Globules model n Proposition 4.1. for y = x one has :

4 Existence of dynamics for globules and linear molecule models

4.1 Globules model

Let us prove that the globules model satisfies the assumptions in proposition 3.4. The set of allowed configurations is : A g = \ 1 ≤i j≤n D i j ∩ \ 1 ≤i≤n D i+ ∩ \ 1 ≤i≤n D i − where D i j = ¦ x ∈ R d n+n , |x i − x j | ≥ ˘x i + ˘ x j © D i+ = ¦ x ∈ R d n+n , ˘ x i ≤ r + © D i − = ¦ x ∈ R d n+n , ˘ x i ≥ r − © The D i j have smooth boundaries on A g and the characterization of normal vectors is easy : at point x satisfying |x i − x j | = ˘x i + ˘ x j

0, the unique unit inward normal vector n

i j x = n is given by : n i = x i − x j 2 ˘ x i + ˘ x j ˘ n i = − 1 2 n j = x j − x i 2 ˘ x i + ˘ x j ˘ n j = − 1 2 every other component vanishes On the boundary of the half-space D i+ , at point x such that ˘ x i = r + , the unique unit normal vector n has only one non-zero component : ˘ n i = −1. Similarly, D i − is a half-space, x belongs to its boundary if ˘ x i = r − , and the unique unit normal vector n at this point has ˘ n i = 1 as its only non- zero component. These vectors do not depend on x and will be denoted by n i+ , n i − instead of n i+

x, n

i −

x. Proposition 4.1.

A g satisfies properties U ES ‚ r 2 − 2r + n p n Œ and U N C    s 1 − r 2 − 2 6 r 2 + n 3 , r 5 − 2 14 r 4 + n 6    . Moreover, the vectors normal to the boundary ∂ A g are convex combinations of the vectors normal to the boundaries ∂ D i j , ∂ D i+ , ∂ D i − , that is, for every x in ∂ A g : N A g x =    n ∈ S d n+n , n = X ∂ D i j ∋x c i j n i j x + X ∂ D i+ ∋x c i+ n i+ + X ∂ D i − ∋x c i − n i − with c i j , c i+ , c i − ≥ 0    Proof of proposition 4.1 We have to check that the assumptions of proposition 3.4 are satisfied for the set A g . Since the D i+ and D i − are half-spaces, the Uniform Exterior Sphere property holds for them with any positive constant formally α i+ = α i − = +∞. For the same reason, the Uniform Normal Cone property holds for D i+ with any constants β i+ and δ i+ , and with l i+ x equal to the normal vector n i+ . This also holds for the sets D i − with any β i − and δ i − , and with l i − x = n i+ . Let us consider x ∈ A g such that x ∈ ∂ D i j , i.e. |x i − x j | = ˘x i + ˘ x j . By definition of n i j

x, for y = x

− ˘x i + ˘ x j n i j

x, one has :

y i = x i − ˘x i + ˘ x j x i − x j 2 ˘ x i + ˘ x j = x i + x j 2 = x j − ˘x i + ˘ x j x j − x i 2 ˘ x i + ˘ x j = y j ˘ y i + ˘ y j = ˘ x i − ˘x i + ˘ x j − 1 2 + ˘ x j − ˘x i + ˘ x j − 1 2 = 2˘ x i + ˘ x j 154 For z ∈ B0, ˘ x i + ˘ x