We pose the task of morphogenetic decoding as a problem in local, cell autonomous inference of position from a morphogen input (Figure 1), where each cell acts as an information/inference channel with the following information flow:

1. ‘reading’ of the morphogen input by receptors on the cell surface,

2. ‘processing’ by various cellular mechanisms such as receptor trafficking, secondary messengers, feedback control, and

3. ‘inference’ of the cell’s position in the form of a transcriptional readout.

Figure 1

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At a phenomenological level, reading of the morphogen input is associated with the binding of the morphogen ligand to various receptors with varying degree of specificity, leading to the notion that the information channel describing positional inference must possess multiple branches. Furthermore, the multiple processing steps associated with compartmentalisation of cellular biochemistry and/or signal transduction modules, for example phosphorylation states, provide the motivation for invoking multiple tiers in the channel architecture. At an abstract level, one may think of the branch-tier architecture of the cellular processing as a bipartite Markovian network/graph Hartich et al., 2014, with a fast direction (involving multiple branches) consisting of ligand-bound and unbound states along with chemical state changes, and a slower direction (involving multiple tiers) consisting of intracellular transport, fission and fusion, characterised by energy-utilising processes or a flux imbalance. A general developmental context with multiple morphogens may involve several such bipartite Markov networks/graphs with different receptors (or branches) in parallel. Some of these receptors could be shared between different morphogens. We refer to signalling receptors as those which transduce a signal upon binding to their specific morphogen ligand and non-signalling receptors as those that participate in the signalling pathway without directly eliciting a signalling response. At the end of processing, each individual cell may pool information from the various branches for the final inference of position, i.e. a transcriptional readout (Figure 2).

Figure 2

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The task of achieving a precise inference is complicated by the noise in morphogen input arising from both production and transport processes, and by the stochasticity of the reading and processing steps; thus the inference must be robust to the extrinsic and intrinsic sources of noise. The use of feedback control mechanisms is a common strategy to bring about robustness in the context of morphogen gradient formation and sensing Averbukh et al., 2017. Motivated by this, in Section ‘Quantitative models for cellular reading and processing’ we consider different feedback controls in conjunction with the tiers and branches. With these three elements to the channel architecture, the task of morphogenetic decoding can be summarised in the following objective.

Figure 1 describes information processing during development across a two dimensional tissue of n_x, n_y cells in $x$ and $y$ directions, respectively. The direction of morphogen gradient is taken to be along $x$, with the morphogen source to the left of $x=0$. Each cell is endowed with a chemical reaction network (CRN) with the same multi-tiered, multi-branched architecture with feedbacks described previously, that reads a noisy input $L(x,y)$ (morphogen concentration) and produces an ‘output’ (biochemical ‘signal’) $\theta (x,y)$ that is also noisy. Here, we choose to construct the noisy morphogen profile in the following manner: for a given position $x\in [0,1]$, cells along the $y$-direction see different amounts of ligand coming from the same input distribution$p(L|x)$,

characterised parametrically by a mean ${\mu }_L(x)$ and standard deviation ${\sigma }_L(x)$. Experimental data can be fit to this distribution Equation 1 (or another distribution suitable for the specific experimental system) to obtain the parameters. Here, we consider an exponentially decaying mean ${\mu }_L$ and standard deviation ${\sigma }_L$.

Alternatively, one could choose a different parametrisation consistent with experimental observations for a morphogen profile with a monotonically decaying mean. The values of $A,\lambda$ chosen for our analysis are listed in Table 1. The corresponding output distribution $p(\theta |x)$ can be used to infer the cell’s position. Since we do not know the precise functional relationship between the output and inferred position, we invoke Bayes rule MacKay and Mac Kay, 2003, as in previous work Tkačik et al., 2015, to infer the cell’s position,

where $p(\theta )={\int }_0^{1}dxp(\theta |x)p(x)$ and $p(x)$ is the prior distribution which we take to be uniform over a tissue of unit length, $p(x)=1$. We quantify precision in the inference by the local inference error, ${\sigma }_X(x)$. For each position $x$, the inferred position $x^{\ast }$ of cells along the $y$-direction is taken to be the maximum a posteriori estimate,

where we use $\tilde{x}$ to differentiate from the true position $x$. From this, the local and average inference error can be computed.

where the average in Equation 6 is over cells in the $y$-direction. The logic behind this definition of the inference error is that development of the tissue relies on the precision in the inference of cells’ positions throughout the tissue. However, there may be tissue developmental contexts, where only the positions of certain regions or cell fate boundaries need to be specified with any precision (as in the case of short-range morphogen gradients like Nodal Liu et al., 2022). The definition of inference error may be readily extended to incorporate such specifications (see Section ‘Choice of objective function’).

We have been motivated to use the maximum a posteriori (MAP) estimate in Equation 5 by its successful use in previous studies in Drosophila embryo Dubuis et al., 2013; Petkova et al., 2019; Tkačik et al., 2015 and, more importantly, that it is a local estimate not requiring the computation of ${\displaystyle p(\theta )}$ (which is independent of ${\displaystyle x}$). We have checked that a different definition of the inference error, which does not use the MAP estimate and takes into account the entire distribution ${\displaystyle p(x^{\ast }|x)}$,

leads to the same qualitative results.

In order to calculate the probability of the inferred position given the output $p(x^{\ast }|\theta )$ and hence the inference error ${\overline{\sigma }}_X$, one needs to know the prior $p(x)$ and the input-output relation giving rise to the output distribution $p(\theta |x)$ in Equation 4. While a uniform prior may be justified by a homogeneous distribution of cells in the developing tissue at the stage considered, the input-output relation needs to be developed using a specific model based on the general channel design principles described previously. Thus, we will take each cell to be equipped with a chemical reaction network (CRN) that has up to two receptor types both of which bind the ligand on the cell surface but only one is signalling competent Hemalatha et al., 2016; Tabata and Takei, 2004. This latter aspect breaks the symmetry between the receptor types and hence the branches, a point that we will revisit in Section ‘Asymmetry in branched architecture: promiscuity of non-signalling receptors’. In multi-tier architectures, the bound states of both the receptors are internalised and shuttled through several compartments. The last compartment allows for a conjugation reaction between the two receptors (as in the case of Wingless and Dpp Hemalatha et al., 2016; Zhu et al., 2020). The signalling states, defined by all the bound states of the signalling receptor, contribute to the output. Within this schema, we consider control mechanisms on the surface receptor concentrations and in the chemical reactions downstream to binding on the surface (i.e. on internalisation, shuttling, conjugation, etc). We formulate the control on processing steps as a feedback/feedforward regulation from one of the signalling species in the CRN. On the other hand, the control of surface receptors is considered in the form of an open-loop control by allowing receptor profiles to vary within certain bounds, as described below. The key parameters are chemical rate parameters describing the rates of various reactions in the CRN, receptor parameters describing the receptor concentration profiles, feedback topology in the CRN that is a combination of actuator and rate under regulation, control parameters describing the strength and sensitivity of the feedback/feedforward. With these parameters specified, an input-output relation, calculated as a tier-wise weighted sum of all signalling states, can then be used to infer the cell’s position by Equation 4.

