Nuclear crowding and nonlinear diffusion during interkinetic nuclear migration in the zebrafish retina

We imaged fluorescently labeled nuclei of whole retinas of developing zebrafish at 2 min intervals, an optimal time period given the difficulty to track nuclei accurately over long times and the increased photobleaching with shorter intervals. We compared movies of retinas imaged at 2 min and at 20 s intervals over a period of 2 hrs and found that the improvement in temporal resolution made no difference to our analyses. This suggests that it is unlikely that there are important intervening movements that might complicate the analysis within each 2-min interval.

To follow the nuclei of all cells within a portion of the retina, we used H2B-GFP transgenic lines with GFP expression exclusively in the nuclei (Figure 1A). In order to achieve the desired temporal resolution without sacrificing image quality, fluorescence bleaching and sample drift must be minimized as much as possible. The retinas of H2B-GFP embryos were imaged using either a single-angle lightsheet microscope (see Figure 1B for a schematic) or an upright two-photon scanning microscope. Both of these methods yield images with minimal bleaching compared to other microscopic techniques (Svoboda and Yasuda, 2006; Stelzer, 2015). However, while the single-angle lightsheet can generate large stacks of images, it is very sensitive to lateral drift due to a small area of high resolution imaging. Therefore, some data sets were produced using two-photon microscopy, which, despite the limitations of scanning time, could produce areas of high-resolution images of sufficient size.

Figure 1

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Both lightsheet and two-photon microscopes produced images of at least half the retina with a depth of at least 50 μm over several hours in 2-min intervals. The images were processed using a suite of algorithms (Amat et al., 2015) to compress them to a lossless format, Keller Lab Block (KLB), correct global and local drift, and normalize signal intensities for further processing. Automated segmentation and tracking, in three dimensions, of the nuclei were carried out through a previously published computational pipeline that takes advantage of watershed techniques and persistence-based clustering (PBC) agglomeration to create segments and Gaussian mixture models with Bayesian inference to generate tracks of nuclei through time (Amat et al., 2014). Two main parameters greatly affect tracking results, overall background threshold and PBC agglomeration threshold. To obtain best automated tracking results, ground truth tracks were created for a section of the retina over 120 min and were compared to tracks generated over a range of these two parameters. The best combination of the two parameters was chosen as the one with highest tracking fidelity and lowest amount of oversegmentation in that interval.

The most optimal combination of parameters yielded an average linkage accuracy, from each time point to the next, of approximately 65%. Hence, extensive manual curation and correction of tracks were required. Tracking by Gaussian mixture models (TGMM) software generates tracks that can be viewed and modified using the Massive Multi-view Tracker (MaMuT) plugin of the Fiji software (Wolff et al., 2018; Schindelin et al., 2012). A region of the retina with the best fluorescence signal was chosen and all tracks within that region were examined and any errors were corrected. The tracks consist of sequentially connected sets of 3D coordinates representing the centers of each nucleus (Figure 1C), with which their movement across the tissue can be mapped over time. For example, Figure 1D shows IKNM of a single nucleus tracked from its birth, at the apical surface of the retina, to its eventual division into two daughter cells.

This process yielded tracks for hundreds of nuclei, across various samples, over time intervals of at least 200 min. We used custom-written MATLAB scripts to analyze these tracks. The aggregated tracks of the main data set, in Cartesian coordinates, for all tracked lineages are shown in Figure 2A. Single tracks for any given time interval can be extracted and analyzed from this collection. In order to transform the Cartesian coordinates of the tracks into an apicobasal coordinate system, we drew contour curves at the apical surface of the retina (see Figure 1A) separating RPC nuclei from the elongated nuclei of the pigmented epithelium. We then calculated curves of best fit (second degree polynomials) in both the XY and YZ planes. Assuming that the apical cortex is perpendicular to the apicobasal axis of each cell, displacement vectors of the nuclei at each time point can be separated into apicobasal and lateral components. Since, in IKNM, the apicobasal motion is that of interest, we used this component for our remaining analyses.

Figure 2

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Figure 2C,D show the speed and position of tracked nuclei of the same data set, over the duration of their cell cycle, for all cells that went through a full cell cycle. While all nuclei behave similarly minutes after their birth (early G1) and before their division (G2), their speed of movement and displacement is highly variable for the majority of the time that they spend in the cell cycle (Figure 2C,D). Most daughter nuclei move away from the apical surface, within minutes of being born, with a clear basalward bias in their speed distribution (Figure 2C). This abrupt basal motion of newly divided nuclei has also been recently observed by others (Leung et al., 2011; Shinoda et al., 2018; Barrasso et al., 2018). However, immediately after this brief period, nuclear speeds become much more equally distributed between basalward and apicalward, with a mean value near 0. Such a distribution is indicative of random, stochastic motion, which in turn leads to a large variability in the position of nuclei within the tissue (away from the apical surface) during the cell cycle (Figure 2B).

Interestingly, except during mitosis, we find an apical clearing of a few microns for dividing cells (Figure 2D). We checked to see if this was an artifact of measuring the distance to nuclear centers due to nuclear shape, as nuclei are rounded during M phase but are more elongated along the apicobasal axis at other times. We found no significant difference between the average length of the nuclear long axis when measured for 50 random nuclei right before their division (5.0 ± 0.7 µm) compared to 50 others chosen randomly from any other time point within the cell cycle (5.3 ± 1.1 µm), indicating that this clearing is likely to have a biological explanation, such as the preferential occupancy of M phase nuclei and surrounding cytoplasm at the apical surface during IKNM. We also performed the same measurements for 25 random nuclei 10 min after division when the average long axis length is significantly decreased by 0.8 fold (3.9 ± 0.5 µm). However, this measurement increased significantly in the following 10 min (4.8 ± 0.7 µm) to become similar to that at M phase.

Previous work has shown that when RPCs are pharmacologically inhibited from replicating their DNA, their nuclei neither enter G2 nor exhibit rapid persistent apical migration that normally occurs during the G2 phase of the cell cycle (Leung et al., 2011; Kosodo et al., 2011). A more surprising result of these experiments is that the stochastic movements of nuclei in G1 and S phases also slow down considerably during such treatment (Leung et al., 2011). It was, therefore, suspected that the migration of nuclei of cells in G2 toward the apical surface jostles those in other phases (Norden et al., 2009). We searched our tracks for evidence of such direct kinetic interactions among nuclei by correlating the speed and direction of movement of single nuclei with their nearest neighbors. These neighbors were chosen such that their centers fell within a cylindrical volume of a height and base diameter twice the length of long and short axes, respectively, of an average nucleus. Figure 3A shows the lack of correlation between the speed of movement of nuclei and the average speed of their neighbors. We further categorized the neighboring nuclei by their position in relation to the nucleus of interest (along the apicobasal axis), their direction of movement, and whether they were moving in the same direction of the nucleus of interest or not. None of the resulting eight categories of neighboring nuclei showed a correlation in their average speed with the speed of the nucleus of interest. Furthermore, we considered the movement of neighboring nuclei one time point (2 min) before or one time point after the movement of the nucleus of interest. Yet, we still found no correlation between these time-delayed and original speeds. These results suggest that there does not appear to be much transfer of kinetic energy between neighboring nuclei, and this is consistent with general considerations of the strongly overdamped character of motion at these length scales.

