Using the above symbols, the fixed boundary conditions then alter to equation(10) ϕ(0)=0,X(0)=0,ϕ0=0,X0=0, equation(11) ϕ(π)=π,   X(π)=0,   Y(π)=0,ϕπ=π,   Xπ=0,   Yπ=0,and the geometric relations can be recast as equation(12) X′=cosϕ,X′=cosϕ, equation(13) Y′=−sinϕ.Y′=−sinϕ. Moreover, an additional boundary condition at the point s = a can be derived as (see Eq. (A6) in Appendix

A) equation(14) ϕ′01−C0=(1+μ)ϕ′02.ϕ′01−C0=1+μϕ′02. It should be mentioned that, although the intrinsic boundary conditions for this problem are fixed, they can be imagined as movable, and then the new variation method about a functional with movable boundary conditions can be put to use [27] and [28]. In fact, the energy functional of Eq. (1) is special in that the undetermined variable a   causes the boundary movement of the system, which should create an additional selleck screening library term during the variation process. At the point s   = a  , the displacement and the slope angle are continuous, namely, X−(A)=X+(A)XA−=XA+, Y−(A)=Y+(A)YA−=YA+, ϕ−(A)=ϕ+(A),ϕA−=ϕA+, but the curvature is abrupt.

In use of the variation TGF-beta Smad signaling principle dealing with movable boundary conditions, one can derive the transversality condition (The detailed derivations are shown in Appendix A) equation(15) ϕ′01−C0=(1+μ)ϕ′02.ϕ′01−C0=1+μϕ′02.When κ  2→ ∞ and C  0 = 0, Eq. (15) degenerates to the situation of a vesicle sitting at a rigid substrate, i.e. ϕ′012=2w, and this solution is consistent with the former results in Refs. [11], [12], [13] and [14]. Without loss of generality, we take λ˜1=0 and C  0 = 0, for the spontaneous curvature doesn’t appear in the governing equations [13] and [22], then Eqs. (7) and (8) can be reduced to equation(16) ϕ″−λ˜2sinϕ=0,0≤A≤A, equation(17) 1+μϕ″−λ˜2sinϕ=0,A≤S≤π,and λ˜2 is a constant. Multiplying ϕ′ to

both sides of Eqs. (16) and (17), the integrations lead to equation(18) dS=dϕ2C1−λ˜2cosϕ,0≤A≤A, equation(19) dS=dϕ2C2−λ˜2cosϕ/1+μ,A≤S≤π,where C1 and C2 are two integration constants. In combination with Eqs. (12), (18) and (19) and the fixed boundary condition X(0) = 0, one has equation(20) ∫0ϕ0cosϕdϕ2C1−λ˜2cosϕ+∫ϕ0πcosϕdϕ2C2−λ˜2cosϕ/1+μ=0. In order to close this problem, Doxorubicin chemical structure the inextensible condition of the elastica is supplemented [29], which reads equation(21) ∫0ϕ0dϕ2C1−λ˜2cosϕ+∫ϕ0πdϕ2C2−λ˜2cosϕ/1+μ=π. Substituting Eqs. (18) and (19) into Eq. (15) yields equation(22) w=μC1−λ˜2cosϕ01+μ=μC2−λ˜2cosϕ0. Therefore, when the dimensionless work of adhesion w and the rigidity ratio μ are given, the total equation set ( (20), (21) and (22)) involving four variables can be solved. A numerical code based upon shooting method has been developed, and the shape equations of the vesicle and the substrate can be solved.