Cellular Reading via surface receptors

In the framework described previously, we consider the morphogen ligand as an external input to the receiving cells, outside the cellular information processing channel. The signal and noise of this external input are captured by the distribution Equation 1. This implicitly assumes that there is no feedback control from the output to the ligand input, that is no ‘sculpting’ of the morphogen ligand profile. We revisit this point in the Discussion. Given a distribution of the morphogen input, we address the local, cell autonomous morphogenetic decoding that allows the cells to tune their reading dynamically.

We subject the local, cellular reading to an open-loop control on total (ligand bound plus unbound) surface availability of the signalling $\psi$ and non-signalling $\varphi$ receptors. This implies that for each evaluation of inference error within the optimisation routine (see Section ‘Performance of the Channel Architectures’), the local surface receptor levels are held constant in time through a chemostat (see Appendix 1). In our analysis, we consider a family of monotonic (increasing or decreasing in $x$ and independent of $y$) receptor profiles, which for convenience we take to be of the Hill form (Figure 3), that is either

(8) ${\displaystyle \text{Monotonically increasing in }x:f_A(x)=A_0+\frac{A_1{x}^a}{A_2^a+x^a}\text{or}}$

(9) ${\displaystyle \text{Monotonically decreasing in }x:f_B(x)=B_0+\frac{B_1}{1+(x/B_2{)}^b}}$

The range of values for these parameters considered in the numerical analysis are listed in Table 1. Therefore, when considering $\psi (x)$ to be monotonically increasing in $x$, we parametrise it with $f_A$. It follows that in a one-branch channel, there are two possibilities: $\psi \in \{f_A,f_B\}$ while in a two-branch channel, there are a total of four possibilities: $(\psi ,\varphi )\in \{f_A,f_B\}\times \{f_A,f_B\}$. This allows us to simulate the ‘reading’ step performed by the cells (see Figure 1b).

Figure 3

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Note that we are not fixing a receptor profile but taking it from a class of monotonic profiles (including a uniform profile), over which we vary to determine the optimal inference (see Section ‘Performance of the Channel Architectures’ below). Further, in the optimisation scheme (Section ‘Performance of the Channel Architectures’), we allow the receptor concentrations to vary over the space of all monotonically increasing, decreasing or flat profiles, and do not encode the positional information in the receptor profiles. Monotonicity implicitly assumes a spatial correlation in the receptor concentrations across cells – we return to this point in the Discussion.

Dynamics of processing in a single-tier channel

In a single tier channel, all processing is restricted to the cell surface. We represent the bound state of the signalling receptor as $R^{(1)}$ and that of the non-signalling receptor as $S^{(1)}$. The conjugated state is represented by $Q^{(1)}$. The CRN for such a system with one and two branches is shown in Figure 4a. Rates associated with these reactions are listed in Table 1. The differential equations that describe the binding, unbinding, conjugation, splitting and degradation reactions of the receptors are given by

(10) ${\displaystyle {\mathrm{\partial }}_t{R}^{(1)}=r_{b}L(\psi (x)-R^{(1)})-(r_u+r_d)R^{(1)}-{\kappa }_C{R}^{(1)}{S}^{(1)}+{\kappa }_S{Q}^{(1)}}$

(11) ${\displaystyle {\mathrm{\partial }}_t{S}^{(1)}={\kappa }_{b}L(\varphi (x)-S^{(1)})-({\kappa }_u+{\kappa }_d)S^{(1)}-{\kappa }_C{R}^{(1)}{S}^{(1)}+{\kappa }_S{Q}^{(1)}}$

(12) ${\displaystyle {\mathrm{\partial }}_t{Q}^{(1)}={\kappa }_C{R}^{(1)}{S}^{(1)}-{\kappa }_S{Q}^{(1)}}$

The steady-state output $\theta$, defined as the sum of all the ligand-bound signalling states, is given by $\theta =R^{(1)}+Q^{(1)}$. Note that to describe the 1-branch system, we simply set all rates $\kappa$ to zero.

Figure 4

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Feedback Control

We consider all rates in the CRN, except the ligand binding and unbinding rates, as potentially under feedback regulation. Any chemical rate $r\in \{r_I,{\kappa }_I,{\kappa }_C,\mathrm{\dots }.\}$ that is under feedback control actuated by the node $R\in \{R^{(1)},S^{(1)},\mathrm{\dots }.\}$ is modelled as.

(18) ${\displaystyle r_{+}=r_0\left(1+\frac{\alpha R^n}{{\gamma }^{-n}+R^n}\right)\text{if under positive feedback}}$

(19) ${\displaystyle r_{-}=\frac{r_0}{1+(\gamma R{)}^n}\text{if under negative feedback}}$

with r₀ as the reference value of the chemical rate in the absence of feedback. The range of values for amplification $\alpha$, feedback sensitivity $\gamma$ and feedback strength $n$ are listed in Table 1. Figure 5 shows the different categories of possible feedback controls. We discuss the heuristics underlying the feedback controls in Appendix 2.

Figure 5

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With the model in place, we address the Objective discussed previously, by studying the performance of different channel architectures, i.e. number of tiers and branches, and feedback topology. We define a vector $\overrightarrow{v}$ belonging to a parameter space ${\displaystyle \mathcal{V}}$ of the channel parameters related to chemical rates, receptor profiles and feedback (see Table 1). While the chemical rates and feedback parameters are the same in all cells, the receptor profile parameters help define the receptor concentrations at each cell position $x,y$. For a given morphogen input distribution $p(L|x)$ and a channel architecture under consideration, the optimisation can be stated as

and implemented by the following algorithm, the details of which are presented in Appendix 3.