Figure 3

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Another hypothesis advanced for the basal drift in IKNM is that the nuclear movements are driven by apical crowding (Kosodo et al., 2011; Okamoto et al., 2013). How apical crowding might result in basal IKNM can be understood by comparing IKNM to a diffusive process. In diffusion, a concentration gradient drives the average movement of particles from areas of high to areas of low concentration. However, despite the average movement being directed, each individual particle’s trajectory is a random walk (Reif, 1965). Similarly, during IKNM a gradient in nuclear concentration is generated because nuclei divide exclusively at the apical surface. If basal IKNM were comparable to diffusion, this nuclear concentration gradient would be expected to result in a net movement of nuclei away from the area of high nuclear crowding at the apical side of the neuroepithelium (Miyata et al., 2014; Okamoto et al., 2013). Indeed, in IKNM each individual nucleus’ trajectory resembles a random walk (Norden et al., 2009). Therefore, for the cells in the G1 and S phases (which account for more than 90% of the cell cycle time in our system), IKNM has, at least on a phenomenological level, the main features of a diffusive process.

To test further whether we can indeed describe IKNM using a model of diffusion, we first asked what would happen to the concentration gradient if we blocked the cell cycle in S phase, which inhibits both the apical movement of the nuclei in G2 and mitosis at the apical surface. If the comparison to diffusion were valid, we expect the blockage to abolish the build-up and maintenance of the concentration gradient. We, therefore, compared the normally evolving distribution of nuclei in a control retina with that measured from a retina where the cell cycle was arrested at S phase using a combination of hydroxyurea (HU) and aphidicolin (AC) (Leung et al., 2011; Icha et al., 2016). These compounds inhibit DNA polymerase and ribonuclear reductase, respectively, to halt DNA replication (Baranovskiy et al., 2014; Singh and Xu, 2016). In the HU-AC-treated retina, we counted the number of nuclei in a three-dimensional section of the tissue containing approximately 100 nuclei, at equal time intervals, starting with 120 min after drug treatment. The delay ensured that almost all cell divisions, from nuclei that had already completed the S phase at the time of treatment, had taken place. These results are shown in Figure 4A,C, in which retinal tissue is approximated as a spherical shell of apical radius a and the rescaled coordinate $\xi =r/a$, where r is the distance from the center of the lens, is presented on the x-axis. As expected from the diffusion model (Figure 4D), over the course of 160 min, the mean of the nuclear distribution moved further toward the basal surface in treated retinas, and the concentration difference between the apical and basal surfaces diminished (Figure 4B,C). In contrast, in control retinas the mean of the nuclear distribution moved toward the apical surface (Figure 4A,C) as the gradient continued to build up. Hence, these results support the suitability of a diffusive model to describe the basal nuclear migration during IKNM.

Figure 4

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To investigate whether a diffusion model provides a quantitative description of IKNM, we focused on the crowding of nuclei at the apical side of the tissue. In mathematical terms, crowding creates a gradient in nuclear concentration c along the apicobasal direction of the retina. If we assume there is no dependence of the nuclear concentration on the lateral position within the tissue then we require a diffusion equation for the nuclear concentration $c(r,t)$ as a function only of the apicobasal distance r and time t. The retina can be approximated as one half of a spherical shell around the lens, and thus we use spherical polar coordinates with the origin of the coordinate system at the center of the lens, the basal surface at $r=b$ and the apical surface at $r=a$ (Figure 5B). We first consider the simplest diffusion equation for this system, in which there is a diffusion constant D independent of position, time, and c itself, namely

Figure 5

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We seek to determine D from the experimental data of the concentration profile $c(r,t)$. Note that in this parsimonious view of modeling we have not included a ‘drift’ term of the kind that is expected to be present at the very late stages of IKNM, when nuclei return to the apical side.

In addition to Equation 1, we must specify the boundary conditions appropriate to IKNM. Since nuclei only divide close to the apical surface of the tissue, we treat mitosis as creating an effective influx of nuclei through the apical boundary. To quantify this influx, we extracted the number of cells $N(t)$ as a function of time. As during the stages of development examined here cells are neither dying nor exiting the cell cycle (Biehlmaier et al., 2001), we assumed that the number of cell divisions is always proportional to the number of currently existing cells. This assumption predicts an exponential increase in the number of cells or nuclei, over time, as was recently confirmed by Matejčić et al., 2018:

where $N_0$ is the initial number of nuclei and $\tau =T_P/\ln 2$, with $T_P$ the average cell cycle length. Figure 5A shows the agreement between the theoretically predicted curve $N(t)$ with the experimentally obtained numbers of nuclei over time. Having obtained $N_0$ and $T_P$ from our experimental data, the predicted curve has no remaining free parameters and thus no fitting is necessary. Using Equation 2, we formulate the influx boundary condition as

with S the apical surface area of our domain of interest. In contrast to the apical side of the tissue, there is no creation (or depletion) of nuclei at the basal side (Matejčić et al., 2018), and hence a no-flux boundary condition,

The position $r=b$ where this basal boundary condition is applied could change throughout tissue development. Matejčić et al., 2018 found that a basal exclusion zone, where nuclei cannot enter due to accumulation of basal actin, exists in the zebrafish retina before approximately 42 hpf. Before this point in development, the no-flux boundary condition is applied at the tissue radius where the nuclear exclusion zone begins, while later in development, the no-flux boundary condition should be applied at the position of the actual basal cell surfaces. Here, we only model early stages of embryonic development well before the disappearance of the basal exclusion zone, therefore the location $r=b$, where we apply our basal boundary condition, is chosen such that we only consider the region of the retinal tissue actually accessible to moving nuclei during IKNM. Thus, taken together, Equations 1, 3 and 4 fully specify this simplest mathematical model of IKNM.

In solving these equations to find the concentration of nuclei $c(r,t)$ in the retinal tissue it is convenient to introduce dimensionless variables for space and time,

and further define the purely geometric parameter $\rho =b/a<1$. The exact solution for the nuclear concentration, whose detailed derivation is given in the Appendix, is

The first terms within parentheses describe the decay over time of the initial condition $c(\xi ,s=0)$. Here, ${\lambda }_i$ are the eigenvalues and $H_i(\xi )$ the eigenfunctions of the radial diffusion problem, and the coefficients $h_i$ are determined from the experimental initial conditions (see Materials and methods). The second terms within the sum and the final term on the right hand side of Equation 6 are constructed such that the solution fulfills the boundary conditions Equation 3 and Equation 4. In the last term, the constant $g_0$ was obtained using the constraint that the volume integral of the initial concentration yields the initial number of nuclei ${\displaystyle N_0}$. ${\displaystyle f_0}$, σ and ${\alpha }_i$ emerge within the calculation of the solution and are specified in the Appendix. Thus, the diffusion constant D in Equations 1 and 6 is the only unknown.

