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Asymmetrical Inheritance of Plasmids Depends on Dynamic Cellular Geometry and Volume Exclusion Effects

The asymmetrical inheritance of plasmid DNA, as well as other cellular components, has been shown to be involved in replicative aging. In Saccharomyces cerevisiae, there is an ongoing debate regarding the mechanisms underlying this important asymmetry. Currently proposed models suggest it is established via diffusion, but differ on whether a diffusion barrier is necessary or not. However, no study so far incorporated key aspects to segregation, such as dynamic morphology changes throughout anaphase or plasmids size. Here, we determine the distinct effects and contributions of individual cellular variability, plasmid volume and moving boundaries in the asymmetric segregation of plasmids. We do this by measuring cellular nuclear geometries and plasmid diffusion rates with confocal microscopy, subsequently incorporating this data into a growing domain stochastic spatial simulator. Our modelling and simulations confirms that plasmid asymmetrical inheritance does not require an active barrier to diffusion, and provides a full analysis on plasmid size effects.


Abstract
The asymmetrical inheritance of plasmid DNA, as well as other cellular components, has been shown to be involved in replicative aging. In Saccharomyces cerevisiae, there is an ongoing debate regarding the mechanisms underlying this important asymmetry. Currently proposed models suggest it is established via diffusion, but differ on whether a diffusion barrier is necessary or not. However, no study so far incorporated key aspects to segregation, such as dynamic morphology changes throughout anaphase or plasmids size. Here, we determine the distinct effects and contributions of individual cellular variability, plasmid volume and moving boundaries in the asymmetric segregation of plasmids. We do this by measuring cellular nuclear geometries and plasmid diffusion rates with confocal microscopy, subsequently incorporating this data into a growing domain stochastic spatial simulator. Our modelling and simulations confirms that plasmid asymmetrical inheritance does not require an active barrier to diffusion, and provides a full analysis on plasmid size effects.

Author Summary
Cellular division is a highly controlled critical process on which all life depends.
Among many other organisms, baker's yeast (Saccharomyces cerevisiae) cells undergo asymmetrical inheritance. As the name suggests, yeast daughter cells inherit more of some cellular components, and less of others, than expected by cellular volume alone. While this process has been shown to be important in reducing the effects of cellular aging in daughter cells, the mechanisms behind this asymmetrical inheritance are still not well understood. It has been proposed that asymmetrical inheritance is achieved through the dynamical geometry of the cell alone, additional diffusion barriers, or a combination of the two. These contrasting views have created a controversy in the field of asymmetric inheritance of nucleoplasmic components. In this study we model the inheritance of a yeast plasmids inside the nucleus by using cell geometries and diffusion rates derived from confocal microscopy data into a new growing domain spatial stochastic simulator. We confirm that strong asymmetrical retention of the plasmid in the mother cell can be achieved without the need for a distinct barrier, and can solely rely on the moving boundaries and cellular geometry of the dividing cell.