We begin with architectures comprising single-tiered channels with one and two branches. Such architectures are similar in design to the classic picture of ligand-receptor kinetics Lauffenburger and Linderman, 1996; Alberts et al., 2017, but also to the self-enhanced degradation models for robustness of morphogen gradients Eldar et al., 2003. Before we proceed, it helps to recall a simple heuristic regarding signal discrimination. Appendix 4—figure 1 illustrates that precision in positional inference requires both that the output variance at a given position be small and that the mean output at two neighbouring positions be sufficiently different.

Let us first consider a minimal architecture of a one-tier one-branch channel without feedback control on any of the reaction rates. The output of this channel, here $R^{(1)}$, is a monotonic, saturating function of the input, with the surface receptor concentration setting the asymptote. As in Appendix 5—figure 1a, if the receptor concentrations decrease with mean ligand input, i.e. increases with distance from source ($f_A$ in Figure 3a; ), the outputs for different input ranges overlap significantly. On the other hand, if the receptor concentrations increase with mean input ($f_B$ in Figure 3b), the outputs overlap to a lesser degree (see Appendix 5—figure 1b). Thus within this minimal architecture, the inference error is optimised when the receptor concentrations increase with the mean input.

Introducing a feedback in this one-tier one-branch architecture, either on receptor levels or degradation rate, only partially reduces the inference errors (Figure 6a, c). As seen in Figure 6d, this is because the surface receptor concentration $\psi$ sets both the asymptote and the steepness of the input-output functions, resulting in significant overlaps between outputs at neighbouring positions. The receptor control introduces a competition between robustness of the output to input noise and sensitivity to systematic changes in the mean input (see Appendix 4).

Figure 6

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Including a non-signalling receptor $\varphi$ via an additional branch in the channel architecture opens up several new possibilities of feedback controls, in addition to providing an extra tuning variable. Now, as opposed to the one-tier one-branch case, an inter-branch feedback control (Figure 7a) results in an input-output relation with a sharp rise followed by a saturation (Figure 7d). By appropriately placing the receptors at spatial locations that receive different input, as shown by black arrow in Figure 7d, one can cleanly separate out the cellular outputs in neighbouring positions. For a detailed description see Appendix 6. This mitigates the above-mentioned tension between robustness to input noise and sensitivity to systematic changes in the mean input to a considerable extent (see Appendix 4—figure 2).

Figure 7

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As seen in Figure 7c, the two-branch architecture with inter-branch feedback leads to a dramatic reduction in the inference errors, to reach one cell’s width precision at most spatial locations in the tissue.

We would like to highlight two unexpected features of the optimised two-branch architecture. (i) The signalling and non-signalling receptors present opposing optimal profiles – a consequence of the negative inter-branch feedback. (ii) The optimal non-signalling receptor decreases away from the source, indicating that the non-signalling receptor ‘reads’ the ligand input, while the signalling receptor increases away from the source, buffering the noise in the output (Figure 7). A heuristic understanding of the opposing optimal receptor profiles is provided in Appendix 7. In contrast, in the one-branch architectures, it is the signalling receptor that does the reading and buffering.

We next investigate the effects of addition of tiers (compartments) on the inference errors. Our optimisation shows there are two distinct optimised two-tier two-branch architectures, one with inter-branch feedback on the internalisation rate of the non-signalling receptors ${\kappa }_I$ and the other on the conjugation rate ${\kappa }_C$, that have comparable inference errors (Figure 8b, c). Both the receptor profiles and the input-output relations of these two optimised two-tier two-branch channels are qualitatively similar (Appendix 8—figure 1).

Figure 8

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It would seem that addition of further tiers, that is more than two, would lead to further improvement in the inference. However, in both these optimised architectures, addition of tiers leads only to a marginal reduction of inference errors (Figure 8a) while invoking a cellular cost. Of course, extensions of our model that involve modification of the desired output could favour the addition of more tiers. For instance, additional tiers could facilitate signal amplification or improvement in robustness to input noise through an increase in signal-to-noise ratio (SNR) Stoeger et al., 2016. Further, by making the output $\theta$ a multi-variate function of the tier index (compartment identity) one can multitask the various cellular outcomes (as in Ras/MAPK signalling Fehrenbacher et al., 2009 or with GPCR compartmentalisation Ellisdon and Halls, 2016).

So far, we have only considered noise due to fluctuations in the morphogen profile, that is extrinsic noise. Given that we are considering a distributed channel, intrinsic noise due to low copy numbers of the reacting species in the CRN will have a significant influence on the inference. As discussed in Section ‘Conceptual framework and quantitative models’ and Appendix 3, we solve the stochastic chemical master equations (CMEs) to compute the output distributions and the positional inference. It is here that we find that the addition of tiers contribute significantly to reducing inference errors. A comparison of the one-tier two-branch and two-tier two-branch channel architectures (Figure 9a and b) optimised for extrinsic noise, shows that in the presence of intrinsic noise, additional tiers lead to significantly lower inference errors (Figure 9c). The large inference errors seen in the one-tier one-branch channel in the presence of intrinsic noise, can be traced to the instabilities of steady-state trajectories of the two signalling species $R^{(1)}$ and $Q^{(1)}$ driven by the non-linear feedback (Figure 9d–f). This effect is more prominent for larger values of ligand concentrations, that is closer to the source at $x=0$. On the other hand, we find that in the two-tier two-branch architecture (Figure 9g–i), the fluctuations in the signalling species are more tempered, the inter-branch feedback leads to a mutual damping of the fluctuations of the signalling species from the two branches. Details of this heuristic argument appear in Appendix 9.

Figure 9

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In summary, we find that the nature of the channel architectures play a significant role in robustness of morphogenetic decoding to both extrinsic and intrinsic sources of noise. Of the three elements to the channel architecture - branches, tiers, and feedback control, we find that a branched architecture can significantly reduce inference errors by employing an inter-branch feedback and a control on its local receptor concentrations. For this, the receptor concentration profiles required to minimise inference errors are such that the concentration of signalling (non-signalling) receptor should decrease (increase) with mean morphogen input. Crucially, in the absence of feedback, performance of the channel diminishes and the optimised receptor profiles both decrease away from the source (Appendix 10—figure 1). Further, we show in Appendix 11—figure 1 that having uniform profiles for the signalling and non-signalling receptors, with or without uncorrelated noise, fares poorly in terms of inference capability. This provides a posteriori justification for the monotonicity in receptor profiles. Addition of tiers can help in further bringing down inference errors due to extrinsic noise, but with diminishing returns. An additional tier, however, does provide a buffering role for feedback when dealing with intrinsic noise. We note that these qualitative conclusions remain unaltered for different morphogen input characteristics, that is input noise and morphogen decay lengths (see Appendix 12).

