Spatiotemporal Analysis of Different Mechanisms for Interpreting Morphogen Gradients

Abstract

During development, multicellular organisms must accurately control both temporal and spatial aspects of tissue patterning. This is often achieved using morphogens, signaling molecules that form spatially varying concentrations and so encode positional information. Typical analysis of morphogens assumes that spatial information is decoded in steady state by measuring the value of the morphogen concentration. However, recent experimental work suggests that both pre-steady-state readout and measurement of spatial and temporal derivatives of the morphogen concentration can play important roles in defining boundaries. Here, we undertake a detailed theoretical and numerical study of the accuracy of patterning—both in space and time—in models where readout is provided not by the morphogen concentration but by its spatial and temporal derivatives. In both cases we find that accurate patterning can be achieved, with sometimes even smaller errors than directly reading the morphogen concentration. We further demonstrate that such models provide other potential benefits to the system, such as the ability to switch on and off gene response with a high degree of spatiotemporal accuracy. Finally, we discuss how such derivatives might be calculated biologically and examine these models in relation to Sonic Hedgehog signaling in the vertebrate central nervous system. We show that, when coupled to a downstream transcriptional network, pre-steady-state measurement of the temporal change in the Shh morphogen is a plausible mechanism for determining precise gene boundaries in both space and time.

Introduction

Cells often need a way of determining spatial position, be it position within the cell (for example, to determine the site of cell division (1–3)) or position within a tissue or organism (for example, to successfully pattern distinct cell types during multicellular development (4–6)). Various mechanisms exist to obtain positional information (7), from short-ranged cell-cell signaling pathways (such as lateral inhibition (8)) to long-ranged morphogen gradients (9–11). A key property of any such mechanism is that it must robustly specify cell fates or boundaries. There has been considerable study of robust spatial positioning in development, with particular focus on the role of morphogen gradients (12,13) and how downstream gene networks can improve precision (14–18).

Morphogens are molecules that differ in concentration from place to place in a predictable way, so that measurements of their concentration (or related quantities) encode the spatial position (19). Perhaps the best-studied example is Bicoid, a transcription factor present in the Drosophila blastoderm, that is produced in the anterior region and which is responsible for patterning of the head and thorax along the anterior-posterior axis (20,21). The high precision of Bicoid patterning under both fluctuations in the external environment and internal noise has been the focus of a great deal of experimental and theoretical work (12,13,22–26). Other well-studied systems include Sonic Hedgehog (Shh) (27,28), Bone Morphogenetic Protein (29,30), and Squint (31) in vertebrate development, and Decapentaplegic (Dpp) (32,33) and Wingless (34,35) in Drosophila wing development, all of which display high levels of precision and could well be controlled by similar underlying biophysical mechanisms.

The formation and shape of a morphogen gradient is influenced by many factors, such as how the morphogen is produced (36,37), how it is degraded (38,39), how it is transported (23,40–43), and how it interacts with other system components (44). The original and conceptually simplest morphogen model involves a single morphogen species that diffuses freely, degrades at constant rate, and is produced/injected at one end of the system (23). At steady state this leads to a decaying exponential gradient that, using various concentration thresholds, can lead to distinct regions as in the French Flag model (19).

Reliable interpretation of the morphogen signal is essential for precise spatial positioning (45), and experimental evidence now suggests that the French Flag model alone is insufficient to explain the observed boundary specification (14,46,47). For example, there is increasing evidence that many morphogen gradients may actually be interpreted before steady state (22,48–52). Theoretical work suggests that such pre-steady-state measurement may, in some circumstances, reduce the effects of fluctuations on the precision of spatial boundaries (22,53), particularly when coupled to a downstream gene network (54). Further, pre-steady-state readout could enable gene responses at different positions to be turned on at different times, providing much greater developmental flexibility.

It is commonly assumed that cells measure the value of the morphogen concentration itself. However, cells can, in principle, measure quantities other than the morphogen concentration, such as the spatial (55) or temporal derivative (56). The temporal derivative is a particularly exciting possibility because the rate of concentration increase has a maximum at finite time, before steady state, so that a single gradient can be used to define both an on- and off-time. Qualitatively similar behavior has recently been suggested in TGF-β (57,58) and Dpp (59). Furthermore, evidence from the vertebrate nervous system and limb buds suggests that the duration of exposure to Shh (rather than just its absolute concentration) also plays an important role in specification (60,61).

Here we study in detail morphogen readout in both steady- and pre-steady state, and the consequences of measuring the spatial or temporal derivative of the morphogen concentration rather than the concentration itself. Using both analytic and in silico approaches, we determine the relative advantages and disadvantages of different interpretation schemes and under which conditions they can be biological useful. In particular, we look at how accurately each mechanism can pattern tissues in the presence of realistic noise, and their relative stability to variations in the underlying parameters. We further discuss the precision of readout at a particular time (and not just position), an important component in any system that interprets morphogens before steady state. We examine whether temporal derivative readout could, in principle, be accurate enough to turn on expression for only a limited time. Finally, we suggest plausible mechanisms for how cells might calculate spatial and temporal derivatives, and apply our results to Sonic Hedgehog (Shh) signaling in vertebrate neural tube patterning, showing that our conclusions are relevant to real biological systems.

Materials and Methods

See the Supporting Material.