To determine the effective diffusion constant D from the data, the experimental distribution of nuclei in the retinal tissue was first converted into a concentration profile. Then, the optimal D-value, henceforth termed $D^{*}$, was obtained using a minimal-${\chi }^2$ approach. The value obtained within the linear model for a binning width of 3 μm and an apical exclusion width of 4 µm is $D_{\text{lin}}^{*}=0.17\pm 0.07$ µm²/min. Using this, we can examine the decay times of the different modes in the first term of Equation 6. The slowest decaying modes are the ones with the smallest eigenvalues ${\lambda }_i$ and we find that the longest three decay times are ${𝒯}_1\approx 1325$ min, ${𝒯}_2\approx 350$ min and ${𝒯}_3\approx 158$ min. This shows that indeed all three terms of Equation 6 are relevant on the timescale of our experiment and need to be taken into account when calculating the concentration profile. The corresponding plots of $c(\xi ,s)$ are shown in Figure 6A–C. As can be seen from this figure, the diffusion model fits the data very well at early times, $t\le 200$ min after the start of the experiment at 24 hpf (see Materials and methods). However, for $t\ge 200$ min the model does not fit the data as well; the experimentally observed nuclear concentration levels off at a value between $4.00\times {10}^{-3}$ µm⁻³ and $4.50\times {10}^{-3}$ µm⁻³ (Figure 6D), an aspect that is not captured by this model of linear diffusion.

Figure 6

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One particular aspect of the biology that the linear model neglects is the spatial extent of the nuclei. In the linear diffusion model, particles are treated as point-like and non-interacting. However, our microscopy images (see Figure 1A) clearly indicate that the nuclei have finite incompressible volumes, so that their dense arrangement within the retinal tissue would lead to steric interactions once the nuclear concentration is sufficiently high. Moreover, the packing density of nuclei can not exceed a maximum value dictated by their geometry. Therefore, we next examined whether accounting for such volume and packing density effects leads to a more accurate theory describing the nuclear distribution during IKNM.

When the diffusion Equation 1 is written in the following form

we can identify the term $c\ln c$ as proportional to the entropy density $𝒮$ of an ideal gas, and its derivative with respect to c as a chemical potential. In an ideal gas, all particles are treated as point-like and without mutual interactions. In order to include the spatial extent of particles (i.e. the spatial extent of nuclei in this case), we must estimate the entropy in a way that accounts for the maximum concentration allowable given steric interactions. This is a well-studied problem in equilibrium statistical physics, in which, purely as a calculation tool, it is useful to consider space as divided up into a lattice of sites. Each of these sites can be either empty or occupied by a single particle. In this ‘lattice gas’ model, the discrete sites assure a minimum distance of approach for particles and thus effectively introduce a particle size and, correspondingly, a maximum particle concentration $c_{\text{max}}$ (Huang, 1987). In this system, a useful approximation to the entropy is

Substituting this expression for the term $c\ln c$ in Equation 7, we obtain the nonlinear diffusion equation

The term 'nonlinear' refers to the mathematical structure of the newly obtained Equation 9. In the mathematical classification, an equation is linear in a certain variable if this variable only appears raised to the power one within the equation. For example, the simplest diffusion Equation 1 is linear in c and all its derivatives with respect to r and t, such as $\partial c/\partial t$. In contrast, in Equation 9 the term $c_{\text{max}}/(c_{\text{max}}-c)$ appears which is proportional to $c^{-1}$. Hence, Equation 9 is said to be nonlinear. The additional nonlinear term in Equation 9 (as compared to Equation 1) is an important aspect of the model as it arose from the introduction of the spatial extent of the nuclei and their maximum possible packing density $c_{\text{max}}$. This effect also has to be taken into account in the boundary conditions. Adjusting the boundary conditions at the apical side accordingly leads to

while the basal boundary condition remains the same as Equation 4. Together, Equation 9 and the boundary conditions in Equations 4 and 10 represent an extension to the diffusion model for IKNM, which now accounts for steric interactions between the nuclei. The maximum concentration $c_{\text{max}}$ incorporated in this model was obtained, as described in the Materials and methods, by considering a range of nuclear radii and the maximum possible packing density for aligned ellipsoids (Donev et al., 2004).

Similar to fitting the linear model, we also need to establish a description of the initial condition. To make both models consistent with each other, we employ the linear model’s initial condition, Equation 6 at $s=0$ with $h_i$ as obtained from Equation 17 (Figure 6A). The concentration profile in the nonlinear model and its derivative were obtained numerically using the MATLAB pdepe solver. Fitting this concentration profile to the data was by means of a minimal-${\chi }^2$ approach as well. When the optimization takes data points up to $t=200$ min into account, we find $D_{\text{nonlin}}^{*}=0.09\pm 0.05$ µm²/min (Figure 6, Table 1). As can be seen, by choosing $c_{\text{max}}$ correctly, an excellent fit to the data can be obtained, particularly to the flattened part of the distribution at later times near the apical side ($\xi \sim 1$), where the linear model fails. These results show that a lattice-gas based diffusion model is indeed suitable to describe time evolution of the nuclear concentration profile of the zebrafish retina during IKNM over several hours of early development.

This diffusion model, with the calculated diffusion constant $D_{\text{nonlin}}^{*}=0.09\pm 0.05$ µm²/min obtained from the nonlinear implementation, allows us to probe the physical and biological considerations that could set its scale. Notably, at low nuclear densities, $c\ll c_{\text{max}}$, the term $c_{\text{max}}/\left(c_{\text{max}}-c\right)$ in Equation 9 tends to unity, and the ordinary diffusion Equation 1 with $D_{\text{lin}}^{*}=D_{\text{nonlin}}^{*}$ is recovered. We can thus make use of its well-known properties for further evaluation. First, we assess whether nuclei in IKNM move due to free equilibrium thermal diffusion in a fluid. If so, the diffusion constant obeys the Stokes-Einstein equation (Einstein, 1905)

where $k_{\text{B}}=1.38\times {10}^{-23}$ JK⁻¹ is the Boltzmann constant, T is the absolute temperature, and ζ is the drag coefficient for the particle, the constant of proportionality between the speed with which it moves and the force applied. For a spherical particle of radius ${\displaystyle \mathfrak{R}}$ in a fluid of viscosity η, the reference value is ${\displaystyle {\zeta }_0=6\pi \eta \mathfrak{R}}$. If we assume that the particles move in water at 25 °C, for which $\eta \approx 9\times {10}^{-4}$ Pas, and if we approximate the nuclei as spheres with ${\displaystyle \mathfrak{R}=3.5}$ µm, corresponding to the maximum nuclear concentration $c_{\text{max}}=4.12\times {10}^{-3}$ µm⁻³ (as in Figure 6), we obtain $D_{\text{thermal}}\approx 4.2$ µm²/min. This value is about 50 times larger than the measured value of $D_{\text{nonlin}}^{*}$, implying that freely diffusing nuclei in water would be vastly more mobile than seen during IKNM.

While the free thermal diffusivity of nuclei serves as a useful reference quantity, nuclei clearly do not move in pure water, nor in an unbounded fluid. The viscosity of the cytoplasm is likely much higher than that of water due to the high number of organelles and polymeric components present; a higher viscosity leads to a lower diffusion constant via the Stokes-Einstein relation (Equation 11). Similarly, the slender shape of the individual cells within pseudostratified epithelia (Norden, 2017) would imply that a considerable amount of energy is required to transport fluid through the narrow region between the nucleus and the membrane.

