Random sub-diffusion and capture of genes by the nuclear pore reduces dynamics and coordinates interchromosomal movement

Hundreds of genes interact with the yeast nuclear pore complex (NPC), localizing at the nuclear periphery and clustering with co-regulated genes. Dynamic tracking of peripheral genes shows that they cycle on and off the NPC and that interaction with the NPC slows their sub-diffusive movement. Furthermore, NPC-dependent inter-chromosomal clustering leads to coordinated movement of pairs of loci separated by hundreds of nanometers. We developed Fractional Brownian Motion simulations for chromosomal loci in the nucleoplasm and interacting with NPCs. These simulations predict the rate and nature of random sub-diffusion during repositioning from nucleoplasm to periphery and match measurements from two different experimental models, arguing that recruitment to the nuclear periphery is due to random subdiffusion, collision, and capture by NPCs. Finally, the simulations do not lead to inter-chromosomal clustering or coordinated movement, suggesting that interaction with the NPC is necessary, but not sufficient, to cause clustering.


INTRODUCTION
In eukaryotes, genomes are spatially organized within the nucleus. Chromosomes occupy distinct subnuclear "territories", heterochromatin is segregated from euchromatin, and individual genes show non-random positioning relative to nuclear structures and other genes (Misteli, 2020). Gene positioning reflects physical interactions of chromosomal loci with nuclear structures like the nuclear lamina, nuclear pore complexes or nuclear bodies and changes in gene expression are often accompanied by changes in gene positioning (Brickner, 2017). The positioning of genes can impact their transcription, mRNA processing, or chromatin modifications.
Much of the work on gene positioning and recruitment to the nuclear periphery has utilized static population measurements such as microscopy, chromatin immunoprecipitation, or HiC. Although these studies have revealed important players necessary for gene positioning to the nuclear periphery, there are questions that cannot be answered using static methods. For example, while some loci interact very stably with the nuclear envelope (e.g., telomeres and centromeres; Heun et al., 2001;Jin et al., 2000), leading to ~85% of cells showing colocalization of these loci with the nuclear envelope, genes that interact with the NPC show lower levels (~50-65%; Brickner and Walter, 2004;Casolari et al., 2004). This has been suggested to reflect transient interaction with the nuclear periphery (Brickner and Walter, 2004), cell-cycle regulation of peripheral localization  or, perhaps, two distinct populations, one that stably associates with the NPC and the other that does not (Brickner and Walter, 2004;Cabal et al., 2006). Likewise, the repositioning of genes from the nucleoplasm to the nuclear periphery is not well-understood. Some data -including the involvement of nuclear actin and myosin -has suggested that repositioning to the NPC could be an active, vectorial process (Guet et al., 2015;Wang et al., 2020). Finally, while inter-chromosomal clustering is a widespread phenomenon (Apostolou and Thanos, 2008;Brickner et al., 2012;Brown et al., 2006;Homouz and Kudlicki, 2013;Kim et al., 2017;Lin et al., 2009;Noma et al., 2006;Schoenfelder et al., 2010;Thompson et al., 2003), relatively few studies have explored the dynamics of clustering over time and it is unclear if clustering reflects a stable physical interaction Dai et al., 2018). High-resolution, quantitative dynamics of chromatin diffusion are required to approach each of these questions.
Chromatin is a mobile polymer and individual loci exhibit constrained or anomalous diffusion (Bystricky et al., 2004;Gasser, 2002;Hajjoul et al., 2013;Heun et al., 2001;W.F Marshall et al., 1997). Chromatin motion can reveal important aspects of the nuclear environment and the biophysical mechanisms that control the spatial organization of the genome. Targeting to the NPC in budding yeast is an intriguing model for such studies because it is inducible, relatively rapid, controlled by well-understood DNA elements, and induces both a change in position and inter-chromosomal clustering.
Here we show that targeting to the nuclear periphery is continuous and dynamic but uniform within the population, suggesting that, within each cell, targeting to the periphery it is a probabilistic process. Localization at the nuclear periphery correlates with more constrained diffusion, as suggested by previous work (Backlund et al., 2014;Cabal et al., 2006). Using mean squared displacement (MSD) and molecular genetics, we pinpoint this effect to the interaction with the nuclear pore complex. The parameters of sub-diffusion derived from MSD of nucleoplasmic loci were used to develop a computational simulation that faithfully recapitulates the behavior of such genes. This simulation was also adapted to model repositioning to the nuclear periphery through random sub-diffusion and capture at the nuclear envelope. The repositioning predicted by the simulation was then compared with several rapid repositioning experiments to determine if targeting to the NPC is vectorial or super-diffusive.