Results

The basic model and its readout

The simplest morphogen models assume that the morphogen concentration, ρ(x,t), depends only on one spatial dimension, x, and time, t. Such one-dimensional models typically capture the essence of morphogen gradient formation and are consistent with experiments in many systems (40,62). The morphogen is assumed to be produced (or injected) at x = 0, and to diffuse and be degraded throughout the whole system (with length L). In the simplest case, degradation is assumed to depend linearly on ρ, so that we can write

$$\frac{\partial \rho }{\partial t}=D\frac{{\partial }^2\rho }{\partial x^2}-\mu \rho ,$$ (1)

where D is the diffusion constant and μ is the degradation rate. Production/injection at x = 0 is included by imposing the boundary condition Dρ′|_x=0 = −J, with J the morphogen production rate. Different boundary conditions make little difference to the profile, especially away from the source region, and so are not considered further here (38,63). For sufficiently long systems (L ≫ λ), the steady-state solution is given by

$$\frac{J\lambda }{D}{e}^{-x/\lambda },$$

where $\lambda =\sqrt{D/\mu }$ is the decay length. See Table 1 for all parameter definitions used in this work. The time taken to reach steady state in this model has been carefully studied (64), allowing pre-steady-state and steady-state behavior to be clearly differentiated.

Table 1.

Summary of model parameters and other quantities

Parameter/Quantity	Meaning
D	diffusion constant
J	production/injection rate per unit area
μ	degradation rate
$\lambda \equiv \sqrt{D/\mu }$	decay length
L	system length
A	area of system perpendicular to x
T	time at which fixed-time scenario is applied
Δx	jump in x over which to calculate ρ′
Δt	jump in t over which to calculate $\dot{\rho }$
Wx	spatial length of averaging window
Wt	temporal length of averaging window
Nx	number of measurements along x in averaging window
Nt	number of measurements along t in averaging window
x∗	threshold position separating regions above and below threshold
t1∗	time when threshold is first exceeded
t2∗	time in $\dot{\rho }$-model when $\dot{\rho }$ drops below threshold
τ ≡ t2∗ − t1∗	total time in $\dot{\rho }$-model for which threshold is exceeded
${\overline{x}}^{\text{∗}},{\sigma }_x$	mean and width of x∗ distribution
${\overline{t}}^{\text{∗}},{\sigma }_t$	mean and width of t∗ distribution
$\overline{\tau },{\sigma }_{\tau }$	mean and width of τ-distribution

In addition to the usual idea that cells read the morphogen concentration itself, we consider the suggestion that spatial or temporal derivatives may instead be measured (55,56). This leads us to focus on three readout models of morphogen interpretation: 1) the traditional ρ-model where the absolute value of ρ is read, 2) the ρ′-model where the spatial derivative is measured, and 3) the $\dot{\rho }$-model where the time derivative is read. The profiles of ρ, ρ′, and $\dot{\rho }$ are shown in Fig. 1, A–C, at various times, with ρ and ρ′ gradually increasing and decreasing respectively to their steady-state values. In contrast, $\dot{\rho }$ behaves quite differently: at any given position, $\dot{\rho }$ initially increases, reaches a maximum, and then decreases back toward zero (Fig. 1 D). Such a maximum exists regardless of specific parameter values, or indeed the specific details of morphogen transport or degradation.

Figure 1.

Profiles of ρ, ρ′, and $\dot{\rho }$ in space and time. (A) The morphogen concentration, ρ(x,t), as a function of x at various times. (B) The spatial gradient of the concentration, ρ′(x,t), as a function of x at the same times. (C) The temporal gradient of the concentration, $\dot{\rho }\left(x,t\right)$, as a function of x. For (A)–(C), the final curve, at t = 10 h, is effectively the steady-state profile. (D) The temporal gradient, $\dot{\rho }\left(x,t\right)$, but now as a function of time at various positions, showing the maximum that occurs at finite time. Parameters: J = 0.3 μm⁻² s⁻¹, D = 1 μm² s⁻¹, and μ = 10⁻⁴ s⁻¹. To see this figure in color, go online.

Cells cannot measure the instantaneous change in ρ, with respect to either space or time. Unicellular systems face a similar problem when measuring, for example, gradients of chemoattractants (65,66). Instead cells must estimate the instantaneous change by either sampling the concentration at multiple points (to estimate ρ′) or at multiple times (to estimate $\dot{\rho }$). Thus, an estimate for the spatial gradient at some position x and time t is given by

$${\rho }^{\prime }\left(x,t\right)\approx \frac{\rho \left(x+\Delta x/2,t\right)-\rho \left(x-\Delta x/2,t\right)}{\Delta x},$$ (2)

where Δx is now a new parameter, the distance over which the spatial gradient is estimated. Although mathematically Δx must be as small as possible to accurately estimate ρ′, this is not relevant for real biological systems. In such systems it is only important that the measured quantity allows precise patterning, not that the quantity corresponds as closely as possible with some mathematical limit. Thus, even if Δx is too large for Eq. 2 to be a good approximation for ρ′, it could still be used to accurately control expression. In this study, we typically take Δx to be on the order of the cell size. In a similar way, the estimate for $\dot{\rho }\left(x,t\right)$ requires the introduction of Δt, the time over which the temporal gradient is estimated, so that

$$\dot{\rho }\left(x,t\right)\approx \frac{\rho \left(x,t\right)-\rho \left(x,t-\Delta t\right)}{\Delta t}.$$ (3)

Due to noise inherent within any biological system, single concentration measurements show significant fluctuations and cannot, by themselves, be used for precise patterning (67). Cells must implement averaging, probably both spatial and temporal, in order to reduce errors (13,16,68). Thus we assume that each ρ-measurement involves multiple measurements within some averaging window of spatial length W_x and time W_t. Within this window, measurements are made at N_x spatial locations and N_t times, so that a total of N_xN_t measurements are made (and averaged) for each measured value of ρ (see Fig. S1 in the Supporting Material).

