Sequential Pattern Formation Governed by Signaling Gradients

Abstract

Rhythmic and sequential segmentation öf the embryonic body plan is a vital developmental patterning process in all vertebrate species. Here we show that a set of coupled genetic oscillators in an elongating tissue that is regulated by diffusing and advected signaling molecules can account for segmentation as a self-organized patterning process. This system can form a finite number of segments and the dynamics of segmentation and the total number of segments formed depend strongly on kinetic parameters describing tissue elongation and signaling molecules. The variety of different patterns formed in our model can account for the variability of segmentation between different animal species.

Morphogenesis, the formation of shapes and patterns in the developing embryo, relies on the tight coordination of cellular actions [1]. During embryonic development, spatial profiles of signaling activity in tissues control the behavior of cells such as proliferation, migration, and differentiation [2–4]. In vertebrates, a vital morphogenetic process is the segmentation of the elongating body axis during which the pre-cursors of the vertebrae are formed [5]. Segments form rhythmically and sequentially from an unsegmented progenitor tissue, the presomitic mesoderm (PSM), see Fig. 1A. During segmentation, the body axis elongates while the PSM continuously changes its length and eventually shortens. After a species-dependent number of segments, the process terminates when the PSM becomes very small [6, 7]. The total segment number can vary from about ten in frog to several hundreds in snake [8]. The temporal progress of segmentation is controlled by oscillations of the cellular concentration levels of functional proteins in the PSM [5, 9]. These oscillations give rise to nonlinear waves that propagate through the PSM [10–17], see Fig. 1B. A new segment is formed with each completed oscillation at the anterior end of the PSM, corresponding to an arriving wave [17]. Hence, in contrast to pattern formation via instabilities of homogeneous states, segmentation is characterized by a spatially inhomogeneous system with the PSM driving the patterning process. In addition, pattern formation takes place in a dynamic medium: the body axis continuously elongates during segmentation while the PSM at the tail of the body axis shortens until segmentation terminates.

FIG. 1.

(A) Schematic depiction of a zebrafish embryo during the segmentation of the body axis. (B) Oscillations of gene activity of several genes (Her1, Her7) manifest themselves as traveling waves of gene expression through the PSM (blue). Arrows indicate the direction of wave propagation. (C) Concentration profiles of the signaling molecules Wnt and FGF with highest concentration at the posterior tip (green) and an opposing gradient of Retinoic Acid (RA, red) having highest concentration in the segments.

To yield robust morphological results, this complex patterning process requires tight integration of spatial and temporal cues. What regulates the integration of tissue growth and patterns of oscillating gene expression in vivo? The elongating body axis exhibits spatial concentration profiles of several signaling molecules which are thought to be involved in guiding the segmentation process [18], see Fig. 1C. Basic principles of segmentation were captured by simplified models in which the PSM length was kept constant [9, 13, 14, 19]. Furthermore, the role of signaling activity within the PSM was studied using different approaches [20–26]. However, how the interplay of signaling activity and genetic oscillations leads to self-organized and robust segmentation of the body axis in a dynamic tissue is not understood.

In this Letter, we present a theoretical description of vertebrate segmentation, in which spatial concentration profiles of signaling activity regulate both growth and the dynamics of cellular oscillators that control segmentation. In our model, local interaction rules lead to robust self-organization of a patterning system that yields the correct spatial morphology and shortening of the segmenting tissue. First, we illustrate the basic ideas using a minimal model based on oscillators and a single signaling activity Q. This describes a simplified scenario of segmentation with time-periodic patterns of oscillatory gene expression and segment formation. We study how the key features of segmentation depend on the kinetic parameters of the signaling system. This model describes sequential pattern formation but does not capture tissue shortening and termination of segmentation. Hence, in a second step, we introduce a model including a second signaling activity R opposed to Q whose interactions with Q leads to dynamic shortening of the segmenting tissue and termination of segmentation after a finite number of segments.

