The first question we want to answer is whether the adhesion between the vesicle and the substrate will occur with any w and μ . In the case that the substrate is stiff enough to resist selleck inhibitor any deformation and the vesicle maintains a circular shape with the radius 1/ϕ˙01=L, the reduced work of adhesion is written as equation(23) w=μ21+μ. Eq. (23) gives the critical condition for the occurring of the adhesion, which is w>μ/[2(1+μ)].w>μ/[2(1+μ)]. Especially, the case of μ → ∞ corresponds
to a vesicle adhering on a rigid substrate, and in this case the critical condition for adhesion is w > 1/2. If w<μ/[2(1+μ)],w<μ/[2(1+μ)], the substrate does not deform and keeps a straight line, then the dimensionless free energy of the system can be calculated as equation(24) E=2ΠLκ1=∫0Aϕ′2dS+1+μ∫Aπϕ′2dS−2wπ−A=π. The vesicle with a circular shape and the horizontal substrate are shown in selleck Fig. 2(a), where there is a singularity at the contact point due to the jump of the curvatures in Eq. (14). Another special case is the vesicle fully enveloped by the very soft substrate. During this situation, the radii of the vesicle
and the substrate are both 1/ϕ˙02=L, and the reduced work of adhesion reads equation(25) w=μ1+μ2.When w is bigger than the critical value in Eq. (25), the vesicle will be fully wrapped by the elastic substrate, and otherwise, this limit state never happens. In fact,
there is also a singular point at the apex of the vesicle due to the curvature jump of Eq. (14). The reduced free energy of this limit state is equation(26) E=π1−μ2=π21−4w+1+8w. Notably, when w < 0.5, the free energy of the vesicle-rigid substrate system is bigger than that Non-specific serine/threonine protein kinase of the vesicle-soft substrate case. Next we will numerically solve the above close-formed governing equation set ((20), (21) and (22)) in the light of shooting method, and demonstrate how the reduced free energy E changes with the variation of the rigidity ratio 1/μ=κ1/κ2.1/μ=κ1/κ2. The curve is shown in Fig. 3, where w is set as 2. This figure manifests strong bifurcation property induced by the nonlinearity of the governing equations. The detailed illustrations are formulated as follows: (1) Firstly, point a corresponds to the state of a vesicle adhering on a rigid substrate. With the increase of the substrate flexibility, there is a bifurcation, i.e. two solutions of the free energy when 0 < κ1/κ2 < 0.18. In what follows, the phase diagram including w and κ1/κ2 is shown in Fig. 5. Line 1 denotes the critical adhesion condition in Eq. (23). Below Line 1, adhesion cannot occur, with the substrate being a straight line and the vesicle being a circle. Similar critical condition was also obtained by Das and Du  for nonzero pressure case. Between Line 1 and Line 2, the substrate takes a concave shape without a point of inflection, which is schematized in Fig. 4(c) (phase I in Fig. 5).