It is important to distinguish two alternative scenarios for how thresholding is implemented. In most models of morphogen-gradient interpretation, it is assumed that at some fixed time T (where T is very large for steady-state models), the morphogen concentration is measured and compared to a threshold value. A particular response is activated only in regions where the measured value is above (or below) the threshold. We refer to this as the “fixed-time scenario”. However, another possibility (which is consistent with experiments on Shh (60)) is that the cell continually monitors the morphogen concentration and activates regions whenever the measured value is above the threshold value. This means that, as the concentration changes, different parts of the system are activated at different times. We call this the “continuous-time scenario”. In this scenario, readout of $\dot{\rho }$ can result in both turning on and later off some particular gene response, without the need for any additional downstream network.

We are interested in precisely how these different models for morphogen interpretation can be used for patterning in the presence of noise and parameter shifts. Previous work has tended to distinguish internal and external noise, with external noise typically referring to a fixed change in injection rate of J → J + δJ, and internal noise to the stochastic nature of morphogen diffusion and degradation within the system. Mathematically, such a distinction is somewhat artificial and here we consider errors to be due to either parameter shifts or noise, with no separation between internal and external sources. Parameter shifts refer to a situation where one or more parameters are altered by a constant amount from their usual fixed value. For example, the injection rate J could be changed by variation in the distance between cells, or the diffusion constant D could be altered due to a mutation within the morphogen. In contrast, noise is an unavoidable stochastic effect that continuously fluctuates and is present both internally and externally: not only is morphogen diffusion and degradation stochastic (67,68), but the injection rate also fluctuates in time (69). We include all these sources of noise in our simulations. Below, we focus on the simulation results and discussion of the effects of parameter variation and noise, with details of analytic calculations provided in the Supporting Material.

The fixed-time scenario

We first consider the situation where morphogen interpretation occurs at a fixed time t = T. Only positions where the concentration exceeds some threshold are activated, which results in distinct regions of activation separated by a threshold boundary. In this scenario only the position of the boundary is of interest, denoted by x^∗, which, due to noise, will follow some distribution with average ${\overline{x}}^{\text{∗}}$ and width σ_x. We are interested in both how ${\overline{x}}^{\text{∗}}$ moves due to parameter shifts and how σ_x is determined by noise.

Parameter shifts

A parameter shift will alter the morphogen concentration, which will in turn affect ${\overline{x}}^{\text{∗}}$. Although some progress can be made analytically (see the literature (22,53,54) and the Supporting Material), more accurate results are obtained by numerically solving for the altered boundary position (see the Supporting Material).

First consider a shift in the production rate, J. The behavior in x is shown in Fig. 2 A, which uses parameters suitable for Bicoid gradient formation. In the Supporting Material, we also consider our results with parameters relevant for Shh in the vertebrate neural tube, although with no changes to our conclusions. In all three readout models, the resulting shift in the threshold boundary decreases as x increases (as shown for the ρ-model in Bergmann et al. (22)). Notably, the error in the ρ′-model is always greater than that in the ρ-model, with the error in the $\dot{\rho }$-model greater still. Thus, if only fluctuations in J were considered, the ρ-model is most precise at specifying boundaries. The behavior in time is shown in Fig. 2 B, showing that the shift in ${\overline{x}}^{\text{∗}}$ increases with t in all three readout models, always with the ρ-model having the smallest shift. As t continues to increase, the ρ- and ρ′-models tend to the same constant value, independent of x, so that both models are equally robust to shifts in J at steady state. As expected, the error in the $\dot{\rho }$-model grows without limit as the system approaches steady state, making this model effectively useless at large times.

Figure 2.

How parameter shifts affect ${\overline{x}}^{\ast }$. (Left plots) Shift in ${\overline{x}}^{\text{∗}}$ against x at t = 40 min. (Right plots) Shift in ${\overline{x}}^{\text{∗}}$ against t at x = 120 μm. (A and B) Effect of changing the production rate, J, by 10%. (C and D) Effect of changing the diffusion constant, D, by 10%, showing that the ${\overline{x}}^{\text{∗}}$ shift can vanish in all three readout models. The effect of changing the degradation rate, μ, is shown in Fig. S3. Parameters: J = 0.3 μm⁻² s⁻¹, D = 1 μm² s⁻¹, and μ = 10⁻⁴ s⁻¹. To see this figure in color, go online.

Now we examine shifts in the diffusion constant, D. Fig. 2 C shows that the behavior as a function of x is quite different to that for shifts in J. In all three readout models, there are now positions where the shift in ${\overline{x}}^{\text{∗}}$ completely vanishes. In the ρ-model this reflects the fact that there are times and positions where ρ(x,t) does not change under a shift in D, i.e., ∂ρ/∂D = 0. This could provide potential benefits to an organism because, for example, any variation in the morphogen that affects D would have almost no effect on the position of the boundary. Such a situation may occur, at least partially, in the subcellular gradient of pom1 in fission yeast (70). Furthermore, the position where the shift in ${\overline{x}}^{\text{∗}}$ vanishes depends on the model in question and so, for a given set of parameters, the ρ′- or $\dot{\rho }$-model may be preferred to the ρ-model. At later times, the boundary shift in the $\dot{\rho }$-model increases without limit, whereas the ρ- and ρ′-models tend to some (different) limit (Fig. 2 D). As time increases, the position of the zero boundary shift gradually increases in all models, tending to x = λ in the ρ-model and x = 2λ in the ρ′-model (see the Supporting Material).

Parameter shifts in the degradation rate have similar effects to those in the production rate, and detailed discussion is provided in the Supporting Material.

