, 2017), so long as , where is the correlation time. We see that, as expected for a time-averaged Poisson process, the variance increases with the mean m and decreases with the number of independent measurements made in the time T. The precision is therefore

We see that the precision increases with the profile amplitude m, the number of independent measurements , and the profile steepness . The transport process influences the precision only via m, not . For a given N, j, and , we find that the precision is maximized at a particular (Figure 3A). The reason is that an exponential profile () has constant steepness but small amplitude, whereas a power-law profile () has low steepness but large amplitude due to its long tail; the optimum is in between.

Figure 3.

Comparing theoretical DT precision to SDC precision for a single cell.

(A) DT precision shows a maximum as a function of shape parameter for any value of the profile lengthscale. (B) Ratio of DT to SDC precision shows a crossover () as a function of profile lengthscale for 1D, 2D, and 3D geometries. Here is the central cell of target cells. For each value of the value of which maximizes precision in the DT model () as seen in A is used.

SDC model

We next consider the SDC case (Figure 1B). Again a single source cell at the origin produces morphogen at rate . However, now morphogen molecules diffuse freely along x with coefficient D and degrade spontaneously at any point in space with rate . The dynamics of the morphogen concentration arewhere the noise terms associated with diffusion, degradation, and production obeyrespectively (Gardiner, 2004; Gillespie, 2000; Fancher and Mugler, 2017; Varennes et al., 2017). Here, the time independent is the steady state mean concentration, with characteristic lengthscale . We imagine a target cell located at x that is permeable to the morphogen and counts the number of morphogen molecules within its volume V. We use this simpler prescription over explicitly accounting for more realistic mechanisms such as surface receptor binding because it has been shown that the two approaches ultimately yield similar concentration sensing results up to a factor of order unity (Berg and Purcell, 1977). For a cell in steady state at position , the integral evaluates to

Importantly, in the model presented here the morphogen molecules can diffuse both into and out of the target cells, thus violating the condition that they can only degrade once in a target cell and disallowing our previous argument depicted in Figure 2. However, because Equation 6 is linear with Gaussian white noise, calculating the time-averaged variance is straightforward: we Fourier transform Equation 6 in space and time, calculate the power spectrum of , and take its low-frequency limit (Appendix 1). So long as , we obtain the same functional form as Equation 5,because diffusion is a Poisson process. This result is distinguished from that of the DT system by the correlation time taking the form

The factor in brackets is always less than one and decreases with . It reflects the fact that, unlike in the DT model, molecules can leave a target cell not only by degradation, but also by diffusion. Therefore, the rate at which molecules are refreshed is larger than that from degradation alone. This effect increases the precision because more independent measurements () can be made.

To understand this effect more intuitively, consider a simplified SDC model in which diffusion is modeled as discrete hopping between adjacent target cells at rate h. The autocorrelation function is (Appendix 2), where I0 is the zeroth modified Bessel function of the first kind. The correlation time is , and we see explicitly that it decreases with both degradation () and diffusion (h). In fact, in the limit of fast diffusion (), the expression becomes . Correspondingly, in the fast-diffusion limit of Equation 9 (), the term in brackets reduces to , and it becomes . These expressions are identical, with playing the role of the hopping rate h, as expected.

Comparing the models

We now ask which model has the higher precision. We first note that while the precision in both models has the same form when expressed in terms of the means and correlation times (Equations 5 and 8), substituting in the corresponding expressions reveals how the precision depends on the various parameters of each model. Specifically, the precision in the DT model is seen to depend on N, j, the product , , and . Importantly, it is independent of so long as the condition is met. This is due to the mean molecule count scaling as (Equation 2) and the number of independent measurements () scaling as , thus causing their product and in turn the precision to be independent of . In the SDC model, the precision depends on j, the product , and . For similar reasons as in the DT model, does not explicitly appear once Equations 7 and 9 are inserted into Equation 8, but it is present implicitly in the definition of .

To properly compare the two models, we equate several variables. Specifically, we wish to compare the precision of each model within a specific system, which requires N and j to be the same in both models. Additionally, and are restricted by the energy and material costs of producing and degrading morphogen molecules while T is restricted by the need of the system to properly develop in a finite amount of time. These restrictions are assumed to be independent of the method of morphogen profile establishment, which implies that both the DT and SDC systems will adopt the same maximal values of and T. While the value of is restricted in a similar manner, explicitly equating it between the two models does not reduce the number of free parameters, as the precision is independent of in the DT model, and is insufficient to fully define the value of in the SDC model. Therefore, we equate the characteristic profile length scale , as this parameter is consistently defined across both models and is also measured in experiments, allowing us to compare our theory with data as discussed below. Finally, we optimize over as it is unique to the DT model.

