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Upregulation of Forces and Morphogenic Asymmetries in Dorsal Closure during Drosophila Development

X. G. Peralta ^*, Y. Toyama ^*, M. S. Hutson , R. Montague , S. Venakides , D. P. Kiehart  and G. S. Edwards ^*

Departments of ^* Physics, Biology, and Mathematics, Duke University, Durham, North Carolina; and Department of Physics and Astronomy, Vanderbilt University, Nashville, Tennessee

Correspondence: Address reprint requests to D. P. Kiehart, E-mail: dkiehart{at}duke.edu.

	ABSTRACT

TOP ABSTRACT INTRODUCTION MATERIALS AND METHODS RESULTS DISCUSSION APPENDIX A: DETERMINATION OF... APPENDIX B APPENDIX C APPENDIX D: ESTIMATE OF... SUPPLEMENTARY MATERIAL ACKNOWLEDGEMENTS REFERENCES

 
Tissue dynamics during dorsal closure, a stage of Drosophila development, provide a model system for cell sheet morphogenesis and wound healing. Dorsal closure is characterized by complex cell sheet movements, driven by multiple tissue specific forces, which are coordinated in space, synchronized in time, and resilient to UV-laser perturbations. The mechanisms responsible for these attributes are not fully understood. We measured spatial, kinematic, and dynamic antero-posterior asymmetries to biophysically characterize both resiliency to laser perturbations and failure of closure in mutant embryos and compared them to natural asymmetries in unperturbed, wild-type closure. We quantified and mathematically modeled two processes that are upregulated to provide resiliency—contractility of the amnioserosa and formation of a seam between advancing epidermal sheets, i.e., zipping. Both processes are spatially removed from the laser-targeted site, indicating they are not a local response to laser-induced wounding and suggesting mechanosensitive and/or chemosensitive mechanisms for upregulation. In mutant embryos, tissue junctions initially fail at the anterior end indicating inhomogeneous mechanical stresses attributable to head involution, another developmental process that occurs concomitant with the end stages of closure. Asymmetries in these mutants are reversed compared to wild-type, and inhomogeneous stresses may cause asymmetries in wild-type closure.

	INTRODUCTION

TOP ABSTRACT INTRODUCTION MATERIALS AND METHODS RESULTS DISCUSSION APPENDIX A: DETERMINATION OF... APPENDIX B APPENDIX C APPENDIX D: ESTIMATE OF... SUPPLEMENTARY MATERIAL ACKNOWLEDGEMENTS REFERENCES

 
Embryonic development comprises dynamic processes that link gene expression to morphogenesis. The processes involve a precisely orchestrated interplay between gene expression, tissue movement, and cell shape changes. The mechanisms that regulate this interplay are under intense investigation (1).

Dorsal closure is an essential stage of Drosophila embryogenesis that requires 2–3 h of spatially coordinated and temporally synchronized tissue movement and serves as a model system for research in developmental biology ((2–4), reviewed in the literature (5,6)). In addition, the cellular and molecular mechanisms involved in dorsal closure are thought to parallel those that characterize wound healing ((2,7), reviewed in the literature (8–10)). During dorsal closure, two advancing flanks of epidermal tissue demarcate an eye-shaped opening of exposed amnioserosa tissue on the dorsal side of the embryo (Fig. 1). As closure proceeds, the advancing flanks of epidermal tissue progressively cover the dorsal opening. At each canthus, i.e., corner of the eye-shaped opening, adhesion between the two advancing flanks forms a seam in a process known as zipping. The canthi anchor the purse-strings and influence their curvature, where each leading edge approximates an arc and the two arcs lie roughly in a plane.

View larger version (48K): [in this window] [in a new window]   FIGURE 1  (a) Geometric parameters that describe dorsal closure, where dashed lines are contours of the leading edges (see text). The arc length LR is indicated by the shaded line. (b,c) Confocal fluorescent images of native closure taken 1700 s apart. (d,e) Images of the single-canthus nicking protocol targeted to the posterior end, also taken 1700 s apart. In frame d, dark patches are evident near the left purse-string and the posterior canthus. Dark patches were observed occasionally and are due to the dorsal surface lying just outside the focal plane; however, scanning the focal plane verified the tissues were continuous. In frame e, the X characters indicate the location and extent of the laser perturbations. Scale bar is 50 µm. The anterior end (A) is to the left in all figures.

The mechanics of morphogenesis have also been investigated and, in some cases, the physical forces responsible for tissue movements have been identified (2,4,10,17–22). Laser perturbations have proven to be useful tools to probe such forces. Previously (2,4), we combined laser microsurgery techniques with biophysical modeling to investigate the forces involved in dorsal closure, where the laser rapidly ablates tissue and our focus is on the response of the remaining tissues and not on wound healing per se. More specifically:

1. We showed that the dynamic geometry of the dorsal opening during closure is a consequence of four biological processes: the contractile force of an actomyosin-rich supracellular "purse-string" located at the leading edge of epidermis; the contractile force due to cortical actomyosin networks in amnioserosa cells; the resistive force due to movement and deformation of the lateral epidermis; and adhesive zipping. We also determined force ladders that specify the relative magnitude of the first three applied forces as compared to the drag force. We found that the vector sum of these applied forces acting on a point of the leading edge is only a small fraction of any one of the individual forces. Although these forces collectively drive a process that is, to a first approximation, quite symmetric, closer inspection makes it clear that various asymmetries are apparent. We contend that quantifying morphogenic asymmetries in native embryos is essential for understanding the spatial and temporal differences in the contributing processes. Moreover, we seek to investigate the extent to which asymmetric forces can vary and still result in successful closure. Ultimately, understanding asymmetry provides a basis for understanding changes that occur to compensate for genetic or laser perturbations.
   
