Experimental determination and mathematical modeling of standard shapes of forming autophagosomes

The formation of autophagosomes involves dynamic morphological changes of a phagophore from a disk-shaped membrane cisterna into a cup-shaped intermediate and a spherical autophagosome. However, the physical mechanism behind these morphological changes remains elusive. Here, we determined the average shapes of phagophores by statistically investigating three-dimensional electron micrographs of more than 100 phagophores. The results showed that the cup-shaped structures adopted a characteristic morphology; they were longitudinally elongated, and the rim was catenoidal with an outwardly recurved shape. To understand these characteristic shapes, we established a theoretical model of the shape of entire phagophores. The model quantitatively reproduced the average morphology and revealed that the characteristic shape of phagophores (i.e., an elongated shape with a catenoidal rim) was primarily determined by the relative size of the open rim to the total surface area. These results suggest that autophagosomal membranes are highly flexible and that the morphological changes during autophagosome formation follow a stable path determined by elastic bending energy minimization. Summary The formation of autophagosomes involves dynamic morphological changes of membrane cisternae. Sakai et al. determined the average shapes of forming autophagosomes by statistically investigating three-dimensional electron micrographs and established a theoretical model that quantitatively reproduces the phagophore shapes.


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Eukaryotic cells have membranous organelles with a variety of characteristic shapes. The 42 autophagosome, which mediates macroautophagy (hereafter simply referred to as 43 autophagy), is unique among the various organelles in that its generation involves 44 characteristic membrane deformation. Autophagy is a bulk degradation process, through 45 which a portion of the cytoplasm is sequestered into autophagosomes and is degraded upon 46 fusion with lysosomes ( Fig. 1) (Mizushima et al., 2011). In this process, a flat cisterna 47 called a phagophore (also called an isolation membrane) grows from an initial disk through is only in its early phase, the morphology of phagophores has been analyzed by 63 approximating parts of a sphere or an ellipse (Knorr et al., 2012;Sakai et al., 2020). 64 However, the actual morphology of phagophores observed in cells is more complex, and 65 previous studies have oversimplified their morphology. 66 In the present study, we statistically and quantitatively investigated the morphology of 67 phagophores. First, we determined the average morphology of more than 100 phagophores 68 obtained from three-dimensional (3D) electron micrographs. The results showed that 69 phagophores were elongated vertically, with a rim curved outward to form a catenoid-like 70 shape. To understand the morphological characteristics, we developed a theoretical model 71 based on the elastic bending energy. The morphology of the phagophore was considered 72 to be in equilibrium at each step (Sakai et al., 2020). The physical parameter, Gaussian 73 modulus, which determines the elastic properties of the phagophore membrane, was

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Quantification of phagophore morphology from 3D-electron micrographs 81 To understand the standard shapes of phagophores, we determined the average morphology 82 of phagophores in mouse embryonic fibroblasts by array tomography, a technique that can 83 be used in 3D electron microscopy (Micheva and Smith, 2007). Images of more than 100-

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To obtain an average shape of structures with different sizes, we normalized the size of 97 each 3D phagophore using standard z-score normalization since the morphologies were 98 almost size-independent (Fig. S1). In z-score normalization, each point ⃗ !,# was converted 99 to a standardized point (⃗ !,# as

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Because phagophores in the electron microscopy images were oriented in various 105 directions, we rotated each structure to be oriented in a uniform direction (along the z-axis).

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The average edge vector of a normalized point cloud, ⃗ ) ! , was used as the orientation of 107 the -th phagophore and was obtained as . (4)

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The four categories of observed shapes were as follows: disk ( ) ! > 1.1), early-cup (0.8 <  One methodological concern was that these characteristic shapes might have resulted from Here, the first term is the elastic bending energy with the total curvature, = 1 + 2 , and 187 Gaussian curvature, = 1 2 , of the inner and outer membranes, which is scale-invariant.

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The spontaneous curvature of the inner and outer membranes was assumed to be zero  obtained by electron microscopy (Fig. 3). Note that < was always negative (Fig. 6A-D), 233 which indicates that membrane area growth decreased the energy of cup shapes [see Eq.

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Next, we calculated the total curvature to quantitatively evaluate the shapes obtained by 237 the theoretical model ( Fig. 6A-D). In all structures, from a disc to a cup, the total curvature 238 was reduced near the rim. The meridional curvature, 1 , and parallel curvature, 2 , were 239 also calculated ( Fig. 6E-H). The meridional curvature was negative near the rim, reflecting 240 the outward curvature of the membrane, while the parallel curvature was positive. As a 241 result of the two competing curvatures, negative and positive, the total curvature near the 242 rim seemed to be smaller. Except for the rim, which was isotropic, the two curvatures were 243 of approximately the same magnitude. 244 We further investigated the change of curvatures during the morphological transition from 245 a disc to a cup (Fig. 6I). As the rim radius decreased, the parallel curvature 2 at the rim 246 increased monotonically because the curvature given by 2 = sin / was approximately 247 inversely proportional to the radial distance . In the middle-and late-cup shapes, the 248 surface near the rim got closer to the rotational axis, and therefore the parallel curvature,  Comparison between theoretical and experimental morphologies 254 We compared the results of our theoretical calculations with our experimental data. The 255 point clouds obtained from electron micrographs (Fig. 3E-H) were superimposed on the 256 theoretically calculated morphologies with the same rim sizes (Fig. 7). At each step, the 257 experimental and theoretical results overlapped well. In particular, the longitudinally 258 elongated shape and the catenoidal rim were well captured by the theoretical calculations.

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The individual structures of various sizes at each step were also compared with the 260 theoretical morphology (Fig. S1). Over a wide range of size scales, the theoretical model  However, there was a slight discrepancy between the theoretical expectation and 268 experimental data, especially at the late-cup stage (Fig. 7). This discrepancy may be   The rim length is considered to be constant on the time scale of mechanical relaxation as 306 we assumed in this study, but it changes with phagophore growth on larger time scales.

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The change in the rim length could be explained by a change in the abundance of curvature 308 generators at the rim that stabilizes the highly curved rim (Sakai et al., 2020).

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A recent cryo-electron tomography study in yeast shows that the intermembrane space   were removed with a razor and by soaking in liquid nitrogen repeatedly.

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To cut ultra-thin serial sections, we used a diamond knife with an ultra-jumbo boat (Ultra where : is the bending rigidity and 4 , ̅ 4 , and 4 are the total curvature, spontaneous 417 curvature, and area, respectively, of the rim. The rim geometry was modeled as a part of a 418 torus with an intermembrane radius (Fig. 5A). In the torus approximation, the total 419 curvature and the surface element of the rim were respectively given by (20) 447 Therefore, the Gaussian modulus, ; , is determined by fitting the experimental phagophore 448 membrane shapes (Fig. 4), where the curvatures are obtained from Eq. (7).

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The Gaussian modulus obtained by fitting the experimental phagophore membrane shapes 450 with a polynomial function was -(0.21 ± 0.05) : , with no significant differences among 451 the disc, early-, middle-, and late-cups (Fig. 5B). The deviation $ defined in Eq. (1), which 452 represents the spatial extent of the point clouds of each morphology (Fig. 3), was used as  and c correspond to the inner membrane, the outer membrane, and the rim, respectively.

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Eventually, autophagosomes and lysosomes fuse to degrade the contents of the former.