Continuum theory of gene expression waves during vertebrate segmentation

Abstract The segmentation of the vertebrate body plan during embryonic development is a rhythmic and sequential process governed by genetic oscillations. These genetic oscillations give rise to traveling waves of gene expression in the segmenting tissue. Here we present a minimal continuum theory of vertebrate segmentation that captures the key principles governing the dynamic patterns of gene expression including the effects of shortening of the oscillating tissue. We show that our theory can quantitatively account for the key features of segmentation observed in zebrafish, in particular the shape of the wave patterns, the period of segmentation and the segment length as a function of time.


Introduction
In all vertebrate animals, the segmentation of the body plan proceeds during embryonic development in a process termed somitogenesis [1]. During somitogenesis, the elongating body axis segments rhythmically and sequentially into somites, the precursors of vertebrae and ribs. Failure of proper segmentation, caused for instance by mutations, can give rise to birth defects such as congenital scoliosis [2]. Somites are formed in characteristic time intervals from an unsegmented progenitor tissue, the presomitic mesoderm (PSM) ( figure 1(A)). The temporal regularity with which somites form has provoked the idea that a biological clock comprised of cellular oscillators coordinates the temporal progress of segmentation in the PSM. The so-called 'clock-and-wavefront' mechanism suggests that a wavefront at the anterior end of the PSM reads out the state of this clock and triggers the formation of a new segment upon each completed clock cycle [3]. Indeed, patterns of oscillating gene expression have been found in the PSM of various vertebrates such as zebrafish, chick, mouse, frog, and snake [1]. These patterns resemble traveling waves sweeping through the PSM and occur as a result of coordinated cellular oscillations in the concentration of gene products ( figure 1(B)). Genetic oscillations are proposed to occur autonomously in single cells as a result of delayed autorepression of specific genes [5,6]. Cellular oscillators mutually couple through Delta-Notch signaling between neighboring cells, which tends to locally synchronize their oscillatory dynamics [7][8][9][10][11]. Local synchronization due to coupling is important to maintain coherent wave patterns by preventing the cellular oscillators from drifting out of phase due to noise in gene expression [12][13][14]. The emergence of traveling waves at the tissue level has been linked to a gradual slowdown of genetic oscillations in the PSM along the body axis [1,13,15,16]. This gradual slowdown corresponds to a spatial profile of intrinsic frequencies of the cellular oscillators.
During segmentation, the waves of gene expression emerge at the posterior of the PSM and travel towards its anterior end, where the new segments are formed ( figure 1(B)). Segment formation occurs upon arrival of a wave at the anterior end of the PSM. This corresponds to the formation of one segment with each completed oscillation cycle at the anterior end [4]. Segmentation is a highly dynamic process: in parallel with segment formation, the body axis elongates while at the same time PSM changes its length as cells leave the PSM at the anterior end to form somites [4,16]. A shortening of the PSM, as observed in Zebra fish moves relative to the waves giving rise to a Doppler effect (figure 1(C)) [4]. The motion of the anterior end relative to the posterior tip leads to an increase of the frequency of oscillations seen by an observer at the anterior end. Since the oscillation frequency at the anterior end specifies the rate of segmentation, this Doppler effect contributes to a decrease of the period of morphological segment formation. In addition to the Doppler effect, the wavelength of the pattern dynamically changes over time. This leads to a modulation of the local frequency and contributes to an increase of the period of segmentation. Together, both effects combine to determine the timing of segment formation. Hence, in addition to the time scale of genetic oscillations, the rate of segment formation is regulated by the time scale set by tissue shortening and the wavelength of the wave pattern. These observations highlight the need to capture the effects of tissue deformation in theories of vertebrate segmentation.
In this paper, we present a minimal continuum theory of vertebrate segmentation based on coupled phase oscillators in a dynamic medium that takes into account local growth and shortening of the oscillating tissue during the segmentation process. In section 2, we introduce our continuum theory of vertebrate segmentation and the key observables that can be obtained from the theory. In section 3, we illustrate the basic mechanism of pattern formation with oscillators using a simplified scenario with constant length of the oscillating tissue. In section 4, we apply our theory to quantitatively describe segmentation in developing zebrafish embryo, taking into account tissue shortening. In section 5, we discuss the factors that regulate the period of segmentation and show how a Doppler effect and a dynamic wavelength effect emerge from the interplay of tissue shortening and changing wave patterns. In section 6, we discuss our findings and give an outlook for further research.