In order to understand the effects of membrane confinement on fluid transport, it is useful to consider a minimal energetic description of the cell shape. That is provided by an energy $ℰ$ that incorporates membrane elasticity, through a bending modulus κ, and surface tension γ,

where $dS$ is the element of surface area and $\mathscr{H}$ is the mean curvature. For a cylindrically symmetric shape given by a function $\delta (z)$, $dS=2\pi \delta \sqrt{1+{\delta }_z^2}$ and

where ${\delta }_z$ stands for $d\delta /dz$, etc. The equilibrium shape of a membrane is that which minimizes Equation 13 subject to constraints such as boundary conditions and/or a given enclosed volume.

As first understood in the context of the so-called ‘pearling instability’ of membranes under externally imposed tension (Bar-Ziv and Moses, 1994; Nelson et al., 1995; Goldstein et al., 1996), narrow necks emerge as characteristic equilibrium structures when the dimensionless ratio $\gamma R_{\mathrm{\infty }}^2/\kappa$ is much larger than unity, where $R_{\mathrm{\infty }}$ is a characteristic tube radius imposed far from the neck (e.g. the nuclear radius ${\displaystyle \mathfrak{R}}$). In this limit, the neck radius is on the order of $\sqrt{\kappa /\gamma }$. For fluid membranes, it is known that $\kappa \sim 20k_{\text{B}}T$ (Helfrich, 1973), while the magnitude of tension (an energy per unit area) is such that the surface energy associated with a molecular area is comparable to thermal energy; $\gamma {\mathrm{\ell }}^2/k_{\text{B}}T\sim 1$, where $\mathrm{\ell }$ is a molecular dimension (e.g. 1 nm). Thus, γ may be as large as $\sim {10}^{-5}$ Jm⁻² and ${\displaystyle \gamma {\mathfrak{R}}^2/\kappa }$ is very large indeed ($\sim {10}^5$).

To illustrate the kinds of shapes that are energetic minima of Equation 12, we show in Figure 7 that which arises when we impose (i) an overall aspect ratio of $\sim 20$ for the cell, as measured by Matejčić et al., 2018, (ii) cell radii of 1.98 and 0.94 µm at the apical and basal sides of the tissue, respectively, as determined from that aspect ratio and the approximate length L of cells in our experiment, and (iii) position of the nucleus at the midpoint of the cell, with a radius ${\displaystyle \mathfrak{R}=3.5}$ µm. The details of calculations are given in the Appendix. As the necks become extremely narrow in the relevant limit, we have taken a smaller value of γ to illustrate the basic effect. Because the gap between the membrane and the enveloped sphere is so thin, we have set the membrane radius equal to that of a sphere with membrane radius ${\displaystyle {\mathfrak{R}}_{\mathrm{t}\mathrm{u}\mathrm{b}\mathrm{e}}}$ over some angular extent and minimized the energy with respect to the position of the last contact point, as detailed in the Appendix.

Figure 7

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The similarity of this shape to those described in the literature suggests that this model is a useful starting point for the discussion of the fluid dynamics of nuclear motion during IKNM. Recently, Daniels, 2019 considered the transport of a sphere through the fluid contained within a cylindrically symmetric tight-fitting tubular membrane with bending modulus κ and surface tension γ, much like the geometry of cells undergoing IKNM. At a finite temperature T, the membrane will exhibit thermally driven shape fluctuations which, as shown by Helfrich, 1978, produce a repulsive interaction with the nearby sphere, swelling the gap. In the limit of large tension (appropriate for a tight-fitting membrane), the calculation simplifies to yield the result

where, for ease of interpretation, we have written the factors within parentheses as a product of two convenient dimensionless ratios. As the nuclear radius is micron-sized, we find ${\displaystyle \gamma {\mathfrak{R}}^2/k_{\text{B}}T\sim {10}^7}$, which in turn implies a drag coefficient ratio on the order of 10⁵ and diffusivities $D_{\text{tube}}\approx \left(1-5\right)\times {10}^{-6}{D}_{\text{thermal}}$. Because of the very close spacing between the membrane and nucleus and the high viscous drag associated with such a geometry, these values are about 3 to 4 orders of magnitude smaller than the measured $D_{\text{nonlin}}^{*}$. This is without considering changes in the cytoplasmic viscosity, which would decrease the value of D even further. Therefore, we conclude that the nuclear movements in IKNM cannot be due to thermal diffusion, but must be actively driven, for example through cytoskeletal transport.

We can turn to a more microscopic interpretation of the value of the diffusion constant. At low nuclear concentrations, when Equation 1 holds, the behavior of individual particles can be described using the overdamped Langevin equation (compare to Lemons and Gythiel, 1997)

where $ℱ(t)$ is a stochastic force. In the standard way, if we average over realizations of the random force $ℱ(t)$ and integrate in time, the mean squared displacement $⟨r{(t)}^2⟩=\mathrm{\Gamma }t/{\zeta }^2$ is obtained, where $\mathrm{\Gamma }=\int 𝑑qQ(q)$, with $Q=⟨ℱ(t^{\prime })ℱ(t^{\prime \prime })⟩$ the correlation function of the stochastic force between time points $t^{\prime }$ and $t^{\prime \prime }$ and $q=t^{\prime }-t^{\prime \prime }$. For systems at densities low enough for Equation 1 to hold, we know further that $⟨r{(t)}^2⟩=6Dt$, leading to the result

expressing the unknown quantity $\mathrm{\Gamma }$ in terms of the measured diffusion constant and the friction coefficient. Using the numerical values quoted above, we find $\mathrm{\Gamma }\approx (1.2\times {10}^{-18}-3.4\times {10}^{-17})$ N²s. As the units of $\mathrm{\Gamma }$ are force² × time, we can estimate the underlying forces if we know their correlation time. As most molecular processes of cytoskeletal components have characteristic time scales of 10 ms to 1 s, we obtain forces in the range of 1–50 nN. This result is compatible with cytoskeletal transport under the assumption that the nucleus is transported either by multiple molecular motors at once, since each molecular motor protein typically exerts forces on the order of several pN, or through typical forces arising from polymerization of cytoskeletal components, which are in the same range (Peskin et al., 1993; Footer et al., 2007).

Having obtained an interpretation of the diffusion constant $D^{*}$ as arising from cytoskeletal transport throughout the cell cycle, and not only during the apicalward movement of the nuclei during G2, we turn to an interpretation of the concentration dependence of IKNM that results from nuclear crowding (Equation 9). To this end, we seek an extension to the stochastic dynamics of individual nuclei (Equation 15) that corresponds to the concentration evolution in the nonlinear diffusion Equation 9. In general, there are two different ways to achieve such a correspondence. In the first, an additional force $F_{\text{external}}$ is introduced into the Langevin Equation 15, which describes the average effect of surrounding nuclei on the individual nucleus in question and is thus concentration-dependent. In the second, we make direct use of the fact that $D_{\text{nonlin}}{c}_{\text{max}}/\left(c_{\text{max}}-c\right)\to D_{\text{lin}}$ as $c\to 0$. Inverting this relationship and applying it to the expression $\mathrm{\Gamma }=6{\gamma }^{2}D$ for the low concentration case, we can also extend the Langevin Equation 15 by making $\mathrm{\Gamma }$ concentration-dependent, that is, $\mathrm{\Gamma }=6{\gamma }^2{D}_{\text{nonlin}}^{*}{c}_{\text{max}}/\left(c_{\text{max}}-c\right)$.