Segmentation governed by a single signaling gradient

We introduce a curved coordinate axis along the embryo, in which x = 0 corresponds to the posterior tip of the PSM, see Fig. 1A. we consider a set of cellular oscillators in the PSM, described by their phase ϕ in the oscillation cycle. Growth of the tissue and frequency of the oscillators is regulated by a signal Q that moves with the cell flow and is degraded, see Fig. 2A. Its spatio-temporal distribution is described by the one-dimensional activity field Q(x,t). Furthermore, we introduce a phase field ϕ(x,t) that represents the local state of the cellular oscillators in the PSM [13, 27]. The dynamic equations for ϕ and Q are given by

$${\partial }_t\varphi +v{\partial }_x\varphi =\omega +\epsilon {\partial }_x^2\varphi ,$$ (1)

$${\partial }_{t}Q+{\partial }_x(vQ)=-kQ+\mu \sigma (x),$$ (2)

FIG. 2.

(A) Interaction structure of the model. Gray arrows represent interactions only present in the extended model. (B) Steady state solutions Q(x) and Ψ(x), given by Eqs. (12) and (8). The lowest panel shows the associated wave pattern p(Ψ) = (1 C cos Ψ)/2. Parameters are k = 1, g = 1, Q* = 0.05, k ₀ = 0.3, w ₀ = 4, ε0 = 0:004, x ₀ = 1 which yield Ω = 26:7, S = 1:28, x = 4:75, W ≃ 2:2 by Eqs. (9), (11), (13), and (14). The numerically obtained value for W is W = 2:73.

where v(x,t) is a velocity field accounting for cell flow, w(x,t) is the intrinsic frequency of the cellular oscillators, ε(x, t) is the coupling strength, μ is the production rate,, and k is the decay rate. For simplicity, we here consider a constant length xo of the source region, $\sigma (x)=\Theta \left(x_0-x\right)$. In our model, the local elongation rate ∂_x v as well as the frequency and coupling strength profiles ω and ε are controlled by Q through the relations

$${\partial }_{x}v={\kappa }_{0}Q/Q^{\ast },{\hspace{1em}v|}_{x=0}=0,$$ (3)

$$\omega ={\omega }_{0}Q/Q^{\ast },$$ (4)

$$\epsilon ={\epsilon }_{0}Q/Q^{\ast }$$ (5)

where k ₀, w ₀, and ε ₀ are a characteristic elongation rate, oscillation frequency, and coupling strength, respectively. Note that the dependence of v on Q makes Eq. (2) a nonlinear equation. The position $\overline{x}(t)$ of the anterior end of the PSM is defined as the point where the level of Q reaches the threshold level Q*,

$$Q(\overline{x}(t),t)=Q^{\ast }$$ (6)

see Fig. 2B. We consider open boundary conditions for the phase field, ${{\partial }_x\varphi |}_{x=0}=0$.

The system of Eqs. (1–5) has a solution with a stationary activity profile $Q(x,t)=Q(x)$) and a time-periodic wave pattern $\varphi (x,t)=\Omega t+\psi (x)$, where Ω is the collective frequency with which the wave pattern corresponding to the phase profile Ψ repeats, see Fig. 2B. The profile of Q generates a spatial profile of intrinsic frequencies through Eq. (4). In the source region, the profile of Q is flat due to the balance of production, decay, and growth. The frequency profile leads to a wave pattern, i.e., different parts of the PSM are out of phase. [11, 13, 27]. The pattern is flat in the posterior and displays a characteristic wavelength in the anterior, in accordance with experiments [17]. The number W of waves in the PSM and the segment length S at formation are given by [27]

$$W=[\psi (0)-\psi (\overline{x})]/2\pi ,\hspace{1em}S=2\pi /\left|{\psi }^{\prime }(\overline{x})\right|.$$ (7a,b)

The number of waves is the total phase difference between posterior tip x = 0 and anterior end $\overline{x}$, and the segment length is given by the local wavelength of the pattern at the anterior end.

