Mathematical models of dorsal closure

Abstract

Dorsal closure is a model cell sheet movement that occurs midway through Drosophila embryogenesis. A dorsal hole, filled with amnioserosa, closes through the dorsalward elongation of lateral epidermal cell sheets. Closure requires contributions from 5 distinct tissues and well over 140 genes (see Mortensen et al., 2018, reviewed in Kiehart et al., 2017 and Hayes and Solon, 2017). In spite of this biological complexity, the movements (kinematics) of closure are geometrically simple at tissue, and in certain cases, at cellular scales. This simplicity has made closure the target of a number of mathematical models that seek to explain and quantify the processes that underlie closure's kinematics. The first (purely kinematic) modeling approach recapitulated well the time-evolving geometry of closure even though the underlying physical principles were not known. Almost all subsequent models delve into the forces of closure (i.e. the dynamics of closure). Models assign elastic, contractile and viscous forces which impact tissue and/or cell mechanics. They write rate equations which relate the forces to one another and to other variables, including those which represent geometric, kinematic, and or signaling characteristics. The time evolution of the variables is obtained by computing the solution of the model's system of equations, with optimized model parameters. The basis of the equations range from the phenomenological to biophysical first principles. We review various models and present their contribution to our understanding of the molecular mechanisms and biophysics of closure. Models of closure will contribute to our understanding of similar movements that characterize vertebrate morphogenesis.

Keywords: Actin; Cell junctions; Cell sheet morphogenesis; Computational; Myosin; Oscillation.

Figure 1

Drosophila embryos undergoing germband retraction (a–d) or dorsal closure and head involution (e–h, anterior is to the left). The end of germband retraction (d) occurs about 9hrs and 20min after fertilization and following a 1–1.5hr gap, the process of dorsal closure requires approximately 2.5–3hrs to complete (at 25° C). Fluorescence is from the genetically encoded green fluorescent protein fused to the F-actin binding domain of Drosophila moesin (called sGMCA, see Kiehart et al., 2000) which labels F-actin. The amnioserosa (AS), the germband (GB), the actomyosin-rich cable or purse string (PS), the lateral epidermis (Lat. Epi) and the canthi are so designated. Arrowheads in c show the irregularly shaped leading edge of the lateral epidermis (see also Figure 2). Arrows in d point to the accumulation of actin at the leading edge of the lateral epidermis. Black arrows in f point to the dorsal ridge (DR) indicating the progress of head involution and the dorsalward movement of the PS, respectively. Scale bar in h is 100 µm. Ultimately, the seam in h disappears as the formed dorsal epidermis becomes seamless (not shown). Reproduced and modified with permission (Kiehart et al., 2000, Figure 1).

Figure 2

High resolution view of transgenic Drosophila embryos expressing a fluorescent tag for F-actin (sGMCA; a–d) or E-cadherin (e–h) during dorsal closure. Posterior and anterior are the same in a–d and e–h and are labeled in a. Arrowheads in a, b and d indicate the accumulation of F-actin into the forming actomyosin rich cable or purse string (labeled in d) which forms near the leading edge of the dorsal most epidermal cells (DME, labeled in a and d) of the lateral epidermis (labeled in a) where it interfaces with the amnioserosa (labeled in a and e). Canthi mark the “corners” of the eye-shaped dorsal opening (labeled in d and f) and are the origin of the seam that marks the joining of two lateral epidermal sheets to form the dorsal epithelium. Ultimately, the seam disappears and the dorsal epithelium that results is seamless. Panel a–d reproduced with permission (Kiehart et al., 2017, Figure 4). Panels e–h kindly provided by Regan Price Moore.