Using both models, we can simulate individual nuclei in the experimental environment they experience during IKNM, namely the time-varying nuclear distribution across the retinal tissue that we found as the solution of the nonlinear model. Simulating several nuclei where each single one corresponds exactly to one nucleus in the experiment gives us a means to replicate the processes that took place in the tissue over a larger period of time. From such a simulation, we can also extract a mean squared displacement curve (MSD curve) that corresponds to the MSD curve calculated from the experimental nuclear trajectories. Of course, because our simulations are based on a stochastic equation, suitable averaging over realizations of the stochastic force are used to obtain statistically significant results.

Figure 8 shows the range of possible MSD curves for simulations of the low concentration model described by Equation 15 and those with the two possible high-concentration extensions, each represented by a shaded area. Shown also is the experimental MSD curve obtained from the very same nuclei used in the numerics. As can be seen, the experimental curve only agrees with the model that assumes a concentration-dependent value of $\mathrm{\Gamma }$, and not the low-concentration model from Equation 15. In addition, the experimental curve does not agree with the possibility of including the effects of surrounding nuclei as an independent, additional force. These results have two implications. First, they lend further support to the notion raised above that IKNM cannot be understood as a single-cell phenomenon. Instead, we can only interpret quantities such as MSD curves of nuclei undergoing IKNM correctly if we explicitly take the surrounding nuclei into account, even if there seems to be no direct energy transfer between nuclei, as shown from our experimental work (Figure 3). Second, the simulation results shown in Figure 8 provide a means to distinguish between different ways in which the neighboring nuclei may act on a moving nucleus. As the experimental MSD curve only agrees with the model that assumes a concentration-dependent stochastic force, among those considered, the results indicate that cells are, in some manner, sensitive to the local nuclear concentration. As we have previously shown, the strength of this stochastic force is compatible with cytoskeletal transport. At high nuclear concentrations (i.e. when nuclei are packed close to the maximum possible packing density), as is the case closest to the apical surface of the retinal tissue, cells may recruit more molecular motors to transport nuclei away from this surface faster, leading to the concentration dependence of the stochastic force.

Figure 8

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The diffusion model may also address mechanistic questions about IKNM in retinas growing under varying experimental conditions. Zebrafish embryos are often grown at different temperatures to manipulate their growth rate (Kimmel et al., 1995; Reider and Connaughton, 2014), but it has been unclear how the nuclei in the retina behave at these different temperatures. To examine this issue, we grew the embryos at the normal temperature of 28.5 °C overnight and then incubated them at lower temperature (LT) of 25 °C or higher temperature (HT) of 32 °C during imaging. We could directly measure the change in average cell cycle length from experimental data and found that in HT, it is 205.5 min, while in LT, it is a much larger 532.78 min. We were then able to use these values in the model to investigate whether the change in temperature influences the processes that determine the effective diffusion constant of the nuclei. The resulting values for $D_{\text{nonlin}}^{*}$ are summarized in Table 1. Based on these values, two-sided t-tests (see Materials and methods) confirmed that there is no significant difference between the D-values obtained from the two normal condition data sets. In contrast, D-values for the LT and HT data sets were significantly different from the normal ones, with $p\le 0.01$. These results indicate, that aside from its effect on cell cycle length, incubation temperature is likely to influence IKNM directly by altering the mobility of nuclei, here represented by the effective diffusion constant D.

All animal works were approved by Local Ethical Review Committee of the University of Cambridge and performed in accordance with a Home Office project license PL80/2198. All zebrafish were maintained and bred at 26.5 °C. All embryos were incubated at 28.5 °C before imaging sessions. At 10 hr post-fertilization (hpf), 0.003% phenylthiourea (PTU) (sigma) was added to the medium to stop pigmentation in the eye.

Images of retinal development for the main data set were obtained using lightsheet microscopy. Double transgenic embryos, Tg(bactin2:H2B-GFP::ptf1a:DsRed) were dechorionated at 24 hpf and screened positive for the fluorescent transgenic markers prior to the imaging experiment. The embryo selected for imaging was then embedded in 0.4% low gelling temperature agarose (Type VII, Sigma-Aldrich) prepared in the imaging buffer (0.3x Daniau's solution with 0.2% tricaine and 0.003% PTU [Godinho, 2011]) within an FEP tube with 25 µm thick walls (Zeus), with an eye facing the camera and the illumination light shedding from the ventral side. The tube was held in place by a custom-designed glass capillary (3 mm outer diameter, 20 mm length; Hilgenberg GmbH). The capillary itself was mounted vertically in the imaging specimen chamber filled with the imaging buffer. To ensure normal development, a perfusion system was used to pump warm water into the specimen chamber, maintaining a constant temperature of 28.5 ºC at the location of the specimen.

Time-lapse recording of retinal development was performed using a SiMView light-sheet microscope (Tomer et al., 2012) with one illumination and one detection arm. Lasers were focused by Nikon 10x/0.3 NA water immersion objectives. Images were acquired with Nikon 40x/0.8 NA water immersion objective and Hamamatsu Ocra Flash 4.0 sCMOS camera. GFP was excited with scanned light sheets using a 488 nm laser, and detected through a 525/50 nm band pass detection filter (Semrock). Image stacks were acquired with confocal slit detection (Baumgart and Kubitscheck, 2012) with exposure time of 10 ms per frame, and the sample was moved in 0.812 µm steps along the axial direction. For each time point, two 330 × 330 × 250 µm³ image stacks with a 40 µm horizontal offset were acquired to ensure the coverage of the entire retina. The images were acquired every 2 min from 30 hpf to 72 hpf. The position of the sample was manually adjusted during imaging to compensate for drift. The two image stacks in the same time point were fused together to keep the combined image with the best resolution. An algorithm based on phase correlation was subsequently used to estimate and correct for the sample drift over time. The processing pipeline was implemented with MATLAB (MathWorks).

Images for the repetition data set and all other conditions were obtained using a TriM Scope II 2-photon microscope (LaVision BioTec). A previously established Tg(H2B-GFP) line, generated by injecting a DNA construct of H2B-GFP driven from the actin promoter (He et al., 2012), was used for all these experiments. Embryos were dechorionated and screened for expression of GFP at 24 hpf. An embryo was then embedded in 0.9% UltraPure low melting point agarose (Invitrogen) prepared in E3 medium containing 0.003% PTU and 0.2% tricaine. The agarose and embryo were placed laterally within a 3D printed half cylinder of transparent ABS plastic, 0.8 mm in diameter, attached to the bottom of a petri dish, such that one eye faced the detection lens of the microscope. The petri dish was then filled with an incubation solution of E3 medium, PTU, and tricaine in the same concentrations as above. For the experiment involving cell cycle arrest, hydroxyurea and aphidicolin (Abcam) were added to the incubation solution right before imaging, to a final concentration of 20 mM and 150 µM, respectively. The imaging chamber was maintained at a temperature of 25 °C, 28.5 °C, or 32 °C, as required, using a precision air heater (The Cube, Life Imaging Services).

