11/2008 for PRE
Supplementary Material
What are the true values of the bending modulus of simple lipid bilayers?
John F. Nagle*, Michael S. Jablin, Stephanie Tristram-Nagle, and Kiyotaka Akabori
Department of Physics, Carnegie Mellon University, Pittsburgh, PA 15213 USA
At the end of section 5 the distinction is mentioned between an input modulus Kθ and the output thermodynamic modulus Kθ+ that is derived from the input energy via statistical mechanics. Here we present a relation between Kθ+ and Kθ for a simple tilt model that is similar to the phenomenological model presented by (May et al., 2004). In their presentation the bending modulus KC does not affect Kθ+ so our simplified model does not include any splay/curvature energy. Our input energy is therefore
E = (1/2)ACKθt2 - ACH·t = (1/2)ACKθtan2θ - ACHtanθcosφ , (S1)
where H is the appropriate theoretical field introduced for the purpose of calculating
1/ Kθ+ ≡ limH=0(∂2lnZ/∂H2)/βAC , (S2)
where β=1/kT and the statistical mechanical partition function is
[pic] . (S3)
The blue curve labeled t = tanθ in Fig. S1 shows that Kθ+ is greater than Kθ by about a factor of 1.5 for typical values of the input Kθ of 35-50 mN/m for monolayers.
It is noteworthy that Eq. (S3) leads to Kθ+ = 0 for Kθ = 0. This contrasts with Kθ+ = 3kT/AC obtained by (May et al., 2004) who employed the small angle approximation. When we replace the defined tilt t = tanθ by t = sinθ in Eq. (S3) we obtain the same value at Kθ = 0 shown by the red dashed curve in Fig. S1. The non-zero value of Kθ+ when Kθ =0 comes about by the imposition of a maximum value t = 1 when θ = π/2, contrary to its accepted definition t = tanθ (May et al., 2004, May et al., 2007, Hamm and Kozlov, 2000) that is unbounded.
It may also be noteworthy that Kθ+ > Kθ when t = tanθ is used. That can be understood by expressing Eq. (S3) in the limit H = 0 in terms of t
[pic] . (S4)
The last factor in the denominator of Eq. (S4) decreases the effective phase space for large t which can be thought of as a negative entropic contribution to the input (free) energy as a function of t, thereby making tilt more difficult. In contrast, when t = sinθ is used, the phase space factor is 1/(1-t2)1/2 so Kθ+ < Kθ for large Kθ; for small Kθ the cutoff tmax = 1 dominates the behavior of Kθ+.
Figure S1. Thermodynamic tilt modulus Kθ+ versus input energy tilt modulus Kθ, both for monolayer values, for the t = tanθ and the t = sinθ calculations.
Bibliography
HAMM, M. & KOZLOV, M. M. 2000. Elastic energy of tilt and bending of fluid membranes. European Physical Journal E, 3, 323-335.
MAY, E. R., NARANG, A. & KOPELEVICH, D. I. 2007. Role of molecular tilt in thermal fluctuations of lipid membranes. Physical Review E, 76.
MAY, S., KOZLOVSKY, Y., BEN-SHAUL, A. & KOZLOV, M. M. 2004. Tilt modulus of a lipid monolayer. Eur Phys J E Soft Matter, 14, 299-308.
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