2. We showed that dorsal closure is robust and surprisingly resilient to laser perturbations, as individual biological processes can be removed without blocking closure. Three specific examples are illustrative (4). First, when the force produced by the amnioserosa is removed by laser incision(s), the leading edges rapidly recoil away from the dorsal midline, and after reaching a maximum displacement, closure resumes. Second, when zipping is laser-inhibited at both canthi, the geometry changes dramatically—the dorsal opening distorts such that the leading edges constrict toward the dorsal midline and subsequently meet to form an interior seam that completes closure. Third, when zipping is inhibited at one canthus, closure progresses approximately on schedule, albeit asymmetrically, as zipping is essentially limited to the unperturbed canthus. Collectively, our observations demonstrated that when one of the cellular processes governing dorsal closure was laser-perturbed, the remaining processes could compensate through alterations in the force(s) they produced. In part, this robust and resilient behavior is due to redundancy—more than one force contributes to closure. We contend that having large applied forces combined with the ability to compensate for perturbations and for native variations in force-producing processes contributes to the stability underlying the robustness and the resiliency of closure. With regard to zipping, our observations suggest that while zipping contributes a key role in coordinating forces driving closure and is essential for seam formation, it is not required for the bulk of tissue movements during closure. Our results suggest that identifying specific processes that are upregulated as a consequence of perturbations can elucidate the dynamics that contribute to this compensation, the relationships between apparently independent biological processes, and their relative importance for closure with the potential to ultimately explain how complicated morphogenic movements result in a properly formed tissue.
   
3. We proposed an empirically derived rate equation to describe the zipping rate that incorporates the leading edges into the seam. This equation depends on representing the dorsal opening as symmetric about the anterior-posterior axis and assigning a single zipping rate constant for both canthi. Application of this model to the dynamic geometry of the dorsal opening of myospheroid embryos combined with a comparison to native and laser-perturbed closure indicated that β_PS integrin has an early, important role in zipping. Extending this quantitative model to accommodate spatial, kinematic, and dynamic antero-posterior asymmetries to successfully quantify morphogenic asymmetries is another open research area.
   

As mentioned earlier, dorsal closure has been characterized as being coordinated in space, synchronized in time, and resilient to laser perturbations; however, the detailed mechanisms responsible for these attributes are not fully understood.

In this study, we extend our use of laser confocal microscopy, a steered UV microbeam, genetic strategies, and quantitative biophysical reasoning to advance our understanding of tissue dynamics during dorsal closure. A key objective is to further assess the resiliency of dorsal closure in response to laser perturbations and to identify the specific tissues where biological processes are upregulated. A second objective is to investigate the asymmetric dynamics of dorsal closure in the context of overall embryonic development for wild-type and mutant embryos. In particular, in this study, we:

1. Quantify spatial and kinematic asymmetries observed during dorsal closure in laser-perturbed wild-type and mutant embryos and compare them to the asymmetries of wild-type, unperturbed closure.
   
2. Interpret these asymmetries in terms of molecular and cellular dynamics.
   
3. Elucidate the regulatory consequences of laser-perturbed closure.
   

	MATERIALS AND METHODS

TOP ABSTRACT INTRODUCTION MATERIALS AND METHODS RESULTS DISCUSSION APPENDIX A: DETERMINATION OF... APPENDIX B APPENDIX C APPENDIX D: ESTIMATE OF... SUPPLEMENTARY MATERIAL ACKNOWLEDGEMENTS REFERENCES

 
Experimental design
The fly genetics and optical design are detailed elsewhere (2,15). All Drosophila melanogaster fly lines were homozygous for sGMCA, a transgene that encodes GFP fused to the actin binding region (GFP-moe) of moesin and is expressed under the control of the spaghetti squash promoter. GFP-moe renders F-actin fluorescent in vivo (2,3). We generated stocks in which bsk² (12) and scb² were balanced with CyO, TwiGal4, and UAS-GFP, and mys¹ was balanced with FM7, TwiGal4, and UAS-GFP (all obtained from Bloomington Stock Center, Bloomington, IN). Hemizygous mys¹ and homozygous scb² and bsk² embryos were distinguished from their siblings through the lack of Twi-driven GFP expression using a GFP-dissecting microscope. Wild-type, bsk (12,13), mys (14), and scb (15) mutant embryos were collected, dechorionated, and subsequently mounted in a modified chamber between a gas-permeable membrane (Teflon) and a glass coverslip that allowed development to proceed while time-lapsed confocal images were collected (24).