Figure 3

Amnioserosal cell areas oscillate (or pulsate, from Solon et al., 2009). a. Upper panel: A schematic showing a dorsal view of an embryo after germband retraction but before the onset of dorsal closure (note, canthi have not yet formed). b. Lower panel: micrograph showing the interface between dorsal most epidermal (labeled DME) cells of the lateral epidermis and amnioserosal cells (labeled AS and corresponding to the boxed region in the upper panel). b. The oscillating areas of individual, colored cells in panel a are shown and are color coded to correspond to oscillations of the marginal cells of the amnioserosa (adjacent to the DME cells of the amnioserosa, upper, reddish traces), the second row of amnioserosa cells (middle, greenish traces) and the third row of amnioserosa cells (lower, bluish traces). Note, oscillations in amnioserosal cells persist longer if they are farther away from the DME cells. c. The average areas of the tracked cells for each row are plotted in panel c. The plots indicate that the marginal cells stop oscillating first, whilst the deepest, third row cells show the largest oscillations in cell area. Third row oscillations persist throughout the duration of the time studied. This further indicates that there is a progression of area loss from the marginal cells toward the middle of the amnioserosa. Reproduced with permission (Solon et al., 2009, Figure 1, panels E–G).

Figure 4

Hand traced schematic of zipping at the posterior canthus. a. At the beginning of the time-lapsed movie these images were traced from, canthi had just formed. Peripheral amnioserosa (PAS) cells are traced in black and three adjacent PAS cells are labeled in a. All other amnioserosa (AS) cells are traced in green. Lateral epidermal cells are traced in red except for two dorsal most epidermal (DME) cells in each panel, which are traced in blue and can be followed from free leading edge (in a) to two cells away from the canthus (in b) to a position six cells into the seam (in c). Three adjacent DME cells are shown in a. Time is from the beginning of the time-lapsed video sequence and is shown in the lower left hand corner of each panel. Reproduced with permission (Lu et al., 2015, Figure 1, panel F).

Figure 5

Dorsal closure occurs in three dimensions. a. A domed amnioserosa characterizes bona fide closure, but models to date are largely two (and in one case, one) dimensional. Cell junctions in the amnioserosa are labeled with green fluorescent protein fused to Drosophila E-cadherin, cell margins in the lateral epidermis are outlined with a red fluorescent protein fused to the F-actin binding fragment of Drosophila moesin. Two orthogonal planes highlight the curvature of the domed amnioserosa in the anterior - posterior (AP) and dorsal-ventral (DV) axes. Scale bar is 50µm. b and c. Cell traces made from z-sections of the amnioserosa and flanking lateral epidermis early (b) and later (c) in closure. Angles are described in Lu et al. (2016). Future models will need to embrace the curvature of the tissues and their constituent cells. Reproduced with permission (Lu et al., 2016, Figure 2 panels A,B and Figure 3 panel C).

Figure 6

The mathematical model of Layton et al. (2009) tracks wild type, native closure with precision, but fails to accurately track closure following surgical removal of the amnioserosa. a. The force balance diagram of Hutson et al. (2003) dictates the dynamics used by Layton et al. (2009) and includes forces from a purse string (the tension T), which is resolved in the direction of dorsal-ward movement as T_κ (not shown), from the lateral epidermis, σ_LEds and from the amnioserosa, σ_ASds. b. The circular arc geometry of Hutson et al. (2003) is consistent with an emergent property of closure in which zipping at the canthi maintains the strict relationship between the rate of change of the width of the dorsal opening (W) and the rate of change of its height (measured as the distance between advancing leading edges, H or the distance between the leading edge and the dorsal midline, h). The result is that the curvature of the leading edge changes only slightly during the course of closure. In Layton et al. (2009) adherence to circular arc geometry was relaxed. c. Plots of height vs. time illustrate the failure of Layton et al. (2009) to accurately predict the morphology of closure following surgical removal of the amnioserosa. Open circles are experimental data from an embryo in which a surgical cut removes the amnioserosa and AS. The model’s prediction for a solely elastic tissue (gray line), for a solely contractile tissue (dashed gray line) and for a tissue that is both elastic and contractile (black line). d and e. The predicted evolution of leading edge morphology is super-imposed on micrographs of closure in a wild type, native (i.e., a non-surgically perturbed) embryo. The red line in d uses a linear force velocity relationship, the yellow line in e uses a hyperpolic force velocity relationship. f and g. The predicted evolution of leading edge morphology is super-imposed on micrographs of an embryo in which the amnioserosa was surgically removed. As in d and e, the red line in f uses a linear force velocity relationship whereas the yellow line in g uses a hyperbolic force velocity relationship. Panels a and b are reproduced with permission (Figure 3 panel A is in Hutson et al., 2003). Panels c–g are reproduced with permission (Figure 8 panel D and Figure 9 panel B are in Layton et al., 2009).

