An allometric relationship between mitotic spindle width, spindle length, and ploidy in Caenorhabditis elegans embryos

Abstract

The mitotic spindle is a diamond-shaped molecular apparatus crucial for chromosomal segregation. The regulation of spindle length is well studied, but little is known about spindle width. Previous studies suggested that the spindle can self-organize to maintain a constant aspect ratio between its length and width against physical perturbations. Here we determine the widths of metaphase spindles of various sizes observed during embryogenesis in Caenorhabditis elegans, including small spindles obtained by knocking down the tpxl-1 or spd-2 gene. The spindle width correlates well with the spindle length, but the aspect ratio between the spindle length and spindle width is not constant, indicating an allometric relationship between these parameters. We characterize how DNA quantity (ploidy) affects spindle shape by using haploid and polyploid embryos. We find that the length of the hypotenuse, which corresponds to the distance from the apex of the metaphase plate to the spindle pole, remains constant in each cell stage, regardless of ploidy. On the basis of the quantitative data, we deduce an allometric equation that describes the spindle width as a function of the length of the hypotenuse and ploidy. On the basis of this equation, we propose a force-balance model to determine the spindle width.

FIGURE 1:

Spindle shape during embryogenesis in C. elegans. (A–H, e–h) Microscopic images of metaphase spindles at four representative cell stages. (A–D) Chromosomes and centrosomes were simultaneously visualized by GFP–histone H2B and GFP–γ-tubulin. (E–H, e–h) Microtubules and chromosomes of spindles with similar sizes as in A–D were visualized by GFP-tubulin and mCherry–histone H2B. Only microtubules (E–H) or both microtubules (green; e–h) and chromosomes (red; e–h) are shown. (A, E, e) The 2-, (B, F, f) 16-, (C, G, g) 50-, and (D, H, h) 200-cell stage embryos. Bar, 5 μm. (I) The length (green diamonds) and width (yellow circles) of the metaphase spindles plotted against the cell length in wild-type embryos. Schematics show the structure of the C. elegans mitotic spindle and illustrate the definitions of spindle width and length used in this study. Microtubules, chromosomes, and the centrosome are shown with green, red, and yellow, respectively. Spindle length was defined as the distance between the centers of two centrosomes visualized by GFP–γ-tubulin. Spindle width is the width of microtubules at the equatorial plane and is identical to the long axis of the GFP-histone–positive region (metaphase plate) in the C. elegans embryo.

FIGURE 2:

Relationship between spindle width and spindle length in C. elegans. (A) Relationship between spindle width and half–spindle length in C. elegans embryos. (B) The calculated aspect ratio of spindle width/length plotted against the spindle length in C. elegans embryos. Color shows the data at different cell stages. Blue circles, 1-cell stage; light blue diamonds, 2- and 4-cell stages; green triangles, 8- and 16-cell stages; yellow rectangles, 28-cell stage; and pink crosses, 50-cell and later stages.

FIGURE 3:

Manipulation of spindle length. Half–spindle length (A), spindle width (B), cell length (D), nuclear diameter (E), and chromosome length (F) of one-cell-stage wild-type (wt, blue), tpxl-1 (RNAi) (pink), and spd-2 (RNAi) (yellow) embryos are shown. Error bar, SD. The half–spindle length and spindle width for both in tpxl-1 (RNAi) and spd-2 (RNAi) embryos are significantly different from those of wild-type embryos (*p < 0.005). These mean values are listed in Supplemental Table S2. (C) Relationship between spindle width and half–spindle length in tpxl-1 (RNAi) and spd-2 (RNAi) embryos. Data from tpxl-1 (RNAi) embryos (pink circles) or spd-2 (RNAi) embryos (yellow triangles) at the one-cell stage are plotted against data of wild-type (gray diamonds) embryos. Blue diamond shows data from only wild-type embryos at the one-cell stage.

