Fig. 5

Spatial profile of HIIS axial field component along NaV1.7’s pore. a Contour map of HIIS axial field component, \(m^{(1)}_{z}({\mathbf{p}},l_{\alpha }({\mathbf{p}}))\), for \({\mathbf{p}}\in P\) and \(\alpha =1,2,\ldots ,K_{\alpha }=800\). Blue- and red-colored contour domains represent configurations of \({\vec{\varvec{m}}}^{(1)}_{z}({\mathbf{p}},l_{\alpha }({\mathbf{p}}))\) with orientation “out” and “in”, respectively. Black lines \(R({\mathbf{p}})\), \(D({\mathbf{p}})\) and \(L({\mathbf{p}})\) indicate geometrical pore characteristics (see “Methods” section). Magenta dashed line \(\nu ({\mathbf{p}})\) depicts the scales at which the PMs-VSs spatial transition takes place. Dashed black lines \(s({\mathbf{p}})\) and \(o({\mathbf{p}})\) account for the upper and lower boundary of the lag and asymptote atom-packing domains. Critical radius \(\xi ({\mathbf{p}}_{crit.})\approx 33.4\) Å is plotted while inflection points line \(\xi ({\mathbf{p}})\) is omitted for clarity. Zero-crossing points of \(m^{(1)}_{z}({\mathbf{p}},l_{\alpha }({\mathbf{p}}))\) collected in \(\Omega ^{(1)}\) (see Additional file 1: S4) correspond to boundaries among contour domains \(T_{1}^{(1)}\), \(T_{2}^{(1)}\), \(T_{3}^{(1)}\), \(T_{4}^{(1)}\) and \(T_{5}^{(1)}\). Black arrows \(\mathrm{[a]}\), \(\mathrm{[b]}\), \(\mathrm{[c]}\), \(\mathrm{[d]}\), and \(\mathrm{[e]}\) are plotted in order to highlight domain boundaries. Two sets of missense SCN9A-gene mutation sites are employed; a pain-related set containing IEM, PPD and SFN mutation sites, and a neutral set containing mutation sites which are not expected to associate with pain disease phenotypes (Additional file 1: S8). Mutation sites highlighted with red color correspond to misclassified events (classification criterion; distance from the SF (see Additional file 1: S9b)). Grey-shaded areas “a1”, “a2”, and “a3” highlight contour map regions where the number of mutation sites maximizes, i.e., mutation sites occupancy rates maximize. ES, SF\(_{crit}\), CC, AG, and IS labels mark the locations of the extracellular side, of the critical pore point \({\mathbf{p}}_{crit.}\), of the central cavity, of the activation gate, and of the intracellular side, respectively. b Traces of normalized (with respect to corresponding maximum values) median distances between pain-related and neutral mutation sites from the critical point, \(\xi ({\mathbf{p}})\), represented as \(\bar{D}^{pain}_{\xi }=\bar{D}_{\xi ({\mathbf{p}})}(V_{pain})\) and \(\bar{D}^{neut.}_{\xi }=\bar{D}_{\xi ({\mathbf{p}})}(V_{neut.})\), respectively, are plotted for \({\mathbf{p}}\in P\) (see Additional file 1: S9b for calculation of \(D_{\xi ({\mathbf{p}})}(V_{pain})\) and \(D_{\xi ({\mathbf{p}})}(V_{neut.})\)). Circles indicate that for \({\mathbf{p}}\approx {\mathbf{p}}_{crit.}\), \(\bar{D}^{pain}_{\xi }\) is globally minimized with \(\bar{D}^{pain}_{\xi }\approx 0.17\), while \(\bar{D}^{neut.}_{\xi }\) exhibits a local maximum with \(\bar{D}^{neut.}_{\xi }\approx 0.93\). c Trace of \(m^{(1)}_{z}({\mathbf{p}},l_{\alpha }({\mathbf{p}}))\) for \({\mathbf{p}}={\mathbf{p}}_{crit.}\) and \(\alpha =1,2,\ldots ,K_{\alpha }=800\). Power-law approximations of \(m^{(1)}_{z}({\mathbf{p}}_{crit.},l_{\alpha }({\mathbf{p}}_{crit.}))\) described by Eq. r1 are plotted in light and dark green color accounting for the first and second part of the inflection domain, i.e., for \(s({\mathbf{p}}_{crit.})<l_{\alpha }({\mathbf{p}}_{crit.})\le \xi ({\mathbf{p}}_{crit.})\) and \(\xi ({\mathbf{p}}_{crit.})<l_{\alpha }({\mathbf{p}}_{crit.})\le o({\mathbf{p}}_{crit.})\), respectively. The mean absolute relative fitting errors (MARFEs) of the power-law approximation for the first and second part of the inflection domain are \(0.09\pm 0.01\) and \(0.15\pm 0.03\), respectively. d Trace of AE, \(U({\mathbf{p}},l_{\alpha }({\mathbf{p}}))\), for \({\mathbf{p}}={\mathbf{p}}_{crit.}\) and \(\alpha =1,2,\ldots ,K_{\alpha }=800\). Modeling approximations of \(U({\mathbf{p}}_{crit.},l_{\alpha }({\mathbf{p}}_{crit.}))\) described by Eq. r2 are plotted in light and dark green color accounting for the first and second part of the inflection domain, i.e., for \(s({\mathbf{p}}_{crit.})<l_{\alpha }({\mathbf{p}}_{crit.})\le \xi ({\mathbf{p}}_{crit.})\) and \(\xi ({\mathbf{p}}_{crit.})<l_{\alpha }({\mathbf{p}}_{crit.})\le o({\mathbf{p}}_{crit.})\), respectively. The MARFEs of the modeling approximation for the first and second part of the inflection domain are \(0.11\pm 0.02\) and \(0.14\pm 0.03\), respectively. Extrapolation of model approximations toward the lag domain, i.e., for \(l_{\alpha }({\mathbf{p}})\le s({\mathbf{p}})\), and toward the asymptote domain, i.e., for \(l_{\alpha }({\mathbf{p}})>o({\mathbf{p}})\), are plotted with dashed light and dark green lines, respectively, and result in a MARFE of \(6.06\pm 16.0\) and \(1.55\pm 6.39\), respectively. Richards model parameters used for modeling AE are \(\{A({\mathbf{p}}_{crit.})=1.03,t({\mathbf{p}}_{crit.})=0.03,s({\mathbf{p}}_{crit.})=18.16,\tilde{q}({\mathbf{p}}_{crit.})=0.47\}\)