. . . . "A hierarchical Bell model, which extends Bell's rate theory of bond rupture to explicitly include the hierarchical architecture of a protein (how many H-bonds break together and how they are arranged), gives a structure–property relationship for the strength of protein domains." . "mathematical-law" . . "Notation basis: The abstract names the model and says it derives from Bell's theory; the Bell force–rate form written here is the standard one the model extends. The hierarchical form (how E_b and x_b scale with b) is in the full text, which is not vendored, so it is not written here." . "The hierarchical Bell model: a structure–property law for protein strength" . "F(v) = \\frac{k_B T}{x_b}\\,\\ln\\frac{v}{v_0} + \\frac{E_b}{x_b}, \\quad E_b,\\ x_b \\text{ set by the number } b \\text{ of H-bonds rupturing together}" . "Theodor Ackbarow et al., “Hierarchies, multiple energy barriers, and robustness govern the fracture mechanics of α-helical and β-sheet protein domains”, Proceedings of the National Academy of Sciences 104, 16410-16415 (2007), doi:10.1073/pnas.0705759104; Abstract. Scope: Protein domains whose strength is set by clustered H-bond rupture" . "Theodor Ackbarow, Xuefeng Chen, Sinan Keten, Markus J. Buehler. Hierarchies, multiple energy barriers, and robustness govern the fracture mechanics of α-helical and β-sheet protein domains. Proceedings of the National Academy of Sciences 104, 16410-16415 (2007). doi:10.1073/pnas.0705759104" . "Theodor Ackbarow, Xuefeng Chen, Sinan Keten, Markus J. Buehler" . "2007"^^ . "PMC2034213" . "doi:10.1073/pnas.0705759104" . "pmid:17925444" . "Hierarchies, multiple energy barriers, and robustness govern the fracture mechanics of α-helical and β-sheet protein domains" . . . "delphisci buehler-protein-materials claim extractor" . . . . "Claude Fable 5.1" . . . . . . . . . . . . "Quoted from the published abstract as returned by Europe PMC (2007-hierarchies.abstract.txt, sha256 4f052f4a8a9ddc7dfecc384493c31e0f00fcd9873ded36486b8618493fdf4e18); the full text was not read (not in Europe PMC open-access subset). Checked by build/check.py." . . . "We derive the hierarchical Bell model, a theory that explicitly considers the hierarchical architecture of proteins, providing a rigorous structure-property relationship." . . "4f052f4a8a9ddc7dfecc384493c31e0f00fcd9873ded36486b8618493fdf4e18" . "Checksum of the vendored text file inputs/2007-hierarchies.abstract.txt, not of the publisher's copy." . . . "Abstract of Hierarchies, multiple energy barriers, and robustness govern the fracture mechanics of α-helical and β-sheet protein domains, as read" . . "buehler-protein-materials" . "2026-11-08T04:38:11.893Z"^^ . . . . . "Theodor Ackbarow et al. 2007 claim: The hierarchical Bell model: a structure–property law for protein strength" . . "RSA" . "MIIBIjANBgkqhkiG9w0BAQEFAAOCAQ8AMIIBCgKCAQEAyFHtPB6p0mCjDozxlOY9CmO3nBm7Te3jFbXyLmM+G1tGe47Da0jYIbhkl9Oo1ZjxmCa3LCwlEqK8gj6hEIcL4ahW30JXh9HycfKVk1bugkzMEV15tJdTzAU3sGMFnbNsn43xY2LFLy45tApkRm5byEiM7oRRrooIdwoQKnP4Wzq/bfAoJ9qvW+MGkelyOTd1jbSt+KfmT7xKmDfWzAHddUlWwMX2QJOLxDaffOteTop6ZRgPiNmPJKA/S62Gac9XtSYT3oDkZ6/nr5cxxNLpOyxTf5FPISiCe11xYO/unYjY+mwnOfA8yhJIyx1l0O9I0vpOpn+E5TvyHJV+oSTrvwIDAQAB" . "Wd7SmHThu+Ae501WWSmHuxoBoY+OtSKYzwJ1BpERtrqI3jX8SsVbqp5g+nOweg3Y8R1NFASADY8nn6H+n5wqb2MTz3L0ERZD6Q+AqxaMEstL8fzm2Ou5wewEOicSUOVrDnM/fbSDsjNjf+d9GuZZzs6bwpBPWDywV8cj6IZhEXJbkPlAY1GfFiixN3nkNGPHxmJuYhrpN0W9H2g3J/XQQK8a85avwWMiv2MIo5stHzpnoi7qeaBQdPlEYlsYLksHZe0RGfUEqa/zpM+u3bsSD9hp7oy6NCOlItYbkvHTkOU9jTYoV7Fl5ToSKVXS/sQvMy13RtLOfssPJRGjKZKVow==" . . .