Green fluorescence was excited using an Insight DeepSee laser (Spectra-Physics) at 927 nm. The emission of the fluorophore was detected through an Olympus 25x/1.05 NA water immersion objective, and all the signals within the visible spectrum were recorded by a sensitive GaAsP detector. Image stacks with step size of 1 μm were acquired with exposure time of 1.35 ms per line averaged over two scans. The images were recorded every 2 min for 10–15 hrs starting at 26–28 hpf. The same post-processing procedure for data compression and drift correction was used on these raw images as on those from lightsheet imaging.

The radial coordinates $r_n$ of nuclei were calculated by subtracting $l_n$ from a, wherein $l_n$ is the distance from the center of a nucleus n to the apical surface and a is the distance from the center of the lens to the apical surface. We estimated a total uncertainty of $\mathrm{\Delta }r=\pm 3$ µm for each single distance measurement of $r_n$. This value is a result of uncertainty in detecting the center of the nucleus and in establishing the position of the apical surface.

Because each nuclear position has an error bar $\mathrm{\Delta }r$, binning the data leads to an uncertainty in the bin count. In order to calculate this uncertainty, we considered the probability distribution of a nucleus’ position. In the simplest case, this probability is uniform within the width of the positional error bar and zero elsewhere. The probability, $p_{n,\text{bin}}$, of finding a given nucleus n within a given bin, is proportional to the size of the overlap of probability distribution and bin. It follows that the expectation value for the number of nuclei within a bin is given as $E(N_{\text{bin}})={\sum }_n{p}_{n,\text{bin}}$. Correspondingly, $\text{Var}(N_{\text{bin}})={\sum }_n{p}_{n,\text{bin}}(1-p_{n,\text{bin}})$ is the variance of the number of nuclei within this bin. Thus, the error bar of the bin count is ${\sigma }_{y,\text{bin}}=\sqrt{\text{Var}(N_{\text{bin}})}$. The nuclear distribution profile $N(r,t)$ is not expected to be uniform or linear, therefore the expectation value $E(N_{\text{bin}})$ does not correspond to the number of nuclei at the center of the bin. Since the position of the expectation value is unknown a priori, it is still plotted at the center of the bin with an error bar denoting its positional uncertainty. Here we assume this error bar to be the square-root of the bin size $\mathrm{\Delta }{r}_{\text{bin}}$, that is, ${\sigma }_{x,\text{bin}}=\sqrt{\mathrm{\Delta }{r}_{\text{bin}}}$.

In order to obtain the experimental nuclear concentration profile $c(r,t)$, and its error bars, from the distribution of nuclei $N(r,t)$, the volume of the retina also has to be taken into account, since $c=N/V$. The total retinal volume within which nuclei tracking took place was estimated directly from the microscopy images. To this end, we outlined the area of observation in each image slice using the Fiji software and multiplied this area with the distance between successive images. Given the total volume, $V_{\text{total}}$, we proceeded to calculate the volume per bin, which depends on the radii at the inner and outer bin surfaces. In general, the volume of a spherical sector is $V_{\text{sector}}=\frac{1}{3}\mathrm{\Omega }{r}_{\text{sector}}^3$, where $\mathrm{\Omega }$ denotes the solid angle. Knowing the apical and basal tissue radii, $r=a$ and $r=b$, one can thus calculate $\mathrm{\Omega }$ as $\mathrm{\Omega }=3V_{\text{total}}/(a^3-b^3)$. This gives the volume of each bin as $V_{\text{bin}}=\frac{1}{3}\mathrm{\Omega }\left(r_{\text{bin,outer}}^3-r_{\text{bin,inner}}^3\right)$, where $r_{\text{bin,outer}}$ and $r_{\text{bin,inner}}$ denote the outer and inner radii of a bin, respectively. Similarly, we calculated the effective surface area S through which the influx of nuclei occurs (see Equation 3) from the solid angle $\mathrm{\Omega }$. This surface area is simply given as $S=\mathrm{\Omega }{a}^2$.

To retrieve the average cell cycle time $T_P$ for each of the data sets, we used two different approaches. In the case of the main data set, sufficient number of nuclear tracks consisting of a whole cell cycle were present. Thus, we directly calculated the average cell cycle duration from these tracks. For the other data sets, we make use of the fact that the number of nuclei follows an exponential growth law depending on $T_P$ (see Equation 2). Knowing the initial number of tracked nuclei $N_0$ for each data set, we obtained $T_P$ from fitting the following equation to the number of nuclei as a function of time in a log-lin plot: $\ln N(t)=\ln N_0+t/\tau =\ln N_0+(\ln 2/T_P)t$. Then $T_P$ was deduced from the slope of this fit.

In order to determine the maximum nuclear concentration $c_{\text{max}}$ for the nonlinear model, we first randomly selected 100 nuclei from our dataset of tracked nuclei and measured the size of their longest diameter in both XY and YZ planes. From these measurements, we established that the size of the principal semi-axis of each nucleus is likely to lie in the range of about 3 µm to 5 µm, where the nuclear shape is regarded to be ellipsoidal. This led to the range of possible maximum concentrations $c_{\text{max}}$, although we did not measure the precise nuclear volume. The lower limit for the nuclear volume is set by the volume of a sphere of radius 3 µm, the upper limit by a sphere of radius 5 µm. Taking into account the maximum possible packing density of nuclei, which for aligned ellipsoids is the same as that of spheres (Donev et al., 2004), $\pi /\left(3\sqrt{2}\right)\approx 0.74$, we obtained a range of ${\displaystyle 1.41\times {10}^{-3}\mu {\mathrm{m}}^{-3}\le {\mathrm{c}}_{\mathrm{m}\mathrm{a}\mathrm{x}}\le 6.55\times {10}^{-3}\mu {\mathrm{m}}^{-3}}$.

We determined the prefactors $h_i$ from the experimental nuclear distribution at the start of the experiment, $c_{\text{exp}}(\xi ,0)$. For convenience, we chose to determine first $\widetilde{h_i}=h_i+{\alpha }_i{f}_0/(\sigma +{\lambda }_i^2)$ and then obtained $h_i$ by subtracting ${\alpha }_i{f}_0/(\sigma +{\lambda }_i^2)$ from the results. The $\widetilde{h_i}$ can be calculated from the data, using Equation 6 for $s=0$, as

where m denotes the m-th binned data point, ${\xi }_m$ its position and $\mathrm{\Delta }{\xi }_{\text{m}}$ the width of bin m. As in Equation 6, the index i denotes the i-th eigenfunction or -mode.

The non-linear concentration profile was determined numerically from the same initial condition as used for the linear model, Equation 6, at $s=0$ with $\widetilde{h_i}$ as in Equation 17. Time evolution of the initial condition, according to Equation 9, was performed using the pdepe solver in MATLAB.

The range of sizes of the nuclear principal semi-axes was used to determine the range of data to be included in our fits. Any data closer than 3 µm to 5 µm from the apical or basal tissue surfaces was not taken into account for fitting because the center of a nucleus cannot be any closer to a surface than the nuclear radius. Thus, all data collection very close to the apical or basal tissue surfaces must have been due to the above-mentioned measurement uncertainties $\mathrm{\Delta }r$.