Fluorescent images were collected with either a Zeiss LSM410 or a LSM510 (488 nm excitation; Carl Zeiss, Minneapolis, MN) laser-scanning confocal microscope using a 40x, 1.3 NA oil-immersion objective or a 25x, 0.8 NA multi-immersion objective, respectively. Scanning time was either 1.08 or 1.14 s and either two or eight scans were averaged to produce an image. Sequences of images were stored for subsequent analysis.

The microbeam uses the third harmonic of a Continuum Q-switched Nd:YAG laser (355 nm; Continuum, Santa Clara, CA), either the YG571C or the Minilite II, which was interfaced to the confocal microscope to dissect the dorsal surface of the embryos with subcellular spatial resolution. It can be steered across the focal plane to make laser incisions of arbitrary shape while acquiring images with the confocal system. The UV laser light was focused onto the dorsal surface, where 4 ns pulses, 300–700 nJ per pulse at 10-Hz repetition rate, produced typical laser lesions of 2 µm in diameter and a lesion depth of <15 µm. The physical mechanism for UV laser lesions in tissue has been attributed to the disruption of chemical bonds, with the subsequent release of both chemical and mechanical energy, resulting in minimal collateral damage to surrounding tissue (25). For sufficient laser intensity, cell lysis is very fast relative to the collection time of the confocal microscope. The signature for ablation in our studies is tissue recoil, which is readily distinguishable from photobleaching effects alone.

Analysis
The contours of the fluorescent leading edges from the confocal images were digitized using a customized active contour algorithm (26) implemented in ImageJ (27). Once digitized, various geometrical parameters of the dorsal opening (Fig. 1 a) were obtained using custom programs in Mathematica (Wolfram Research, Champaign, IL) and analyzed further. W is the canthus-to-canthus length, L_R is the arc length of the right leading edge (top in the image), and h_i is the height as i is indexed through the digitized points. w_A (w_P) is the length of seam formed on the anterior (posterior) end, _A,R is the anterior angle between the right leading edge and the dorsal midline, and _Pos is the total, posterior, angle between the two leading edges given by _{Pos P,R} + _P,L. Similarly, _A,L, _P,R, and _P,L (not shown) are defined by the dorsal midline and the remaining three segments of leading edges near the canthi while _{Ant A,R} + _A,L. In the mathematical model, we treat the purse-string as a cable under tension, T, so a segment of the leading edge is subjected to a force per unit length from both the lateral epidermis, _LE, and the amnioserosa, _AS. The Appendices include detailed accounts of determination of the centroid, dynamic geometry of the dorsal opening, the zipping rate model and comparison to experiment, the asymmetric zipping rate model, analysis of mechanical jump experiments, determination of the recoil velocity, and an estimate of the increase in the force produced by the amnioserosa in the presence of a localized constriction. All results are reported as mean ± SE (n) where n is the number of trials.

	RESULTS

TOP ABSTRACT INTRODUCTION MATERIALS AND METHODS RESULTS DISCUSSION APPENDIX A: DETERMINATION OF... APPENDIX B APPENDIX C APPENDIX D: ESTIMATE OF... SUPPLEMENTARY MATERIAL ACKNOWLEDGEMENTS REFERENCES

 
Mechanics of the dorsal opening during native closure
Spatial asymmetries
While the dorsal opening is approximately symmetric about the antero-posterior (A-P) and left-right (L-R) axes, it frequently exhibits asymmetric teardrop geometry. To quantify this A-P asymmetry, we evaluated the total angle at each canthus _Ant and _Pos (Fig. 1 a). Typically _Ant < _Pos (Fig. 2 a) and averaging over the duration of dorsal closure _Ant,avg is 86(±12)% of _Pos,avg (n = 7) (Table 1). Fig. 2 b shows that the trend in both angles is to decrease as closure progresses.

View larger version (28K): [in this window] [in a new window]   FIGURE 2  Geometric parameters of the dorsal opening for (a–c) a wild-type, unperturbed embryo, and (d–f) a scab mutant (see text). In panels a and d, each circle represents a time point. Solid lines in panel c are the result of the fit based on Eq. 5 and detailed in Appendix B. A fit was not possible in panel f due to complications in implementing Eqs. B-17–B-19 in the analysis; specifically, the data set was too sparse for numerical convergence.

(1)

where we define the x axis to lie along the dorsal midline and the (fixed) origin to be W(0)/2 where W(0) refers to the first image analyzed (Fig. 3 a), x and y are length increments and A is the area of the opening. Two moments are shown in Fig. 3 a; for example, the distance x_i times the area h_ix and the distance h_j times the area x_jy. The points (x_i, h_i) trace out the two leading edges as i is stepped from 1 to N = 100, as do the points (x_j, h_j) as j is stepped from 1 to N = 100. The coordinates x_c and y_c, respectively, quantify the A-P and L-R geometric asymmetries. Averaging x_c and y_c over the duration of dorsal closure imaged for each embryo and then averaging over n embryos, we obtain x_c,avg = 1.3 ± 0.3 µm and y_c,avg = 0.0 ± 0.3 µm (n = 7), with maximum values within 5 µm and 0.6 µm, respectively. This indicates a small but significant A-P asymmetry with the head of the teardrop located posteriorly throughout dorsal closure (Fig. 3 b), and essentially no L-R asymmetry.