Interestingly, general properties can be discussed independent of knowledge of the full solution. We here consider the case in which coupling only provides a minor correction to the phase pattern. As an approximation, we set ε = 0 in Eq. (1). For the phase profile ϕ, we thus obtain [27]

$$\psi (x)={{{\displaystyle \int }}^{}}_0^x\frac{\omega (y)-\Omega }{v(y)}\text{d}y.$$ (8)

The condition v(0) = 0 determines the collective frequency as

$$\Omega =\omega (0)={\omega }_0{Q}_0/Q^{\ast }$$ (9)

where Q ₀ = Q(0). Consequently, the segment length is given by $S=\left(2\pi \overline{v}/{\omega }_0\right){\left(Q_0/Q^{\ast }-1\right)}^{-1}$, where $\overline{v}=v(\overline{x})$. The velocity v at the anterior end and the posterior concentration Q ₀ can be obtained from the stationary profile Q(x), which, according to Eq. (2), satisfies

$$kQ=\mu \sigma (x)-\frac{\text{d}}{\text{d}x}(vQ).$$ (10)

Eliminating Q in Eq. (10) by using Eq. (3), we obtain an equation for the elongation rate dv/dx. Integrating from 0 to x yields $\overline{v}=\mu x_0\lambda /Q^{\ast }$ for the case $\overline{x}\ge x_0$, where $\lambda ={\left(1+k/{\kappa }_0\right)}^{-1}$. The posterior concentration Q ₀ is determined by Eqs. (10) and (3) at the posterior boundary x = 0 as $Q_0/Q^{\ast }=\left[{\left(k^2+4{\kappa }_0\mu /Q^{\ast }\right)}^{1/2}-k\right]/2{\kappa }_0$. Hence, the segment length is given by

$$S=\frac{\mu \lambda T_0}{Q_0-Q^{\ast }}{x}_0,$$ (11)

where $T_0=2\pi /{\omega }_0$. Since $\text{d}\lambda /\text{d}{\kappa }_0>0$ and $\text{d}{Q}_0/\text{d}{\kappa }_0<0$, the segment length S increases with increasing elongation rate. Note that S is proportional to the length x ₀ of the source region, which is the only length scale in the system.

The full solution for the steady state of Q is given by

$${Q|}_{x<x_0}=Q_0$$ (12)

$${Q|}_{x>x_0}=\frac{(1-\beta )Q_0}{1+W{\left(-{\beta }^{-1}{\text{e}}^{-{\beta }^{-1}\left(1+{(1-\beta )}^2\left(x-x_0\right)/x_0\right)}\right)}^{-1}}$$

where $\beta =1+\left(k/{\kappa }_0\right)\left(Q^{\ast }/Q_0\right)$ and W is the principal branch of the Lambert W function, defined by the relation $W(z){\text{e}}^{W(z)}=z$ [28]. The growth field generated by Q through Eq. (3) corresponds to a velocity field, where cells move anteriorly and reach their maximum speed at the anterior end of the tissue. From Eq. (12), the PSM length $\overline{x}$ can be obtained using the definition Eq. (6),

$$\overline{x}=\frac{\beta }{{(1-\beta )}^2}(\beta +\lambda -\log \beta \lambda -2)x_0.$$ (13)

The number of waves W can be obtained using Eqs. (8) and (4) and approximating the velocity field by the velocity at the anterior end, $v(x)\simeq \overline{v}$,

$$W\simeq \frac{\Omega }{2\pi }\left(\frac{\overline{x}}{\overline{v}}+\frac{1-\beta }{k}\right).$$ (14)

Dynamic solutions of the minimal model with constant PSM length are shown in Fig. 2B (see also Supplemental Movie 1). The stationary profiles for Q(x) and Ψ(x) describe the periodic wave-like pattern of gene expression that is determined by a signaling gradient. The waves propagate in a dynamic tissue that expands with a velocity profile determined by Eq. (3).