The simulation matched the observed behavior of loci in cells, suggesting that repositioning from the nucleoplasm to the nuclear periphery does not require an active mechanism.
Finally, we monitored the dynamics of inter-chromosomal clustering. Unlike pairs of simulated paths, genes that exhibit clustering remain near each other for tens of seconds and show correlated movement. Simulated interaction with the NPC, while sufficient to recapitulate the chromatin dynamics of individual loci, is not sufficient to recapitulate this correlated movement. Therefore, we propose that inter-chromosomal clustering relies on a distinct physical interaction between genes that can extend hundreds of nanometers.

Chromatin targeting to the NPC is continuous and dynamic.
The localization of genes at the nuclear periphery can be followed in live yeast cells by tagging chromosomal loci of interest with an array of 128 Lac operators in a strain expressing GFP-Lac repressor (GFP-LacI) and quantifying its colocalization with mCherry-marked nuclear envelope ( Figure 1A; Brickner and Walter, 2004;Egecioglu et al., 2014;Robinett et al., 1996;Straight et al., 1996). In static confocal microscopy experiments, repositioning of inducible genes such as HIS4 or INO1 to the periphery leads to an increase in the fraction of cells in which the locus colocalizes with the nuclear envelope from that expected for a random distribution (~30%) to ~50-65% (Figure 1B & D;Brickner and Walter, 2004;Egecioglu et al., 2014). However, artificially tethering chromatin to the nuclear envelope leads to ~85% colocalization with the nuclear envelope (Brickner and Walter, 2004). This suggests that localization to the nuclear periphery reflects either dynamic and continuous targeting to the NPC or two distinct populations of cells, one that exhibits stable association with the nuclear envelope and the other that does not. To distinguish between these possibilities, we quantified peripheral localization of three LacO-tagged loci over time in individual cells: the inducible genes HIS4 and INO1, as well as the negative control URA3, which localizes in the nucleoplasm (Figure 1B & D;Brickner et al., 2019;Brickner and Walter, 2004;Randise-Hinchliff et al., 2016).
To avoid the complication that targeting of many genes to the NPC is lost during S-phase , cells were synchronized using nocodazole and released into G1 for 30 minutes before scoring colocalization with the nuclear envelope every 10s over 10 minutes. In complete media (i.e., uninducing conditions), all three genes showed similar patterns: episodic, brief colocalization with the nuclear envelope ( Figure 1C, E & F). However, under inducing conditions (-histidine for HIS4 or -inositol for INO1), the pattern changed. Both HIS4 and INO1 showed longer periods of colocalization with the nuclear envelope ( Figure 1C, F & J), while URA3 was unaffected ( Figure 1E). The pattern was consistent across the population and the fraction of cells in which HIS4 or INO1 colocalized with the nuclear envelope at each time point ( Figure 1H) was in close agreement with the fraction of time spent colocalized with the nuclear envelope in each cell ( Figure 1I). This argues against two distinct populations and instead suggests that targeting to the NPC is continuous and dynamic over time, increasing the duration of colocalization with the nuclear envelope.
Targeting to the NPC is mediated by transcription factors binding to cis-acting elements that function as DNA zip codes (Ahmed et al., 2010;Brickner et al., 2019;Light et al., 2010).
For example, the Gene Recruitment Sequence GRS1 from the INO1 promoter binds to the Put3 transcription factor to mediate targeting to the NPC (Brickner et al., 2012). Likewise, the Gcn4 binding site (GCN4 BS) from the HIS3 promoter mediates targeting to the NPC . Such zip codes are both necessary and sufficient to cause peripheral localization; inserting them near URA3 causes repositioning to the nuclear periphery (e.g. Figure 1D; Ahmed et al., 2010;Randise-Hinchliff et al., 2016). The association of URA3:GRS1, which shows unregulated targeting to the periphery, with the nuclear envelope over time resembled that of active HIS4 and INO1 ( Figure 1G-J). Thus, DNA zip code-mediated targeting to the NPC is sufficient to produce continuous and dynamic association with the nuclear envelope.

URA3:GRS1,
Chromatin sub-diffusion is suppressed by interaction with the nuclear pore complex.