We now observe how the precision of the DT and SDC models compare in a representative system. Figure 3B shows as a function of profile length for a cell in the center () of a line of target cells, where for each we use the that maximizes as seen in Figure 3A. Since and T have been equated between the two models, is independent of both. We see that for short profiles the DT model is more precise () whereas for long profiles the SDC model is more precise (). This effect holds for a single source cell providing morphogen for a 1D line of target cells as well as for a 1D line of source cells with a 2D sheet of target cells and a 2D sheet of source cells with a 3D volume of target cells. For the DT model, the 2D and 3D cases are identical to the 1D case as we assume that cytonemes extend perpendicular to the source cells; for the SDC model see Appendix 1.

The reason that the SDC model is more precise for long profiles is that long profiles correspond to fast diffusion, which increases the refresh rate as discussed above. Conversely, the reason that the DT model is more precise for short profiles is that it has a larger amplitude. It also has a smaller steepness, but the larger amplitude wins out. Specifically, whereas the SDC amplitude falls off exponentially, , for sufficiently small the DT amplitude falls off as a power law, . The steepness of the SDC profile is constant, while the steepness of the DT profile also scales like . Thus, the product of the ratio of amplitudes and the square of the ratio of steepnesses, on which depends, scales like . For small , the exponential dominates over the cubic for the majority of j values. Consequently, the DT model has the higher precision.

Comparison to data

We now test our predictions against data for various morphogens. In Drosophila, the morphogen Wingless (Wg) is localized near cell protrusions such as cytonemes (Huang and Kornberg, 2015; Stanganello and Scholpp, 2016), and the Hedgehog (Hh) gradient correlates highly in both space and time with the formation of cytonemes (Bischoff et al., 2013), suggesting that these two morphogen profiles are formed via a DT mechanism. Conversely, Bicoid has been understood as a model example of SDC for decades (Driever and Nüsslein-Volhard, 1988; Gregor et al., 2007a; Houchmandzadeh et al., 2002). Similarly, Dorsal is spread by diffusion; however, its absorption is localized to a specific region of target cells via a nonuniform degradation mechanism, making it more complex than the simple SDC model (Carrell et al., 2017). Finally, for Dpp there is evidence for a variety of different gradient formation mechanisms (Akiyama and Gibson, 2015; Müller et al., 2013; Wilcockson et al., 2017).

In zebrafish, the morphogen Fgf8 has been studied at the single molecule level and found to have molecular dynamics closely matching the Brownian movement expected in an SDC mechanism (Yu et al., 2009). Similarly, Cyclops, Squint, Lefty1, and Lefty2, all of which are involved in the Nodal/Lefty system, have been shown to spread diffusively and affect cells distant from their source (Müller et al., 2013; Rogers and Müller, 2019). This would support the SDC mechanism, although Cyclops and Squint have been argued to be tightly regulated via a Gierer-Meinhardt type system, thus diminishing their gradient sizes to values much lower than what they would be without this regulation (Gierer and Meinhardt, 1972; Rogers and Müller, 2019).

For all these morphogens, we estimate the profile lengthscales from the experimental data (Kicheva et al., 2007; Wartlick et al., 2011; Gregor et al., 2007b; Liberman et al., 2009; Yu et al., 2009; Müller et al., 2012) (Appendix 3). Figure 4A shows these values and indicates for each morphogen whether the evidence described above suggests a DT mechanism (red), an SDC mechanism (blue), or multiple mechanisms including DT and SDC (white). We see that in general, the three cases correspond to short, long, and intermediate profile lengths, respectively, which is qualitatively consistent with our predictions.

Figure 4.

Comparing theory and experiment.

(A) values for morphogens estimated from experiments, colored by whether experiments support a DT (red), SDC (blue), or multiple mechanisms (white). (B) Data from A overlaid with color from theory using values of a and N estimated from experiments. Color indicates percentage of cells for which SDC is predicted to be more precise with dots signifying 0% and 100%, dashed lines signifying 25% and 75%, and solid lines signifying 50%. The axis is normalized by , the value of at which 50% of cells are more precise in SDC.