Introduction
Organisms need to properly manage cellular aging processes to ensure population survival. Complex multicellular organisms do so by following a number of strategies, including the separation of a germ line early in development or the maintenance of pluripotent stem cells. Many strategies are not available to single--cell organisms, but they efficiently reduce the impact of aging through asymmetric cell division, wherein cellular components are not divided equally between mother and daughter cells during mitosis. This allows for titrating away cellular components such as extra chromosomal DNA, aged organelles and mis--folded proteins that could potentially shorten the life of subsequent cell generations.
For Saccharomyces cerevisiae, there are a limited number of divisions an individual cell can undergo before becoming senescent [1] . This process of replicative aging is undeniably complex, with many environmental and genetic factors influencing it. However, recent studies have demonstrated that asymmetrical inheritance of cellular components, where cellular components tend to remain disproportionally in the mother cell, are a key factor in yeast aging. Numerous experiments have demonstrated that the asymmetrical inheritance of mitochondria [2] and plasmids, such as autonomously replicating extrachromosomal rDNA circles (ERCs) [3] and autonomously replicating ARS domain containing plasmids (ARS plasmids) [4] , are linked to replicative aging (cf. reviews in [5][6][7] ).
There are notable differences between ERCs and ARS plasmids. Although both are self--replicating pieces of circular DNA, ERCs arise spontaneously from the resolution of Holiday junctions and appear ubiquitous throughout Eukaryota (reviewed in [5,6] ). In S. cerevisiae, ERCs arise from the 9.1kb locus containing 150 repeats of the rRNA precursors [3] . These plasmids accumulate in yeast cells over their life and can reach >500 copies [3] . In contrast, ARS plasmids are not naturally occurring, but are the result of modifying the naturally occurring 2μm plasmid [8] . In comparison to ERCs, the ARS plasmids tend to have lower copy numbers in the range of 10--100 [9] .
Although the existence of circular DNA asymmetrical inheritance has been known for a long time, there are still considerable gaps in the understanding of the mechanisms underlying this asymmetry. The current debate, regarding a potential mechanism for this asymmetrical inheritance, has two main postures: (1) either a morpho--kinetic passive diffusion model, with asymmetry being generated due to time and cell morphology [10][11][12] ; or (2) a barrier division model, where physical agents such as cellular machinery actively restrain DNA transmission [13][14][15] .
Many contemporary studies of plasmid inheritance have focused on only a single plasmid type [10,[13][14][15] . The lack of direct comparisons between ERCs or ARS plasmids compounds the difficulty in comparing these studies. Thus, it becomes necessary to highlight several factors that may be crucial to understanding the differences between observations and studies. First, S. cerevisiae mutants with extended mitosis length show a lower level of plasmid asymmetrical inheritance [10] . This can be intuitively expected, given lengthier time frames effectively allow for diffusion of more cellular components than shorter time frames. More recently, it was reported that ERCs are bound to the nuclear pore complex via interaction with the master regulator complex SAGA [13] . The authors of [13] also note that, even in the absence of nuclear pore binding, passive diffusion continues to play a critical role in asymmetrical inheritance, attributing this effect to a potential diffusion barrier. However, the movement of nuclear pores from the mother cell to the daughter themselves further complicates the diffusion of nuclear pore fused plasmids [16][17][18] . For instance, by using photo--switchable florescence microscopy, it was shown that nuclear pores effectively diffuse from the mother to the daughter nuclear lobe during anaphase [16] . Moreover, there is evidence for the active transport, mediated by the protein Nsp1, of the nuclear pores from mother to daughter [18] . In addition, computational models have suggested that moving boundaries can have an important effect [10] . However, these conclusions were derived by comparing the equilibration time of diffusing particles in static geometries to the overall length of mitosis. The question remains as to the magnitude of this effect, which can only be investigated by building a framework where moving boundaries are explicitly modelled. By the same token, most previous studies have focussed on treating plasmids as point particles [10,15] , rather than particles with volume, as is actually the case. Given that the region between daughter and mother nuclei can narrow in late anaphase, it stands to reason that the role of plasmid size in driving asymmetric cell division is understated in all existing models. Lastly, there might be large heterogeneity within nuclear division profiles, which in turn could lead to heterogeneous transmission rates.
The effects of particle size, moving cell boundaries and cell--to--cell variability, on plasmid diffusion, have remained largely unstudied. In this paper, we aim to incorporate these factors into the morpho--kinetic model, by using mathematical and computational modelling based on cell geometries and plasmid motion as obtained from confocal microscopy. We have used measured S. cerevisiae cellular geometries as a basis for modelling the geometry within which the inheritance of a single plasmid happens. In addition we tracked individual plasmids to compare experimental data with these simulation results.
By incorporating dynamical changes to cell morphology, explicitly into a spatial stochastic simulation framework, and plasmid diffusion rates, we explore the opposing postures of the debate from a quantitative perspective.