We next examined how interaction of genes with the NPC impacts the dynamics of diffusion. Chromosomal loci exhibit constrained sub-diffusion, which can be quantified using mean squared displacement analysis (MSD; W. F Marshall et al., 1997). For comparison, we tracked the movement of the less mobile nuclear envelope-embedded spindle pole body (SPB) and a much more mobile cytoplasmic particle (the µNS viral capsid; Munder et al., 2016). While µNS was highly diffusive, the SPB showed very limited displacement at this time scale, reflecting both slow diffusion within the membrane and movement of the whole nucleus ( Figure  2B). We assessed the MSD of eleven nucleoplasmic loci not associated with the NPC and two telomeres tethered to the nuclear envelope ( Figure 2B; Table S1). These loci exhibited a range of intermediate MSD between these two extremes, with the nucleoplasmic loci showing greater mobility than tethered telomeres and telomeres showing greater mobility than the SPB ( Figure   2B). Simultaneously acquiring images of chromosomal loci and the SPB to correct for nuclear movement significantly reduced the time resolution. Given that nuclear movement was much less than chromosomal movement at these time scales, it could be ignored. We also determined the MSD of chromosomal loci in 3D. Although this gave very similar results (Table   S1), the quality of the data was lower because of the longer time interval (>1s). For these reasons, we limited our movies for MSD analysis to 40s at 210ms resolution (200 x 0.21s) in a single focal plane and calculated MSD for time intervals up to 4s ( Figure 2B).
The nucleoplasmic loci showed a range of mobility by MSD, perhaps reflecting nearby physical interactions with the nuclear envelope. Indeed, the initial mean squared displacements (t = 0.21s) correlated with the distance to the nearest nuclear envelope tethering point (either centromeres or telomeres; MSD ~ log10(bp); R 2 = 0.65; Figure 2C). Thus, diffusion of chromatin is influenced over a range of distances by stable interactions with the nuclear envelope , complicating direct comparison of MSD between loci.
To quantify the effect of local interaction with the NPC on chromatin sub-diffusion and to avoid this complication, we exploited genes that show conditional association with the NPC. We compared the MSD of INO1, HIS4 and URA3 under either uninducing and inducing conditions (± histidine and ± inositol). URA3 showed no change in MSD under these conditions ( Figure   2D). However, both HIS4 and INO1 showed significantly reduced mobility upon induction ( Figure 2E & F), confirming that repositioning to the nuclear periphery correlates with reduced chromatin sub-diffusion.
Shifting cells to the inducing conditions leads to both repositioning to the nuclear periphery and transcriptional activation. The change in MSD of INO1 and HIS4 could be due to the effects of transcription, the interaction with the NPC, or both. To distinguish the effects of transcription from the effects of interaction with the NPC, we also measured the effect of DNA zip codes on the MSD of URA3. Single copies of zip codes from the promoters of INO1 (URA3:GRS1; Figure 3A) or HIS4 (URA3:Gcn4 BS; Figure 3B) were integrated beside URA3.
These elements promote interaction with the NPC outside the context of a transcribed promoter.  (Greenberg et al., 1982a(Greenberg et al., , 1982b. In both mutants, the INO1 MSD resembled that of active INO1 ( Figure 3E & F), suggesting that interaction with the NPC is the principal cause of the decrease in sub-diffusion.

Simulating chromatin sub-diffusion and repositioning to the nuclear periphery
Using parameters from the MSD analysis, we developed a simulation of chromatin subdiffusion (https://github.com/MCnu/YGRW). Passive sub-diffusion of a segment of chromatin results from forces affecting the chromatin segment both directly (e.g., the viscoelastic potential of the polymer and boundary collision) and indirectly (forces and membrane tethering along distal regions of the same chromatin polymer; Figure 2C). MSD data for a Rouse polymer like chromatin reflects a relationship MSD(t) = G(t a ) for any time interval t (Socol et al., 2019).