System noise

Although pre-steady-state readout can give reduced errors for parameter shifts, it is critical to also include system noise (53). Unlike parameter shifts that, at least in principle, could be avoided by careful system design, noise is an integral, inherent part of any biological system that cannot be removed (16). The addition of noise means that the boundary position fluctuates from organism to organism, described by an average ${\overline{x}}^{\text{∗}}$ and a width σ_x. Because the average is determined by the parameters and the width by noise, we are interested here in the size of σ_x. As in Tostevin et al. (67), approximate analytic expressions for σ_x can be derived (see the Supporting Material). However, these can be unreliable in relevant parameter regimes, and so we developed a full computational simulation, which includes noise from diffusion, degradation, and injection (see the Supporting Material). These simulations also allow us to include spatial and temporal averaging in a more realistic manner.

As in previous work (53,67), we found that averaging is essential for all readout models: with N_x = N_t = 1 (i.e., with no averaging), ρ and its derivatives fluctuate so wildly that positioning is completely impossible. However, with sufficient averaging (N_xN_t ≳ 1000), all three models can position boundaries with σ_x < 5% of ${\overline{x}}^{\text{∗}}$. For the same level of averaging, the ρ′- and $\dot{\rho }$-models normally have greater errors than the ρ-model (Fig. 3). This is expected because calculating ρ′ and $\dot{\rho }$ requires two measurements of the concentration, rather than just one for ρ itself, with a corresponding increase in the uncertainty. However, at early times both the gradient readout models can have smaller errors than the ρ-model (Fig. 3 B). For the ρ′- and $\dot{\rho }$-models there is an interplay between averaging (determined by W_x, W_t, N_x, and N_t) and the gradient jumps (determined by Δx and Δt). For example, the increased error due to reducing the temporal averaging window W_t can be compensated by an increase in the gradient jump Δt.

Figure 3.

Fluctuation in boundary position, σ_x, due to noise. (Solid circles) Numerical simulations; (lines) analytic approximations (see the Supporting Material). (A) σ_x against x at t = 2 h. (B) σ_x against t at x = 200 μm. Parameters: J = 0.3 μm⁻² s⁻¹, D = 1 μm² s⁻¹, μ = 10⁻⁴ s⁻¹, A = 20 μm², Δx = 52.5 μm, Δt = 1001 s, W_x = 22.5 μm, W_t = 100 s, N_x = 9, and N_t = 50. To see this figure in color, go online.

In the ρ-model, σ_x monotonically increases with x at all times, going approximately like

$$\frac{\sqrt{\rho }}{\left|{\rho }^{\prime }\right|}$$

(53). This behavior is quite different for the ρ′- and $\dot{\rho }$-models (Fig. 3 A). Both models lead to large errors for large values of x, but both can exhibit minima at nonzero positions. As shown in the Supporting Material, σ_x in the ρ′-model is approximately proportional to

$$\frac{\sqrt{\rho }}{{\rho }^{\text{″}}}.$$

At early times, this function has a minimum at nonzero x, whereas at late times the minimum is at the origin. This agrees well with the full simulation, with the case shown in Fig. 3 A corresponding to a sufficiently late time, so that there is no nonzero minimum. Similarly, the $\dot{\rho }$-model error is approximately proportional to

$$\frac{\sqrt{\rho }}{\left|{\dot{\rho }}^{\prime }\right|},$$

which has a minimum at nonzero x at all times (Fig. 3 A).

The behavior of σ_x can also be studied as a function of time. For the ρ-model, σ_x is initially large and decreases to its steady-state value at large times. However, this behavior is not monotonic: at all positions the minimum error occurs at finite time (54). We find identical behavior for the ρ′- and $\dot{\rho }$-models (Fig. 3 B), with the effect even more pronounced in the $\dot{\rho }$-model (where σ_x becomes infinite at steady state). Thus, measurements in pre-steady state can result in increased readout precision at appropriate measurement times.

The continuous-time scenario

Unlike the fixed-time scenario (where quantities are measured at some time T), the continuous-time scenario involves continuous measurements at all positions and times. Now both the position x^∗ and the activation time t^∗ are relevant measures. Further, in the $\dot{\rho }$-model, we can define both the time that a region is activated, t₁^∗, and the later time when the same region is deactivated, t₂^∗. This naturally leads to the concept of the expression time, τ ≡ t₂^∗ − t₁^∗, the total time for which the threshold is exceeded. We are interested in both the averages (${\overline{t}}^{\text{∗}}$, $\overline{\tau }$) and widths (σ_t, σ_τ) of the activation and expression times. For simplicity, we will ignore the boundary position, x^∗, in this section and concentrate only on t^∗ and τ. Our conclusions hold for any reasonable choice of x^∗.

Fluctuations and shifts in the activation time

First we consider how t^∗ is shifted by a change to the injection rate, J. For the ρ- and ρ′-models, the behavior in space is similar to that for the fixed-time scenario, with a gradual decrease as x increases (Fig. 4 A). However, the $\dot{\rho }$-model is completely different, with a sharp peak occurring at an intermediate position. As explained in the Supporting Material, this position occurs where $\dot{\rho }$ has a maximum, i.e., at positions and times where x² = 2Dt(2μt + 1), and is an unavoidable consequence of measuring $\dot{\rho }$. In fact, such a sharp peak occurs in the $\dot{\rho }$-model for all parameter shifts and when variation due to noise is considered.

Figure 4.

