Oblique circle method for measuring the curvature and twist of mitotic spindle microtubule bundles

Abstract

The highly ordered spatial organization of microtubule bundles in the mitotic spindle is crucial for its proper functioning. The recent discovery of twisted shapes of microtubule bundles and spindle chirality suggests that the bundles extend along curved paths in three dimensions, rather than being confined to a plane. This, in turn, implies that rotational forces, i.e., torques, exist in the spindle in addition to the widely studied linear forces. However, studies of spindle architecture and forces are impeded by a lack of a robust method for the geometric quantification of microtubule bundles in the spindle. In this work, we describe a simple method for measuring and evaluating the shapes of microtubule bundles by characterizing them in terms of their curvature and twist. By using confocal microscopy, we obtain three-dimensional images of spindles, which allows us to trace the entire microtubule bundle. For each traced bundle, we first fit a plane and then fit a circle lying in that plane. With this robust method, we extract the curvature and twist, which represent the geometric information characteristic for each bundle. As the bundle shapes reflect the forces within them, this method is valuable for the understanding of forces that act on chromosomes during mitosis.

The chirality of the mitotic spindle provides a mechanical response to forces and depends on microtubule motors and augmin

“…The bundle tracing method is an extension of the approach developed previously 28 , where individual bundles are manually traced by following the bundle contour in the end-on view of the spindle (Figure S1A, right; Methods). Subsequently, each bundle is analyzed so that a plane is fitted to the points representing the bundle, and a circle that lies within this plane is fitted to these points and used to calculate the curvature and twist of individual bundles 43 (Figure S1A, right; Methods).…”

“…To quantify spindle twist, we used 3 complementary approaches: visual assessment, optical flow, and bundle tracing (Figure S1A; Methods). As it is still an open question in the field what method is the most appropriate to measure spindle twist 28,29,43 , visual assessment is useful as a quick and rough estimate of the twist and as a control for automated or semi-automated methods. In the visual assessment method, the spindle is observed end-on and the rotation of microtubule bundles around the pole-to-pole axis is estimated visually.…”

“…We convert the imaging plane (z-plane) to its corresponding z-coordinate by multiplying with the distance between successive planes set during image acquisition (0.5 μm) and by a factor of 0.81 to correct for the refractive index mismatch 28 . Next, to describe the shape of a microtubule bundle, we use the Oblique circle method 43 . We first position the spindle so that the pole-to-pole axis is aligned with our coordinate system, i.e., we untilt the spindle.…”

“…S1A, right; Methods). Subsequently, each bundle is analyzed so that a plane is fitted to the points representing the bundle, and a circle that lies within this plane is fitted to these points and used to calculate the curvature and twist of individual bundles (Ivec et al, 2021) (Fig. S1A, right; Methods).…”

“…We convert the imaging plane (z-plane) to its corresponding z-coordinate by multiplying with the distance between successive planes set during image acquisition (0.5 µm) and by a factor of 0.81 to correct for the refractive index mismatch (Novak et al, 2018). Next, to describe the shape of a microtubule bundle, we use the method described in (Ivec et al, 2021). We first fit a plane to the points representing the bundle, and then we fit a circle that lies in this plane to the same points.…”