Gamma (G) describes the fundamental diffusion coefficient, while an a exponent less than one reflects a hallmark for sub-diffusive movement: each step vector is anti-correlated with both the previous and subsequent steps (Lucas et al., 2014). Published work (Hajjoul et al., 2013)  trajectories that approximate chromatin sub-diffusion (Lucas et al., 2014). FBM produces nonindependent steps across time, allowing us to impart the anti-correlation between individual steps that is characteristic of chromatin sub-diffusion. For each trajectory, two numeric arrays for the x and y dimensions of movement (Dietrich and Newsam, 1997) were generated based on an expected covariance matrix and the a = 0.52. This array produces a stochastic time series of vectors with an anti-correlation structure functionally identical to that observed for chromatin movement. Finally, these vectors were scaled according to the experimentally derived G value and Hurst exponent (a/2; Mandelbrot and Ness, 1968). Starting from random positions within the nucleus, the resulting array of discrete step lengths describes a single, twodimensional sub-diffusive particle trajectory. This simple and rapid approach generates Paths generated by FBM suffer from one significant shortcoming. In an enclosed volume, FBM will deplete occupancy of particles near the boundary over time, resulting in a biased distribution (Figure 4 -Figure supplement 1C). This phenomenon has also been reported by others (Vojta et al., 2020) and is not consistent with observations that chromosomal loci, unless associated with the nuclear envelope, localize at the nuclear periphery at a frequency expected from a random distribution (Brickner and Walter, 2004;. This may reflect a fundamental difference between sub-diffusion of particles and the apparent subdiffusion of a segment of chromatin. We explored several methods to avoid depletion at the nuclear periphery and found that the following was effective: steps that would have taken the locus beyond the boundary were replaced with steps to the boundary along the same vector and, upon interaction with the boundary, the normalized, correlated noise for future steps was  Figure 4G) that most closely matched that of URA3:GRS1. We refer to this modified simulation as simulation + zip code. The fit of the simulation to the mean MSD for URA3 and of the simulation + zip code to the mean MSD for URA3:GRS1 was excellent (Pearson's C 2 sums of 0.001 and 0.003, respectively, for t from 0.21 -4s). Together, these two relatively simple simulations capture important aspects of chromatin sub-diffusion and gene positioning at the nuclear periphery.

Chromatin repositioning is achieved by random sub-diffusion and capture
Chromosomal loci can undergo long-range, directed movement (Miné-Hattab and Rothstein, 2013), raising the possibility that repositioning from the nucleoplasm to the nuclear periphery could be an active process. To explore this question, we first determined the behavior of the simulation, which does not possess active, vectorial movement. Initiating the default simulation of chromatin movement and the simulation + zip code from random positions within the nucleus, we followed the percent of the population showing localization within 150nm of the nuclear edge over time. As expected, the random initial positions resulted in ~28% peripheral localization ( Figure 5A). For the nucleoplasmic simulation, this was unchanged over time.
However, the simulation + zip code resulted in stable repositioning to the nuclear periphery within ~ 2 minutes ( Figure 5A). Therefore, rapid repositioning to the nuclear periphery can occur without any directed, active movement.
To compare these simulations to experimental results, we applied live-cell tracking during repositioning from the nucleoplasm to the periphery. One challenge with such experiments is that the time required for genes to reposition when cells are shifted from uninducing to inducing conditions is gene-specific and can be quite slow (e.g., t1/2 ~ 30min; Brickner et al., 2012Brickner et al., , 2007Randise-Hinchliff et al., 2016). This suggests that the rate-limiting step for repositioning often reflects the regulation of transcription factors that mediate repositioning, rather than the rate-limiting step for movement to the periphery . To overcome this complication, we developed two approaches to maximize the rate of repositioning from the nucleoplasm to the nuclear periphery. First, we arrested cells bearing URA3:GRS1-LacO with a-factor mating pheromone, which disrupts peripheral localization by inhibiting Cdk, which phosphorylates Nup1 and is required for peripheral targeting . Upon release from a-factor arrest, URA3:GRS1 repositioned to the nuclear periphery within ~15 minutes ( Figure 5C).
As a complementary approach, we used an optogenetic switch to recruit URA3 to the nuclear periphery. Tethering of a 27 amino acid "positioning domain" from the Gcn4 transcription factor (PDGCN4) near URA3 using the LexA DNA binding domain (DBD) is sufficient to target URA3:LexABS to the nuclear periphery (Brickner et al., 2019). Cryptochrome 2 (CRY2) and Cryptochrome Interacting protein CIB1 from A. thaliana undergo rapid dimerization when exposed to 488nm light (Benedetti et al., 2018). In a strain having both the LacO array and the LexA binding site at URA3, CRY2-LexA DBD was co-expressed with CIB1-PDGCN4 to generate a light-induced peripheral targeting system ( Figure 5D; Brickner et al., 2019). LexA DBD-Gcn4 served as a positive control and a mutant CIB1-pdGCN4 that is defective for peripheral targeting served as a negative control. Cells were arrested, synchronized in G1 and illuminated with 488nm light for 1s pulses every 10s over 10 minutes. Illumination resulted in rapid, PDGCN4dependent repositioning to the nuclear periphery within ~7.5 minutes ( Figure 5E). Thus, both the biological and the opto-genetic stimuli led to rapid repositioning to the nuclear periphery with kinetics comparable to the simulation.