In principle, the full solution for $c(\xi ,s)$ is composed of infinitely many modes. However, in practice, we truncated this series and only included the first eight modes in our fits. This is due to the fact that we have a finite set of data points, so adding too many modes could lead to over-fitting. Fits with a wide range of numbers of modes were found to result in the same optimal D-values.

For fitting, we first rescaled the data in accordance with the non-dimensionalization of the theoretical variables r and t (see Equation 5). Thus we obtain $c_{\text{exp}}(\xi ,s)$ from $c_{\text{exp}}(r,t)$. Then both models were fitted to the experimental data using a minimal-${\chi }^2$ approach. The goodness of fit parameter ${\chi }^2={\sum }_m{\left(c_{\text{exp}}(\xi ,s)-c(\xi ,s)\right)}^2/{\sigma }_m^2$, where ${\sum }_m$ denotes the summation over all bins m. Since binning resulted in uncertainties ${\sigma }_{y,\text{bin}}$ and ${\sigma }_{x,\text{bin}}$ in the y- and x-directions, both had to be taken into account when calculating ${\sigma }_m$ and ${\chi }^2$. The combined contribution of x- and y- uncertainties is: ${\sigma }_m^2={\sigma }_{y,m}^2+{\sigma }_{y,\text{indirect},m}^2$ with ${\sigma }_{y,\text{indirect},m}={{\sigma }_{x,m}\left(\mathrm{d}c(\xi ,s)/\mathrm{d}\xi \right)|}_{\xi ={\xi }_m}$ (Bevington and Robinson, 2003). In our fits, the value ${\chi }^2$ was calculated for a large range of possible diffusion constants D, from $D=0.01$ µm²/min to $D=10$ µm²/min. By finding the value of D for which ${\chi }^2$ became minimal for a given data set and time point, we established our optimal fit.

The minimal-${\chi }^2$ approach furthermore enabled us to determine the optimal binning width $\mathrm{\Delta }{r}_{\text{bin}}$ or $\mathrm{\Delta }{\xi }_{\text{bin}}$ and width of data exclusion for the fits. In order to do so, fits of the normal data set were performed for different data binning widths and exclusion sizes of 3 µm to 5 µm. For each of these fits the ${\chi }^2$-value and the number of degrees of freedom ν, that is, the number of data points minus the number of free fit parameters (here number of data points minus 1), were registered. From ${\chi }^2$ and ν, we calculated the reduced ${\chi }^2$ value, ${\chi }_{\nu }^2={\chi }^2/\nu$ (Bevington and Robinson, 2003). Using ν and ${\chi }_{\nu }^2$, the probability $P_{\chi }({\chi }^2;\nu )$ of exceeding ${\displaystyle {\chi }^2}$ for a given fit can be estimated, which should be approximately 0.5 (Bevington and Robinson, 2003). Therefore, we found our optimal data binning width of 3 µm to 4 µm as the width that resulted in a $P_{\chi }({\chi }^2;\nu )$ as close to 0.5 as possible for all the different time points when fitting the nonlinear model. The exact choice of exclusion width was found not to influence the fitting result for the nonlinear model.

In addition to finding the optimal D-value for individual time points, we also modified the minimal-${\chi }^2$ routine to find the value of D that fits a whole data set (i.e. all time points simultaneously) in the best possible way. In order to do so, we summed the ${\chi }^2$-values obtained for each D over all time points, in this way producing a ${\sum }_t{\chi }^2(D)$-curve. The minimum of this curve indicates $D^{*}$ for the whole time series. Furthermore, dividing ${\sum }_t{\chi }^2(D)$ by the number of time points included in the optimization yields an average ${\chi }^2$- and reduced ${\chi }^2$-value corresponding to this $D^{*}$. In addition, the width of this time averaged curve at ${\chi }^2={\chi }_{\text{min}}^2+1$ indicates the standard deviation of the optimal D-value, ${\sigma }_D$. By approximating the minimum with a quadratic curve, we obtain an estimate for this standard deviation as ${\displaystyle {\sigma }_D={\mathrm{\Delta }}_D\sqrt{2\left({\chi }_{D^{\ast }-{\mathrm{\Delta }}_D}^2-2{\chi }_{D^{\ast }}^2+{\chi }_{D^{\ast }+{\mathrm{\Delta }}_D}^2\right)}}$ (Bevington and Robinson, 2003) where ${\mathrm{\Delta }}_D$ is the step size between individual fitted D-values, here ${\mathrm{\Delta }}_D=0.01$ µm²/min. Lastly, based on the average reduced ${\chi }^2$-values, we also compared several $c_{\text{max}}$-values for each data set to find the fit with probability $P_{\chi }({\chi }^2;\nu )$ the closest to 0.5 in each case.

All fits were performed using custom MATLAB routines. Horizontal error bars were plotted using the function herrorbar (van der Geest, 2006).

The average nuclear radius used to calculate the friction coefficient and thermal diffusion coefficient of IKNM nuclei was the radius corresponding to the maximum concentration $c_{\text{max}}$ obtained from the fitting procedure.

Among all the nuclei tracked in the experiments, we selected those nuclei where tracking data was available beginning right from cell division and also over a sufficiently long period of time to cover a substantial part of the cell cycle (at least 75 time steps, i.e. 150 min). For these nuclei, we extracted their respective birth times within the experiment from the full tracks and sorted the nuclei accordingly. The first 40 nuclei were chosen for further analysis, as these were nuclei with a minimum of 150 min of tracking data completely within the first 200 min of experiments, corresponding to the time frame used for D-optimization in the non-linear diffusion model. The exact distribution of their birth times was stored for use in the individual nuclei simulations.

Further, the nuclear tracks of the chosen 40 nuclei were transformed from being a function of experimental time to being a function of cell cycle time by simply subtracting a nucleus individual birth time from the experimental time for each step of its tracking data. Then the experimental mean squared displacement curve was calculated from the so obtained cell-cycle-dependent tracks.

Here, we give more information on the numerical calculation of cell shapes. Further details can be found elsewhere (Herrmann, 2020). Minimization of the elastic energy (Equation 12) leads to the equilibrium condition on the shape, expressed in terms of the mean curvature $\mathscr{H}$ and the Gaussian curvature $𝒦$ (Zhong-can and Helfrich, 1989),

where, for an axisymmetric shape $\delta (z)$,

and $\mathrm{\Delta }$ is the Laplacian operator,

The resulting shape equation is fourth order in z-derivatives and thus requires four boundary conditions. Given the symmetry of the system, we solve for the shape in the left half of the domain $z=(0,L/2)$ and impose $\delta (0)={\delta }_a$ and ${\delta }_z(0)=0$ at the apical surface. Imposing boundary conditions like ${\displaystyle \delta (L/2)=\mathfrak{R}}$ and ${\delta }_z(L/2)=0$ at the top of the nucleus usually leads to solutions that are incompatible with the presence of the nucleus (the resulting membrane shapes would cut through the nucleus). Therefore, we further divide the domain $z=(0,L/2)$ into a region away from the nucleus and a region where the membrane is in close contact with it. In the latter region, we assume the membrane to be bent into a spherical arc around the nucleus, leaving a small equilibrium gap as estimated by Daniels, 2019. The contact point $z_{\mathrm{contact}}$ between the two regions is adjusted until the membrane radius and its derivative are continuous through the contact point. The membrane shape away from the nucleus is then found using the MATLAB bvp5c solver. As can be seen from energy minimization using (Equation 12), the solution in each case turns out to be the one for which $z_{\mathrm{contact}}$ has been chosen such that the resulting $\mathscr{H}$ in $z\in [0,z_{\mathrm{contact}}]$ is equal to ${\displaystyle {\mathcal{\mathscr{H}}}_{\mathrm{c}\mathrm{i}\mathrm{r}\mathrm{c}\mathrm{l}\mathrm{e}}=-1/{\mathfrak{R}}_{\mathrm{t}\mathrm{u}\mathrm{b}\mathrm{e}}}$ for $z\to z_{\mathrm{contact}}$, where ${\displaystyle {\mathfrak{R}}_{\mathrm{t}\mathrm{u}\mathrm{b}\mathrm{e}}}$ is the radius of the membrane arc around the nucleus.