How parameter shifts and noise affect t^∗. (Left plots) Shift/noise against x at t = 2 h. (Right plots) Shift/noise against t at x = 100 μm. (A and B) Shift in ${\overline{t}}^{\text{∗}}$ due to changing the production rate, J, by 10%. (C and D) Shift in ${\overline{t}}^{\text{∗}}$ due to changing the diffusion constant, D, by 10%. The effect of changing the degradation rate, μ, is shown in Fig. S4. (E and F) Width of t^∗ distribution, σ_t, due to noise. (Solid circles) Numerical simulations; (lines) analytic approximations (see the Supporting Material). Parameters as in Fig. 3. To see this figure in color, go online.

The behavior in time is now opposite to that in the fixed-time scenario (Fig. 4 B). As time increases, it is now the ρ- and ρ′-models where the shift in ${\overline{t}}^{\text{∗}}$ increases without limit. Conversely the $\dot{\rho }$-model now tends to a finite value (see the Supporting Material). Although conceptually interesting, this is unlikely to be biological relevant because, at large times, $\dot{\rho }$ becomes vanishingly small: a small change in ${\overline{t}}^{\text{∗}}$ is irrelevant if $\dot{\rho }$ itself is too small to be accurately measured. However, despite this, it is noteworthy that there are positions and times where the $\dot{\rho }$-model has a smaller error than the other two models.

The results of a shift in the diffusion constant, D, are shown in Fig. 4, C and D. As in the fixed-time scenario, all models contain positions and times where the movement of the boundary vanishes. This leads to interesting behavior in the $\dot{\rho }$-model, with both a zero due to ∂$\dot{\rho }$/∂D = 0 and a maximum due to $\dot{\rho }$ = 0. All three models tend to a constant value as x → ∞. As with shifts in J, the errors in the ρ- and ρ′-models continually increase as time increases (despite the appearance in Fig. 4 D), whereas the $\dot{\rho }$-model tends to a finite constant. In addition to regions where the $\dot{\rho }$-model has the smallest error, there are now also regions where the ρ′-model is the most accurate.

Shifts in the degradation constant cause similar changes to shifts in J, and further discussion is given in the Supporting Material.

Including noise results in a distribution for t^∗, characterized by its width σ_t. The ρ-model is already interesting with, as shown in the Supporting Material, σ_t approximately proportional to

$$\frac{\sqrt{\rho }}{\dot{\rho }}.$$

This should be contrasted with the equivalent result for σ_x, where the $\dot{\rho }$ in the denominator is replaced with ρ′ (53,67). The behavior in space is similar to that for the fixed-time scenario (Fig. 4 E), with the greatest σ_t at large x, although now the minimum is at nonzero x. The behavior in time (Fig. 4 F) is such that the greatest errors occur at very small and very large times. As with σ_x, the minimum error occurs at intermediate times, but now with a much more pronounced minimum, suggesting that, for positioning in time, pre-steady state may be much more accurate.

The ρ′-model behaves similarly to the ρ-model, with a minimum in σ_t occurring at intermediate positions and times (Fig. 4, E and F). Although the ρ-model normally has smaller errors, the ρ′-model becomes more accurate at large positions and small times.

Finally, we consider the $\dot{\rho }$-model, where

$${\sigma }_t\propto \frac{\sqrt{\rho }}{\left|\ddot{\rho }\right|}$$

(see the Supporting Material). The spatial behavior of σ_t is complex, with a maximum at some intermediate position (where $\dot{\rho }$= 0), an infinite error at x = ∞, and two minima either side of where $\dot{\rho }$ = 0 (Fig. 4 E). Which minimum is the global minimum depends on the value of t. Similar behavior is observed when σ_t is plotted against time (Fig. 4 F). Again there are two minima, but now the true minimum occurs at the earlier time for all positions (see the Supporting Material). The error in the $\dot{\rho }$-model is usually greater than that in the ρ-model; whether the ρ′- or $\dot{\rho }$-model has the smaller error depends on the values of Δx and Δt.

Fluctuations and shifts in the expression time

An interesting possibility is that the $\dot{\rho }$-model could be used to both turn on expression at time t₁^∗ and turn it off later at time t₂^∗, leading to expression for only a time τ ≡ t₂^∗ − t₁^∗. For some fixed $\dot{\rho }$-threshold, the value of t₁^∗ and t₂^∗ (and hence τ) will depend on the position x. Further, some positions (where $\dot{\rho }$ never reaches the threshold) will never turn on, so that τ = 0.

In the above results, we always considered how quantities (boundary shifts and widths) either depended on x at some fixed time or on t at some fixed position. This involved continually adjusting the relevant threshold. For example, for the x dependence of the ρ-model, the critical threshold concentration above which a region is turned on was always chosen to ensure that the different positions were all considered at the same time. For τ, this is not possible, because adjusting the threshold affects both t₁^∗ and τ. We can choose to fix one, but not both. Here we choose the second option, with the $\dot{\rho }$-threshold continually tuned to ensure τ is constant, with t₁^∗ necessarily varying.

The shifts in τ due to parameter variation are shown in Fig. S5. All shifts increase without limit as x → ∞, with (as usual) a minimum in the diffusion case where the error completely vanishes. The behavior as a function of τ is similar for all three parameters: as τ increases, the shift in τ initial decreases, before reaching a minimum and then increasing, either without limit (for μ) or to finite values (for J and D).