Having established that these two approaches lead to rapid peripheral localization, we then used particle tracking to define the nature of the movement during this transition. URA3, URA3:GRS1 or URA3:LexABS were tracked for 5 minutes at 0.5s resolution (600 frames) during repositioning. For each movie, the position and time of initial colocalization with the nuclear envelope was recorded (if observed). While peripheral colocalization of URA3:GRS1 and URA3:LexABS + CIB1-PDGCN4 represents -at least some of the time -interaction with the NPC, peripheral colocalization of the negative controls does not. Therefore, we expected that if an active process directs genes to the nuclear periphery, the positive and negative controls should show differences in the step velocities, time of arrival or directness of the path preceding arrival at the nuclear periphery. For comparison, we also determined each of these parameters for paths generated by the default simulation and the simulation + zip code, which include no directed movement. The mean velocities for the simulations and experimental controls were statistically indistinguishable, ranging from 0.163 ± 0.10 µm s -1 to 0.207 ± 0.13 µm s -1 ( Figure   5F; n = 6077-9724 steps per strain), suggesting that the speed of movement was not increased during peripheral targeting. We did not observe significantly more large steps in the experimental movies than in the negative control movies ( Figure 5F). The mean arrival time prior to initial contact with the nuclear envelope was also similar between the simulations and the experimental movies, ranging from 105 ± 49s to 133 ± 54s ( Figure 5G; n = 27 -40 cells per strain), consistent with the predictions from the simulation. Finally, to assess if any of the loci underwent processive, vectorial movement during translocation, we measured the radial deviation (q) of each step from a direct path to the ultimate contact point at the nuclear envelope ( Figure 5H). Random sub-diffusion should produce an average q of ~ p/2 = 1.57 radians, while directed movement would produce an average of 0. The simulations were close to random and, while the experimental loci appear slightly more directed than random, the positive and negative controls were indistinguishable ( Figure 5I). Taken together, these results indicate that repositioning of chromatin from the nucleoplasm to the nuclear periphery is likely a passive process that relies on random sub-diffusion and collision with the NPC.

Dynamics of inter-chromosomal clustering
Genes that are targeted to the yeast NPC can exhibit inter-allelic or inter-genic clustering with genes that are targeted to the NPC by the same transcription factor (Brickner et al., 2015(Brickner et al., , 2019(Brickner et al., , 2012Kim et al., 2019Kim et al., , 2017Randise-Hinchliff et al., 2016). Loss of nuclear pore proteins or transcription factors that bind to DNA zip codes disrupts clustering (Brickner et al., 2012). Clustering has been observed using microscopy as a significant shortening of the distances between two loci in the population (Brickner et al., 2012) or using biochemicalgenomics methods such as HiC (Kim et al., , 2017. To explore the dynamics of interchromosomal clustering, we tracked the positions and inter-genic distances of well- frames that negatively regulate Gcn4 expression (uORFmt; Mueller et al., 1987;Mueller and Hinnebusch, 1986), led to high-level, constitutive inter-allelic clustering of HIS4 ( Figure 6B; Randise-Hinchliff et al., 2016), while loss of Nup2 disrupted all clustering ( Figure 6B). Finally, URA3, the simulated nucleoplasmic paths, and the simulated peripheral paths showed no clustering. Thus, NPC-and TF-dependent clustering can be observed over time and the simulated interaction with the NPC is not sufficient to produce clustering.
We also assessed the stability of clustering over time. The lifetimes of clustering (i.e., time two loci remain within 550nm) increased from ~ 5s for unclustered loci to 20-50s upon clustering ( Figure 6C). Similarly, the fraction of the total time points in which clustering was observed reflected the strength of clustering ( Figure 6D). Because inter-chromosomal clustering persists for relatively long periods of time, it likely reflects a physical interaction.