Segmentation governed by two opposed signaling gradients

The minimal model does not capture PSM shortening over time and therefore leads to an infinite sequence of segments (≃infinite snake’) [9, 13]. PSM shortening and thereby a structure with a finite number of segments can be achieved by introducing a second signaling gradient R opposed to Q which triggers additional degradation of Q, see Fig. 2A. The dynamic equations for the phase field and the concentrations Q and R are given by

$${\partial }_t\varphi +v{\partial }_x\varphi =\omega +\epsilon {\partial }_x^2\varphi$$ (15)

$${\partial }_{t}Q+{\partial }_x(vQ)=E{\partial }_x^{2}Q-kQ-k^{\prime }RQ+\mu \sigma (x),$$ (16)

$$\begin{array}{lll}{\partial }_{t}R+{\partial }_x(vR)\hfill & =\hfill & D{\partial }_x^{2}R-hR-h^{\prime }QR\hfill \\ \hfill & \hfill & +\left[v+v^{\prime }\rho (\varphi )\right]\Theta \left(Q^{\ast }-Q\right)\hfill \end{array}$$ (17)

For the signaling activities, we consider no-flux boundary conditions at the posterior tip, ${{\partial }_{x}Q|}_{x=0}={{\partial }_{x}R|}_{x=0}=0$, and open boundary conditions for the phase field, ${{\partial }_x\varphi |}_{x=0}=0$. Here, v is the basal production rates of R, v’ is the production rate of R in the formed segments, D and E are diffusion constants, h and k are decay rates, and h’ and k’ indicate the degree of mutual degradation of Q and R. The dependence of the production rate of R on $\rho (\varphi )=(1+\cos \varphi )/2$ leads to production in the center of the formed segments.

The mechanism proposed here is capable of describing the self-organized dynamics of segmentation and length decrease of the PSM. As initial condition, we start with a steady state, in which Q and R form opposing gradients in the absence of phase dynamics, see Fig. 3A. This corresponds to no segments and a PSM of finite length with a flat phase profile. Figs. 3A,B show snapshots of a numerical solution of Eqs. (15–17) with (3–5) for different time points (see also Supplemental Movie 2). As soon as waves leaves the PSM, i.e., as they enter the region where Q < Q*, they become sources of R according to Eq. (17), see Fig. 3A. The thus elevated levels of R diffuse into the PSM and lead to an increased degradation of Q in the vicinity of the anterior end (at Q = Q*) and the profile of Q shortens towards the posterior, see Figs. 2C and 3A. Consequently, the PSM shortens, see Fig. 3C. Finally, the segmentation process terminates with the PSM reaching zero size, x = 0.

FIG. 3.

(A) Snapshots of the time evolution of the system Eqs. (15–17) and (3–5). Spatial distributions of Q (green), R (red) and p(0) (blue) at different points in time (from top to bottom: t = 0,0.5, 2,11.33). The dashed line marks the threshold level Q * which also sets the x-axis scale for concentration levels. The colored top panel shows a density plot representation of $\rho (\varphi )=(1+\cos \varphi )/2$ showing the corresponding wave pattern. (B) Velocity profile v (solid curve) as determined from Eq. (3) and $\overline{v}$ (dashed line) as velocity reference for the respective time points in A. In A and B, the shaded area marks the PSM region where Q > Q *. (C) Time evolution of the PSM length $\overline{x}$, defined through Eq. (6), the number of waves w, the segment length s at time of formation, and the segment number n. Parameters are E = 0.05, k = 1, k’ = 150, u = 1, D = 1, h = 5, h’ = 40, v = 0.5, v’ = 3, Q* = 0.05, k ₀ = 0.15, w ₀ = 4, ε0 = 0.004, x ₀ = 1.

To discuss the dynamic features of this model, we define the time-dependent number of waves and the segment length, $w(t)=[\varphi (0,t)-\varphi (\overline{x}(t),t)]/2\pi$ and $s(t)=2\pi /\left|{\partial }_x\varphi (\overline{x}(t),t)\right|$, respectively. Furthermore, we define the number of formed segments as the number of completed oscillations at the anterior end, $n(t)=\varphi (\overline{x}(t),t)/2\pi$. Fig. 3C shows these functions together with the time evolution of the PSM length. The time dependence of these quantities capture key features of the segmentation process as seen in experiments, such as the decrease in PSM length, the formation time of segments, and the decrease of segment length over time [17, 27]. The non-monotonic time dependence of the number of waves is not observed in experiments and is related to the constant source length, which we here chose for simplicity.