We now examine the effect of noise, which leads to a distribution for τ. Our simulations show that this distribution is approximately normal (Fig. S6). The width of the distribution, σ_τ, is a combination of the errors in the turn-on time, t₁^∗, and the turn-off time, t₂^∗. Because noise goes like $\sqrt{\rho }$ (see the Supporting Material) and ρ monotonically increases in time, the larger error always arises from the turn-off time. For the $\dot{\rho }$-model to be useful in behaving as a switch, σ_τ must be sufficiently small. In fact, with insufficient averaging, huge errors in τ (>500%) render the whole mechanism implausible. However, our full simulation shows that reasonable (σ_τ < 0.1τ) errors can be attained with realistic levels of temporal and spatial averaging. For example, using parameters derived from Bicoid, it is possible to get σ_τ < 15% of τ (see the Supporting Material). Therefore, with biologically accessible parameters, the $\dot{\rho }$-model can indeed be used for accurate positioning of boundaries while also providing an on/off mechanism.

We now investigate how σ_τ depends on the spatial position and expression time. For fixed τ, the value of t₁^∗ (and hence t₂^∗) must increase with increasing x. The resulting σ_τ increases rapidly with increasing x (Fig. 5 A), suggesting that measuring τ near x = 0 incurs the smallest error. The behavior in τ at fixed position is more interesting (Fig. 5 B). For small τ, both t₁^∗ and t₂^∗ must be chosen near the time when $\dot{\rho }$ vanishes. Because $\ddot{\rho }$ is small at these times, this implies relatively large errors and hence large σ_τ. For large τ, t₁^∗ is almost zero and t₂^∗ gets very large, again leading to large σ_τ. In between these extremes there is a minimum corresponding to an optimal τ, which occurs when t₂^∗ is near to the second minimum in Fig. 4 F. If the relative error, σ_τ/τ, is plotted instead of σ_τ, exactly the same behavior is found. Thus, interestingly, for a given position, there is an optimum expression time. Equivalently, to measure a given τ at a given position, there is an optimum choice of parameters that minimize the error due to noise.

Figure 5.

The width of the τ-distribution due to noise. (Solid circles) Numerical simulations; (lines) analytic approximations (see the Supporting Material). (A) σ_τ against x for τ = 2000 s. (B) σ_τ against τ at x = 100 μm. Parameters as in Fig. 3. To see this figure in color, go online.

Possible biological mechanisms for calculating derivatives

It is interesting to speculate how real organisms might calculate the derivatives ρ′ and $\dot{\rho }$, which is potentially more difficult than measuring the morphogen concentration itself. First consider $\dot{\rho }$, which requires measuring and comparing ρ at different times. This is technically challenging, but could be achieved by using a second component. If initial morphogen interpretation results in a state-change of the second component (such as phosphorylation), then, with the state-change modulating future morphogen interpretation, the cell could estimate temporal changes in ρ and hence its time derivative.

However, it is also possible that $\dot{\rho }$ is calculated using a transcriptional network. Perhaps the simplest such network involves two components, X and Y, with X produced directly by morphogen and inhibited by Y, and Y produced by X (see Fig. 6 A) (58). If only X degrades (so that the concentration of Y continually increases), then the system is described, at a fixed position, by

$$\begin{array}{l}\frac{dX}{dt}=\beta \rho \left(t\right)-2\sqrt{\beta \gamma }X-\gamma Y,\hfill \\ \frac{dY}{dt}=\beta X,\hfill \end{array}$$ (4)

where β and γ are rates, and the choice of parameters corresponds to critical damping (see the Supporting Material). With β = 1 s⁻¹ and γ = 0.003 s⁻¹, the value of γX is close to the value of $\dot{\rho }$, with most of the fluctuation arising from the variation in ρ rather than from the inability of the network to estimate $\dot{\rho }$ (see Fig. 6 B). In the Supporting Material, by including the degradation of Y, we consider a more realistic version of this model. It is worth noting that this system is not unique; other networks (such as Nakajima et al. (71)) can also be used for temporal sensing.

Figure 6.

Network for measuring $\dot{\rho }$ and application to Sonic Hedgehog signaling. (A) Two-component network that, with suitable parameters, leads to X ∝ $\dot{\rho }$. (B) Example X and Y output from the XY network at x = 80 μm, with ρ taken from the full simulation. (Red dashed lines) Analytic expressions for ρ and $\dot{\rho }$. (C) Schematic of the Shh gene network with three mutually repressing transcription factors. The Shh readout G ∝ $\dot{\rho }$ affects the production rates of Olig2 and Nkx2.2. (D) Example output in time for Shh model at x = 100 μm. (Left axis) P, N, and O concentrations. (Right axis) Value of G. (E) Comparison, in Shh model, of σ_τ as a function of τ for $\dot{\rho }$ readout and Olig2 readout, showing that, for large τ, measuring Olig2 can lead to smaller errors. All data points are from numerical simulations. Parameters as in Fig. 3. To see this figure in color, go online.

Now consider how a cell might measure ρ′, which requires the comparison of ρ at two spatially separated points. In principle this might be possible using a network similar to that in Eq. 4, but extended to include a spatial component and diffusion. However, eukaryotic cells are well known to be able to accurately measure chemical gradients during chemotaxis, by comparing receptor occupancy at opposite ends of the cell (72). It has been shown that cells can reliably interpret spatial gradients across the length of a cell (73). Alternatively, within a cell, very large macromolecules could potentially determine ρ′ by measuring (perhaps by conformational change) the variation in bound morphogens along their length. Although none of these potential mechanisms have yet been observed for morphogens, the fact that the spatial derivative may well play an important role during growth and development (55,74) suggests that an experimental search for such mechanisms could well be fruitful.

