Physical model of the dynamic instability in an expanding cell culture

Physical model of the dynamic instability in an expanding cell culture 1 Supplementary Information Variation of an arbitrary curved membrane, of variable length, in two-dimensions We derive the forces at the outer surface of the cell culture, by treating it as a one-dimensional membrane with an effective bending modulus and tension coefficient. The Helfrich free energy of this membrane is given in Eq.1, and is used to derive the local restoring forces by the usual variation method. Since the overall contour length is not constant in our system, the variation of the coordinates has to be taken with respect to their index u along the contour, which is constant. In these terms the curvature H appearing in the free energy (Eq.1) is written as (standard differential geometry) H = ẋy ̈ − ẍẏ√ ẋ2 + ẏ2 3 (S1) where the ˙ symbol denotes differentiation with respect to the index of the point along the contour u, and the free energy i