View larger version (16K): [in this window] [in a new window]   FIGURE 3  (a) Contour of the leading edges during native closure showing examples of hix and xjy used in the calculation of the centroid xc. Points are the result of the active contour algorithm and indicate the stepping procedure (see text). (b) Contours of the leading edges during native, unperturbed closure, where the dashed contours were taken 1700 s after the initial solid contours. Circles indicate xc at the earlier (solid) or later (open) times. (c) Contours of the single-canthus nicking protocol targeted to the posterior canthus, where the initial contour is just before the first laser incision. (d) Contours of the single-canthus nicking protocol targeted to the anterior canthus, where the initial contour is 900 s after the first laser incision. The vertical line indicates the location of W(0)/2 and the vertical ticks locate the canthi. Scale bar is 20 µm.

Dynamic geometry of the dorsal opening
We summarize the derivation of dW/dt, based on a model for the dynamic geometry of the dorsal opening, confirming the previous empirically derived rate equation (Eq. 2 in (4)) and placing it on a firmer geometric foundation (Appendix B). The model idealizes the opening as two intersecting arcs symmetric about both the A-P and L-R axes, each constrained to remain a circular arc while allowing L, W, , the height at the symmetry point h, and the curvature to vary in time. We obtain dW/dt and dh/dt to analytically track the dynamic geometry, yielding

(2)

and

(3)

Experimental values indicate that the first term on the right-hand side of Eqs. 2 and 3 is approximately an order-of-magnitude larger than the second term. Consequently, dividing Eq. 2 by Eq. 3 we find, to a good approximation (justified in Appendix B), that

(4)

where k_z –2 dh/dt. Therefore, when applied to the dorsal opening, 1/tan is a consequence of circular arc geometry and k_z –2v_native, where v_native = dh/dt.

Our analysis shows that (t) is not strictly constant; the dorsal opening measurably rounds up during the later stages of native dorsal closure (Appendix B). Inspection of Eqs. 2 and 3 indicate that both dW/dt and dh/dt are relatively insensitive to changes in curvature due to the d(ln )/dt dependence, which is a direct consequence of circular arc geometry.

Zipping rate constants
To quantify the asymmetry in the rates of seam formation, each canthus was treated independently by relaxing any constraints on A-P and L-R symmetry (Appendix B),

(5)

where we have separated the anterior (k_z,A, _A,R, _A,L) and posterior (k_z,P, _P,R, _P,L) dependencies. Integrating numerically from the time the first image was taken to the time of the last image, we obtained k_z,A and k_z,P and accounted for the experimental observations, i.e., curves through the data in Fig. 2 c. In six of the cases analyzed, k_z,A > k_z,P (in one case k_z,A = k_z,P to within 1%) and, averaging over seven embryos, k_z,A > k_z,P by 45(±20)% (Table 2).

Dorsal closure in basket, myospheroid, and scab mutant embryos
We confirmed that bsk, mys, and scb mutants fail to complete closure (12–17). Here we report the initial point of failure in all cases as the detachment between the lateral epidermis and the amnioserosa on the anterior side of the dorsal opening (bsk, n = 10; mys, n = 11; scb, n = 6) (Fig. 4; see Supplementary Material, Movies 1–3). For each of these mutants, W, L_Avg (L_R + L_L)/2, and the ratio w_A/w_P have critical values that quantitatively predict the onset of failure (Table 3). Across mutants, the critical values for W and L_Avg agree to within 4% while those for w_A/w_P agree to within 12%. In contrast, the critical values for h, w_A, w_P, _Ant, and _Pos show variations that range from 34% to 105% across mutants. In Discussion, below, we will interpret these results in the context of head involution, another developmental process that occurs concomitant with dorsal closure.

View larger version (29K): [in this window] [in a new window]   FIGURE 4  Dorsal closure in (a) bsk, (b) mys, and (c) scb mutant embryos shown at the onset of failure. The arrows locate the site of failure. Scale bar is 50 µm.

Upregulation during laser perturbed dorsal closure
Single-canthus nicking protocol
We targeted the microbeam to block zipping by nicking amnioserosa tissue close to either canthus with the aims of altering the inherent A-P asymmetry and elucidating changes that result from this perturbation. Zipping was blocked while leaving the leading edges intact, as was previously not the case (4). A nick was 10-µm long, took 1 s, and was repeated approximately every 30 s (Fig. 1, d and e).

Confocal images analyzed by using Eq. 5 indicate an upregulation in k_z,A when targeting the posterior end, and in k_z,P when targeting the anterior end (Fig. 1, d and e; Fig. 3, c and d; Table 2). The upregulation was asymmetric: when the embryos were nicked on the relatively slowly zipping posterior canthus, k_z,A increased by 14(±10)% with respect to the anterior value during native, unperturbed closure; when the embryos were nicked on the relatively rapidly zipping anterior canthus, k_z,P increased by 49(±22)% with respect to the analogous posterior value. In addition, the dorsal opening constricts as a consequence of the single-canthus nicking protocol applied to the anterior (67%; n = 6) or the posterior (25%; n = 8) canthus. An example is shown in Fig. 3 d (dashed contour). This constriction strongly suggests an upregulation of the force produced by the amnioserosa, which will be considered further in Discussion, and motivated the double-canthi nicking and edge-cut protocol described below.