Application to Sonic Hedgehog signaling

We now apply our temporal gradient readout model to Shh signaling, which is critical in patterning the central nervous system in vertebrates. Rather than examining how the cell might estimate derivatives, we now assume that the cell has already calculated $\dot{\rho }$ and investigate how the downstream network affects the accuracy of patterning. Shh emanates from the notochord and spreads along the dorsoventral axis of the neural tube, inducing the floor plate and its subsequent differentiation into five domains with distinct neuronal subtypes (75–78). Recent work has shown how the interpretation of the Shh gradient involves a downstream transcriptional network (17). A mathematical model of a reduced version of this network, with just three transcription factors (Pax6, Olig2, and Nkx2.2), was able to reproduce much of the observed dynamics including the robustness of patterning to fluctuations in the Shh gradient (17).

This mathematical model considers the concentrations of Pax6 (P), Olig2 (O), and Nkx2.2 (N), and their various interactions. Each species is degraded (with rates μ_P, μ_O, and μ_N) and produced (with rates proportional to α_P, α_O, and α_N). In addition, the production rates depend on the other species, such that the production of Pax6 is inhibited by Olig2 and Nkx2.2, the production of Olig2 repressed by Nkx2.2, and the production of Nkx2.2 inhibited by Pax6 and Olig2. This leads to the gene regulatory network shown in Fig. 6 C. These inhibitory effects are introduced via Hill-like behaviors, with Hill coefficients h_i and thresholds X_critY (for species Y with inhibitor X).

The influence of Shh is included by its effect on the production rates. In Balaskas et al. (17), this is achieved via the concentration of an extra component, Gli (G), which acts as an intermediate between the Shh concentration and its effect on P, O, and N. Unlike the Shh concentration, which monotonically increases with time, the Gli concentration initially increases before reaching a maximum and decreasing to zero. This is not dissimilar to the $\dot{\rho }$ profile, which motivates setting G ∝ $\dot{\rho }$. Although the real Gli profile may be more complex than this, it is not unreasonable to assume it is related to the rate of change of Shh (11). This agrees with previously measured Gli profiles (17,60,79). As with P, O, and N, the effect of G is represented via a Hill function, with coefficient n_1,2 and threshold G_critO,N. Our parameter values are given in Table S1 in the Supporting Material.

This leads to a coupled set of ordinary differential equations given by

$$\begin{array}{l}\frac{dP}{dt}={\alpha }_P\frac{1}{1+{\left(\frac{N}{N_{\text{critP}}}\right)}^{h_1}+{\left(\frac{O}{O_{\text{critP}}}\right)}^{h_2}}-{\mu }_{P}P,\hfill \\ \frac{dO}{dt}={\alpha }_O\frac{G^{n_1}}{G_{\text{critO}}^{n_1}+G^{n_1}}\frac{1}{1+{\left(\frac{N}{N_{\text{critO}}}\right)}^{h_3}}-{\mu }_{O}O,\hfill \\ \frac{dN}{dt}={\alpha }_N\frac{G^{n_2}}{G_{\text{critN}}^{n_2}+G^{n_2}}\frac{1}{1+{\left(\frac{O}{O_{\text{critN}}}\right)}^{h_4}+{\left(\frac{P}{P_{\text{critN}}}\right)}^{h_5}}-{\mu }_{N}N,\hfill \end{array}$$ (5)

where we set G = 10⁴ $\dot{\rho }$. (The prefactor, which could easily be removed by rescaling the other parameters, allows parameter values to be used that are similar to those used in Balaskas et al. (17).)

The effect of noise is critical in understanding the feasibility and stability of this system. Only constant-variance white noise around a constant G was considered in Balaskas et al. (17). Here, by coupling the above network to our full simulation, we are able to study the system in the presence of realistic noise arising from fluctuations in morphogen diffusion, degradation, and injection. To do this, we numerically solve the system in Eq. 5 with the parameters given in Table S1 and with G given by the output of our full simulation. As in Balaskas et al. (17), we take P = α_P/μ_P and O = N = 0 as the initial condition. Typical output at x = 100 μm is shown in Fig. 6 D. It is notable that the fluctuations in P, O, and N are much smaller than those in $\dot{\rho }$ itself, suggesting, as further confirmed in Balaskas et al. (17), that the transcriptional network provides robustness against fluctuations in the morphogen gradient.

We can now ask how this network alters the readout noise in the $\dot{\rho }$-model and, in particular, how this affects the accuracy of the expression time, τ. Whereas previously expression in the $\dot{\rho }$-model was turned on/off when $\dot{\rho }$ passed a critical value, this role is now played by one of the transcription factors. Because in Fig. 6 D only Olig2 rises and drops like $\dot{\rho }$, we use Olig2 as this readout. This involves introducing a new parameter, a threshold for O, so that expression is on/off when O is above/below the threshold. By using our full simulation, we can measure the expression time both in the original model (with $\dot{\rho }$ the trigger), ${\tau }_{\dot{\rho }}$, and in the new version (with Olig2 the trigger), τ_O, and then the fluctuation in these times, σ_τ.

As with ${\tau }_{\dot{\rho }}$, measuring τ_O with suitable parameters can lead to acceptable (<10%) errors for τ. Further, a similar dependence of σ_τ on τ is found: large and small τ are associated with large errors, with an optimum σ_τ corresponding to some intermediate τ (red data points in Fig. 6 E). However, interestingly, whether measuring $\dot{\rho }$ or O leads to smaller σ_τ depends on τ itself. For small τ, the wide peak in the O profile relative to that for $\dot{\rho }$ (Fig. 6 D) leads to the smallest errors when it is $\dot{\rho }$ that is measured (as, for example, at τ = 2 h in Fig. 6 E). Conversely, for larger τ, the fact that Olig2 fluctuates less than $\dot{\rho }$, means that measuring O leads to smaller σ_τ (as, for example, at τ = 4 h in Fig. 6 E). This suggests that shorter expression times are better measured by reading $\dot{\rho }$, whereas longer expression times benefit from reading Olig2. In addition, it follows that the transcriptional network allows much longer expression times to be measured with reasonable errors.

