Correction to: Scientific Reports https://doi.org/10.1038/s41598-020-59030-2, published online 10 February 2020

In the Results section under subheading ‘Mathematical model of nanomechanical lipid bilayers: comparison between measurements and calculations’,

“The surface tension ($$\upsigma$$) is a function of the area strain ($${\upalpha }$$) of the membrane ($${\upalpha }=\left(\mathrm{A}-{\mathrm{A}}_{0}\right)/{\mathrm{A}}_{0}$$) with a resting area of $${\mathrm{A}}_{0}$$, and the function integration of $$\upsigma$$ with respect to $${\upalpha }$$ gives the surface strain energy density.”

“The surface tension ($$\upsigma$$) is the function of the area strain of the membrane $${\upalpha }=\frac{\left({A}^{res}-{A}_{0}^{res}\right)}{{A}_{0}^{res}}=\frac{\left({\upphi }_{0}^{res}-{\upphi }^{res}\right)}{{\upphi }^{res}}$$ where $${A}^{res}$$ is the area of the lipid membrane reservoir determined by $${\mathrm{r}}_{\mathrm{cr}}$$-$${\mathrm{r}}_{\mathrm{ct}}$$, $$\Omega$$, and $${\mathrm{r}}_{\mathrm{cb}}$$ (see the next paragraph and Fig. S3 in Supplementary Information online); $${A}_{0}^{res}$$ is the area of the lipid reservoir at the resting reference configuration; $${\upphi }^{res}$$ is the uniform lipid number density; and $${\upphi }_{0}^{res}$$ is $${\upphi }^{res}$$ at the resting reference configuration. The function integration of $$\upsigma$$ with respect to $${\upalpha }$$ gives the surface strain energy density.”

In the Methods section under subheading ‘Finite element modeling for lipid membranes’,

“The vesicle area strain is $${\upalpha }=\left(\mathrm{A}-{\mathrm{A}}_{0}\right)/{\mathrm{A}}_{0}= \left({\upphi }_{0}-\upphi \right)/\upphi$$ with resting (i.e., initial) area $${\mathrm{A}}_{0}$$ or initial lipid density $${\upphi }_{0}$$.”

“The vesicle area strain is $${\upalpha }=\frac{\left({A}^{res}-{A}_{0}^{res}\right)}{{A}_{0}^{res}}=\frac{\left({\upphi }_{0}^{res}-{\upphi }^{res}\right)}{{\upphi }^{res}}$$ where $${A}^{res}$$ is the area of the vesicle membrane determined by $${\mathrm{r}}_{\mathrm{vc}}$$, $$\Omega$$, and $${\mathrm{r}}_{\mathrm{cb}}$$ (see Fig. S3 in Supplementary Information online); $${A}_{0}^{res}$$ is the resting reference area of the vesicle in the spherical configuration; $${\upphi }^{res}$$ is the uniform lipid number density; and $${\upphi }_{0}^{res}$$ is $${\upphi }^{res}$$ at the resting reference configuration.”

$${\mathrm{G}}_{\mathrm{a}}={\int }_{{\Omega }_{\mathrm{a}}}\left[\begin{array}{c}{\overset{\lower0.5em\hbox{\smash{\scriptscriptstyle\frown}}}{\text{A}}}\\ {\overset{\lower0.5em\hbox{\smash{\scriptscriptstyle\frown}}}{\text{B}}}\\ {\overset{\lower0.5em\hbox{\smash{\scriptscriptstyle\frown}}}{\text{C}}}\\ {\overset{\lower0.5em\hbox{\smash{\scriptscriptstyle\frown}}}{\text{D}}}\\ {\overset{\lower0.5em\hbox{\smash{\scriptscriptstyle\frown}}}{\text{E}}}\end{array}\right]\left[\begin{array}{c}{\mathrm{N}}_{\mathrm{ss}}{\left(\mathrm{s}\right)}_{\mathrm{a}}\mathrm{sin}{\uptheta }_{\mathrm{a}}\\ {\mathrm{N}}_{\mathrm{s}}{\left(\mathrm{s}\right)}_{\mathrm{a}}\mathrm{sin}{\uptheta }_{\mathrm{a}}\\ {\mathrm{N}}_{\mathrm{ss}}{\left(\mathrm{s}\right)}_{\mathrm{a}}\mathrm{cos}{\uptheta }_{\mathrm{a}}\\ {\mathrm{N}}_{\mathrm{s}}{\left(\mathrm{s}\right)}_{\mathrm{a}}\mathrm{cos}{\uptheta }_{\mathrm{a}}\\ \mathrm{N}{\left(\mathrm{s}\right)}_{\mathrm{a}}\mathrm{cos}{\uptheta }_{\mathrm{a}}\end{array}\right]\mathrm{ds}=0$$
(16)