Simulations of nuclear trajectories for each of the three Langevin-type models were performed using a custom Python 3 routine. Time discretization of the stochastic differential equations was achieved via the Euler-Maruyama method. Simulations were performed using 0.2 min time steps and were checked against those with smaller time steps to ensure that this choice was sufficiently small.

In each run of a simulation, 40 nuclei were simulated and their birth times within the simulation were chosen to be the same birth times as those obtained from the nuclei within the experiments. Each nucleus was simulated for a total of 150 min, corresponding to the chosen experimental data. The value for the diffusion constant in these simulations was set to be the previously obtained value $D_{\text{nonlin}}^{*}$. For simulations with nuclear concentration-dependent Langevin equations, $c_{\text{max}}$ and the average nuclear concentration field $c(r,t)$ were similarly extracted from the results of the previous fits using the non-linear diffusion equation. Herein, $c(r,t)$ was provided for each time step of the simulation. As $c(r,t)$ can only be provided for discrete spatial coordinates r but the Langevin-type simulations were continuous in the spatial coordinate r, c was averaged over the values at the two closest spatial points whenever a nucleus' position did not exactly coincide with a point where the value for c was provided.

The resulting simulated nuclear trajectories were treated in the same way as the experimentally obtained ones. That is, the nuclei's birth times were subtracted from the trajectories to obtain cell cycle dependent tracks. Then, the mean squared displacement curve was calculated from the resulting set.

For each model, the same simulation was repeated 2500 times to obtain the range of distributions of the resulting mean squared displacement curves. For each cell cycle time step, the minimum and maximum of the mean squared displacement values out of all 2500 repetitions were calculated to obtain the areas depicted in Figure 8.

To compare results between data sets, the values $D^{*}$ and corresponding ${\sigma }_D$ from the overall fits were considered. It should be noted that these values were not obtained by averaging several data sets of the same experimental condition but instead each value results from one data set only. However, the sample size for each data set was set to 100 because 100 time points were taken into account for each overall optimization. These time points might not be completely uncorrelated, limiting the predictive power of the t-test. Two sided tests, specifically unequal variances t-test, also known as Welch’s t-test, (Precht and Kraft, 2015), were performed in order to determine whether samples differ significantly from each other.

Source code files 1, 2, 3, 4, 5, 6 are Matlab files containing the tracking data, as follows.

After rescaling space and time as in Equation 5 and introducing $\rho =b/a<1$, Equation 1 and the boundary conditions Equation 3 and Equation 4 read

where we have defined $f_0=a{N}_0/DS\tau$ and $\sigma =a^2/D\tau$. We transform this homogeneous differential equation with inhomogeneous boundary conditions into the problem of solving an inhomogeneous differential equation with homogeneous boundary conditions by writing $c(\xi ,s)$ as a sum of two contributions, 

where we require $\varphi (\xi ,s)$ to satisfy the inhomogeneous boundary conditions

These conditions are satisfied if $\varphi (\xi ,s)$ has the form

where $g_0$ is a constant of integration to be determined later. The remaining problem to solve for $\psi (\xi ,s)$ is

with homogeneous boundary conditions

We can further write $\psi (\xi ,s)$ as the sum of two contributions,

where ${\psi }_h$ is the general solution of the homogeneous problem

and ${\psi }_p$ is a particular solution of the full inhomogeneous problem Equation 26. The full solution of the homogeneous problem is given as a series of linearly independent eigenfunctions, each of the form

where the eigenvalues λ can be found from simultaneous solution of the boundary conditions,

which yields the transcendental relation

for which each eigenvalue ${\lambda }_i$ is a solution corresponding to one of the linearly independent eigenfunctions (only ${\lambda }_i>0$ need to be taken into account). We can further deduce from the Equation 30 that $B_i={\beta }_i{A}_i$, where

and we normalize the obtained expression for $W_i(\xi )$ from Equation 29

with

Thus, the homogeneous solution ${\psi }_h$ is

with prefactors $h_i$ to be determined from the initial condition.

In order to find a particular solution of the inhomogeneous problem, we first rewrite Equation 26 as

Now, we express $ℛ(\xi ,s)$, as well as the unknown inhomogeneous solution ${\psi }_p(\xi ,s)$ in terms of the normalized eigenfunctions $H(\xi ,s)$ of the homogeneous problem,

and

Substituting these forms into Equation 36, and noting that each term in the series must vanish separately we obtain

From the form of $ℛ(\xi ,s)$ it follows that $R_i(s)={\alpha }_i{f}_0{e}^{\sigma s}$ with some purely numerical prefactors ${\alpha }_i$, so we expect $C_i(s)\propto p_i{e}^{\sigma s}$ and find 

Finally, we determine the ${\alpha }_i$ by reconsidering Equation 37. We multiply both sides by ${\xi }^2{H}_j(\xi )$, where $H_j(\xi )$ is one specific but arbitrary eigenfunction of the homogeneous problem, and then integrate over the whole volume V. By the orthogonormality of these eigenfunctions, we obtain

and all the ${\alpha }_i$ can be calculated explicitely. Thus, the full solution of the linear problem is 

The constant $g_0$ can now be calculated from the requirement that $\int c(\xi ,s=0)dV=N_0$. Here, we make use of the fact that $\int H_i(\xi ){\xi }^2𝑑\xi =0$ if ${\lambda }_i$ satisfies Equation 31, thus 

The funders had no role in study design, data collection and interpretation, or the decision to submit the work for publication.

We thank Kevin O’Holleran and Martin Lenz at Cambridge Advanced Imaging Centre for their help and support in imaging zebrafish retinas. We also thank Oliver Y Feng, Timothy J Pedley, Michael E Cates and Salvatore Torquato for helpful advice and input. This work was supported by the Cambridge Wellcome Trust PhD Programme in Developmental Biology, the Cambridge Commonwealth, European and International Trust, and Natural Sciences and Engineering Research Council of Canada (AA); the Engineering and Physical Sciences Research Council, a Helen Stone Scholarship at the University of Cambridge through the Cambridge Commonwealth, European and International Trust, and the Cambridge Philosophical Society (AH); Established Career Fellowship EP/M017982/1 from the Engineering and Physical Sciences Research Council (REG); and Wellcome Trust Investigator Award (SIA 100329/Z/12/Z) (WAH).

© 2020, Azizi et al.

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