Effect of Moving Boundaries and Heterogeneity
Confocal microscopy showed a heterogeneous profile of nuclear geometries over the course of anaphase. plasmids being actually retained in the mother, as also evidenced by the difference between the proportions in Fig.  2. This effect was solely produced by the dynamic changes of cell geometry, as there was no diffusion barrier incorporated in the study, as previously done for nuclear pores, nucleoplasmic and nuclear membrane proteins in [15] . Simulations of point particles showed that the final transmission of plasmids to the daughter nucleus varied from a minimum 12.1 +/-- 3.8% (Fig.  2b) to a maximum of 26.2 +/-- 5.1% ( Fig.  2g (with 95% confidence intervals), with a mean of 21.4%. Representing the plasmid as a 50 nm sphere altered these percentages to 3.2 +/-- 2.0% (Fig.  2f) to a maximum of 20.0 +/--4.6% (Fig. 2j), with a mean of 13.4%. In all simulations of point particles and particles with 50 nm radius, plasmid transmission to the daughter nucleus was found to be significantly less that the proportion of volume represented by the daughter nucleus at a 95% confidence interval. These ranges are consistent with expectations from experimental studies [10,14] . However, these specific experimental measures require further thought, as will be explained in the Discussion section. The effect of increasing plasmid volume generally decreases the transmission rate to the daughter nucleus. However, we also observe geometry specific effects. These simulation results suggested that changing plasmid size should effect transmission of the plasmid from mother to daughter, in turn affecting the plasmid loss rate. The pAA4 plasmid contains 256 tandem lacO binding sites thereby allowing the reporter protein, LacI fused to GFP, to detect their cellular localisation. Thus, when a strain containing both the pAA4 plasmid and the GFP fusion is grown, the numerous GFP fusion molecules will bind the plasmid. The resulting plasmid with GFP bound could have either a more relaxed or compact configuration. The mitotic stability, the transmission percentage of a plasmid in selective conditions, and the plasmid loss rate, the rate at which a dividing cell produces a plasmid free daughter, was calculated for two strains containing the pAA4 plasmid: GA180, a wild type strain, and GA1320, GA180 containing the GFP--LacI reporter and NUP49--GFP. We found no statistically significant difference between mitotic stability between the two strains, but the plasmid loss rate was significantly lower in the GFP strain (see Table 1). The latter suggests a more compact configuration of pAA4 with GFP bound.

Discussion
The work here demonstrates that heterogeneity, in terms of cell morphology within a S. cerevisiae population, greatly influences the inheritance of individual autonomously replicating plasmids. Moreover, the asymmetry of this inheritance is also influenced by the moving boundaries of the cell, the rate of diffusion and the size of the diffusing plasmid / particle. The effect of moving boundaries is best exemplified in Fig. 3. In regions of parameter space where we expect moving boundaries to have a minimal effect (i.e. where plasmids are modelled as points with high diffusion coefficients), numerical simulations demonstrate an asymmetric distribution of plasmids towards the mother. Moreover, this effect was observed in all measured cell geometries. For our measured geometries, particles diffusing with speeds of D = 0.001− 0.009µm 2 / s were found to be sensitive to changes in diffusion constant. This is in line with previously reported values [12,13] .
It is worth mentioning the work in [12] incorporates the effects of moving boundaries analytically, by using a narrow--escape approximation to calculate the distribution of escape times from the mother to a daughter. Here, we additionally considered several empirically--determined geometries, and included plasmid size into our investigations, complementing and extending the results in [12] . Further work in generalising the analytical techniques in [12] to encompass these factors could lead to interesting further work.
Owing to the size, behaviour, compartmentalisation and function of the numerous cellular components, it is unlikely there is a single mechanism driving asymmetrical inheritance of these cellular components. The analytical model of diffusion--driven transfer in [12] demonstrated there is a threshold passive diffusion rate with which cellular components remain within the mother cell during cell division. Therefore, smaller particles, moving at speeds near or faster than this threshold, may require additional mechanisms to remain in the mother cell during division. This does not preclude the existence of barriers, such as the one hypothesized in [15] , but there are as yet few concrete suggestions as to what might constitute such barriers. The effect of cell heterogeneity, diffusion coefficients and particle size were all found to be mutually dependent, as shown in Fig. 3 and 4. Generally, increasing plasmid size and decreasing diffusion coefficients decreased plasmid transmission to the daughter lobe. This is consistent with previous empirical studies that linked increasing diffusion constants with increasing plasmid inheritance [19] . However, there were geometry--specific exceptions.
Particularly, for biologically plausible plasmid sizes, there were measured nuclear geometries where increasing diffusion coefficients had no meaningful effect on the final transmission of plasmids ( Fig. 3f and 3g). We anticipate that in these scenarios, the geometric constraints were limiting factors in the transmission of plasmids, since these cells formed especially narrow bridges between the daughter and mother lobes in early anaphase. This is consistent with previous studies, namely those with static boundaries that explored the manipulation of the nuclear bridge width [10] . However, there were also cells that showed stronger sensitivity to increasing diffusion coefficients (e.g. Fig. 3d).
These cells had a wider bridge between the mother and daughter lobes throughout anaphase. Thus, these findings caution against using idealised geometries for such studies. However, it should be emphasised that yeast mitosis and cellular geometries are very variable, heavily influenced by numerous factors including growth media, temperature and genetic background. As such, our aim of simulating numerous measured nuclear geometries was to demonstrate that this variation in mitosis could have an important effect on plasmid segregation. Furthermore, there is a great sensitivity of transmission rates to plasmid sizes over biologically relevant parameters (see Fig. 4). These Presently there are three methods for measuring plasmid loss rate: 1) a pedigree based system that relies on cell dissections each generation [20] ; 2) a microscope approach that tracks plasmid movement from mother to daughter cell [13] ; and, 3) plasmid loss rates based on whole populations of cells and the likelihood of dividing cells to lose the plasmid each generation [10,21] . While