Before comparing the theoretical results with experiments, we comment on the implications for the cellular control of the signalling $\psi$ and non-signalling $\varphi$ receptors. In the two-branch architecture, the symmetry between the signalling and non-signalling receptors is broken by the inter-branch feedback and the definition of output $\theta$, the latter taken to be a function only of the signalling states $R^{(k)}$ and $Q^{(k)}$ (Section ‘Conceptual framework and quantitative models’, purple nodes in Figure 7a and Figure 8b and c). What are the phenotypic implications of this asymmetry? In Appendix 13—figure 1, we plot the contours of average inference errors ${\overline{\sigma }}_X$ in the $\psi -\varphi$ plane around the optimal point. We compute the eigenvalues of the local curvature of ${\overline{\sigma }}_X(\mathrm{\Delta }\psi ,\mathrm{\Delta }\varphi )$ around the optimal point ($\mathrm{\Delta }\psi =\mathrm{\Delta }\varphi =0$). The difference in the magnitudes of these eigenvalues, as discussed in Appendix 13, immediately describes stiff and sloppy directions Transtrum et al., 2015 along the $\psi$ and $\varphi$ axes, respectively. This implies that while the signalling receptor is under tight cellular control, the control on the non-signalling receptor is allowed to be sloppy. A similar feature is observed in the contour plots for the robustness measure $\chi$ (defined as the ratio of coefficients of variation in the output to that in the input). Appendix 4—figure 3 shows that for any given input distribution, reduction in output variance requires a stricter control on $\psi$, while the control on $\varphi$ can be lax.

This sloppiness in the levels of non-signalling receptor would manifest at a phenotypic level in the context of multiple morphogen inputs as in the case of Drosophila imaginal disc Tabata and Takei, 2004. Participation of the same non-signalling receptor in the different signalling networks would imply its promiscuous interactions with all ligands. The signalling receptors, therefore, are specific for the various ligands while the non-signalling receptor, being promiscuous, is non-specific. This, as we see below, is the case with the Heparan sulfate proteoglycans (HSPGs) such as Dally and Dally-like protein (Dlp) that participate in the Wingless (Wg) and Decapentaplegic (Dpp) signalling networks Lin and Perrimon, 2000; Romanova-Michaelides et al., 2021.

The above section and Appendix 13 motivate us to study the changes in the inference error upon perturbations of all the channel parameters. We therefore discuss the nature of optima in terms of the local geometry of the fidelity landscape around the optimum, and the geometry of the low inference error states. We work with the case of the optimised one-tier two-branch channel (shown in Figure 7a with optimum channel parameters listed in Table 2, Table 3) in presence of extrinsic noise.

To address the geometry of the local fidelity landscape around the optimum, we compute (i) percent changes in inference error ${\overline{\sigma }}_X$ due to perturbations in channel parameters (Figure 10a), and (ii) the eigenspectrum of the Fisher information metric (FIM, Figure 10b). The FIM $g_{\mu \nu }$ is evaluated in the log-parameter space as Transtrum et al., 2015.

where, ${\displaystyle \overrightarrow{v}\in \mathcal{V}}$ is the channel parameter vector, and $x_i,y_j$ are the indices of cells that run along the $x$- and $y$-directions. As shown in Figure 10a, we see that the inference error does not change significantly (up to 20% change with most parameters), that is it remains within ${\overline{\sigma }}_X\le 2.2\%$. Varying the feedback strength $n$, however, drives a much stronger deviation from the minimum. Similarly, as seen from the heat map (Figure 10b), eigenvectors with the larger eigenvalues (index 1–6) have an appreciable component of the feedback parameters $\gamma ,n$. This implies that variation of the feedback parameters from the optimum would result in significant changes in the inferred positions. Perturbing conjugation ${\kappa }_C$ and splitting ${\kappa }_S$ rates simultaneously (see eigenvector 16) does not produce any notable change to the inferred positions (eigenvalue $\sim {10}^{-13}$). Further, perturbations to channel parameters other than the feedback parameters (eigenvectors 7–16) produce marginal changes in inferred positions.

Figure 10

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Moving now from a local to global analysis of the fidelity landscape, we run the optimisation algorithm (Section ‘Performance of the Channel Architectures’) on the one-tier two-branch channel architecture with 2¹⁶ space-filling initial points in the 16-dimensional parameter space of this architecture. We then define the low inference error states as those channel parameters ${\overrightarrow{v}}^{\text{opt}}$ that yield ${\overline{\sigma }}_X\le 2\%$. This cutoff, which equals $\lceil \frac{{\overline{\sigma }}_X}{0.01}\rceil$, corresponds to declaring as equivalent all the inference errors ${\overline{\sigma }}_X$ that lie between one and two cells’ widths. Consistent with the local analyses, we find that the frequency distribution of optimal feedback parameters $\gamma ,n$ is narrowly distributed about the global optimum (Figure 11a). As shown in Figure 11a, the parameters corresponding to forward and backward rates are skewed towards the upper and lower bounds of the allowed parameter range, respectively. We see that the optimal binding rates in the non-signalling branch (Figure 11a) are more broadly distributed across the permissible range than the optimal binding rates in the signalling branch, which are concentrated towards the upper bound of the permissible range. This again reflects the promiscuity of the non-signalling receptors as described in Section ‘Asymmetry in branched architecture: promiscuity of non-signalling receptors’. All other optimal parameters corresponding to degradation rates, minimum and maximum receptor values and steepness of the receptor profiles, show a very broad spread over this range (Appendix 14—figure 1). To explore the topography of the low inference error landscape, we evaluate the components of the ‘position vectors’ of these minima ${\overrightarrow{v}}^{\text{opt}}$ in the parameter space ${\displaystyle \mathcal{V}}$ along the eigenvectors of the Hessian of ${\overline{\sigma }}_X$, defined as

where $M$ stands for the entire morphogen profile and we have assumed a Euclidean metric. As shown in Figure 11b and c, components of the ‘position vector’ of the minima ${\displaystyle {\overrightarrow{v}}^{\text{opt}}\in \mathcal{V}}$ lie predominantly along the sloppy directions of the Hessian that is along the eigenvectors with small eigenvalues. This suggests that geometry of the low inference error landscape resembles a deep valley, which is shallow along the several sloppy directions and steep along the few stiff directions.