Control of wave patterns and morphologies

We now show that we can obtain a variety of different oscillation patterns and morphologies by changing the dynamics of signaling. As an example, we vary two key parameters: the diffusion constant D of the signaling component R, and the elongation rate k ₀, which sets the scale of cell flow velocity. We define the total time T of segmentation as the time at which the PSM has shortened to zero. The average number of waves in the PSM is denoted by $W={\langle w(t)\rangle }_{0\le t\le T}$, the minimal segment length is $S=\underset{}{{\min }_{0\le t\le T}}s(t)$, and the total segment number is N = n(T). Fig. 4 shows these observables as a function of D and k ₀. We find that the total time of segmentation diverges at the boundary shown in Fig. 4A. Within a parameter range (gray), segments are generated in a time-periodic manner without end (≃infinite snake’) corresponding to the simplified theory Eqs. (1,2). Biologically relevant parameters are found in the green region, where the total time T of segmentation is finite. This occurs if advection described by k ₀ is small enough that R can diffuse sufficiently far into the PSM to degrade Q. In this parameter region, the number of waves W decreases with stronger advection, while the segment length S increases, see Figs. 4B,C. The latter trend can be understood through the simplified model, see Eq. (11) and below. Interestingly, the total segment number N displays a non-monotonic behavior as a function of the elongation rate k ₀ with the smallest number of segments formed for intermediate values of k ₀ see Fig. 4D.

FIG. 4.

Key observables as a function of the diffusion constant D of R and the elongation rate k ₀. (A) Regions in the parameter space in which segmentation terminates after a finite number of segments (green) and where the system attains a steady state at finite PSM length, forming infinitely many segments (gray). The black dot marks the parameter set shown in Fig. 3. (B) Average number of waves, (C) total number of segments, (D) minimum segment length. The other parameters are as given in the caption of Fig. 3.

Discussion

In this paper, we have discussed the segmentation of the vertebrate body plan as a self-organized patterning process combining an elongating tissue with coupled genetic oscillators controlled by signaling gradients. Using a minimal model, we have illustrated how the biochemical properties of signaling regulate the key features of segmentation. An extended model with two opposing signaling gradients provides a mechanism leading to self-organized termination of segmentation with a finite number of segments. By varying biochemical parameters of the signaling factors, a variety of different patterns and morphologies can be generated. Despite the simplicity of the model, the self-organized patterning system described here can account for the differences in the segmentation process between different species, e.g., in fish, mouse, chick, snake, and frog. Among these species, the observed number of segments ranges from about ten to several hundred, the number of waves ranges from one to five [8]. Note that the principle of self-organization proposed here does not depend on whether the posterior signaling molecule is diffusible or not. In particular, frequency and growth profiles could emerge either from diffusion or from advection or both [29, 30].

Which signaling molecules regulate segmentation in vivo? The PSM exhibits posterior protein concentration profiles of FGF (fibroblast growth factor) and Wnt and an opposing anterior profile of RA (Retinoic Acid) [8, 31–37], see Fig. 1C These signaling molecules are thought to be involved in regulating cell fate during segmentation, i.e., maintaining cells in an oscillatory state within the PSM and triggering segment formation upon arrival at its anterior end [18]. Experimentally controlled reduction of Wnt signaling, for instance, leads to a shorter average PSM length, faster PSM shortening, and longer segments [37], all observations consistent with our model. Furthermore, our model could be tested by experimentally induced up- or downregulation of Retinoic Acid in a similarly quantitative and dynamic assay system. How signaling activity regulates the frequency of the cellular oscillators and cell fate on a molecular level is yet unknown and a detailed understanding remains a challenge for future experimental and theoretical research.