Yeast Strains, Growth & Manipulations
The stains used in this study are summarized in Table 2. Yeast media was made as described in Sherman [22] , and S. cerevisiae strains were transformed using the lithium acetate method. Strains GA180 & GA1320 were transformed with the pAA4--lacO plasmid [23] . This plasmid contains a 256--tandom array of lacO binding sites that allow binding of a LacI--GFP fusion reporter protein. GA1320 + pAA4 GA180, NUP49--GFP LacI--GFP, pAA4--lacO [24] Microscopy For imaging, S. cerevisiae cells were grown in SC low fluorescence media overnight at 30°C and 400 μl of this culture was transferred to 5ml of fresh SC--URA--HIS low fluorescence media and grown for a further 4 hours at 30°C. Cells were immobilised on 0.7% low melting temperature agarose. The agarose squares were then inverted in 35mm µ--Dishes (Ibidi 81158). All microscopy was conducted at 30°C.

Cell Geometry
Cell geometry was visualised using a Zeiss LSM--780 microscope with a 100x oil emersion objective. GFP was visualised using a 488 wavelength Argon laser at 1% intensity. Two hundred 20 slice z--stacks, separated by 0.4µm, were taken at 30--second intervals. The FIJI installation of ImageJ was used for image analysis. Geometric coordinates were generated in ImageJ for growing cells, and nuclear lengths were generated accordingly (See Fig. 1 for geometry comparisons and S8 for geometry measurements). Ten individual cells were arbitrarily selected and imaged. These initial 10 nuclear geometries were used as the basis for the simulations. Subsequently, 50 additional cells were measured to examine how representative the original selection was. Although it is difficult for simple metrics to capture the complexity of dividing nuclei, several metrics were chosen and corresponding histograms were generated, with which we compared the initial selection to the subsequent 50 geometries (See Fig. S7). The metrics chosen aim to capture the size of both the mother and daughter cells and ratios between them. Based on this analysis our 10 cells are reasonably representative.

Plasmid Diffusion
Estimates of plasmid diffusion were obtained by using images generated on a Zeiss ELYRA PS.1 microscope with a 100x oil emersion objective. Plasmids in cells that contained a single bright spot were tracked. However, it is worth noting that these single plasmid spots may be made up of several plasmids (cf. Plasmid Loss Rate). Two hundred time steps with six slice z--stacks, separated by 0.5µm, were taken at 5--second intervals. GFP was visualised using a 488-wavelength Argon laser at 1% intensity with 200ms exposure. The FIJI installation of ImageJ was used for image analysis and bleach correcting [25] .
Each image was stabilised using the plugin Correct 3D Drift [26] . Individual cells were extracted from the image into new flies and movement of the plasmid was tracked using the plugin Manual Tracking with local barycentre correction.
Tracking consisted of movement of the plasmid from time point to time point.
However, tracking was halted when plasmid movement was localised to the nuclear membrane, and restarted once it moved away. This was due to technical limitation is discerning between the GFP tagged plasmids and the GFP tagged Nup49. Moreover, at 0.5μm, the z--slices were too far apart to accurately track plasmid movement through the z--axis. Therefore, plasmid movement was tracked on a single Z plane; when the plasmid changed Z position the tracking was stopped and restarted. This limits the data collected but reducing the distance between Z--slices resulted in greatly increased bleaching of GFP.
Furthermore, assuming the diffusive motion is isotropic, the loss of resolution in the Z coordinate should lead to no difference in the estimated diffusion coefficient if the resulting data is treated as a two dimensional Brownian path.
Between 50--100 plasmid jumps were recorded for each plasmid. A total of 25 plasmids were individually tracked from cells in both anaphase and interphase (See Supplementary Fig. 1 for microscopy examples). Individual n is the number of generations, derived from Longtine et al. [21] .
A microscopy approach to plasmid loss was obstructed by plasmid clumping. Although the plasmids are often present in high numbers within a cell, there are numerous cells that have a single bright spot. This plasmid clumping behaviour has also been observed in previous studies [10,13] . It was found that in many of anaphase cells examined this single bright spot would split into two equally bright spots in late anaphase, with one entering the daughter.
In Silico Simulations Diffusion of plasmids within dividing nuclei were simulated using tailor-made particle--tracking methods designed to incorporate moving boundary and plasmid size effects [28] . Nuclear geometries were approximated by two prolate spheroids connected by a cylindrical bridge, where the midpoint of the bridge was considered as the boundary between the mother and daughter nuclei.
Relevant geometric parameters were measured by hand, one by one, from images obtained by confocal microscopy. For simulated times between confocal images, geometric parameters were linearly interpolated so as to allow for a smoothly changing simulated geometry (see Supplementary Video S4). The initial position of particles was sampled from a uniform distribution over the volume of the nucleus. At each time step, particles were propagated according to a Gaussian kernel. For particles propagated onto or outside the nuclear membrane, the particles were immediately remapped to their closest point within the nucleus. The numerical method is described in more detail in Supplementary Text S5 and [28] . For each simulation run, a single plasmid was simulated per cell. While we anticipate that there is significant potential for volume exclusion effects via multiple factors, e.g. via plasmid--chromosome, plasmid--RNA or plasmid--transcription factor interactions, such interactions cannot be investigated through currently available data. Thus we interpret these effects to be incorporated into the effective measured diffusion constants of plasmids. In this way, consistency is maintained between our observations from confocal microscopy and the simulated model system.