To make the comparison quantitative, we estimate the values of cell radius a and cell number N from the experimental data (Kicheva et al., 2007; Gregor et al., 2007a; Liberman et al., 2009; Yu et al., 2009; Kimmel et al., 1995) (Appendix 3) in order to calculate from our theory in each case. The background color in Fig. Figure 4B shows the percentage of cells within each system for which we predict that the SDC model is more precise as a function of . The data points in Fig. Figure 4B show the values of from the experiments, also normalized by , the value of at which 50% of cells are more precise in SDC, from the theory. For each morphogen species, we assume a 1D system for simplicity as we have checked that considering higher dimensions yields negligible differences to the results presented in Figure 4B. We see that our theory predicts the correct threshold: the morphogens for which the evidence suggests either a DT or an SDC mechanism (red or blue) fall into the regime in which we predict that mechanism to be more precise for most of the cells, and the morphogens with multiple mechanisms (white) fall in between. This result provides quantitative support for the idea that morphogen profiles form according to the mechanism that maximizes the sensory precision of the target cells.

Discussion

We have shown that in the steady-state regime, the DT and SDC models of morphogen profile formation yield different scalings of readout precision with the length of the profile and population size. As a result, there exist regimes in this parameter space in which either mechanism is more precise. While the DT model benefits from larger molecule numbers and no added noise from the transport process, the ability of molecules to diffuse into and away from a target cell in the SDC model allows the cell to measure a greater number of effectively unique molecules in the same time frame. By examining how these phenomena affect the cells’ sensory precision, we predicted that morphogen profiles with shorter lengths should utilize cytonemes or some other form of direct transport mechanism, whereas morphogens with longer profiles should rely on extracellular diffusion, a prediction that is in quantitative agreement with measurements on known morphogens. It will be interesting to observe whether this trend is further strengthened as more experimental evidence is obtained for different morphogens, as well as to expand the theory of multicellular concentration sensing to further biological contexts.

Despite the quantitative agreement between our theory and experiments, it is clear that the models presented here are minimal and thus cannot be directly applied to all systems. This is exemplified by morphogen such as Dorsal, which due to aforementioned diffusive spreading and nonuniform degradation mechanism clearly does not strictly follow either model. Additionally, the SDC model can be violated if the diffusion of morphogen through a biological environment is hindered by the typically crowded nature of such environments, leading to possibly subdiffusive behavior (Ellery et al., 2014; Fanelli and McKane, 2010). For the DT model, we explicitly ignored the dynamics of the cytonemes themselves due to the growth rate of the cytonemes being sufficiently fast so as to traverse the entire system size in significantly less time than is required for the cells to integrate their morphogen counts over (Bischoff et al., 2013; Chen et al., 2017). This assumption is problematic if cytonemes continue to behave dynamically after reaching the source cell. In particular, the process of cytonemes switching between phases of growing and retracting can introduce super-Poissonian noise sources to the morphogen count within the target cells. Super-Poissonian noise can also be introduced by relaxing the assumption that the morphogen molecules behave independently of each other as this was a critical component of our proof that the molecule count in the target cells is statistically identical to a birth-death process. Finally, it is conceivable that a hybrid system which utilizes a combination of diffusion and directed transport could be developed. However, we are unaware of any experimental study into such a possibility and as such have not considered it in this particular work. It will be interesting to explore the implications of each of these complications in future works.

Materials and methods

Methods are described in Appendices 1, 2, and 3 along with more detailed evaluations of 2D and 3D SDC systems and further discussion of data used in Figure 4. Additionally, code used to generate the plots seen in Figure 3 and Figure 4 can be found at Fancher, 2020.

In the interests of transparency, eLife publishes the most substantive revision requests and the accompanying author responses.

Acceptance summary:

During embryonic development, morphogen gradients provide positional information to differentiating cells. Experiments have demonstrated that the precision by which these gradients are created and read out is often very high, raising the question of the fundamental limits by which nuclei and cells can obtain positional information.

Fancher and Mugler compare two mechanisms for gradient formation, direct-transport and synthesis-diffusion-clearance, and show that the former is favoured for steep gradients and the latter at shallow gradients, in good qualitative agreement with observations.

