Turning and twisting motion of chromosomes as an accelerating force to align homologous chromosomes during meiosis

Homologous sets of parental chromosomes must pair during meiosis to produce recombined sets of chromosomes for their progeny. This is accompanied by nuclear oscillatory movements. This study aimed to elucidate the significance of these movements with a model, wherein external force was applied to the oscillating nucleus and via hydrodynamic interactions within the nucleus. Simulations revealed that a major force for aligning homologous chromosomes is length-dependent sorting during chromosomal turning and twisting, which occur when the nucleus reverses the direction of its movement.

Meiosis is an important process for eukaryotic organisms to generate inheritable haploid gametes from a diploid cell. During meiosis, homologous (paternal and maternal) chromosomes recombine; these recombined chromosomes are inherited by their progeny. For recombination of homologous chromosomes, side-by-side alignment of homologous loci along the chromosome is necessary [1]. During this alignment of homologous chromosomes, telomeres form a cluster beneath the nuclear envelope in various eukaryotic organisms [2]. Unicellular fission yeast Schizosaccharomyces pombe is a useful model system for studying the alignment of homologous chromosomes during meiosis [3,4]. A meiotic cell of S. pombe contains 3 pairs of homologous chromosomes (chromosomes 1, 2, and 3 of 5.6, 4.6, and 3.5 Mb, respectively) [5]. During meiotic prophase, the nucleus elongates, oscillates between the cell poles, and shows drastic deformation, so-called "horsetail movement" [6], during which telomeres remain clustered at the spindle pole body (SPB) at the leading edge of the nuclear movement [6]. Telomere clustering and nuclear movements reportedly promote the alignment of homologous chromosomes [7,8,9].
To elucidate mechanisms underlying the alignment of homologous chromosomes, computer simulation strongly facilitates the examination and prediction of biological events under conditions that are never accomplished empirically. Various theoretical models of the pairing of homologous loci during meiosis have been proposed on the basis of telomere clustering [10,11,12,13]. However, the role of the nuclear envelope, which confines the chromosomes, remains unknown in these models.
Studies are required to consider the role of the nuclear envelope in models of the meiotic nucleus of S. pombe, since its nucleus migrates and undergoes striking deformation during meiosis. During the horsetail movements, the nuclear envelope receives drag forces from the cytoplasm and from the nucleoplasm, while each chromosome receives drag forces only from the nucleoplasm. Thus, models of the meiotic nucleus of S. pombe would help describe the motion of the deformable nuclear envelope, direct collisions between the nuclear envelope and chromosomes, effects of the cytoplasm on the nuclear envelope, and effects of the nucleoplasm on chromosomes and the nuclear envelope, in addition to telomere clustering at the SPB. This study aimed to investigate the role the horsetail nuclear movements in the alignment of homologous chromosomes via simulations of an initial model displaying the aforementioned nuclear membrane dynamics.
Model construction: A model of chromosomes confined within the nuclear envelope of S. pombe cells ( Fig. 1(a)) was developed using the following methods. We described each chromosome as a chain and the nuclear envelope as a one-layered shell-both comprising particles with an excluded volume ( Fig. 1(b)). A meiotic nucleus of S. pombe contains 3 homologous pairs of chromosomes; thus, the model contains 6 chains in total confined in the shell. To account for the effects of the SPB on chromosomal movement, we constructed a virtual sphere around the SPB ("the SPB sphere"), confining a limited area of the shell (nuclear envelope) extending toward the space inside the shell ( Fig. 1(b)). To simulate telomere movement, which are clustered near the SPB during the horsetail movement, we assumed that ends of the chromosome chains move along the surface of the SPB sphere inside the shell.
To simulate the horsetail movements, we assumed that the SPB sphere oscillates periodically in the cell. Here, the drag force from the cytoplasm is exerted exclusively on the nuclear envelope. Since the cytoplasm can be considered a static intracellular fluid, we assumed that this drag force is proportional to the velocity of the nuclear envelope. However, the drag forces from the nucleoplasm to the nuclear envelope and chromosomes depend on the intranuclear velocity profile of the nucleoplasm, which is primarily determined on the basis of velocity profiles of chromosomes and the nuclear envelope. Thus, drag forces from the nucleoplasm could be described by the hydrodynamic interaction among particles in the chains and the shell.
These methods allowed for the generation of an appropriate model involving the effects indispensable for simulating the S. pombe meiotic nucleus.