Figure 11

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The objective function as defined in Equation 7 gave equal weight to inference errors at all positions $x$ along the tissue, driving the inference error to reduce at all positions simultaneously. In certain developmental contexts, the objective could be to partition the tissue into cell identity segments (reviewed in Briscoe and Small, 2015). In such a case, the partition boundaries would need to be sharp Gregor et al., 2007a that is only the errors at the segment boundaries would need to be minimised. We show that even with this choice of objective function, the qualitative results for the optimal channel architectures remain unaltered. We define the inference error for a tissue with $N_p$ segmented cell identities as.

where $\delta$ and $\mathrm{\Theta }$ denote the Kronecker-delta and Heaviside-theta functions respectively, ${\xi }_i$ is the position of the boundary between the i-th and (i+1)-th segments, and $g$ is a function that maps position (actual or inferred) to a segment, that is $g:[0,1]\to \{1,2,\mathrm{\dots },N_p\}$ with $N_p$ as the total number of segments.

We optimise one-tier one-branch, one-tier two-branch and two-tier two-branch channel architectures for the inference error as defined in Equation 23 with $N_p=4$ and equally spaced boundaries located at positions ${\xi }_1=0.25,{\xi }_2=0.5,{\xi }_3=0.75$ along the $x$-axis. As before, this optimisation suggests that an additional branch aids in reducing the inference errors due to extrinsic noise (compare Figure 12b and d), with similar opposing receptor profiles as in Section ‘Branched architecture with multiple receptors provides accuracy and robustness to extrinsic noise’. Tiers play only a moderate role in reducing the inference errors further in a two-branch channel (compare Figure 12d and f). However, just as with the previous objective function, an additional tier provides substantial robustness to intrinsic noise as shown in Figure 13c.

Figure 12

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Figure 13

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The phenomenology of the morphogen reading and processing of Wg in the wing imaginal disc of Drosophila melanogaster Hemalatha et al., 2016 suggests a one-to-one mapping to the two-tier two-branch channel defined above, thus providing an ideal experimental system for a realisation of the ideas presented here (Figure 14a, b).

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Figure 14—source data 1

CV of intensity measurements of CAAX GFP and endocytosed Wg, in control and myr-Garz-DN, as a function of distance from producing cells in individual samples of wing imaginal discs.

https://cdn.elifesciences.org/articles/79257/elife-79257-fig14-data1-v1.xlsx

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Wingless (Wg) is secreted by a line of cells (1–3 cells) at the dorso-ventral boundary and forms a concentration gradient across the receiving cells Neumann and Cohen, 1997. Receiving cells closer to the production domain show higher Wg signalling while those farther away have lower Wg signalling Neumann and Cohen, 1997. Several cell autonomous factors influence reading and processing of the morphogen Wg in the receiving cells. Binding of Wg to its signalling receptor, Frizzled-2 (DFz2), initiates signal transduction pathway and nuclear translocation of $\beta$-catenin which further results in activation of Wg target genes (reviewed in Clevers and Nusse, 2012). In addition to the signalling receptor, binding receptors such as Heparin Sulphate Proteoglycans (HSPGs) – Dally and Dlp also contribute to Wg signalling Baeg et al., 2001; Franch-Marro et al., 2005. Further, the two receptors follow distinct endocytic pathways Hemalatha et al., 2016: while, DFz2 enters cells via the Clathrin Mediated Endocytic pathway (CME), Wg also enters cells independent of DFz2, possibly by binding to HSPGs, through CLIC/GEEC (CG) endocytic pathway. The two types of vesicles, containing Wg bound to different receptors, merge in common early endosomes Hemalatha et al., 2016. However, only DFz2 receptors in their Wg-bound state, both at the cell surface and early endosomes, are capable of generating a downstream signal leading to positional inference through a transcriptional readout Tsuda et al., 1999. This phenomenology is faithfully recapitulated in our two-tier two-branch channel architecture (Figure 14b) in which DFz2 and HSPG receptors play the role of the two branches. The conjugated state ‘Q’ represents a combination of the readings from the two branches, possibly realised by the co-receptors HSPGs that bind Kirkpatrick et al., 2006; Capurro et al., 2008 and present Hemalatha et al., 2016 diffusible ligands to signalling receptors (either on the cell surface or within endosomes).

Since an experimental measurement of positional inference error poses difficulties, we measure the cell-to-cell variation in the signalling output for a given position $x$ as a proxy for inference error (Appendix 4—figure 3). Larger the variation, higher is the inference error. This is calculated as coefficient of variation (CV, Appendix 15) in the output across cells in the $y$-direction (Figure 14c).

Let us first discuss the results from the theoretical analysis. The optimised two-tier two-branch channel (Figure 14b) shows that the magnitude and the fluctuations in the coefficients of variation are small, with a slight increase with position (blue, Figure 14f). This is consistent with the low inference error associated with the optimised channel (Figure 8b). Upon perturbing this channel via removal of the non-signalling branch, the magnitude and fluctuations in the signalling output variation increases significantly (orange, Figure 14e). This qualitative feature of the coefficient of variation in the optimised two -tier two-branch channel is replicated in the Wg measurements of wild type cells.

In the experiments, we first established the method by determining the CV of a uniformly distributed signal, CAAX-GFP (expressed using ubiquitin promoter), and observed that the CV of CAAX-GFP is relatively uniform in $x$, the distance from Wg producing cells (Figure 14d). In order to study the steady state distribution of Wg within a cell and within the endosomes, we performed a long endocytic pulse (1 hr) with fluorescently labelled antibody against Wg Hemalatha et al., 2016; Prabhakara et al., 2022. Following this, we estimated the CV of the Wg endocytic profile as a function of $x$ (Figure 14f, and Figure 14—figure supplement 1).