During single-canthus nicking, x_c moved toward the targeted canthus at about four times the speed observed during unperturbed closure (Fig. 3, b–d; Table 2). x_c/t reverses its direction and is 21(±15)% larger when the anterior canthus is targeted relative to targeting the posterior canthus. We considered whether induced changes in teardrop geometry, asymmetric rates of seam formation, and/or contraction of the purse-strings could account for the change in x_c/t. While it seems unlikely that the effects we observe are due to alterations in teardrop geometry, the average angle of the unperturbed canthus does increase by up to 20% relative to native dorsal closure (data not shown). However, this promotes x_c movement away from the targeted canthus, contrary to experimental observations. Consequently, this indicates that seam formation and possibly contraction of the purse-strings dominate the angular contribution in determining x_c during single-canthus nicking.

Double-canthi nicking and edge-cut protocols
The double-canthi nicking protocol alternately targeted the microbeam to the amnioserosa in the proximity of each canthi to completely block zipping (Fig. 5 a). Nicks were 10-µm long, required 1 s, and pairs of nicks were repeated every 30 s. As a consequence, a local constriction forms after 40 min such that the leading edges assume a region of reverse curvature (Fig. 5, a and b). In seven of these embryos, the constriction formed closer to the anterior end, and in one case, it formed near W/2.

View larger version (77K): [in this window] [in a new window]   FIGURE 5  Determination of vrecoil in the presence of a localized constriction due to the double-canthi nicking protocol (a–c) and in native, unperturbed closure (d–f). In panels a and d, confocal fluorescent images with free body diagrams superimposed. Scale bar is 50 µm. In panels b and e, digitized contours showing the recoil of the purse-strings after an edge-cut protocol (dashed lines), where the innermost contour is just before the edge-cut and subsequent contours progress from inside to outside in 10.8 s steps. The total time is 140 s. In panels c and f, vrecoil after edge-cuts. Vertical arrows represent vrecoil and correspond to the far-left axis. Bold lines indicate the contour of the purse-strings just before the edge-cut where height corresponds to the far-right axis. The X characters in panels a–c indicate the location and extent of the laser perturbations. In panel c, the dotted lines indicate the boundary of the intact amnioserosa.

The recoil velocities determined by the edge-cut protocol are shown for embryos with (Fig. 5 c) and without (Fig. 5 f) a localized constriction, where the constriction was a consequence of double-canthi nicking (Fig. 5 a). Inspection of Fig. 5 c indicates: that v_recoil increases in a localized region that correlates with the constriction and the central band of intact amnioserosa; that local maxima (minima) in v_recoil are correlated with local minima (maxima) in h(x); and that there is a decrease in v_recoil toward the canthi. The average recoil velocity measured after an edge-cut following a constriction was v_recoil,1 = 3131 ± 215 nm/s (n = 8), as determined by selecting the maximum v_recoil within the region for each embryo. The average recoil velocity measured after an edge-cut in the absence of double-canthi nicking was v_recoil,2 = 2280 ± 104 nm/s (n = 6), as determined by averaging over a 20-µm window centered on the canthus-to-canthus midpoint. Both v_recoil,1 and v_recoil,2 far exceed v_native = dh/dt = –6.0 ± 0.3 nm/s (n = 8). Using the t-test at a significance level of 0.05, v_recoil,1 > v_recoil,2 (t = 3.20; DoF = 12; P = 0.004). Since v_recoil,1/v_recoil,2 is 1.37 ± 0.16, this comparison indicates that the response to the double-canthi nicking protocol caused the force per unit length produced by the amnioserosa _AS to increase by 37(±12)%. Appendix D analyzes the increase in _AS necessary to compensate for the reversal of the curvature as seen in Fig. 5, a and d, where the free body diagrams represent the forces just before the edge-cut. This analysis estimates _AS to increase by 60(±6)%, which compares favorably to the analysis of v_recoil. In the Discussion, we will consider molecular mechanisms that may trigger this upregulation in response to both the single-canthus and double-canthi nicking protocols.

The revised determination of _AS due to the edge-cut protocol necessitated a recalculation of the force ladder: 510:380:130:1 _LE:_AS:T:bv_native 490:380:110:1, where _LE (_AS) is the force/unit length of the lateral epidermis (amnioserosa), T is the magnitude of the tension in the purse-string, and bv_native is the viscous drag/unit length (see Fig. 5 d). This updated force ladder is in qualitative agreement with that previously reported (4).