Continuum theory of coupled oscillators in a dynamic medium
Here we introduce a theory that aims to describe the wave patterns in the PSM and the dynamic features of segmentation that result from these wave patterns. The wave patterns and the timing of segmentation have previously been quantified in transgenic zebrafish embryos, in which oscillating genes have been tagged with a fluorescent marker protein [4]. Waves can be traced by introducing a one-dimensional coordinate x along the curved embryonic body axis and measuring the fluorescent intensity level along this axis over time (figures 1(B) and 2(A)). Since these wave patterns are a tissue-level phenomenon and phase differences between neighboring cellular oscillators are typically small, we here choose a coarse-grained continuum description of the oscillatory medium. We describe the local state of oscillation by a phase field x t , . f ( ) Our theory combines three key ingredients involved in pattern formation during vertebrate segmentation: (i) autonomous oscillators with a spatial profile x w ( ) of intrinsic frequencies [13,15], (ii) local oscillator coupling with strength ε [10,13], and (iii) a cell velocity field v(x) capturing deformation and elongation of the segmenting body axis [17,18]. The dynamic equation for the phase field f is given by [13]  t The intrinsic frequency of the oscillators is described by a position-dependent frequency profile x . w ( ) Motion of the cellular oscillators is described by an advective term where v is the cell velocity. In previous work, we have considered a constant velocity v. Local oscillator coupling with strength ε is described by a term that tends to even out local phase differences and thus describes the oscillators' tendency to locally synchronize [19]. We impose open boundary conditions, which corresponds to the situation where there are no oscillators beyond the posterior tip.
In order to describe a shortening PSM, we consider the simple case where the frequency and the velocity profile are rescaled with tissue length where U and V are spatial profiles that are adjusted to the variant length x t ( ) of the PSM, 0 w is the maximum frequency at the posterior tip x = 0, and v 0 is a typical velocity.
Phase waves travel in an anterior direction if the frequency profile attains its maximum frequency at the posterior tip x = 0 and decays in an anterior direction [13,15]. For simplicity, we consider that oscillations have ceased beyond the wavefront and therefore choose the following frequency profile figure 2(B), where x x x =¯denotes a non-dimensional position coordinate and k −1 is a characteristic (nondimensional) length scale of the profile. The function U has the boundary values U 0 1 = ( ) and U 1 s = ( ) ( figure 2(B)).
The velocity field in the segmented region can be estimated from experiments by tracking the velocity of segment boundaries, see appendix A. Choosing the boundary condition v 0 0, = ( ) a simple choice for the velocity profile consistent with the quantified data is figure 4(B). The velocity gradient v corresponds to local growth rate with a profile v x qv x e qx x 0 ¶ ¶ = -(¯)t hat takes its maximum value at the posterior tip x = 0 and decays over the characteristic length scale x q. The choice of the functional forms for U and V are motivated by experimental observations as they give rise to the type of wave patterns observed in experiments with waves moving in anterior direction and slowing down as they approach the anterior end, see section 3.
The number of waves that simultaneously sweep through the PSM is a key observable that can be measured in experiments [1]. In terms of the phase field f, the number of waves K(t) is given by Hence, K 2p is the total phase difference between the posterior tip x = 0 and the anterior end x x =¯of the PSM. A new segment is formed after each completed oscillation cycle at the anterior end x x =¯ [4]. Accordingly, the number of formed segments at time t is given by and the rate of segment formation is N t d d .The length S of the formed segments at the time t of their formation is given by the wavelength of the pattern at the anterior end, and obeys x t x S t t , , In the case where x f ¶ ¶ does not vary strongly over the length S, the segment length can be approximated as

Time-periodic patterns
We first discuss time-periodic patterns to illustrate how the properties of the wave pattern depend on the parameters of our theory. Such patterns occur for constant PSM length, x t x .
) the system attains a timeperiodic state after transient dynamics. This time-periodic state can be expressed in the form [13,19] x t t x , , where Ω is the collective frequency and the spatiotemporal pattern x t sin , f ( )is fully characterized by the timeindependent phase profile x .
y ( ) The rate of segment formation N t d d , defined through equation (7), is given by N t d d 2p = W and hence given by the collective frequency. Using the time-periodic ansatz equation (9) in equation (1), the phase profile ψ obeys the ordinary differential equation It is instructive to consider the case of weak coupling, in which the coupling term provides only a minor correction to the collective frequency and the phase profile ( figure 3(B)). Neglecting 2 e y ( ) in equation (10), we find the collective frequency , 0 w W  the maximum of the frequency profile at the posterior tip. The phase profile ψ can then be approximated as   Figure 3(B) shows the approximation equation (11) together with the phase profile obtained from a numerical solution of equation (1) including the effects of coupling. The number of waves that simultaneously sweep through the PSM is given by K The length S of formed segments is constant and given by equation (8) as and defined the collective period T 2 . p = W This relationship is wellknown from the clock-and-wavefront model [3,13]. Note that in the case of a velocity profile it only holds approximately and only for time-periodic solutions. The phase velocity v of the waves can be obtained as the velocity of a point x * with constant phase, x t t , [20]. Differentiating this relation with respect to time yields the phase velocity v t x , ) ( )| ( ) which exists at any position x. Using equations (9) and (11) The phase velocity v x( ) is always positive and larger than v(x) because This implies that the waves move in anterior direction and faster than the underlying medium moves away from the tip.

