2025 Theses Doctoral

Mathematical Modeling and Data-Driven Analysis of Embryo Development

Zhu, Hongkang

Embryo development is a highly coordinated process where genetic regulation and mechanical forces interplay to drive the transformation of a single cell into a complex, multicellular organism. It involves many fundamental processes such as cell division, cell differentiation, and morphogenesis. Morphogenesis, the shape changes of tissue, results from collective cell movement, growth, proliferation, and shape changes, guided by genetic and mechanical cues. Despite the comprehensive data obtained from experimental measurements and advanced imaging, the physical mechanisms underlying morphogenesis are poorly understood, a quantitative cell shape pattern that describes morphogenesis has yet to be discovered, and the coupling between cytoskeleton that generates stress and shape changes has not been quantitatively demonstrated.

To address these unsolved questions, we utilized a powerful combination of first-principles modeling and empirical, data-driven approaches. Chapter 1 presents our mathematical model of Drosophila ventral furrow formation, which incorporates actomyosin contractile stress and viscous tissue responses. With all model parameters fitted from experiment, our model quantitatively explained numerous experimental observations in wild-type and genetically perturbed embryos, which were not fully explained by other models assuming elastic tissue responses. Our model revealed that the tissue-scale contraction in ventral furrow formation is driven by the curvature of the multicellular myosin profile. We also demonstrated that the pulsatile time-dependence of myosin acts as a protective mechanism for tissue contraction, suppressing cell-to-cell myosin fluctuations through a low-pass filter effect. This is crucial because tissue contraction is highly sensitive to even small myosin fluctuations, which would otherwise lead to significant inhomogeneous contractions.

Chapter 2 details our data-driven approach to studying Drosophila ventral furrow formation, utilizing time-lapse 3D data from light sheet microscopy. We developed computational algorithms to systematically parameterize over 28,000 cell shapes, designed interpretable cell shape features, and employed unsupervised learning to classify cell shape evolution trajectories. By mapping these classes onto the embryo, we extracted the first quantitative cell shape pattern in the Drosophila embryo. This pattern unveiled key physical mechanisms underlying embryo development, including how mechanical stresses propagate, how cell packing is influenced by embryo curvature, and the stochastic nature of apical constriction during tissue contraction.

Chapter 3 explores the coupling between actomyosin density and shape changes. We developed a mathematical model of the actomyosin cortex, using partial differential equations to describe the evolution of actomyosin density on a deformable surface, which is represented through differential geometry. Our model revealed that although under physiological conditions, the cell cortex is observed to maintain a homogeneous density and shape, this stability is challenged by two factors: increased cortical tension, which is mechanical in nature, and an elongated aspect ratio, which is a geometric feature. Higher cortical tension disrupts this homogeneity, leading to patterned actomyosin density and multiply furrowed shape. In contrast, an elongated aspect ratio drives constriction through a mechanism we named active Rayleigh instability, a modified form of the Plateau-Rayleigh instability. Furthermore, friction plays a crucial role in protecting the homogeneous state by preserving a large region of homogeneity in the state diagram of the cortex. When friction is reduced, this homogeneous region shrinks significantly, making the cortex more vulnerable to destabilization caused by increased tension and an elongated aspect ratio.