	DISCUSSION

TOP ABSTRACT INTRODUCTION MATERIALS AND METHODS RESULTS DISCUSSION APPENDIX A: DETERMINATION OF... APPENDIX B APPENDIX C APPENDIX D: ESTIMATE OF... SUPPLEMENTARY MATERIAL ACKNOWLEDGEMENTS REFERENCES

 
We have identified key regulatory features and provided further phenomenological characterization of morphogenesis in dorsal closure by applying new analytical tools to quantify the observed asymmetries during closure in wild-type, laser, and genetically perturbed embryos. We have derived dW/dt based on the dynamic geometry of intersecting circular arcs, identifying the correlation k_z –2 v_native, finding that progress toward closure is relatively insensitive to changes in , and confirming our previous phenomenological assignment of the 1/tan dependence. In wild-type, unperturbed embryos, A-P asymmetries are manifest in measurements of the canthus angles (_Ant < _Pos), the location and movement of the centroid, the rates of seam formation (w_A > w_P), and the zipping rate constants (k_z,A > k_z,P). The location of the centroid indicates no significant L-R asymmetries. In bsk, mys, and scb embryos, the A-P asymmetries are opposite that of wild-type embryos, including _Ant > _Pos and w_A < w_P, and we observe that they invariably fail at their anterior ends during a critical time in dorsal closure as measured by W, L_Avg, and w_A/w_P. In these mutants, _Ant is larger and w_A is smaller relative to wild-type embryos. In the case of scb, there is a significant decrease in dh/dt, which lies along the L-R axis, relative to wild-type embryos. In addition, we observe two examples of upregulation that promote closure in response to laser perturbation in wild-type embryos: in response to laser inhibition of zipping at one canthus, there is an upregulation of the zipping rate constant at the other canthus, and when zipping is inhibited at both canthi, there is an upregulation of the force produced by the amnioserosa.

A mathematical feature of circular-arc geometry of the dorsal opening is the relative insensitivity of dW/dt and dh/dt to changes in . While it is an open research question as to whether this insensitivity relaxes any constraints for precise coordination and synchronization of the biological processes that produce forces and zipping during native closure, we have observed relatively large differences between k_z,A and k_z,P in wild-type, unperturbed closure. This is in contrast to both the relatively smaller differences in the canthus angles and the remarkable uniformity in dh/dt, where the canthi are moving anchors that influence the curvature of the purse-strings. The biological processes that account for k_z –2v_native remain elusive and it is intriguing to consider whether this correlation is attributable solely to biological processes per se or to an additional mathematical property of circular-arc geometry. While circular-arc geometry is well understood mathematically, its application to describe the dorsal opening where the two leading edges are losing length both via zipping and via contraction is the subject of ongoing research.

The A-P asymmetries that we have observed suggest asymmetries in the molecular and/or cellular processes responsible for canthus dynamics. In the past, zipping processes have been attributed to filopodia and/or adhesion proteins (3,4,28–30) and possible explanations for the asymmetry in the zipping rate constants include an asymmetry in the number, distribution, or force-producing capabilities of these factors. In contrast to wild-type closure, bsk, mys, and scb phenotypes have a reversed and more pronounced A-P asymmetry. How these processes relate to previously characterized A-P asymmetries of the developing embryo that affect dorsal closure (31) is not clear, as are the biological processes responsible for the reduction of dh/dt in scb but not in bsk and mys genotypes. Homsy et al. (17) provide evidence that bsk and other components of the N-terminal jun kinase signaling pathway mediate zipping at least in part through their regulation of mys and scb expression, so the similarities described here are not surprising. On the other hand, differences in phenotype are likely due to defects in the expression of other jun transcriptional targets and differences in the perdurance of maternally loaded integrin (32). At this time, the detailed biological processes responsible for zipping are not known and any explanation of either the intrinsic or regulatory asymmetries in k_z,A and k_z,P would be speculative.

Both laser protocols produce lesions in the amnioserosa, apparently affecting biological process(es) that act as sensors. The lesions change the magnitude of stresses in the remaining amnioserosa, the purse-strings, and the lateral epidermis as well as releasing biochemical debris from damaged cells. These consequences of laser perturbation could affect adhesion molecules, signaling molecules, the contractile components, and mechanically gated channels in the various tissues, therefore triggering a chemical (33,34) and/or mechano-sensing response (35). In any event, the downstream effect of this response is the upregulation of the zipping rate constants in the single-canthus nicking protocol or the force of the amnioserosa in the double-canthi nicking protocol. Both compensatory processes occur in tissue remote from the amnioserosa targeted by the microbeam and therefore we do not consider them to be a local wound-healing response.

There is evidence that the mechanisms for these two regulatory processes are related through _AS. Targeting the microbeam to amnioserosal cells substantially reduces _AS near the canthus and inhibits zipping locally, where neither the seams nor the purse-strings were nicked. When a single-canthus was nicked, k_z at the other canthus is upregulated. Apparently _AS near the canthus influences zipping in two ways, via canthus angle and zipping rate constants. Following both nicking protocols, we observe localized constrictions of the leading edges that, in the case of double-canthi nicking, correlate with regions of increased _AS. The increase in _AS likely generalizes to single-canthus nicking and k_z may be proportional to local values of _AS. The upregulation of both the force produced by the amnioserosa and the zipping rate constant highlight the resiliency of dorsal closure to laser perturbations to ensure timely closure of the dorsal opening.