Selection of Simulation Time
Step Choosing a suitable time--step, δ t , is of crucial importance. For simulations in free space, the mean--squared displacement provides a method by which we can choose a suitable time--step. For a particle with diffusion constant, D diffusing freely in a d--dimensional Euclidean space, the mean--squared displacement is given by x 2 = 2dDt , with the mean--squared displacement along a given direction given by x 2 = 2Dt . Thus, for a desired spatial resolution, sres, to be obtained from a simulation, this motivates the choice of time--step to be δ t = s res 2 2D , with the result that s res ∝ δ t following. When considering the action of moving boundaries at small time scales, for most physical situations (including those we consider here, since we interpolate between geometries linearly), the movement of the boundary is linear with time. Consequently, any motion imparted on a diffusing particle in a short time scale by a moving boundary should be limited to s bound ∝δ t . Thus, provided time steps are small enough, the effect of moving boundaries on the spatial resolution of a particle simulation should be negligible relative to diffusive motion.
Here, to determine a suitable choice of δ t , we considered the worst--case scenario for our simulation methodology: that of small diffusion constants, where the movement of boundaries is largest compared to that of diffusive motion. We considered a range of time--steps which were varied by a parameter  calculated according to the Freedman--Diaconis method [27] . Gaps in the histogram occur due to the plasmid jumping over discrete pixels. Owing to the difficulty in accurately estimating diffusion coefficients from this data, our study investigated a range of diffusion rates in simulations.  Text S5: A brief overview of the simulation methodology, with references to more detailed studies of the correctness of the procedure used. The methodology is described in detail in [1]. We provide a brief overview here, and refer readers there for a full discussion with validation of the technique.
A general yeast nuclear geometry was represented by two oblate spheroids connected by a cylindrical bridge. Each oblate spheroid was truncated in such a way such that the cylindrical bridge would adjoin it without creating any gaps.
Thus, we assume a rotationally symmetric domain for the nuclear geometry. The size of each oblate spheroid could thus be described by two numbers describing Page 34 of 36 the major and minor axes. The cylinder adjoining them could be described by the cylindrical length and diameter. The figure below illustrates a sample geometry projected onto a 2D plane. The parameters describing the geometry were measured by confocal microscopy at 30 second intervals. In order to interpolate between the geometries smoothly, the parameters were linearly interpolated between the 30 second time steps.
The geometries were constructed in such a way so that the centre of mass of the geometry was at the origin at every time step.
At every time step, a particle was propagated according to a Gaussian kernel. In case the particle would overlap with the boundary, a vector connecting the particle position to the closest point on the boundary was constructed. The particle was then moved in the direction of this vector such that the particle would be completely contained within the boundary once more.
The midpoint of the bridge was taken to be the demarcation zone between the mother and the daughter.