Global existence for a bulk/surface model for active-transport-induced polarisation in biological cells

Abstract

We consider a coupled bulk/surface model for advection and diffusion of interacting chemical species in biological cells. Specifically, we consider a signalling protein that can exist in both a cytosolic and a membrane-bound state, along with a variable that gives a coarse-grained description of the cytoskeleton. The main focus of our work is on the well-posedness of the model, whereby the coupling at the boundary is the main source of analytical difficulty. A priori L pestimates, together with classical Schauder theory, deliver global existence of classical solutions for small data on bounded, Lipschitz domains. For two physically reasonable regularised versions of the boundary coupling, we are able to prove global existence of solutions for arbitrary data. In addition, we prove the existence of a family of steady-state solutions of the main model which are parametrised by the total mass of the membrane-bound signal molecule.

Turing Patterning in Stratified Domains

“…A second class of model considers bulk-surface coupling, where one component is confined to the boundary of the main bulk domain, and reactants flow between the two regions, such as in the case of proteins diffusing in the cytoplasm and binding on the cell membrane (Rätz and Röger 2014;Madzvamuse et al 2015;Spill et al 2016;Cusseddu et al 2018;Paquin-Lefebvre et al 2018;Frey et al 2018). There is substantial recent interest in such models, from very theoretical results on existence and fast reaction-limiting behavior (Rätz 2015;Anguige and Röger 2017;Hausberg and Röger 2018), to spike dynamics (Gomez et al 2018) and a myriad of applications to understanding cell polarity (Thalmeier et al 2016;Kretschmer and Schwille 2016;Geßele et al 2020). One particularly well studied example is the pole-to-pole Min protein oscillation in E. coli, which has the biological function of guiding the cell division machinery to midcell (Kretschmer and Schwille 2016).…”

“… 2018 ). There is substantial recent interest in such models, from very theoretical results on existence and fast reaction-limiting behavior (Rätz 2015 ; Anguige and Röger 2017 ; Hausberg and Röger 2018 ), to spike dynamics (Gomez et al. 2018 ) and a myriad of applications to understanding cell polarity (Thalmeier et al.…”

Turing Patterning in Stratified Domains

“…A second class of model considers bulk-surface coupling, where one component is confined to the boundary of the main bulk domain, and reactants flow between the two regions, such as in the case of proteins diffusing in the cytoplasm and binding on the cell membrane [72,58,75,21,68,38,27]. There is substantial recent interest in such models, from very theoretical results on existence and fast-reaction limiting behaviour [71,1,41], to spike dynamics [33] and a myriad of applications to understanding cell polarity [77,54,38,30]. One particularly well studied example is the pole-to-pole Min protein oscillation in E. coli, which has the biological function of guiding the cell division machinery to midcell [54].…”

Spherical Caps in Cell Polarization

“…The formation of such cap-like patterns can rely on autocatalytic membrane recruitment of a diffusible cytoplasmic protein. The molecular and cellular bases for this mechanism have been established in several cases, motivating an impressive number of mathematical models that are being used to explore the quantitative requirements for the formation of robust localized patterns (5,8,(11)(12)(13)(14)(15)(16)(17)(18). The simplest of such models has a single chemical species that interconverts between membrane and cytoplasmic states (13,14,17).…”

“…This model and many of its relatives considered in the context of cell polarization belong to a broad class of reaction-diffusion systems with spatially segregated compartments of different dimensionality (15,(18)(19)(20)(21)(22). In combination with diffusion on the membrane, autocatalytic membrane recruitment can result in the formation and progressive expansion of a domain with high membrane concentration (11,(13)(14)(15)(16)(17)(18)(19)(20)23,24). However, it is counteracted by total mass conservation and fast cytoplasmic diffusion (12)(13)(14).…”