We propose that the anterior end of the dorsal opening is subject to relatively greater stresses due to head involution, which may contribute to the A-P asymmetries and would account for the observed anterior failure in the bsk, mys, and scb embryos. Head involution (Fig. 6; see Supplementary Material, Movie 6) is a developmental process that occurs concomitant with the end stages of dorsal closure and after its completion (28). During head involution, a region of the anterior-most tissue folds into the interior of the embryo while the adjacent lateral epidermis moves anteriorly to cover it. Cells closer to the anterior end of the embryo move a larger distance relative to those located more posteriorly (see arrows in Fig. 6)—we suggest the magnitude of movement correlates with the magnitude of the stresses in the embryo. Consider this correlation in light of the following observations. The dorsal opening in wild-type embryos generally has a teardrop geometry with the sharper angle located anteriorly, which reduces the extent of leading edge-amnioserosa interface exposed to relatively larger stresses. In addition, k_z,A > k_z,P results in the dorsal opening as a whole distancing itself from the region of larger stresses as it translates posteriorly across the dorsal surface as measured by x_c. Furthermore, both the single-canthus nicking and double-canthi nicking protocols promote the movement of the opening away from the anterior end of the embryo by the formation of constrictions closer to the anterior end of the opening, possibly reducing the vulnerability to failure. In the single-canthus nicking protocol there is an upregulation of the zipping rate constants, with k_z,P increasing proportionately more than k_z,A. While upregulated zipping at the unperturbed canthus promotes the completion of closure approximately on schedule, the asymmetric upregulation of k_z,P and k_z,A promotes movement of the dorsal opening away from the anterior end of the embryo, reducing any vulnerability to the inhomogeneous stresses caused by head involution. In contrast, the reversal of the A-P asymmetries in bsk, mys, and scb genotypes exacerbate the vulnerability to relatively greater stresses at the anterior end due to the geometry and location of the opening, which we contend leads to failure during head involution.

View larger version (136K): [in this window] [in a new window]   FIGURE 6  Confocal fluorescent images of native closure showing head involution. The arrows track the movement of anterior and posterior epidermal tissue boundaries. Images were taken 46 min apart. Scale bar is 50 µm.

How forces contribute to morphogenesis throughout phylogeny is key to understanding the development of all metazoans, including humans (1). Moreover, it is a first step in understanding and subsequently ameliorating the suffering that is associated with such developmental defects as spina bifida, cleft palate, congenital heart or kidney diseases. In addition, because various features of dorsal closure are recapitulated in wound healing, our studies provide insight into the cellular and molecular processes that produce the forces responsible for this important biological process as well.

	APPENDIX A: DETERMINATION OF THE CENTROID

TOP ABSTRACT INTRODUCTION MATERIALS AND METHODS RESULTS DISCUSSION APPENDIX A: DETERMINATION OF... APPENDIX B APPENDIX C APPENDIX D: ESTIMATE OF... SUPPLEMENTARY MATERIAL ACKNOWLEDGEMENTS REFERENCES

 
The centroid of an enclosed surface in Cartesian coordinates, r_c = (x_c, y_c), is sensitive to asymmetries in the geometry. The movement with time of r_c gives information about how the enclosed area changes. The general expressions for r_c are (37)

(A-1)

where A is the total area. Discretizing in Eq. A-1 and applying them to the dorsal opening, we obtain the expressions

(A-2)

where x and y are length increments. In Fig. 3 a, the digitized coordinates (x_i, h_i) trace out both of the leading edges as i is stepped from 1 to N = 100, as do the digitized coordinates (x_j, h_j) as j is stepped from 1 to N = 100. N = 100 is an empirically derived number of points needed to describe the leading edges with the active contour algorithm (see Analysis), although not so many as to become burdensome during the numerical analysis. An example of the segmentation h_ix and x_jy is shown in Fig. 3 a. x is found by taking W(0) divided by the smaller number of points along the right or left leading edge. y was found in an analogous manner but with the maximum height h in the first image taken. A comparison of the centroid's position with the canthus-to-canthus midpoint gives direct information about the A-P (a)symmetry of the dorsal opening.

	APPENDIX B

TOP ABSTRACT INTRODUCTION MATERIALS AND METHODS RESULTS DISCUSSION APPENDIX A: DETERMINATION OF... APPENDIX B APPENDIX C APPENDIX D: ESTIMATE OF... SUPPLEMENTARY MATERIAL ACKNOWLEDGEMENTS REFERENCES

 
Dynamic geometry of the dorsal opening
Previously we have reported a quantitative model that successfully accounts for many experimental observations of the geometry of the dorsal opening (4). This model was based on several mathematical assumptions that are well justified experimentally: 1) The shape of each leading edge was assumed to be a circular arc; 2), The height at the symmetry point, h(t), in Fig. 7 decreases linearly with time; 3), The rate of zipping at the canthi was accounted for with an empirical rate equation having the biologically reasonable attribute that the rate of zipping quickened as the angles at the canthi became more acute. Overall, the model provides a quantitative, albeit abstract, summary of the coordination in space and synchronization in time that is a consequence of four cellular processes that are responsible for the dynamics of dorsal closure.

