IMA Journal of Applied Mathematics (2013) Page 1 of 15 doi:10.1093/imamat/hxt030 A mathematical model for the keratin cycle of assembly and disassembly Chengjun Sun Department of Mathematics, University of Manitoba, Winnipeg, MB, Canada R3T 2N2 Rudolf Leube Institute of Molecular and Cellular Anatomy, RWTH Aachen University, 52074 Aachen, Germany Reinhard Windoffer Institute of Molecular and Cellular Anatomy, RWTH Aachen University, 52074 Aachen, Germany and Stéphanie Portet ∗ Department of Mathematics, University of Manitoba, Winnipeg, MB, Canada R3T 2N2 ∗ Corresponding author: portets@cc.umanitoba.ca [Received on 17 May 2012; revised on 29 April 2013; accepted on 29 April 2013] We formulate a mathematical model to study the in vivo keratin organization in terms of repartition of the keratin material between three different structural states: soluble proteins, filament precursors and filamentous state. The model describes a cycle of assembly and disassembly of keratin material. A generic nucleation function and structural stability of filament precursors are assumed. It is established that the three structural states always coexist and that the repartition of keratin material never exhibits any cyclic behaviour. Under some conditions, it is shown that the choice of the nucleation function does not affect the qualitative behaviour of the system. However, it might change the stable repartition of the keratin material in the cell. Keywords: intermediate filaments; cytoskeleton assembly dynamics; index theory; globally asymptoti- cally stable. 1. Introduction The cytoskeleton in epithelial cells is composed of three major filament systems: actin-based micro- filaments, tubulin-based microtubules and keratin-based intermediate filaments. Keratin intermediate filaments differ from the other filament types by their lack of polarity, their high degree of phosphoryla- tion, compositional diversity, extreme elasticity and their propensity to spontaneously assemble in vitro without any co-factors (reviews in Herrmann et al., 2007; Kim & Coulombe, 2007; Magin et al., 2007). Very little is known about the regulation of the in vivo keratin assembly needed to establish, maintain and functionally adapt the complex and evolutionary conserved network arrangements observed in dif- ferent epithelial cell types (e.g. Carberry et al., 2009; Iwatsuki & Suda, 2010; Oriolo et al., 2007). Live cell imaging has revealed a surprisingly high degree of network plasticity, most notably in the context of various stress paradigms (Beriault et al., 2012; Sivaramakrishnan et al., 2009; Strnad et al., 2001, 2002; Windoffer et al., 2004; Woll et al., 2007). Based on such observations in various cell culture systems, a model has been put forward to explain the time-dependent regulation of keratin networks (Leube et al., 2011; Windoffer et al., 2011). Its main tenet is that keratin filaments are subject to continuous c  The authors 2013. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved. IMA Journal of Applied Mathematics Advance Access published May 24, 2013 by guest on May 27, 2013 http://imamat.oxfordjournals.org/ Downloaded from