$${\mathrm{G}}_{\mathrm{a}}={\int }_{{\Omega }_{\mathrm{a}}}{\left[\begin{array}{c}{\overset{\lower0.5em\hbox{\smash{\scriptscriptstyle\frown}}}{\text{A}}}\\ {\overset{\lower0.5em\hbox{\smash{\scriptscriptstyle\frown}}}{\text{B}}}\\ {\overset{\lower0.5em\hbox{\smash{\scriptscriptstyle\frown}}}{\text{C}}}\\ {\overset{\lower0.5em\hbox{\smash{\scriptscriptstyle\frown}}}{\text{D}}}\\ {\overset{\lower0.5em\hbox{\smash{\scriptscriptstyle\frown}}}{\text{E}}}\end{array}\right]}^{\mathrm{T}}\left[\begin{array}{c}{\mathrm{N}}_{\mathrm{ss}}{\left(\mathrm{s}\right)}_{\mathrm{a}}\mathrm{sin}{\uptheta }_{\mathrm{a}}\\ {\mathrm{N}}_{\mathrm{s}}{\left(\mathrm{s}\right)}_{\mathrm{a}}\mathrm{sin}{\uptheta }_{\mathrm{a}}\\ {\mathrm{N}}_{\mathrm{ss}}{\left(\mathrm{s}\right)}_{\mathrm{a}}\mathrm{cos}{\uptheta }_{\mathrm{a}}\\ {\mathrm{N}}_{\mathrm{s}}{\left(\mathrm{s}\right)}_{\mathrm{a}}\mathrm{cos}{\uptheta }_{\mathrm{a}}\\ \mathrm{N}{\left(\mathrm{s}\right)}_{\mathrm{a}}\mathrm{cos}{\uptheta }_{\mathrm{a}}\end{array}\right]\mathrm{ds}=0$$
(16)

“For this purpose, a symmetric and positive-definite Jacobian matrix $$\overline{\text J}$$ of the residual vector $$\overline{G}$$ is defined as follows:”

“For this purpose, the Jacobian matrix $$\overline{\text J}$$ of the residual vector $$\overline{G}$$ is defined as follows:”

“Based on a general pursuit of finer elements in the methods, terms of $${\text{G}}_{\text{a}}$$ associated with the membrane area strain $${\upalpha }$$ i.e., $${\text{T}}_{\upalpha }$$ (see Appendix A in Supplementary Information online) are assumed to be constant values, which provides great simplicity for expanding Eq. (19).”

“By assuming the infinitesimal lateral oscillation of the membrane area (also see next paragraph), $${\text{T}}_{\upalpha }$$ (see Appendix A in Supplementary Information online) is assumed to be constant in expanding Eq. (19), which provides great simplicity.”

“Finally, by substituting an initial guess for the solutions $$\overline{\text{d}}_{0}$$ (i.e. d1, d2, …, dn and λ) to the Jacobian matrix $$\overline{\text{J}}$$ and the residual vector $$\overline{\text{G}}$$, the solution vector $$\overline{\text{d}}$$ for j + 1th Newton’s iteration can be calculated from $$\overline{\text d}_{{\text j} + 1} = \overline{\text d}_{\text j} + \cdot \overline{\text J}\left( {\overline{\text d}_{\text j} } \right)^{ - 1} \cdot \overline{\text G}\left( {\overline{\text d}_{\text j} } \right).$$

“Finally, by substituting an initial guess for the solutions $$\overline{\text{d}}_{0}$$ (i.e. d1, d2, …, dn and λ) to the Jacobian matrix $$\overline{\text{J}}$$ and the residual vector $$\overline{\text{G}}$$, the solution vector $$\overline{\text{d}}$$ for j + 1th Newton’s iteration can be calculated from $$\overline{\text d}_{{\text j} + 1} = \overline{\text d}_{\text j} - \overline{\text J}\left( {\overline{\text d}_{\text j} } \right)^{ - 1} \overline{\text G}\left( {\overline{\text d}_{\text j} } \right)$$

“With given boundary values, this iterative process is continued until the difference of two subsequent solution vectors converges to a certain tolerance.”