A "morphogenetic action" principle for 3D shape formation by the growth of thin sheets

How does growth encode form in developing organisms? Many different spatiotemporal growth profiles may sculpt tissues into the same target 3D shapes, but only specific growth patterns are observed in animal and plant development. In particular, growth profiles may differ in their degree of spatial variation and growth anisotropy, however, the criteria that distinguish observed patterns of growth from other possible alternatives are not understood. Here we exploit the mathematical formalism of quasiconformal transformations to formulate the problem of ``growth pattern selection'' quantitatively in the context of 3D shape formation by growing 2D epithelial sheets. We propose that nature settles on growth patterns that are the `simplest' in a certain way. Specifically, we demonstrate that growth pattern selection can be formulated as an optimization problem and solved for the trajectories that minimize spatiotemporal variation in areal growth rates and deformation anisotropy.  The result is a complete prediction for the growth of the surface, including not only a set of intermediate shapes, but also a prediction for cell displacement along those surfaces in the process of growth. Optimization of growth trajectories for both idealized surfaces and those observed in nature show that relative growth rates can be uniformized at the cost of introducing anisotropy. Minimizing the variation of programmed growth rates can therefore be viewed as a generic mechanism for growth pattern selection and may help to understand the prevalence of anisotropy in developmental programs.

Methods
Analysis of growing appendage in Parhyale hawaiensis

The recording of the transgenic Parhyale embryo with a construct for heat-inducible expression of a nuclear marker (H2B-mRFPruby) was generated using multi-view lightsheet fluorescence microscopy (LSFM) with 7.5 minute time intervals beginning 3 days after egg lay (AEL). More details regarding data acquisition and pre-processing can be found in [1].

Our analysis focused on a period of dramatic outgrowth in the T2 appendage from 95 − 109h AEL and utilized tissue cartography methods to generate coarse-grained flow patterns on cells on the growing limb [2, 3]. Down-sampled data volumes were effectively denoised using Ilastik [4] by training a classifier to distinguish tissue from background. The result of this step was a pixel probability map for each time point (with high values in tissue regions and low values in background regions).

Segmented nuclei positions from [1] were then used to help distinguish the limb from surrounding tissue structures. Alpha shapes of the sparse nuclei locations were generated using MATLAB’s alphaShape function with a sufficiently high hole threshold to ensure that the resulting surfaces were water tight. The pixel probability maps were then multiplied by an exponentially decaying radial basis function of the distance to the alpha surface to suppress probability away from the actual limb. The original tracking from [1] was only logged every 45 minutes, so we linearly interpolated between these locations to take advantage of the full time resolution of the data set. The processed probability maps were then segmented using an active contour method [5]. The result was a binary level set indicating the location of all limb specific tissue. A point cloud approximating the mid-surface of the limb was obtained using a weighted locally optimal projection point cloud (WLOP) simplification algorithm [6].

In order to produce surface triangulations, we first estimated the boundary of the mid-surface point cloud using a custom modified version of the algorithm in [7]. Next, we found the point that was farthest from the boundary by calculating geodesic distances directly on the point cloud [8]. We then used the vector heat method [9] to compute the logarithmic map around this point. The logarithmic map is a local parameterization about a point, where for each point on the surface the magnitude of the log map gives the geodesic distance from the source, and the polar coordinate of the log map gives the direction at which a geodesic must leave the source to arrive at the point. This enabled us to embed the points in 2D, construct a Delaunay triangulation, and then lift the triangulation back into 3D. Triangulated time-dependent surfaces were then mapped conformally into the unit disk using a custom-implementation of the discrete Ricci flow [10]. The conformal degrees of freedom in the time-dependent parameterization were pinned by finding an optimal Möbius transformation that matched the neighborhood structure of nuclei locations at subsequent times without explicit reference to nuclei identity [11].

Once pulled back into the plane, an updated tracking for the nuclei was performed in 2D using a custom built MATLAB GUI enabling the reconstruction of nuclear lineages and cell tracks. The 3D displacement vectors between identified nuclei at subsequent times constituted a sparse set of surface velocities at isolated points. We once again employed the vector heat method to extend these velocities to the entire surface and then smooth them. These velocities were then used to compute the components of the growth tensor (i.e. the time derivative of the Lagrangian metric tensor) with respect to the instantaneous virtual isothermal parameterization of the surface. The infinitesimal change in deformation anisotropy γ from time t to t+1 could then be directly extracted from the growth tensor. We then computed the corresponding update to the 2D quasiconformal parameterization by feeding γ into a custom implementation of the Beltrami Holomorphic Flow algorithm [12]. The complete material flow was then assembled by iteratively propagating the surface mesh at the final time backwards along these quasiconformal mappings in the plane and pushing the resulting 2D parameterizations forward into 3D using the instantaneous conformal mappings and a natural neighbor interpolation scheme [13].

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[2] I. Heemskerk and S. J. Streichan, "Tissue cartography: compressing bio-image data by dimensional reduction", Nature Methods 12, 1139 (2015).

[3] N. P. Mitchell and D. J. Cislo, "Tubular: tracking in toto deformations of dynamic tissues via constrained maps", Nature Methods 20, 1980 (2023).

[4] S. Berg, D. Kutra, T. Kroeger, C. N. Straehle, B. X. Kausler, C. Haubold, M. Schiegg, J. Ales, T. Beier, M. Rudy, K. Eren, J. I. Cervantes, B. Xu, F. Beuttenmueller, A. Wolny, C. Zhang, U. Koethe, F. A. Hamprecht, and A. Kreshuk, "ilastik: interactive machine learning for (bio)image analysis", Nature Methods 16, 1226 (2019).

[5] T. Chan and L. Vese, "Active contours without edges", IEEE Transactions on Image Processing 10, 266 (2001).

[6] H. Huang, D. Li, H. Zhang, U. Ascher, and D. Cohen-Or, "Consolidation of unorganized point clouds for surface reconstruction", ACM Transactions on Graphics 28, 1 (2009).

[7] C. Mineo, S. G. Pierce, and R. Summan, "Novel algorithms for 3d surface point cloud boundary detection and edge reconstruction", Journal of Computational Design and Engineering 6, 81 (2019).

[8] K. Crane, C. Weischedel, and M. Wardetzky, "The heat method for distance computation", Commun. ACM 60, 90 (2017).

[9] N. Sharp, Y. Soliman, and K. Crane, "The vector heat method", ACM Trans. Graph. 38 (2019).

[10] W. Zeng and X. D. Gu, "Ricci Flow for Shape Analysis and Surface Registration", SpringerBriefs in Mathematics (Springer New York, New York, NY, 2013).

[11] H. Le, T.-J. Chin, and D. Suter, "Conformal Surface Alignment with Optimal Möbius Search", in 2016 IEEE Conference on Computer Vision and Pattern Recognition (CVPR), Vol. 2016-Decem (IEEE, 2016) pp. 2507–2516.

[12]  L. M. Lui, T. W. Wong, W. Zeng, X. Gu, P. M. Thompson, T. F. Chan, and S.-T. Yau, "Optimization of Surface Registrations Using Beltrami Holomorphic Flow", Journal of Scientific Computing 50, 557 (2012).

[13]  R. Sibson, "A brief description of natural neighbor interpolation", in Interpreting Multivariate Data, edited by V. Barnett (John Wiley & Sons, New York, 1981) pp. 21–36.