Regulation of Early Zebrafish Embryogenesis by Calcium Signaling and Dachsous1b Cadherin

Early animal embryogenesis entails a dynamic combination of embryonic cleavages, axial patterning, and gastrulation movements to shape a basic body plan. The underlying molecular signaling responsible for regulating this process remains poorly understood. In this thesis work, I first review recent progress in understanding of gastrulation movements in various model organisms brought by advances in imaging techniques. The externally developing and optically translucent zebrafish embryo is an ideal model organism to study vertebrate embryonic development by in vivo imaging. The objective of my thesis research is to leverage experimental advantages in the zebrafish model to uncover novel regulators and elucidate the molecular mechanisms involved in early vertebrate embryogenesis. Calcium signaling has been implicated in the control of many aspects of embryonic development. However, the spatiotemporal dynamics of calcium signaling during embryogenesis are not well characterized. By generating stable transgenic zebrafish lines ubiquitously expressing GCaMP6s, a genetically encoded calcium indicator, I demonstrated higher activities of calcium signaling during cleavage and blastula stages compared to previous reports. In addition, I showed that superficial dorsal-biased calcium signaling during blastula and gastrula stages was strongly correlated with and dependent on the dorsal organizer establishment. In the developing gastrulae, I directly visualized calcium activity in the dorsal forerunner cells and showed it was modulated by Nodal signaling in a cell non-autonomous manner. The GCaMP6s transgenic lines revealed with unprecedented spatiotemporal resolution the dynamic calcium signaling during early zebrafish embryogenesis and provide a superior tool for future studies. In zebrafish, mutations in atypical cadherin dachsous1b/dchs1b cause pleiotropic embryonic defects, including abnormal cleavages. Using the GCaMP6s transgenic reporter to examine the furrow-associated calcium activity in zebrafish dchs1b mutants, I showed that abnormal cleavages in dchs1b mutants were due to furrow progression defects during cytokinesis. These defects were likely caused by misregulated microtubules, as in vivo imaging of fluorescently marked microtubules during cleavage stages revealed reduced microtubule dynamics and impaired midzone microtubule assembly in dchs1b mutants. I further identified Ttc28 cytoplasmic protein as a molecular link between Dchs1b and microtubule dynamics. My biochemical experiments revealed that Dchs1b physically interacts via its intracellular domain with the tetratricopeptide repeat domain of Ttc28, and controls its subcellular distribution. Moreover, genetic inactivation of ttc28 resulted in increased microtubule dynamics and suppressed the microtubule defects in dchs1b mutants, suggesting a mechanism through which Dchs1b controls embryonic cleavages. In the last part of my thesis, I aimed to determine whether the chemokine ligand Ccl19.a1, a potential upstream regulator of calcium signaling, is required for axial patterning in zebrafish. I demonstrated that TALEN-generated ccl19a.1 mutations produce mildly dorsalized phenotypes and partially suppress the ventralized ichabod/ctnnb2 mutant phenotypes to influence axis formation, providing a genetic evidence for Ccl19.1 acting as a negative regulator of β-catenin and axis formation. Together, my work make several advances in understanding early vertebrate embryogenesis: it characterizes dynamic calcium signaling during zebrafish embryogenesis with a superior spatiotemporal resolution, reveals that Dchs1b regulates microtubule dynamics and embryonic cleavages by interacting with Ttc28 and regulating its subcellular distribution, and provides genetic evidence that Ccl19a.1 is necessary to limit β-catenin activity and consequently axis formation in zebrafish

Regularity of singular set in optimal transportation

In this work, we establish a regularity theory for the optimal transport problem when the target is composed of two disjoint convex domains, denoted $\Omega^*_i$ for $i=1, 2$. This is a fundamental model in which singularities arise. Even though the singular set does not exhibit any form of convexity a priori, we are able to prove its higher order regularity by developing novel methods, which also have many other interesting applications (see Remark 1.1). Notably, our results are achieved without requiring any convexity of the source domain $\Omega$. This aligns with Caffarelli's celebrated regularity theory \cite{C92}

Optimal (partial) transport between non-convex polygonal domains

In this paper, we investigate the optimal (partial) transport problem between non-convex polygonal domains in $\mathbb{R}^2$. In the case of the complete optimal transport problem, we demonstrate that the singular set is either a finite set, or, except for a finite number of points, is locally a 1-dimensional smooth curve. As for the optimal partial transport, we establish that the free boundary is smooth except for a finite number of singular points

C2,αC^{2,\alpha} regularity of free boundaries in optimal transportation

In this paper we establish the $C^{2,\alpha}$ regularity for free boundary in the optimal transport problem in all dimensions.Comment: In this updated version, we are able to obtain the $C^{2,\alpha}$ regularity for free boundary in the optimal transport problem in all dimension

Global regularity of optimal mappings in non-convex domains

In this paper, we establish a global regularity result for the optimal transport problem with the quadratic cost, where the domains may not be convex. This result is obtained by a perturbation argument, using a recent global regularity of optimal transportation in convex domains by the authors.Comment: accepted for publication in SCIENCE CHINA Mathematic