Fig. 5: Model description of phagophore and Gaussian modulus. | Nature Communications

Fig. 5: Model description of phagophore and Gaussian modulus.

From: Experimental determination and mathematical modeling of standard shapes of forming autophagosomes

Fig. 5

a A schematic description of the phagophore. a and b represent the inner and outer membranes, respectively, which are approximated by two paralleled axisymmetric membranes. c represents the rim which is approximated as a line. \(z\) is the axis of symmetry, \(s\) is the length along the contour measured from the origin, and \(\theta\) is the angle between the tangent to the contour and the \(x\)-axis. \({M}_{m}\) is the meridional moments. \({J}_{m}\) and \({J}_{p}\) are the two principal curvatures. \(J\) and \(K\) are the total and Gaussian curvatures, respectively, with J > 0, K > 0 for convex surfaces and \(J\approx 0,\) \(K < 0\) for catenoidal surfaces. b Gaussian modulus obtained by fitting the experimental shapes, where \({\kappa }_{b}=20{k}_{B}T\). The membrane area, \(A=4\pi \int f\sqrt{1+{\left({f}^{{\prime} }\right)}^{2}}{dx}\), is also obtained from the polynomial fitting. c–f Membrane shapes obtained from the bending energy with different values of the Gaussian modulus, \({\kappa }_{G}=0\) (red), \(-0.5{\kappa }_{b}\) (green), and \(-{\kappa }_{b}\) (blue), for different values of the rim radius, \(l=0.8\) (c), \(l=0.6\) (d), \(l=0.4\) (e), and \(l=0.2\) (f). The unit of length was non-dimensionalized by the length, \(\sqrt{A/2\pi }\). Source data are provided as a Source Data file.

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