Finally, we asked if pairs of loci that exhibit clustering show coordinated movement. To quantify the degree of coordination, we determined both the correlation of step sizes by each locus and the average difference in step angles made by each locus over 40s movies (200 x 0.21s; Figure 7A  . While the movement of loci that were > 0.55µm (orange dots) was uncorrelated, the subset of loci that were ≤ 0.55µm (purple dots) showed correlated movement, both in terms of step size and angle. We quantified this behavior using the slope and R 2 of the scatter plots (Figure 7). Unclustered control loci gave slopes ~ 0 and R 2 ≤ 0.1 (e.g., Figure 7D

DISCUSSION
Tracking yeast NPC-associated chromatin over time revealed frequent exchange between the nucleoplasm and periphery (Figure 1), suggesting that the interaction with the NPC is continuously re-established and that the population averages reflect this dynamism, rather than distinct, stable subpopulations. In other words, localization to the nuclear periphery is less well described as tethering than as a change in the steady-state positioning through continuous binding and dissociation. As interaction with the NPC enhances transcription (Ahmed et al., 2010;Brickner et al., 2019Brickner et al., , 2016Brickner et al., , 2012Capelson et al., 2010;Jacinto et al., 2015;Liang et al., 2013;Pascual-Garcia et al., 2014;Taddei et al., 2006), it is intriguing that the periodic and transient interaction with the NPC is reminiscent of the widespread phenomenon of transcriptional "bursting" (Femino et al., 1998;Rodriguez and Larson, 2020). Transcriptional bursting leads to heterogeneity in the transcription between cells within a population (Zenklusen et al., 2008) and disrupting the interaction of the GAL1-10 promoter with the NPC leads to a decrease in the number of cells expressing these genes without affecting the amount of transcript produced at the site of transcription . Perhaps interaction with the yeast NPC functions with other transcriptional regulators to simulate transcriptional bursts.
Exploring this connection will require assessing the dynamics of chromatin positioning and transcription simultaneously in live cells.
Chromatin undergoes anomalous sub-diffusive movement during interphase (Hajjoul et al., 2013;W. F. Marshall et al., 1997). The physical interaction between chromatin and the NPC, though transient, reduces chromatin sub-diffusion (Figure 2; Backlund et al., 2014;Cabal et al., 2006), independent of changes in transcription (Figures 2 & 3). Using the parameters derived from MSD, we developed computational simulations for yeast chromatin sub-diffusion in the nucleoplasm and at the nuclear periphery. The exponent a -0.52 for yeast chromatin (Hajjoul et al., 2013) -reflects anticorrelation between successive steps and can be modeled as fractional Brownian motion (FBM, a.k.a. overdamped fractional Langevin motion; Lucas et al., 2014). Sub-diffusion of yeast chromosomal loci is determined by the elastic response from the chromatin polymer and the viscous interaction between the polymer and the nucleoplasm.
While we do not explicitly simulate the total chromatin polymer or other nuclear occupants, FBM captures their net effects, recapitulating the MSD behavior of a nucleoplasmic locus (Figure 4).
However, the FBM model leads to exclusion near boundaries, leading to non-random positioning of loci, a phenomenon that is not consistent with experimental observations. This likely reflects the fact that, while the motion of a segment of chromatin can be modeled as an FBM particle, it is part of a polymer and is not an FBM particle. Our solution to this shortcoming of the FBM model, allowing collisions with the nuclear periphery to reset the calculated path (see detailed explanation in Methods), produced localization patterns and MSD behaviors that are consistent with experimental observations. However, additional theoretical and experimental work will help clarify the biological and physical significance of this modification.
To simulate the interaction of chromatin with the NPC, we allowed loci in an area within 50nm of the nuclear boundary to "bind" to the nuclear periphery, assuming the mobility of the SPB. The width of this annulus is roughly equal to the height of the NPC nuclear basket (Vallotton et al., 2019), whose components are required for chromatin association with the NPC (Ahmed et al., 2010). We independently optimized the probability of binding and of remaining bound by comparing the positioning and MSD of simulated paths with that conferred by a DNA zip code. This simple modification of the simulation was able to reliably recreate the peripheral localization and constraint on chromatin sub-diffusion caused by interaction with the NPC (Figure 4). Thus, the work described here provides a straight-forward and powerful theoretical framework for modeling the biophysical nature of gene positioning through associating with any stable nuclear structure.
Recruitment of genes to the NPC during transcriptional activation occurs over a wide range of time scales, depending on the stimulus and target gene , making it difficult to establish if it represents active or vectorial movement. Our simulated trajectories offer an important insight; starting from random positions within the center of the yeast nucleus, individual loci reached the nuclear periphery within ~2 minutes by random subdiffusion ( Figure 5G). Allowing interaction with the NPC led to a shift in the localization of loci to the periphery within the population on a similar time scale ( Figure 5A). This time scale is comparable to the experimental models for NPC targeting (Figure 5), arguing that active mechanism(s) are unnecessary to explain the observed rate of repositioning. More importantly, experimental analysis of the speed and vector of individual steps preceding contact with the nuclear envelope showed non-vectorial sub-diffusive movement that was indistinguishable from that captured by the simulation (Figure 5). Importantly, there was also no difference between experimental cells and negative control cells for these components. These results indicate that zip code-dependent gene localization results from stochastic, sub-diffusive chromatin movement, random collision with the NPC, leading to dynamic binding. The recently discovered role for actin and Myo3 in targeting of INO1 to the NPC (Wang et al., 2020), raises an important question: how do these factors impact peripheral targeting through a sub-diffusive mechanism?
