The role of mechanics in the growth and homeostasis of the intestinal crypt
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ORIGINAL PAPER
The role of mechanics in the growth and homeostasis of the intestinal crypt A. A. Almet1,2,3 · H. M. Byrne1 · P. K. Maini1 · D. E. Moulton1 Received: 21 May 2020 / Accepted: 3 November 2020 © The Author(s) 2020
Abstract We present a mechanical model of tissue homeostasis that is specialised to the intestinal crypt. Growth and deformation of the crypt, idealised as a line of cells on a substrate, are modelled using morphoelastic rod theory. Alternating between Lagrangian and Eulerian mechanical descriptions enables us to precisely characterise the dynamic nature of tissue homeostasis, whereby the proliferative structure and morphology are static in the Eulerian frame, but there is active migration of Lagrangian material points out of the crypt. Assuming mechanochemical growth, we identify the necessary conditions for homeostasis, reducing the full, time-dependent system to a static boundary value problem characterising a spatially heterogeneous “treadmilling” state. We extract essential features of crypt homeostasis, such as the morphology, the proliferative structure, the migration velocity, and the sloughing rate. We also derive closed-form solutions for growth and sloughing dynamics in homeostasis, and show that mechanochemical growth is sufficient to generate the observed proliferative structure of the crypt. Key to this is the concept of threshold-dependent mechanical feedback, that regulates an established Wnt signal for biochemical growth. Numerical solutions demonstrate the importance of crypt morphology on homeostatic growth, migration, and sloughing, and highlight the value of this framework as a foundation for studying the role of mechanics in homeostasis. Keywords Crypt · Homeostasis · Morphoelasticity · Elastic rod · Growth
1 Introduction The crypts of Liehberkühn are a canonical example of biochemistry and biomechanics combining to maintain tissue homeostasis within a highly deformed morphology. These test-tube-shaped invaginations renew and maintain a protective epithelial layer, called the intestinal epithelium, for the small intestine and colon. In the context of disease, colonic cancer originates in the crypts (Humphries and Wright This paper is dedicated to Credessa Jade Moulton, born the same week the paper was accepted. * D. E. Moulton [email protected] 1
Wolfson Centre for Mathematical Biology, Mathematical Institute, University of Oxford, Andrew Wiles Building, Radcliffe Observatory Quarter, Woodstock Road, Oxford OX2 6GG, UK
2
NSF‑Simons Center for Multiscale Cell Fate Research, University of California, Irvine, Irvine, CA, USA
3
Department of Mathematics, University of California, Irvine, Irvine, CA, USA
2008), while during inflammation, crypts facilitate rapid regeneration of the epithelium (Seno et al. 2009). Therefore, proper crypt function is crucial to a healthy gut. Deciphering the numerous genetic and biochemical signalling pathways governing crypt homeostasis has been the focus of a significant amount of research. Mathematical and computation
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