Decision letter after peer review:

Thank you for submitting your article "Diffusion vs. direct transport in the precision of morphogen readout" for consideration by eLife. Your article has been reviewed by three peer reviewers, one of whom is a member of our Board of Reviewing Editors, and the evaluation has been overseen by Marianne Bronner as the Senior Editor. The reviewers have opted to remain anonymous.

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

We would like to draw your attention to changes in our revision policy that we have made in response to COVID-19 (https://elifesciences.org/articles/57162). Specifically, we are asking editors to accept without delay manuscripts, like yours, that they judge can stand as eLife papers without additional data, even if they feel that they would make the manuscript stronger. Thus the revisions requested below only address clarity and presentation.

Summary:

The authors present theoretical results concerning two canonical, alternative mechanisms for patterning of cells by morphogen gradients. The issue is that cells know about their positions in the organism, and thus their intended cell fates, from measurements of morphogen concentrations. There will necessarily be uncertainty due noise in this concentration, and evolution has presumably favored mechanisms that reduce this noise (in particular relative to signal – namely the average difference in morphogen levels between adjacent cells). The first mechanism is direct transport (DT) of morphogen from a source cell to target cells via specific channels (e.g. cytonemes). The second mechanism is simple diffusion with degradation, which they call synthesis-diffusion-clearance (SDC). By rigorously formulating noise models for these different schemes, the authors draw several interesting conclusions. First, the transport process associated with DT does not in itself contribute to the noise. This follows from the assumption that morphogen molecules are independent so that the details of transport only yield a the net rate of arrival of molecules at the target cell. More importantly, the authors discover that there is a cross-over as a function of profile length from DT as the preferred mechanism to SDC as the preferred mechanism. The authors do an excellent job of presenting intuitive arguments for this and other results. (In this case, the essential disadvantage of the DT mechanism is that cells can only glean information from the morphogens that they absorb, whereas in the SDC mechanism cells can sense all molecules that pass by them.) The authors follow up this theoretical study with an analysis of multiple developmental morphogen systems. The results are impressively consistent with their theoretical predictions concerning which mechanism has higher signal-to-noise given the profile lengths. Overall, this is an exemplary paper: it identifies an important unrecognized question, rigorously formulates and solves theoretical models to address the question, and carefully applies the results to gain original insights into well-studied developmental systems.

Essential revisions:

1) We find the notation in the paper is unnecessarily complicated. For example, all the overbars in Equation 1 are not needed if the quantities are defined to be the mean. Otherwise, we have symbols with three labels, the bar, the j and +/-. Notation in a biology journal must be very carefully chosen for maximum simplicity and clarity. Likewise, the δ functions in these equations could more clearly be shifted to statements about boundary conditions, which would be more intuitive. Remember, very few biologists understand δ functions!

2) The argument presented in the third paragraph of the subsection “Direct Transport Model” is a key one, but we find that it could be explained more clearly. Perhaps by reference to a more detailed diagram?

3) The lack of parallel construction between the presentations of the two models is odd. The DT model is presented as a set of deterministic ODEs for mean concentrations, while the diffusive model (itself an averaged equation) has noise terms added. We would have expected the statement about the noise effect in the DT model to be deduced from some underlying master equation of which the presented model is simply the first moment(s), but that higher order moments are necessary to calculate to draw conclusions about the model.

4) Stepping back from the calculations we are struck by the following question. If, as stated: "We see that regardless of the form of p(τ), the probability of a morphogen molecule entering the target cell in any given small time window δt is simply βδt. This result holds regardless of the mechanism by which morphogen molecules go

from the source cell to the target cell, as the only effect such a mechanism can have is on p(τ)", then why can't we substitute a diffusive mechanism and deduce the same thing?

5) The authors assume that once a morphogen molecule has reached the end of the cytoneme, it is immediately absorbed by the target cell. How realistic is this assumption and how important is it? Would relaxing it change the results qualitatively?

6) In comparing the precision of the two models, it would be good to discuss in a bit more detail how the comparison is made: which parameters are kept the same, which are optimized over, which are varied systematically, and why is this choice the most natural one? For example, the authors chose to study the precision ratio as a function of the gradient length, while optimizing over the DT shape φ. Βut we could also imagine a comparison in which the production and degradation rates are kept the same for both models (because production and degradation are energetically costly), but that over all other parameters the precision is optimized. Would that give a qualitatively different result? Why would the gradient range λ be of special importance (the comparison is made on the footing of equal gradient range)? Should not only the precision matter (such that it would be fine that the gradients in the DT and SDC model are different)? In other words, how do the maximal precisions that can be reached given reasonable constraints (such as production and degradation rates) compare, as a function of the distance to the source?

