Curvature-Dependent Elastic Properties of Liquid-Ordered Domains Result in Inverted Domain Sorting on Uniaxially Compressed Vesicles

Abstract

Recent tumor growth models are often based on the multiphase mixture framework. Using bifurcation theory techniques, we show that such models can give contour instabilities. Restricting to a simplified but realistic version of such models, with an elastic cell-to-cell interaction and a growth rate dependent on diffusing nutrients, we prove that the tumor cell concentration at the border acts as a control parameter inducing a bifurcation with loss of the circular symmetry. We show that the finite wavelength at threshold has the size of the proliferating peritumoral zone. We apply our predictions to melanoma growth since contour instabilities are crucial for early diagnosis. Given the generality of the equations, other relevant applications can be envisaged for solving problems of tissue growth and remodeling. Introduction.-Soft tissue growth models in ideal geometries have shown shape instabilities with a special focus on morphogenesis of living systems [1][2][3]. Anisotropic or inhomogeneous growth and constraints due to boundaries are responsible for these shape instabilities. Most models explore the mechanical properties with perhaps an excessive simplification of the growth process itself. On the contrary, after several decades since the pioneering work of Greenspan [4], tumor growth models have explored all the facets of possible elementary biological processes known to date [5] with different modeling approaches at different scales: continuum models at the tissue scale, discrete models at the cells scale, and eventually hybrid continuum-discrete models [6]. The recent review by [6] reports more than 550 citations but few of them explore the possibility of shape instabilities except in simplified models [4,7]. However, models based on nonlinear coupled partial-differential equations (PDEs) require numerical investigations [8,9] with lack of understanding of the parameters at the origin of shape instabilities.Nevertheless, these instabilities are often correlated with the beginning of aggressive neoplasia: they are of utmost importance for melanoma where symmetry and regularity are key criteria in clinical methods [10]. Our aim is then to identify the main biophysical interactions at the origin of these contour instabilities. It has already been shown, both in vitro [11,12] and theoretically [13], that mechanical interactions between cells and their microenvironment are crucial for tumor evolution and can determine some macroscopic tumor properties [11].We discuss here how they can control tumor shape evolution as well, dictating the condition for the development of contour instabilities on a circular growing tumor, possibly giving rise to the patterns observed in vivo. We consider a continuous multiphase model describing the tumor as a two component mixture [6]. Although simplificative, such a model already contains the main

Through the Lipopolysaccharide Glass: A Potent Antimicrobial Peptide Induces Phase Changes in Membranes

“…6G, H (Table S1), suggests that significant structural changes to lipid bilayers are not responsible for the antimicrobial action of PMB1, but rather that modifications to the diffusive properties of these lipids disrupts Gram-negative membranes. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 window that corresponds to the minimum of the PMF profile calculated at 310 K (see 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 drastically alter the physical properties of bilayers including elastic moduli, shape, mechanical stress and tension, [53]…”

Expansion of the fusion stalk and its implication for biological membrane fusion

“…We applied a parameterized model of cholesterol that has been shown to capture the liquid disorder to liquid order (lipid rafts) phase transition (39). With this simulation model, we have previously demonstrated the enhanced affinity of both PE lipids and cholesterol for the inner and thus negatively curved monolayer of a 20-nm-sized vesicle (34,40).…”

Exploiting Lipid Permutation Symmetry to Compute Membrane Remodeling Free Energies

“…Moreover, despite the widespread use of CH models for highly curved structures (e.g., an hourglass-shaped fusion stalk) [22,23], it is unclear whether or not the harmonic approximation of constant κ is sufficiently accurate in this regime. The first simulations to test this approximation at higher curvatures [6,24] used heavily coarse-grained lipid models, with conflicting results; however, two recent studies have reported anharmonic behavior for binary [25] and ternary (raft-forming) lipid mixtures [26]. We therefore apply our proposed method to compute bending free energies and κ up to very high curvatures to rigorously test the range of validity of the harmonic approximation.…”

“…This nonlinearity is accounted for by adding a quartic bending term ðμ=4ÞH 4 to the integrand of (2) [6,26,[35][36][37] and integrating to obtain a sum of quadratic and quartic energy terms,…”

“…[For details of the change of coordinates ξ → ðH 2 ; H 4 Þ and of computing κ and μ, see the Supplemental Material [29].] Finally, from κ and μ, a curvature dependent, effective bending modulus can be defined as κ eff ¼ κ þ ðμ=2ÞH 2 or, equivalently, κ eff ¼ κð1 þ ½R μ H 2 Þ, where the distance R μ ¼ ffiffiffiffiffiffiffiffiffiffi μ=2κ p indicates the length scale where κ increases [6,26].…”