FIGURE 4:

Manipulation of DNA quantity and calculation of the hypotenuse length of the spindle. (A) Microscopic images of embryos with different ploidy at the two-cell stage. The control embryos are those with a diploid genome. RNAi for klp-18, ani-1, or mei-1 occasionally induced haploid and polyploid (DNA in excess of the diploid genome) embryos. The images are of two-cell-stage embryos at metaphase treated with RNAi for klp-18. Bar, 5 μm. (B) Relationship between spindle width and length in embryos with different ploidy. The data of each embryo are shown using different symbols. Blue diamonds, wild-type diploid; yellow circles, klp-18 (RNAi) haploid; yellow open circles, mei-1 (RNAi) haploid; green triangles, klp-18 (RNAi) polyploid; and green open triangles, ani-1 (RNAi) polyploid. (C) The spindle width (pink), half–spindle length (light blue), and hypotenuse length (orange) at the 2-, 4-, 8-, and 16-cell stages are shown. Symbols and error bars represent the mean values and SD of each parameter, respectively. The data of haploid (1N) and polyploid (>2N) embryos include the results of klp-18 (RNAi), ani-1 (RNAi), and mei-1 (RNAi). Statistical differences between the data from haploid or polyploid embryos and those from diploid embryos at each cell stage are shown by asterisks (*p < 0.005; **p < 0.05). The plotted values are listed in Supplemental Table S3. (D) Microscopic images and schematic figure of the metaphase spindle. The bipolar spindle was assumed to be described by two cones, each attaching at the base plane. The half–spindle length, spindle width, and hypotenuse length of the spindle correspond to the height (light blue), base (pink), and hypotenuse (orange) of each triangle. (E) Microscopic images and schematic figures of prometaphase. The premature spindle was also assumed to be described by two cones, each attaching at the base plane. Spindle parameters were defined as in D. (F) The mean values of spindle width, half–spindle length, and hypotenuse length from the 2- to 100-cell stages are shown. The statistical difference between data from prometaphase (PM) and metaphase (M) at each cell stage is shown by asterisks (*p < 0.005; **p < 0.05).

FIGURE 5:

An allometric relationship between spindle width, hypotenuse length and ploidy. (A) Double-logarithmic plot between spindle width and hypotenuse length in wild-type diploid embryos. (B) The spindle width is plotted against the 0.58 power of hypotenuse length in diploid (blue diamonds) and haploid (yellow circles) embryos. Regression lines for the data of each diploid and haploid embryo are shown. (C) The slopes of each regression line shown in C are plotted against ploidy in a double-logarithmic graph. (D) The spindle width is plotted against P^0.36 × HL^0.58. A regression line is shown for all data from diploid (blue diamond) and haploid (yellow circle) embryos. (E) The value of ploidy in polyploid embryos was calculated using the relationship described in D. With this value, the spindle width was recalculated and plotted (green triangles) as in D. A regression line is shown for data only from polyploid embryos. (F) R² calculated using plots of SW against P^α × HL^β and listed for variations in the exponents of α and β. The background colors represented the strength of R².

FIGURE 6:

Force-balance model to set the spindle width. (A) Schematic of our proposed model. The force balance between a squeezing force (orange arrow) and expanding force (blue arrow) determines the spindle width. In generating the squeezing force (left inset), the microtubule attached to the chromosome is assumed to act as would an elastic rod. When the microtubule is bent by moving the chromosomes to the outside (i.e., increasing the chromosome number), the microtubule tends to straighten. Because the other end of the microtubule is tethered tightly at the centrosome, the force generated by straightening of the microtubule induces the chromosome to move to the inside the spindle, thus compressing the spindle width. For generating an expanding force (right inset), interchromosome repulsion is assumed. When chromosomes are tightly packed (i.e., with increasing chromosome numbers), the chromosomes tend to move away from each other due to this repulsive force. This movement induces an expansion of the metaphase plate. (B) The calculated P^0.5 × HL^0.75 values, based on our proposed model, plotted against spindle width for diploid (blue diamonds) and haploid (yellow circles) embryos. The regression line and its equation are described on the graph. R² = 0.64.