We assessed the CV of endocytosed Wg under two conditions: one, where the endocytic pulse of Wg is captured by the two branches and two tiers (control condition), and another, where we disengage one of the tiers by inhibiting the second endocytic pathway using a genetically expressed dominant negative mutant of Garz, a key player in the CG endocytic pathway Gupta et al., 2009. This perturbation has little or no effect on the functioning of the CME or the levels of the surface receptors that are responsible for Wg endocytosis (Hemalatha et al., 2016; Prabhakara et al., 2022). As predicted by the theory (Figure 14e), CV in the control shows a slight increase with position (Figure 14f) with fluctuations about the mean profile being small. In the perturbed condition, with the CG endocytic pathway disengaged, we find the CV shows a steeper increase with $x$ and has larger fluctuations about the mean profile.

In principle, the coefficient of variation of the output is affected by all the microscopic stochastic processes that intersect with Wg signalling network in the wing imaginal disc and in the ligand input. Therefore, one has to be careful about interpreting the changes in the coefficient of variation of the output, based on the such perturbation experiments. Notwithstanding, this qualitative agreement between theory and experiment is encouraging.

Taking an information theoretic and systems biology approach, we have addressed the issue of accurate and robust morphogenetic decoding of position. For convenience, we summarise our key results in a point-wise manner:

1. The main result is that given a noisy morphogen input, cells in a developing tissue can achieve low inference error of their positions by deploying a more elaborate multi-branch multi-tier channel architecture with feedback control. This ensures a separation between the reading of the morphogen input and buffering against noise.

2. Having a combination of signalling and non-signalling receptors in the channel can significantly improve the performance of cells in their positional decoding.

3. For a monotonically decaying morphogen input, the signalling and non-signalling receptors exhibit spatially varying profiles with the signalling receptor increasing away from the source and the non-signalling receptor decreasing away from the source. This implies that the non-signalling receptor ‘reads’ the morphogen input, while the signalling receptor buffers against noise.

4. The performance of the multi-tier multi-branch channels is enhanced by having a feedback from the signalling branch to the non-signalling branch. Along with control on the levels of signalling receptor, this inter-branch feedback provides buffering against extrinsic noise.

5. Having a multi-tier architecture (cellular compartmentalisation) tempers the effects of intrinsic noise in the channel by stabilising fluctuations of the output at steady-state.

6. The optimisation shows that the characteristics of the signalling receptor are tightly controlled, whereas those of the non-signalling receptor are flexible. This implies that the signalling receptor is specific whereas the non-signalling receptor is promiscuous.

7. Analysis of the geometry of the fidelity landscape reveals that the channel parameters corresponding to feedback, binding rates and profile of the signalling receptor are stiff, while the rest of the channel parameters are sloppy (elaborated in Section ‘Geometry of the inference error landscape: implications for control’).

8. The efficacy of inter-branch feedback control is enabled by having a conjugated state corresponding to a confluence of the signalling and non-signalling branches in a common compartment.

9. Our analysis demonstrates how local, cell autonomous control can facilitate the optimisation of a tissue-level task, here morphogenetic decoding of cellular position.

Our theoretical predictions are compared with experimental observations from Wg morphogen system of Drosophila wing imaginal disc. We first show that Wg signalling in the experimental system is equivalent to a two-tier two-branch channel. In the experiments, we use signal-to-noise ratio (SNR) of the output as a proxy for robustness of inference. Perturbation of the architecture, i.e. removal of the non-signalling branch, results in reduction of SNR. In a forthcoming manuscript, we will provide a detailed verification of the predicted opposing receptor profiles.

We have explored the local geometry of the fidelity landscape around the optimum, and the global geometry of the low inference error states, by perturbing channel parameters and concentration profiles of the receptors.

The local geometry of the fidelity landscape is studied using the Fisher information metric. This shows that steepest variation in the inference error comes from moving along the feedback parameters while perturbations to other channel parameters produces only marginal changes. Further, we explore the global geometry using the spectrum of the Hessian of the inference error. We find that the topography of the low inference error landscape resembles a ravine or a deep valley, which is shallow along the several sloppy directions and steep along the few stiff directions, the latter being predominantly along the feedback parameters. This dimensional reduction appears to be a recurring feature of such high-dimensional optimisation Transtrum et al., 2015; Yadav et al., 2022.

Such a geometrical approach also provides insight on the differences between the signalling and the non-signalling receptors, which shows up in the extent to which they influence inference errors in the neighbourhood of the optimum. Slight changes in the signalling receptor away from the optimum lead to a sharp increase in inference error while similar changes in the non-signalling receptor do not affect the inference errors significantly. This gives rise to the notion of stiff and sloppy directions of control - with non-signalling receptor placed under sloppy control. In a context with multiple morphogen ligands setting up the different coordinate axes (e.g. Wg, Dpp and Hh in imaginal discs Lin, 2004; Lin and Perrimon, 2000), the non-specific receptor can potentially facilitate cross-talks between them. A sloppy control on non-specific receptor would allow for accommodation of robustness in the outcomes of the different morphogens. This could potentially be tested in experiments.

We end our discussion with a list of tasks that we would like to take up in the future. First, the information processing framework established here is very general. Obvious extensions of our models, such as adding more branches, tiers and chemical states, will not lead to qualitatively new features. However, one may alter the objective function – for instance, in the case of short range morphogens like Nodal Liu et al., 2022, only the positions of certain regions (closer to the morphogen source) or cell fate boundaries need to be specified with any precision. To this end, we have analysed another objective function which partitions the tissue into cell identity segments. The qualitative features of the optimised channel architectures remain unaltered. Depending on the developmental context, one might explore other objective functions. This would be a task for a future investigation.

Next, our optimisation study ignores cellular costs due to compartmentalisation, additional receptors and implementation of feedback controls, and thus possible trade-offs between cellular economy and precision in inference. Nevertheless, the observation that addition of extra tiers beyond two provides only marginal improvements to inference, already suggests a balance between precision and cellular costs.

Third, our theoretical result that the optimised surface receptor profiles are either monotonically increasing or decreasing from the morphogen source, suggests that the surface receptor concentrations are spatially correlated across cells. Such correlations could have a mechanochemical basis, either via cell surface tension that could in turn affect internalisation rates Thottacherry et al., 2018 or inter-cellular communication through cell junction proteins Garcia et al., 2018 or from adaptive feedback mechanisms between the output and receptor concentrations Barkai and Leibler, 1997. We emphasize that in the current optimisation scheme, we have allowed the receptor concentrations to vary over the space of all monotonically increasing, decreasing or flat profiles, and have not encoded the positional information in the receptor profiles.

