The state space of the whole system is = ( of the state space is called a configuration and describes the global state of the system. and cell turnover replacing polarized cells by initially unpolarized cells. We show that a persistent Naringenin global orienting signal determines the final mean polarity orientation in this stochastic model. Combining numerical and analytical approaches, we find that neighbour coupling retards polarity pattern reorganization, whereas cell turnover accelerates Tcfec it. We derive a formula for an effective neighbour coupling strength integrating both effects and find that the time of polarity reorganization depends linearly on this effective parameter and no abrupt transitions are observed. This allows us to determine neighbour coupling strengths from experimental observations. Our model is related to a dynamic 8-Potts model with annealed site-dilution and makes testable predictions regarding the polarization of dynamic systems, such as the planarian epithelium. the plane, termed planar cell polarity (PCP) [4]. PCP controls fundamental processes during embryonic development and tissue regeneration in many species including actin filament orientation, convergenceCextension, tissue reshaping, sensory organ formation, wing hair orientation, directional tissue growth and animal locomotion [4C6]. Mechanistically, PCP and the resulting planar tissue polarity integrate two general classes of inputs. (i) Global cues provided by the slope of tissue-scale gradients. These can consist of ligand concentration profiles [7,8], gene expression gradients [9] or mechanical shear stress [10]. (ii) Local cues are provided by cellCcell coupling. The alignment of cell polarization vectors among neighbouring cells propagates anisotropies from tissue boundaries or mutant clones and is mediated by the differential distribution of PCP and/or Fat/Dachsous components across cell/cell interfaces [4,11C13]. These mechanisms are universally found across many species and tissues. In most contexts, both inputs act synergistically to establish and maintain planar tissue polarity [14]. Theoretical studies of the collective phenomena of PCP confirmed that cellCcell neighbour coupling fosters a uniform polarity response of all cells to noisy and non-monotonous tissue-scale signals [6,15C23]. In particular, weak and even transient biases stemming from a polarized boundary or graded signal suffice to orient an entire epithelium when present from the of PCP dynamics in initially unpolarized cells [20,21]. Understanding of the underlying principle can be gained from statistical physics: the in the following, for the study of planar polarity formation and maintenance in biological tissues. Second, it elucidates the transient dynamics approaching the asymptotic state. We propose that new insight into polarity pattern formation can be gained from analysing the particular transient dynamics of polarity reorganization when Naringenin an initially coherent polarity pattern is confronted with an opposing instructive signal. We therefore ask, how the contradiction between inputs is resolved and how the time requirement for conflict resolution depends on parameters, especially the cell birth and death rates. The biological inspiration for our approach is the experimentally inducible inversion of global body plan polarity in the planarian [27,28]. The regeneration of a second head instead of a tail (figure 1the response of planar tissue polarity to dynamically changing global inputs, whereas cell turnover it. Finally, we establish a relation of the system parameters that determines the time requirement for polarity reorganization. We close with a discussion of these results. 2.?Mathematical model of cell polarity and Naringenin turnover 2.1. Model definition We define an IPS?[41C43] model for tissue polarity dynamics at the cellular level, which incorporates polarity alignment with respect to a global signal and to neighbours’ polarity vectors as well as cell turnover. The model cells occupy the nodes of a finite two-dimensional lattice that represents the epithelial tissue subjected to initially conflicting signals, as for instance the grey-shaded area in figure 1cell, which determines the cell’s polarity orientation, is abstracted as one unit vector per cell pointing towards the highest membrane accumulation of a selected PCP component. The directions of the unit vectors are discretized yielding the eight states 2.1 Naive cells, in planaria resulting from progeny Naringenin of stem cells immigrating into the epithelium and presenting no polarity information for an initial period of time, are represented by a ninth, has a polarization state which is an element of = 0, ,.