On the Necessary Conditions for Non-Equivalent Solutions of the Rotlet-Induced Stokes Flow in a Sphere: Towards a Minimal Model for Fluid Flow in the Kupffer’s Vesicle

Yunay Hernández-Pereira, Adán O. Guerrero, Juan Manuel Rendón-Mancha and Idan Tuval
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Yunay Hernández-Pereira: Centro de Investigación en Ciencias, Instituto de Investigación en Ciencias Básicas y Aplicadas, Universidad Autónoma del Estado de Morelos, Av. Universidad 1001, Chamilpa, Cuernavaca 62209, Morelos, Mexico
Adán O. Guerrero: Laboratorio Nacional de Microscopía Avanzada, Instituto de Biotecnología, Universidad Nacional Autónoma de México, Mexico City 62210, Mexico
Juan Manuel Rendón-Mancha: Centro de Investigación en Ciencias, Instituto de Investigación en Ciencias Básicas y Aplicadas, Universidad Autónoma del Estado de Morelos, Av. Universidad 1001, Chamilpa, Cuernavaca 62209, Morelos, Mexico
Idan Tuval: Mediterranean Institute for Advanced Studies (CSIC-UIB), 07190 Esporles, Balearic Islands, Spain

Mathematics, 2019, vol. 8, issue 1, 1-11

Abstract: The emergence of left–right (LR) asymmetry in vertebrates is a prime example of a highly conserved fundamental process in developmental biology. Details of how symmetry breaking is established in different organisms are, however, still not fully understood. In the zebrafish ( Danio rerio ), it is known that a cilia-mediated vortical flow exists within its LR organizer, the so-called Kupffer’s vesicle (KV), and that it is directly involved in early LR determination. However, the flow exhibits spatio-temporal complexity; moreover, its conversion to asymmetric development has proved difficult to resolve despite a number of recent experimental advances and numerical efforts. In this paper, we provide further theoretical insight into the essence of flow generation by putting together a minimal biophysical model which reduces to a set of singular solutions satisfying the imposed boundary conditions; one that is informed by our current understanding of the fluid flow in the KV, that satisfies the requirements for left–right symmetry breaking, but which is also amenable to extensive parametric analysis. Our work is a step forward in this direction. By finding the general conditions for the solution to the fluid mechanics of a singular rotlet within a rigid sphere, we have enlarged the set of available solutions in a way that can be easily extended to more complex configurations. These general conditions define a suitable set for which to apply the superposition principle to the linear Stokes problem and, hence, by which to construct a continuous set of solutions that correspond to spherically constrained vortical flows generated by arbitrarily displaced infinitesimal rotations around any three-dimensional axis.

Keywords: left–right asymmetry in vertebrates; Kupffer’s vesicle; rotlet-induced internal Stokes flow (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2019
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