7) And in making this comparison, does the SDC model exhibit an optimal diffusion constant that maximizes the precision? Understanding this is probably important, because when, in the current comparison, the SDC model is more precise, it is because of the higher protein clearance rate due to diffusion. Yet, a higher diffusion constant also lowers the steepness of the gradient, decreasing the precision,

8) The key mechanism by which the SDC model can become superior is indeed diffusion. Could adding diffusion to the DT model improve the performance of the latter model? Is there evidence for diffusion in experimental systems that employ directed transport?

9) In DT systems, is it clear that molecule transport is independent? How could correlated transport be included in the DT model?

10) Also, in the DT model could noise be reduced by cells measuring arrivals of particles rather than averaging over internal concentration of particles? This would presumably only change the resulting expression for precision by a numerical factor, but could this factor shift the balance in favor of DT in some cases?

Essential revisions:

1) We find the notation in the paper is unnecessarily complicated. For example, all the overbars in Equation 1 are not needed if the quantities are defined to be the mean. Otherwise, we have symbols with three labels, the bar, the j and +/-. Notation in a biology journal must be very carefully chosen for maximum simplicity and clarity. Likewise, the δ functions in these equations could more clearly be shifted to statements about boundary conditions, which would be more intuitive. Remember, very few biologists understand δ functions!

We have made the suggested revisions to the notation by removing the bar notation entirely. Instead, we express stochastic variables as time dependent and their steady state means as time independent. We have also removed the δ functions from the dynamic equations for the DT model (Equation 1) and expressed them instead as boundary conditions. We have left the δ functions in the equations for the SDC model though, as our method of incorporating Langevin noise terms requires the use of δ functions rather than simple boundary conditions. Finally, we have changed all + and − labels to be superscripts so as to make the notation more consistent across variables.

2) The argument presented in the third paragraph of the subsection “Direct Transport Model” is a key one, but we find that it could be explained more clearly. Perhaps by reference to a more detailed diagram?

We have added more text to better clarify the necessary assumptions needed to make this argument as well as created a new figure (Figure 2) to provide a visualization.

3) The lack of parallel construction between the presentations of the two models is odd. The DT model is presented as a set of deterministic ODEs for mean concentrations, while the diffusive model (itself an averaged equation) has noise terms added. We would have expected the statement about the noise effect in the DT model to be deduced from some underlying master equation of which the presented model is simply the first moment(s), but that higher order moments are necessary to calculate to draw conclusions about the model.

We have added a paragraph just before the DT section of our results to address this potential source of confusion. Specifically, our method in both models is to solve for certain moments of the master equation. In the DT model, we only require the first moment to fully characterize the statistical properties of the morphogen molecule count in the target cells, while in the SDC model we solve for the second moment via the known properties of the noise terms introduced into the equation for the first moment.

4) Stepping back from the calculations we are struck by the following question. If, as stated: "We see that regardless of the form of p(τ), the probability of a morphogen molecule entering the target cell in any given small time window δt is simply βδt. This result holds regardless of the mechanism by which morphogen molecules go

from the source cell to the target cell, as the only effect such a mechanism can have is on p(τ)", then why can't we substitute a diffusive mechanism and deduce the same thing?

Our SDC model violates a key assumption of the argument depicted in the new Figure 2 by allowing molecules to leave the target cell after entering it. We have added additional text both in the description of the argument itself as well as the SDC section of our results to reinforce this.

5) The authors assume that once a morphogen molecule has reached the end of the cytoneme, it is immediately absorbed by the target cell. How realistic is this assumption and how important is it? Would relaxing it change the results qualitatively?

While it is difficult to speak to the realism of this assumption, as we are unaware of any experimental study that investigates the dynamics of molecules specifically at the ends of cytonemes, we can say that it is not particularly important. Relaxing this assumption would only alter the specifics of the transport mechanism, which as we argue with our new Figure 2, does not change the noise properties.

The molecule count in the target cells in the DT model would still behave identically to a birthdeath system. The only way relaxing this assumption would qualitatively change our results would be if it caused a significant alteration to the mean profile.

