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


The original version of this Article contained errors.


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.”


now reads:


“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 \({\upphi }_{0}\).”


now reads:


“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)

now reads:

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

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


now reads:


“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).”


now reads:


“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 \(\overline{\text d}_{{\text j} + 1} = \overline{\text d}_{\text j} + \cdot \overline{\text J}\left( {\overline{\text d}_{\text j} } \right)^{ - 1} \cdot \overline{\text G}\left( {\overline{\text d}_{\text j} } \right).\)


now reads:


“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.”


now reads:


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


These errors have now been corrected in the PDF and HTML versions of the Article.