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Article

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

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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
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Laboratorio Nacional de Microscopía Avanzada, Instituto de Biotecnología, Universidad Nacional Autónoma de México, Mexico City 62210, Mexico
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Mediterranean Institute for Advanced Studies (CSIC-UIB), 07190 Esporles, Balearic Islands, Spain
*
Author to whom correspondence should be addressed.
Mathematics 2020, 8(1), 1; https://doi.org/10.3390/math8010001
Received: 14 November 2019 / Revised: 12 December 2019 / Accepted: 16 December 2019 / Published: 18 December 2019
(This article belongs to the Special Issue Mathematical Biology: Modeling, Analysis, and Simulations)
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. View Full-Text
Keywords: left–right asymmetry in vertebrates; Kupffer’s vesicle; rotlet-induced internal Stokes flow left–right asymmetry in vertebrates; Kupffer’s vesicle; rotlet-induced internal Stokes flow
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MDPI and ACS Style

Hernández-Pereira, Y.; Guerrero, A.O.; Rendón-Mancha, J.M.; Tuval, I. 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. Mathematics 2020, 8, 1. https://doi.org/10.3390/math8010001

AMA Style

Hernández-Pereira Y, Guerrero AO, Rendón-Mancha JM, Tuval I. 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. Mathematics. 2020; 8(1):1. https://doi.org/10.3390/math8010001

Chicago/Turabian Style

Hernández-Pereira, Yunay, Adán O. Guerrero, Juan M. Rendón-Mancha, and Idan Tuval. 2020. "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" Mathematics 8, no. 1: 1. https://doi.org/10.3390/math8010001

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