Finally, we have considered the morphogen ligand as an external input to the receiving cells, outside the cellular information processing channel. There is no feedback from the output to the receptors and thus no ‘sculpting’ of the morphogen ligand profile. Morphogen ligand profiles (e.g. Dpp Romanova-Michaelides et al., 2021) are set by the dynamics of morphogen production at the source, diffusion via transcytosis and luminal transport, and degradation via internalisation. These cellular processes are common to both the reading and processing modules in our channel architecture. This would suggest a dynamical coupling and feedback between reading and ligand internalisation, which naturally introduces closed-loop controls on the surface receptors and a concomitant sculpting of the morphogen profile.

The cellular processes involved in a general two-branch channel architecture, described in Section "Quantitative models for cellular reading and processing" and Figure 4, with any number of tiers $n_T$ are: production of receptors, binding-unbinding of ligand with the receptors, intracellular transport between cellular compartments, conjugation and splitting in the final tier, and degradation of the chemical species involved. The mass action kinetics for these cellular processes are written as the following set of first-order ordinary differential equations (ODEs):

$R_j^{(k)}$ denotes species in the first branch, associated with the signalling receptor, in tier $k$ and in bound ($j=1$) or unbound ($j=0$) states. Likewise, $S_j^{(k)}$ denotes the same for the non-signalling receptor. $Q^{(n_T)}$ denotes the conjugated species in the final tier. Chemical rates related to the first branch are denoted by $r$ while those in the second branch are denoted by $\kappa$. The processes of production, binding, unbinding, internalisation, recycling, conjugation, splitting and degradation are represented by the subscripts $p,b,u,I,R,C,S,d$, respectively (Table 1). The superscript on degradation rate indicates the index of the tier in which the degradation takes place. The superscript over internalisation rate stands for the tier-index of the species internalised. Superscript over the recycling rate follows that of the corresponding internalisation rate.

As described in Section ‘Quantitative models for cellular reading and processing’, we consider local cell-autonomous control on the total cell surface receptor concentrations, in a manner akin to the open-loop control Stengel, 1994, such that the sum of free and bound signalling receptors is $\psi (x)$ and that of the non-signalling receptor is $\varphi (x)$. This implies that the open-loop control actuated on the production (or secretion) of the cell surface receptors balances the net flux of receptors away from the surface. Consider the dynamics of $R_0^{(1)}$ and $R_1^{(1)}$ in Eq. A, with now an open-loop control realised through $r_p(t)$ on $R_0^{(1)}$.

At any position $x$, if the total receptor concentration $R_0^{(1)}(x,t)+R_1^{(1)}(x,t)=\psi (x)$ are to stay constant in time (through a chemostat), $r_p(t)$ must satisfy

We emphasize that here we do not consider a closed-loop version of the receptor control. However, it can be realised through an integral control Goodwin et al., 2001, also known as perfect adaptation Barkai and Leibler, 1997. With open-loop-like control on total surface receptor concentrations (Section ‘Quantitative models for cellular reading and processing’), $R_0^{(1)}$ and $S_0^{(1)}$ are replaced by $\psi -R_1^{(1)}$ and $\varphi -S_1^{(1)}$, respectively. We then drop the subscripts from $R_1^{(k)}$ and $S_1^{(k)}$ to simplify notation.

To keep the notation simple, we avoid explicitly indicating superscripts over chemical rate symbols when presenting the dynamical equations in the main text (Equations 10–17). In the numerical analysis, however, the degradation, internalisation and recycling rates of different tiers are treated separately. The steady-state solutions of the above equations, without feedback controls on chemical rates, are given by,

One-tier channel:

Two-tier channel:

For channels with ${\displaystyle n_T>2}$, the solutions are written in the following recursive form Channel with $n_T$ tiers:

where ${\tilde{r}}_d^{(k)}$ are in turn evaluated recursively as

The general form of Equations 26–28 holds upon introduction of inter-branch feedbacks (refer Figure 5) and therefore the set of ODEs (Equation 25) can be solved analytically by appropriately replacing the rates under feedback with the Hill-form functions discussed in Section ‘Quantitative models for cellular reading and processing’. Cases with other types of feedbacks need to be solved numerically (Appendix 3).

In writing these dynamical equations, we have made two assumptions, simply as a matter of convenience. One may note that only the ligand bound states of receptors are transported between tiers. This is done with the consideration that residence times of unbound receptors within a compartment, other than the cell surface (first tier), is very small that is receptors in enclosed compartments re-bind the ligand quickly after unbinding due to small volume of the compartment. Further, the open loop control on receptors need not be strictly on surface receptors (tier 1). In principle, the control could be on the production rate of receptors. This does not change the solution in any substantial manner. For instance, in the case of two tiers, the solution would be

and the rest of the species would follow hierarchically as in the case with surface receptor control (see Equation 28). As long as the deviations in the denominator,$\frac{r_d{r}_I}{r_R+r_d}$ and $\frac{{\kappa }_d{\kappa }_I}{{\kappa }_R+{\kappa }_d}$ remain small compared to r_u, and $\psi \sim \frac{r_p}{r_d^{(1)}},\varphi \sim \frac{{\kappa }_p}{{\kappa }_d^{(1)}}$, the steady solutions for the two cases (control on surface receptors versus total receptors) will not differ significantly.

The figure below illustrates the effect of feedback parameters in Equations 18; 19 discussed under Section ‘Quantitative models for cellular reading and processing’.

Appendix 2—figure 1

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Considering the choices of chemical rate under control $r\in \{r_I,r_R,r_d,{\kappa }_I,{\kappa }_R,{\kappa }_d\}$ and the actuating nodes $R\in \{R^{(1)},R^{(2)},\mathrm{\dots }\}$, there are a large number of possible feedbacks for any given channel architecture. However, categorising the different feedbacks according to their effect can reduce the redundancies and aid numerical analysis. The steady-state output of a channel without the feedback controls on chemical rates is instructive in this categorisation. For instance, consider the output $\theta$ for a two-tier two-branch channel with $n_T=2$.