View larger version (13K): [in this window] [in a new window]   FIGURE 7  Schematic representation of the setting sun model. Two intersecting circular arcs representing native dorsal closure at earlier (left) and later (right) stages. Shown in bold are the arclength L/2; the half-horizontal opening W/2; the half-vertical opening at the symmetry point h; the half-opening angle ; and the radius r.

Moving beyond this idealized image of a setting sun, during dorsal closure the length of the leading edges are reduced through two processes: active contraction of the purse-string and incorporation of the leading edges into the seam through zipping, a process that lengthens the seams as closure progresses (Figs. 1 c and 2 c).

Zipping rate model and comparison to experiment
Consider two intersecting circular arcs as shown in Fig. 7, which corresponds to a geometric model for the dorsal opening that is symmetric before both the A-P and L-R axes. The following three relations hold:

(B-1)

(B-2)

(B-3)

As derived in the Supplementary Material, the time derivative of Eq. B-2 is given by

(B-4)

where, by definition, the curvature 1/r and the following ratio R_W will prove to be of particular interest when it is small relative to 1:

(B-5)

Similarly, the time derivative of Eq. B-3 and the ratio R_h are given by

(B-6)

and

(B-7)

Equations B-4 and B-6 generally describe changes in the geometry of a curve subject to the constraint that the curve remains a circular arc, where L, , h, and may be functions of time. Dividing Eq. B-4 by Eq. B-6 and recognizing that dh/dt = v_native, which has been shown empirically to be essentially constant during native closure, we obtain

(B-8)

When d(ln )/dt = 0, i.e., the constant curvature that characterizes the "setting sun," then R_W = R_h = 0, as seen in Eqs. B-5 and B-7. Consequently, Eq. B-8 exactly becomes

However, d(ln )/dt = 0 is not strictly realized experimentally. Instead, when the condition

(B-9)

is satisfied, which does not necessarily imply that R_W and R_h are both 0 as will be discussed below, then Eq. B-8 is well approximated by

(B-10)

Equation B-10 should be compared to Eq. 5, where w_A and w_P are the lengths of seam formed at the anterior and posterior ends. For the idealized geometry considered here, = _A,R = _A,L = _P,R = _P,L (Fig. 7) and the anterior and posterior zipping rate constants are equal, i.e., k_z = k_z,A = k_z,P. Consequently,

(B-11)

Comparing this result with Eq. B-10 indicates k_z –2 v_native, which is a geometric correlation between the zipping rate constant (k_z) and the rate of movement of the symmetry point of the leading edges (v_native). The experimental value for v_native is –6.0 ± 0.3 nm/s (n = 7) and hence the value for –2 v_native falls between the experimental values of k_z,A = 15.3 ± 0.7 nm/s and k_z,P = 10.6 ± 1.7 nm/s (n = 7), which were reported in the main text. This analysis provides a geometric foundation for the 1/tan dependence in the formula for the rate of zipping.

We now compare this model with experimental results to justify the approximations. First we determine the curvature from experimental observations of native dorsal closure. In the following, the figures present the results for one embryo, while the averages were taken from four embryos (each with two leading edges). Experimental observations indicate that the leading edges form shallow circular arcs during native dorsal closure (Fig. 1), where the aspect ratio 2h/W decreases from 0.4 to 0.2. To quantify we fitted the digitized leading edges, excluding points near the two canthi where the curvature deviates from the arc geometry as the free leading edge joins the zipped seam. Consequently, was determined by fitting 90% of the points in the central region at each time. As a test of the circular arc assumption, we first fitted the digitized leading edge with a polynomial s(x), where was determined analytically at the symmetry point x_sym using the expression (37)

(B-12)

We then fitted the digitized leading edge with an arc of a circle. The two fitting procedures yielded curvatures that agreed to within 7(±2)%, providing experimental support for the circular arc assumption inherent in this model. Fig. 8 a summarizes the experimentally determined -values of the two leading edges for a typical embryo. is approximately, but not strictly, constant during much of dorsal closure, i.e., up to 3000 s in this case. At later times, increases in magnitude, i.e., there is a small but measurable rounding up of the dorsal opening, in this case after 3000 s, as shown in the inset to Fig. 8 a. Consequently, experimental observations indicate d(ln )/dt 0 and strictly speaking the setting sun model does not apply, although it can be a useful approximation.

View larger version (15K): [in this window] [in a new window]   FIGURE 8  Time dependence of various parameters that describe changes in the geometry of the dorsal opening. The solid (open) circles are for the right (left) leading edge. (a) Curvature () of the leading edge. (Inset) fitted arcs at 0, 3000, and 5000 s, where the extent of the upper arc corresponds to the length of the leading edge at 0 s. The arc at 3000 s essentially is superposed on the arc at 0 s. Tick marks correspond to the lengths of the leading edge at 3000 s (upper arc) and 5000 s (lower arc), respectively. (b) Arclengths of the leading edges L and canthus-to-canthus distance W (crosses). (c) Ratio of 1+RW to 1+Rh. (d) dL/dt for the right (solid line) and left (dashed line) leading edges.