Dynamic patterns in a shortening tissue
We now consider the more realistic situation where the oscillating tissue changes its length as is the case for the PSM in developing vertebrate embryos. Here we focus on the spatiotemporal pattern of the oscillating gene Her1. The patterns of this gene product can be observed in vivo by a fluorescent label that is introduced in the transgenic zebrafish line Looping [4]. In zebrafish, the PSM substantially shortens during segmentation [4]. The time dependence of the PSM length x t ( ) can be well captured by the function [4] x t x x t t tanh . 14 Figure 4(A) shows this function with parameters given in table 1 together with experimental data points from [4]. Here, t = 0 corresponds to the formation time of the 7th segment. We now discuss our model taking into account this time dependence of the PSM length. Figure 5(A) shows a kymograph of a numerical solution to equations (1)-(3) using equation (14). The experimentally obtained phase profile from [4] is shown in figure 5 4(B)), see appendix A. The remaining parameters were obtained from fits of the theoretical phase profile to the experimental wave pattern shown in figure 5(B) (for fit procedures see appendix B). An alternative way to display the wave pattern is to introduce the time-dependent phase profile x t x t t , , 0 , , figure 6(A). Note that for time-periodic solutions this becomes the time-independent phase profile defined in equation (9). Figure 6(A) reveals that the wavelength of pattern decreases over time as wave peaks are moving closer together. Furthermore, it can be seen that the number of waves in the PSM decreases over time as the anterior end cuts off one wave peak while the PSM is shortening. The fact that the number of waves in the PSM changes over time shows that the phase profile does not simply scale with the PSM length. Figure 6(B) shows the number of waves as a function of the number of formed segments both from numerical solutions of the phase model and from experiments as presented in [4]. The number of waves substantially decreases during segmentation, which is captured well by the theory ( figure 6(B)). The discrepancy between the solid line in figure 6(B) and the experimental data for segments N 18  suggests that the scaling frequency and velocity profiles, equations (2) and (3), are too simple to capture the wave patterns at late times.
Our theory can also quantitatively account for the features of morphological segment formation. Figures 6(C) and (D) show a comparison of our theory to experiments for the formation time and segment length as a function of the segment number N, respectively (for details see appendix A). The segment length S shows a non-monotonic behavior with largest segments being formed around the 12-segment mark, a behavior also found in wildtype zebrafish [21]. This demonstrates that our theory can quantitatively account for the dynamic features of vertebrate segmentation.

Period of segmentation
A fundamental feature of segmentation is that segments are formed rhythmically and sequentially. Which factors determine the period of morphological segment formation? From the definition equation (7)  where Ω P is the posterior frequency, Ω D is a Doppler contribution and Ω W is a 'dynamic wavelength' contribution. These frequencies are defined by where the phase profile ψ is defined in equation (15). The contribution Ω P is the local frequency at the posterior tip of the tissue at x = 0. The contribution Ω D results from a Doppler effect where x t d d is the speed of the moving observer (the anterior end) traveling into a wave with wavelength ) The contribution Ω W is caused by the change of the phase profile ψ over time, which corresponds to a dynamic change of the wavelength.
Using our theory, we can derive an explicit relation between Ω A and Ω P for the simple case of linear shortening of the PSM, x t v d d , where In equation (19), the factor v v 1 0 +¯describes the Doppler effect with the speed v¯of the moving observer (the anterior end) and the cell velocity v 0 . The factor 1 -D describes the effects caused by changing phase profile due to the shortening of the frequency profile with the PSM length. Hence, this term describes the dynamic wavelength effect. Because 0, D > this factor opposes the Doppler effect. . The average anterior frequency Ω A is thus larger than the posterior frequency Ω P . The Doppler effect and the dynamic wavelength effect can be discussed in the context of classical wave physics.