This form of the output suggests that feedbacks from $R^{(1)}$ (term outside the bracket) on any of the rates in $\frac{r_I}{r_R+r_d}$ and $\frac{{\kappa }_C}{{\kappa }_S}\frac{{\kappa }_I}{{\kappa }_R+{\kappa }_d}$ could cross-correlate $R^{(1)}$ with the other signalling species $R^{(2)}$ and $Q^{(2)}$. Thus, a negative feedback on $r_I$ (${\kappa }_I$) would be qualitatively equivalent to a positive feedback on $r_R$ (${\kappa }_R$). Beside helping to reduce the number of CRNs that need to be considered, this observation also allows one to forego the numerical analysis of a CRN with feedback on $r_R$ from $R^{(1)}$ that may have non-unique solutions. Similarly, feedbacks from different but related actuators could be interchanged. For example, the feedback on ${\kappa }_I$ from $R^{(2)}$, i.e. ${\kappa }_I\to \frac{{\kappa }_I}{1+{({\gamma }^{\prime }{R}^{(2)})}^n}$, could be written in terms of the feedback on ${\kappa }_I$ from $R^{(1)}$, i.e. ${\kappa }_I\to \frac{{\kappa }_I}{1+{(\gamma R^{(1)})}^n}$, by a simple transformation on feedback sensitivity ${\gamma }^{\prime }=\frac{r_I}{r_R+r_d}\gamma$. Note that Equation 31 also indicates the preferred direction of feedbacks, that is with the signalling receptors as actuators.

Optimisation over the channel parameter vector ${\displaystyle \overrightarrow{v}\in \mathcal{V}}$, belonging to a parameter space ${\displaystyle \mathcal{V}}$, is highly non-convex (see Section ‘Performance of the Channel Architectures’). Moreover, the derivative of ${\overline{\sigma }}_X$ with respect to $\overrightarrow{v}$ is ill-defined due to finite sampling of input distributions. With these considerations, we chose a derivative-free (gradient independent) optimisation algorithm viz. Pattern Search algorithm Hooke and Jeeves, 1961. The implementation of this algorithm in MATLAB can be found as patternsearch function under the Global Optimization Toolbox.

With this pattern search algorithm, we perform the optimisation routine as described in Section ‘Performance of the Channel Architectures’ in two rounds. First, with a certain number of initial points $n_{init}=32$ in the parameter space ${\displaystyle \mathcal{V}}$, we determine the advantageous feedback topologies (CRNs) that is, that give lower inference errors. In the next round, we go through the same optimisation routine for the advantageous feedback topologies determined through the the first round, but now with $n_{init}=320$ to converge to the true global minimum. Using parallel computing clusters, the cost of this computation in terms of real running time can be brought down to 12–48 hr.

Ideally, this optimisation ought to be done with extrinsic and intrinsic noises considered together. However, we restrict optimisation to the case of extrinsic noise as the computational cost associated with steady-state solutions of chemical master equations (CMEs), in the case of intrinsic noise, is rather high. Therefore, we evaluate the inference error due to intrinsic noise only of those CRNs that were optimised in the context of extrinsic noise previously. For this, we solve the CMEs using adaptive explicit-implicit tau leaping algorithm Sandmann, 2009 to determine the steady-state outputs and thus inference error. Implementation of this algorithm requires the definition of some numerical parameters that (i) help switch between Gillespie and implicit/explicit tau-leaping algorithms ($n_a,n_d$), (ii) decide the number of reactions when Gillespie algorithm is selected (n_b), and (iii) various threshold ($n_c,ϵ$). The values chosen for these are $n_a=10$, $n_b=10$ and $n_b=100$ if implicit tau-leaping algorithm was used in the previous step, $n_c=10$, $n_d=100$, $\delta =0.05$ and $ϵ=0.1$.

Precision in positional inference requires both that the output variance at a given position be small (i.e. the output is robust to the noise in input) and that the mean output at two neighbouring positions be sufficiently different (the output is sensitive to systematic changes in mean input).

Appendix 4—figure 1

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With this heuristic understanding of an ideal output, we define two local measures: (i) $\chi$ measures the noise in the output of a cell cohort as compared to the noise in the received input. We define it as the ratio of coefficient of variation in the output $\theta$ to the coefficient of variation in the input $L$ for the same cohort of cells.

Thus, robustness of the output increases with decreasing values of $\chi$; (ii) $\xi$ measures the sensitivity of the output to a systematic change in the input. We define it as the ratio of relative change in the mean output $⟨\theta ⟩$ to the relative change in the mean input ${\mu }_L$ for the same cell cohort.

The angular brackets in the above equation denote averaging over cells belonging to the same cohort. Thus, higher sensitivity implies higher values of $|\xi |$. Precision in positional inference, in terms of these two local measures, implies simultaneous minimisation of $\chi$ and ${|\xi |}^{-1}$.

We calculate $\chi$ and $\xi$ at all positions $x$ in the various optimised channels. We find that addition of a branch allows for better robustness and alleviates the tension between sensitivity to systematic change in mean input and robustness to input noise (Appendix 4—figure 2).

Appendix 4—figure 2

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Furthermore, a one-tier two-branch channel with an inter-branch feedback control (Figure 7a of the main text) shows a preference towards certain concentrations of the two receptors, which provide the cellular output robustness to input noise, i.e. minimise $\chi$. To maintain lower values of $\chi$ (higher robustness), the signalling receptors $\psi$ must decrease with increasing mean of the input ${\mu }_L$ (Appendix 4—figure 3). Increasing the non-signalling receptors $\varphi$ as a function of mean input ${\mu }_L$ helps separate the mean outputs $⟨\theta ⟩$ at neighbouring positions and thus aids in increasing the sensitivity (Appendix 4—figure 4). This is consistent with the receptor profiles in the optimised one-tier two-branch channel (Figure 7b of the main text).

Appendix 4—figure 3

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Appendix 4—figure 4

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Here, we try to understand why having an appropriate spatial gradient of receptors helps in separating the outputs in nearby cells, thus facilitating accurate positional inference. This is best done in the simplest case of a minimal channel that is with one tier and one branch. This channel ‘reads’ the ligand input through binding of the ligand on only one, signalling receptor.

Appendix 5—figure 1a shows that setting receptor conentrations to decrease with mean ligand input creates an unfavourable scenario. Low receptor availability for higher ligand concentrations ensures a saturation of the output at lower values (blue). On the other hand, higher receptor availability for ligand concentrations is impractical as the output remains low despite increase in potential saturation point (purple). This causes large overlaps of the outputs at neighbouring positions. On the other hand, Appendix 5—figure 1b shows that the output are better separated with receptor profiles that increase with the mean ligand input and thus would enable a better positional inference (Appendix 4).