Natural tristability of a confined helical filament with anisotropic bending rigidities

Abstract

We find that when c0R∼0.5\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$c_0 R\\sim 0.5$$\\end{document} and τ0R<0.11<c0R\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ au _0R<0.11<c_0R$$\\end{document}, confining a helical filament with anisotropic bending rigidities within a cylindrical tube of radius R can create a natural tristable status which is consisted of two low-pitch helices and one high-pitch helix, where a helical filament is referred to as a filament that has both an intrinsic curvature (c0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$c_0$$\\end{document}) and an intrinsic twist rate (τ0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ au _0$$\\end{document}). The formation of the tristable status also requires that the filament has a significant difference between two bending rigidities and a large twisting rigidity. The relative heights of two low-pitch helices in a tristable status are close to zero, and the smaller the intrinsic twisting angle, the smaller the difference between these two heights. Moreover, at a large intrinsic twisting angle, two low-pitch helices display a large energy difference, and the energy difference increases with decreasing τ0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ au _0$$\\end{document}. Meanwhile, the relative height of the high-pitch helix is always close to that of a straight line. Finally, at some specific intrinsic parameters, the tristable status can include an isoenergic status with two helices of the same energy but distinct pitches.