Perhaps, like actin (Kapoor et al., 2013), Myo3 impacts the function of chromatin remodeling complexes or histone modifying enzymes, which regulate binding of transcription factors to DNA zip codes . Alternatively, perhaps actin/Myo3 act at the NPC to facilitate capture. A better biochemical and biophysical understanding of these processes will illuminate such possible roles.
Interaction with nuclear pore proteins plays a conserved role in promoting transcription.
However, while interaction of yeast genes with nuclear pore proteins occurs at the nuclear periphery in association with the NPC, many genes in mammalian cells and Drosphila interact with soluble nuclear pore proteins in the nucleoplasm (Capelson et al., 2010;Liang et al., 2013;Light et al., 2013). Sub-diffusion for mammalian chromatin (which has been suggested to be less mobile than in yeast; Chubb et al., 2002) in a nucleus with a radius of 5µm would make it impossible (on a biologically meaningful timescale) for loci in the center of the nucleus to reach the periphery. In larger nuclei, recruitment of nuclear pore proteins to sites of action, regardless of their position, likely overcomes this obstacle.
Inter-chromosomal clustering is a widespread phenomenon in eukaryotes (Bantignies et al., 2011;Brickner et al., 2012;J. M. Brown et al., 2008;Cook and Marenduzzo, 2018;Gehlen et al., 2012;Haeusler et al., 2008;Noma et al., 2006;Ramos et al., 2006;Taddei et al., 2009;Thompson et al., 2003;Xu and Cook, 2008). Genes that are targeted to the NPC by shared transcription factors exhibit inter-chromosomal clustering (Brickner et al., 2012;Kim et al., 2019Kim et al., , 2017. Such clustering requires transcription factor(s) and nuclear pore proteins (Brickner et al., 2012;Chowdhary et al., 2017;Kim et al., 2019) but is also mechanistically distinguishable from targeting to the NPC . Clustering persisted for 20-40s ( Figure 6) and led to correlated movement between pairs of loci that were within 550nm (Figure 7). Importantly, independently correlating step size and step angle is sensitive to correlations among pairs of loci in a subset of the cells in the population. Such correlated movement, averaged over the entire population, would be more difficult to appreciate. This may explain why previous work tracking movement of pairs of active GAL1-10 alleles in yeast found little correlation in aggregate (Backlund et al., 2014).
Pairs of paths generated by either the simulation or the simulation + zip code do not lead to inter-chromosomal clustering, consistent with the observation that genes targeted to the NPC by different transcription factors do not exhibit clustering (Brickner et al., 2012). Therefore, while clustering requires transcription factors and interaction with the NPC, it represents a distinct physical interaction. Surprisingly, correlated movement was observed between loci separated by hundreds of nanometers, suggesting that it represents a large molecular complex, or more likely, an environment. Physical interactions that lead to phase-separation could encompasses groups of genes to create a (perhaps transient) nuclear sub-compartment (Hult et al., 2017). This is reminiscent of superenhancers, which exist within phase-separated droplets (Hnisz et al., 2017;Sabari et al., 2018) and are strongly associated with nuclear pore proteins (Ibarra et al., 2016). It is possible that phase separation is facilitated by multivalent interactions between natively unstructured nuclear pore proteins, which can form phase separated droplets in vitro (Frey et al., 2006;Frey and Gorlich, 2007). Such conditional phase separation would be regulated and specified by transcription factors, and potentially other transcriptional complexes such as Mediator or RNA polymerase II, to functionally compartmentalize the nucleus.

Chemicals, reagents, and media
All chemicals were purchased from Sigma-Aldrich unless otherwise noted. Media components were from Sunrise Science Products and a-factor was from Zymo Research. Yeast and bacteria media and transformations were as described (Burke et al., 2000;Wood, 1983).