Figure 7

The Almeida et al. (2011) model for dorsal closure. a. Panel a depicts the model’s rectangular simulation domain D_i, with the irregular, ellipsoidal dorsal opening pictured and the two zipping domains at either end of the dorsal opening, Z_i, depicted in blue. Force vector fields operate on boundaries as shown. Details for other labels are provided by Almeida et al. (2011) Reproduced from Almeida et al., 2011, Figure 3. b. Examples of computed solutions as shown in Figure 4 of Almeida et al. (2011). c. Images of dorsal closure in an embryo expressing spastin, which inhibits zipping. The top four panels are from the left hand panels in Figure 11 of Almeida et al. (2011), the next four are the three left hand panels and the bottom panel from Figure 12 of Almeida et al. (2011). Red traces show that the model nicely tracks the evolution of closure in these genetically perturbed embryos. All figures are reproduced with permission (Almeida et al., 2011).

Figure 8

The model of Solon et al. (2009) used to simulate oscillations in the amnioserosa during early dorsal closure stages. a. The geometry of the system includes 70 to 80 amnioserosa cells (outlined in black) surrounded by an elastic epidermis depicted in red. b. The distribution of elastic elements and the geometry of the model cell sheet is shown. Elastic springs shown in red connect neighboring vertices (gray dots) and each vertex with the center (black dot) of the hexagonal cell. Black lines show cell boundaries. Reproduced with permission (Solon et al., 2009, Figure 7 panel A).

Figure 9

The geometric basis of the model of Wang et al. (2012) is similar to that of Solon et al. (2009). a. It consists of a model amnioserosa consisting of 81 hexagonal cells - each cell has six edges and six spokes. The inset shows a cell with a shaded local triangle adjacent to a spoke, ij. Two such local triangles are used in their model to assess local area changes due to contraction of the spokes. b. Traces of normalized cell areas (thin traces for cells 1 and 2 in panel a) and normalized tissue area (thick trace) show the evolution of area loss with time. Reproduced with permission (Wang et al., 2012, panel a is from Figure 1 and panel b is from Figure 6).

Figure 10

The one-dimensional model of Dierkes et al. (2014). a. A schematic shows a cross section of the apical cortex of an epithelial cell with membrane (black line), actin filaments (green lines), actin monomer (green dots) and bipolar myosin filaments (red). Myosin filaments become concentrated when the cell contracts. They exchange with a reservoir with rates k_on and k_off. b. Depicts a schematic of a minimal model of a mechanochemical oscillator which includes a spring (top, squiggle) with spring constant K; a contractile element (middle, circle with arrows) that account for network contraction due to the hydrolysis of ATP by myosin and its motor activity associated with an actin filament; and a dashpot (bottom, with drag coefficient μ). The tension T(c) produced by the contractile element is dependent on the concentration of active myosin. T_e is external tension opposing deformation of the contractile unit. c and d. Example of three trajectories of a contractile unit depicted in two different ways, phase plane (panel c) and as a function of time (panel d). The blue trajectory is stable (non oscillatory), the green trajectory “blows up” (unstable) and red indicates sustain oscillations. Reproduced with permission (Dierkes et al., 2014, Figure 1a (panel a) and Figure 2a, 2c and 2d (panels b, c and d, respectively)).

Figure 11

a. The two-dimensional model geometry of Dureau et al. (2016). Red and purple dots indicate cell vertices and center of masses, respectively. Green lines indicate apical junctions between adjacent cells. b. Spring-dampers connect adjacent vertices (depicted in green, representing cortical visco-elasticity) and vertices and cell centers (depicted in purple and representing medioapical array visco-elasticity). The model tests the effectiveness of different kinds of visco-elastic linkages. c. A reverse contrast confocal image shows cell junctions, presumably labeled with fluorescent cadherin or one of its binding partners. d. A segmented image showing cell junctions (gray), cell centers purple, and multicellular junctions in red and green. Reproduced with permission (Dureau et al., 2016, Figure 1, panels a and b, and Figure 2, panels c and d).