Doppler effect Consider a wave equation in one-dimension
where u x t , ( )is the amplitude of the wave and c is the wave propagation speed. We consider a wave-emitting source with frequency ω and amplitude u 0 at x = 0 through the boundary condition A simple solution to equation (21) satisfying the boundary and initial conditions (22) and (23) is The phase pattern of this wave is x t t qx , .
which corresponds to Ω D in equation (18). Note that v c 1 , w W = + (¯) which is the usual expression for the Doppler effect of a moving observer [22]. The wave pattern described by equation (24) is shown as a kymograph in figure 8(A). This pattern can be used to illustrate the Doppler effect by considering an observer at rest (dashed white line) compared to an observer moving towards the source (solid white line). The moving observer crosses more wave peaks as compared to the observer at rest during the same time interval and hence observes a higher frequency.  figure 8(B). This pattern can be used to illustrate the dynamic wavelength effect by considering two observers at rest with different positions. An observer at rest that is more distant from the source (solid white line) crosses a smaller number of wave peaks compared to an observer closer to the source (dashed white line). Hence, the observer more distant from the source observes a smaller frequency.

Dynamic wavelength effect
Doppler effects are commonly found in wave physics. However, the dynamic wavelength effect is more unconventional. A time-dependent index of refraction as illustrated here occurs, e.g., in gases ionized by laser pulses due to a spatially and temporally inhomogeneous distribution of free electrons [23,24].

Discussion
In this paper, we have introduced a continuum model of coupled phase oscillators in a dynamic medium to capture the dynamics of vertebrate segmentation. For simplicity, we have considered frequency and velocity profiles that scale with the time-dependent PSM length. Note that the phase profile itself does not scale in contrast to an earlier proposal [25]. Extending previous work [3,13,15,19,26], our approach takes into account tissue deformation due to growth of the embryonic body axis and the change of the PSM length over time. This enables us to quantitatively account for the morphological features of segmentation such as the timing of segment formation and the length of newly formed segments as observed in developing zebrafish embryos. The frequency and velocity profiles that scale with PSM length capture well the time-dependence of the experimentally observed wave patterns. The parameters obtained from the fit to the experimental data suggest that the frequency profile at the anterior end jumps from a finite value to zero. Such a behavior could, e.g., be caused by a Hopf bifurcation. Indeed, if the cellular oscillations pass a Hopf bifurcation from the oscillating state to the non-oscillating state when reaching the anterior end of the PSM, this would give rise to a frequency jump. Moreover, our theory describes the experimentally observed Doppler and dynamic wavelength effects, which regulate the timing of segment formation [4]. In particular, our results imply that the rate of segmentation in zebrafish is faster than the fastest local oscillation frequency found anywhere in the system. This remarkable behavior is due to the interplay of wave patterns and tissue shortening. The Doppler and dynamic wavelength effects observed in zebrafish are a result of the shortening of the PSM and the corresponding decrease in the local wavelength of the wave pattern. We predict these effects in general to occur also in other species. However, the signs and their role during different developmental stages could vary. The signaling pathways and the cellular processes that regulate and mediate the shortening of the PSM, the elongation of the body axis, and the specification of the frequency profile are as yet unknown and remain open challenges for future experimental and theoretical research. version of figure A1(A) smoothened with a moving average of width 12 pixels. Subsequently, the local intensity minima, which correspond to the positions of the segment boundaries, are determined with a peak-finding algorithm. The result is shown in figure A1(B). In the next step, nearby points are connected to obtain time series of the segment boundaries' positions. The resulting traces are shown in figure A1(C). For each segment boundary, we perform linear fits of the boundary position at early and late times to determine its velocity. To obtain a velocity profile, we compute the average position of each segment boundary and assign the velocity of the corresponding boundary to it ( figure 4(B)). The velocity profile within the PSM is inaccessible with the available dataset.
The segment length at the time of segment formation (figure 6(D)) was determined from these time-lapse microscopy movies following the procedure described in [21]. Specifically, the segment length was obtained by determining the distance between two successive indentations of the PSM at the anterior end of the tissue (figure A2).

Appendix B. Fits of theoretical phase profiles to experimental data
We use the shape parameters σ, k, and q of the frequency and velocity profiles as fit parameters. To generate dynamic patterns in our theory, we start the system at time t 0 0 < with x t , 0 0 f = ( ) to create initial conditions at t = 0. We fit the calculated patterns x t , f ( ) for t 0  to the experimental data, using σ, k, q, and t 0 as fit parameters. The time t = 0 corresponds to the formation time of the 7th segment. The experimental phase map The explicit form of u j ( ) can now be found using initial and boundary conditions. We evaluate equation (C.4) at x = 0 using open boundary conditions, x 0 is time-independent. In the second line of equation (C.10), we have integrated by parts and introduced Δ given by equation (20). Using the result (C.10) and x v = -¯in equation (C.7), we finally obtain The posterior frequency Ω P = t x 0 f ¶ ¶ = ( )| can be obtained using equations (C.3) and (C.6), which yields . P 0 w W = Thus, we can interpret equation (C.11) as a relation between Ω A and Ω P . This completes the derivation of equation (19).