Model implementation:
We assumed that chromosomes 1, 2, and 3 consist of ܰ particles (n = 1, 2, or 3) and the nuclear envelope consists of M particles with radii = r.
The motion of the i-th particle is given as where the i-th particle is a part of the chain corresponding to n-th chromosome, and and R S indicate the position of the center and the radius of the SPB sphere.
The potential contribution from the nuclear envelope ܷ ௦ is given as follows: where the i-th particle is a part of the nuclear envelope. indicates the initial distance between the centers of the i-th and j-th particles constructing the nuclear envelope ‫ܮ(‬ , ௌ , the initial distance between the centers of the i-th particle and the SPB sphere). Here, the initial structure of the nuclear envelope is indicated as a spherical shell with a thickness of one particle layer constructed by the rings along with y-axis with various radii consisting of particles ( Fig. 1(c)), and initial position of the center of the SPB sphere is denoted as radius of the cell as described below). We used an unharmonic potential in the 1st term of ܷ ௦ , which provides tension between particles in the nuclear envelope since the nuclear envelope seems to be able to deform easily.
The second term of ܷ ௦ indicates the binding of the SPB to the nuclear envelope, where particles constituting the nuclear envelope and confined in the SPB sphere were connected tightly to the center of the SPB sphere. The third term of ܷ ௦ provides the force to sustain the volume of the cell nucleus where and V indicate the position of the center of mass and the volume of the nucleus, respectively. Here, V is denoted by indicates the excluded volume effect of the cell wall, where the cellular shape was assumed to be a parallel cylinder along the z-axis, with a radius R C . Here, the central axis of this cylinder is denoted as ܷ denotes the potential of the excluded volume effects among particles and the SPB sphere as follows: In the third term of equation (1), indicates the force of hydrodynamic effects from the j-th particle to the i-th particle, providing a drag force to each particle from the nucleoplasm. Herein, we employ the lubrication approximation [14,15], which is denoted as follows: denoted the radius of the particle and a relative unit vector from the j-th particle to the i-th particle , respectively. Nucleoplasm viscosity was assumed the same as the cytoplasm viscosity ߟ in this model.

Parameters for simulations:
In the simulations, we assumed a particle radius Here, the inner diameter of the initial spherical structure of the nuclear envelope was assumed to be , the initial distance between the center of the SPB sphere and that of the nucleus was assumed to be , and the radius of the SPB sphere was assumed ܴ ௦ ൌ 1 0 ‫ݎ‬ ( Fig. 1(b)). In this case, the diameter of the cross-section where nucleus and the SPB sphere intersect was The initial configurations of the chromosomes were randomized in this spherical shell.  To obtain a pictorial view of chromosomal movements, we plotted positions of the chromosomes as probability distributions along the x-y plane (Fig. 4). The plots were obtained from the 34 (= N 3 ) nearest particles from the SPB sphere (N 3 / 2 = 17 nearest particles from each end) along each of the 6 chromosomes (Fig. 4a).  However, when the SPB sphere exhibits 1-d oscillatory motion involving no twisting motion of chromosomes during turning, some homologous chromosome pairs often stay separated from each other (Fig. 4 (d); separation of non-homologous chromosomes was observed exceptionally only in one out of eight simulations).
Furthermore, separation of non-homologous chromosomes was not apparent when the SPB sphere exhibits a linear motion involving no chromosomal turning or twisting ( Fig.   4 (e)). These results suggest that chromosomal turning and twisting motion is crucial for shuffling of chromosomes for appropriate alignment of homologous chromosomes.
Non-homologous chromosomes are considered to separate owing to differences in chain lengths among non-homologous chromosomes. During the horsetail movement, the SPB movement reverses its direction periodically. Chromosomes receive drag forces from the nucleoplasm at every turn of the SPB, while no drag force is exerted on chromosomes and the nuclear envelope if SPB migrates linearly with a constant velocity. During reversal of SPB, the region distal from the SPB in longer chromosomes tends to be dragged by the slow (nearly stopped) motion of the rear region of the bent nucleus, while that in the shorter one tends to follow the SPB motion (Fig. 5). Thus, the relative positions of longer and shorter chromosomes tend to separate gradually from each other owing to reversal of the iterative direction of the SPB movement, resulting in the separation of non-homologous chromosomes. Owing to this separation, homologous chromosomes are aligned and stabilized during the periodic reciprocation of the SPB.

Discussion:
Homologous chromosome pairing has received increasing attention in studies in physics, and several models have been proposed to describe its underlying mechanisms [10,11,12,13,16]. In the present simulations, separation of non-homologous chromosomes was achieved when the SPB sphere shows 3-d oscillations in the cell, but not when the SPB sphere exhibits only a linear motion. This result is further substantiated by previous experimental evidence regarding S. pombe: the efficiency of homologous recombination decreases drastically when the nucleus 0 does not exhibit the horsetail movement but just elongates [17]; the iterative reversals of the direction of the SPB movement during the horsetail movement are essential for the alignment of homologous chromosomes via separating non-homologous chromosomes [7,9]. Our simulation reveals that a major force for the alignment and sorting of homologous/non-homologous chromosomes is the nucleoplasmic flow generated owing to SPB reversal and deformation of the nuclear envelope, involving direct collision and indirect hydrodynamic interactions between chromosomes and the nuclear envelope, which were previously unclear. The present model is the first, to our knowledge, to consider both direct collisions and indirect hydrodynamic interactions as essential mechanisms for the alignment of homologous chromosomes.
This study was partly supported by the Platform Project for Support in Japan Agency for Medical Research and Development (to AA); MEXT KAKENHI Grant Number JP 17K05614 (to AA), JP18H05528 (to TH) and JP18H05533 (to YH).