  Intracellular protein patterns regulate many vital cellular functions, such
as the processing of spatiotemporal information or the control of shape
deformations. To do so, pattern-forming systems can be sensitive to the cell
geometry by means of coupling the protein dynamics on the cell membrane to
dynamics in the cytosol. Recent studies demonstrated that modeling the
cytosolic dynamics in terms of an averaged protein pool disregards possibly
crucial aspects of the pattern formation, most importantly concentration
gradients normal to the membrane. At the same time, the coupling of two domains
(surface and volume) with different dimensions renders many standard tools for
the numerical analysis of self-organizing systems inefficient. Here, we present
a generic framework for projecting the cytosolic dynamics onto the
lower-dimensional surface that respects the influence of cytosolic
concentration gradients in static and evolving geometries. This method uses a
priori physical information about the system to approximate the cytosolic
dynamics by a small number of dominant characteristic concentration profiles
(basis), akin to basis transformations of finite element methods. As a proof of
concept, we apply our framework to a toy model for volume-dependent interrupted
coarsening, evaluate the accuracy of the results for various basis choices, and
discuss the optimal basis choice for biologically relevant systems. Our
analysis presents an efficient yet accurate method for analysing pattern
formation with surface-volume coupling in evolving geometries.