6) In comparing the precision of the two models, it would be good to discuss in a bit more detail how the comparison is made: which parameters are kept the same, which are optimized over, which are varied systematically, and why is this choice the most natural one? For example, the authors chose to study the precision ratio as a function of the gradient length, while optimizing over the DT shape φ. Βut we could also imagine a comparison in which the production and degradation rates are kept the same for both models (because production and degradation are energetically costly), but that over all other parameters the precision is optimized. Would that give a qualitatively different result? Why would the gradient range λ be of special importance (the comparison is made on the footing of equal gradient range)? Should not only the precision matter (such that it would be fine that the gradients in the DT and SDC model are different)? In other words, how do the maximal precisions that can be reached given reasonable constraints (such as production and degradation rates) compare, as a function of the distance to the source?

We have significantly altered our description of how we compare the two models to better clarify our process here. Specifically, we now discuss in much greater detail which parameters are equated between the two models and why. The reviewers are correct to point out that the production and degradation rates (β and ν) should be equated due to their energetic restrictions. While we now present this as our rationale for equating β, we also discuss why the explicit value of ν is not relevant so long as holds true. Thus, while equating β and ν between models is consistent with our procedure, it still leaves the profile lengthscale λˆ as a free parameter. Because the profile lengthscale is a frequently measured parameter across a variety of developmental systems, and our goal here is not only to understand the physical mechanisms behind developmental precision but also to quantitatively compare to data, we do not optimize over λˆ. Instead we imagine that the profile lengthscale is set by a functional objective of the developing system, such as where a boundary is to form, and we ask, given this constraint, which model has the higher precision.

7) And in making this comparison, does the SDC model exhibit an optimal diffusion constant that maximizes the precision? Understanding this is probably important, because when, in the current comparison, the SDC model is more precise, it is because of the higher protein clearance rate due to diffusion. Yet, a higher diffusion constant also lowers the steepness of the gradient, decreasing the precision,

Indeed, there is an optimal value of the diffusion constant (and thus λˆ) in the SDC model, for the precise reasons outlined by the referee. However, this optimal value depends on the choice of cell position: a smaller diffusion coefficient maximizes precision at nearby cells, whereas a larger diffusion coefficient maximizes precision at faraway cells. For the same reasons as outlined in the previous response, we do not presuppose the position at which the maximal precision is desired. Instead, we vary λˆ, which allows us to compare the two models systematically and then compare the results to the experimental data.

8) The key mechanism by which the SDC model can become superior is indeed diffusion. Could adding diffusion to the DT model improve the performance of the latter model? Is there evidence for diffusion in experimental systems that employ directed transport?

We are unaware of any experimental study into a possible hybrid system that incorporates both diffusion and directed transport. Nonetheless, we have added a brief comment on such a possibility in our Discussion section.

9) In DT systems, is it clear that molecule transport is independent? How could correlated transport be included in the DT model?

To our knowledge, there have not been detailed experimental studies into the correlation between dynamics of individual morphogen molecules within cytonemes. Introducing correlated transport would violate a key assumption of our argument presented in our new Figure 2 and would create nonlinear and most likely unsolvable noise terms. As such, we consider the investigation into correlated transport to be beyond the scope of this particular work.

10) Also, in the DT model could noise be reduced by cells measuring arrivals of particles rather than averaging over internal concentration of particles? This would presumably only change the resulting expression for precision by a numerical factor, but could this factor shift the balance in favor of DT in some cases?

Measuring the arrival rate of molecules in the DT model would at best reduce the variance by a factor of 2, as the noise from degradation would be eliminated but the noise from arrival would remain. While this would increase the precision of the DT model, it would not do so significantly enough, and our results remain almost entirely unchanged by its inclusion. We have added a footnote in the manuscript to address this possibility.

Morphogen	Organism	λ (µm)	a (µm)	N
Bicoid	Drosophila	100 ± 10	2.8	90
Fgf8	Zebrafish	197 ± 7	10	40
Lefty2	Zebrafish	150 ± 25	10	40
Lefty1	Zebrafish	115 ± 20	10	40
Dpp	Drosophila	20.2 ± 5.7	1.3	100
Dorsal	Drosophila	22.5 ± 5	2.8	35
Squint	Zebrafish	65 ± 10	10	40
Cyclops	Zebrafish	30 ± 5	10	40
Hh	Drosophila	8 ± 3	1.3	100
Wg	Drosophila	5.8 ± 2.04	1.3	100