Discussion

We have provided a detailed analysis of morphogen-gradient precision by 1) fully considering parameter shifts and noise in morphogen production, diffusion, and degradation; 2) considering three biologically relevant modes of morphogen readout; and 3) exploring the precision of boundary specification in time as well as in space. Because the issue of when readout is performed is still uncertain in many systems, we performed our calculations both in steady state and in pre-steady state. We found that, with sufficient averaging, pre-steady-state interpretation can give precise boundary specification with all our readout models. In fact, pre-steady-state readout can give smaller errors than those in steady state (even when noise is taken into account). Although this was previously known for the ρ-model (54), we have performed full simulations with realistic noise, and shown that pre-steady-state measurement can also be more accurate in derivative readout models.

Under parameter shifts, derivative readout mechanisms (where the spatial or temporal derivative of the morphogen concentration is measured) can have smaller errors than direct interpretation of the concentration. For example, in steady state, interpretation of the spatial gradient is more stable to shifts in the degradation rate than measurement of the concentration itself. However, when noise is included (without parameter shifts), direct interpretation of the morphogen concentration normally has smaller errors than the other readout models. This simply reflects the fact that at least two concentration measurements are required to calculate spatial or temporal derivatives, whereas only one is needed for the concentration itself. Of course, this does not mean that the other models are ruled out, simply that more spatial and temporal averaging is required to achieve the same accuracy.

We applied our temporal derivative model to Sonic Hedgehog signaling in the vertebrate neural tube, where Gli could well be related to the rate of change of Shh, showing that a realistic downstream transcriptional network can reduce noise in the Shh concentration. In particular, much longer expression times can be accurately measured by coupling to the downstream network. This is further evidence that the typical assumption of direct interpretation of morphogen concentration is unnecessary.

Although shifts in the morphogen production rate and degradation rate always move the position of the threshold boundary, we found that, at certain positions, a shift in the diffusion constant makes no difference to the boundary position. This means that, with the correct parameter choice, organisms can protect against shifts in the effective diffusion constant (such as variations in the local environment (42) or a mutation in the morphogen). In particular, in steady state, this effect occurs at x ≈ λ in the direct concentration readout model and at x ≈ 2λ in the spatial derivative readout model, where λ is the decay length. This predicts that organisms that directly read the concentration at steady state will use parameters such that λ equals the boundary position, x_b. In fact, this is already known to be approximately true for some carefully quantified systems, such as Bicoid and Dpp. Further, it may even be possible to distinguish systems that measure the concentration from those that measure its derivative by using the measured value of λ: we predict that, on average, x_b/λ will be twice as large for derivative readout compared with that for direct readout. However, we must be careful because protecting against shifts in the diffusion constant is only one aspect of the problem; systems will also buffer against shifts in other parameters and against noise.

It is important to point out that we have only considered a simple, idealized morphogen system. For example, our models are only one-dimensional. This allows for analytic progress but neglects the geometry of real organisms. Real morphogens move in two or three dimensions and often diffuse in the presence of obstacles such as cells (42). Although it would be interesting to include these effects in our analysis, they are unlikely to alter our general conclusions about the relative benefits of measuring concentration versus its derivatives. We have also not considered the role that cell movement can play in sharpening boundaries (46). Furthermore, with increased quality of experimental data, it is becoming increasingly clear that many morphogen gradients do not form via a simple SDD mechanism (41–43,80,81). For example, nonlinear morphogen degradation can lead to power-law rather than exponential morphogen profiles (38). The approach developed here can also be applied to such models, and qualitatively it is likely that similar results would be found. For example, quadratic decay (with μρ replaced by μρ² in Eq. 1) leads to ρ ∼ 1/x² in steady state. Although this is distinct to the ρ ∼ e^−x/λ profile studied here, it is likely that similar issues (such as the fact that ρ′ and $\dot{\rho }$ measurements incur extra noise) influence patterning. Further, other profiles, such as the sharp profiles considered in Lander (81), are likely to affect all three readout models equally, so that our overall conclusions are unchanged.

We have restricted this study to models that only consider readout of the concentration and its first derivatives (ρ, ρ′, $\dot{\rho }$). Further work could consider other possibilities, such as higher derivatives (ρ″, $\ddot{\rho }$, and ${\dot{\rho }}^{\prime }$, etc.), although these are unlikely to be biologically plausible due to the high level of averaging that would be required. In addition, it would be interesting to study further possible readout mechanisms, such as the time-integral ${\int }_0^T\rho dt$, which would essentially be equivalent to a system that measures ρ and averages over the longest possible time window (i.e., over all times t > 0). Our work here suggests that such systems may be able to achieve even greater precision than the standard ρ-model.

With the increased quality of experimental measurement, it is now apparent that real morphogens are rarely (if ever) interpreted as described in the French Flag model, where thresholds are based purely on direct, local interpretation of the morphogen concentration. The measurement instead of derivatives of the concentration, both spatial and temporal, can sometimes confer certain advantages such as greater robustness and flexibility. Cells are likely to make use of every option available to them in order to pattern tissues accurately, and so the fact that accurate positioning can be achieved with morphogen concentration derivatives means that such mechanisms are likely to play important roles during development.

Author Contributions

D.M.R. and T.E.S. conceived and designed the project; D.M.R. performed all calculations and simulations; and D.M.R. and T.E.S. wrote the article.