Yeast culturing
Yeast cultures were inoculated from a YPD agar plate into synthetic dextrose complete (SDC) or drop out media and rotated at 30˚C for ≥ 18h, diluting periodically to maintain the cultures at OD600 < 0.8. Before microscopy, cultures were diluted to ≤ 0.1 OD/mL and treated with 2ng/mL of nocodazole for 2h. Cultures were then pelleted, washed, and resuspended in SDC to release from M-phase into G1-phase for 10 minutes. Cells were then pelleted again, concentrated, applied to a microscope slide, and covered with a glass coverslip for imaging.
For experiments involving mating pheromone, 100μM a-factor was added to the cultures following release from nocodazole arrest for at ≥ 30 minutes. To release from pheromone arrest, cells were pelleted, washed into SDC and mounted for microscopy.

Microscopy
Confocal microscopy was performed in the Northwestern University Biological Imaging Facility. Tracking microscopy was performed on a Leica Spinning Disk Confocal Microscope (Leica DMI6000 inverted microscope equipped with Yokogawa CSU-X1 spinning disk and Photometrics Evolve Delta512 camera), and static localization experiments ( Figures 1B, 1D, 3, 5C & 5E) were performed on a Leica TCS SP8 Confocal Microscope.
For both single locus/particle MSD and multiple loci tracking, the same acquisition protocol was used. GFP-LacI /LacO spots in G1-phase cells were imaged every 210ms for 200 frames in a single z-plane with a minimum of 40 biological replicates per experimental condition.
Cells that did not remain immobilized or whose loci underwent no movement were excluded from our analysis. For peripheral targeting dynamics experiments ( Figure 5F, G, & I), LacI-GFP/LacO128 arrays in G1-phase cells were imaged every 500ms for 600 frames and Pho88-mCherry was imaged every 10s to determine position with respect to the nuclear periphery (D'Urso et al., 2016;Egecioglu et al., 2014). Figures 1B, 1D, 3, 5C & 5E) were acquired as z-stacks encompassing the full yeast cell, and 30-50 cells were scored per biological replicate as described Brickner and Walter, 2004;Egecioglu et al., 2014). Each strain and condition included at least three biological replicates. To activate light-induced recruitment, cells imaged in Figure 4C were scanned with the 488nm laser every 10 seconds.

Particle tracking and data analysis
Tracking was performed using the ImageJ plugin MTrackJ. To accommodate clustering experiments (which typically have two or more fluorescent particles per nucleus), MTrackJ's region of tracking tool was utilized to ensure the signals from individual loci were tracked separately. Tracking data was output as a comma separated text file and analyzed with R scripts available via GitHub. (https://github.com/MCnu/R_sim_scripts). Repositioning analysis in

Fractional Brownian motion simulations
We model the dynamics of chromosomal loci in the cellular nucleus via a discrete-time random walk with continuously varying step size. This simulation is governed by Fractional Brownian Motion (FBM), which gives rise to anomalous diffusion of the locus. Anomalous diffusion is distinct from Brownian diffusion due to a nonlinear mean-squared displacement over time, with distinct behaviors for the super-diffusive (a > 1) vs. sub-diffusive (a < 1) regimes.
Free fitting our MSD measurements for 23 different loci/conditions, we found an average a = 0.52 (not shown), matching that determined in previous work (Hajjoul et al., 2013). Therefore, for the simulations, we used a = 0.52. Following previous work (Lucas et al., 2014), we present Fractional Langevin dynamics simplified by the assumption of overdamping (i.e., no inertial term) and no driving force. In FBM, the statistical noise is a stationary Gaussian process with a mean equal to zero and a nonzero anticorrelation between successive steps (Meyer et al., 1999). This property is exploited to allow random vector generation with a given correlation structure (Dietrich and Newsam, 1997). We draw values for each simulated dimension of movement to generate the entire time series for a trajectory. We re-scale the vectors to an appropriate magnitude for given time units equal to t using a G parameter provided by non-  (Guggenberger et al., 2019;Vojta et al., 2020Vojta et al., , 2019Wada and Vojta, 2018). These studies agree with our findings that in the sub-diffusive regime, 'depletion' occurs at the boundary (Figure 4,Supplement 1C & D). This depletion at the periphery is rationalized by the fact that because successive steps are anticorrelated, a step which would take the particle over the boundary is likely to be followed by one which would take it away from it. Such depletion is not observed in experimental distributions of control and noncontrol specimens. It is possible that the physicochemical landscape of the periphery or the region near the periphery involve many interactions which have the effect of attracting the chromatin locus to the periphery, but such effects are not present in the aforementioned studies (which do not consider transient binding interactions with a hard wall). Because our particle is actually a segment of a much larger polymer, we instead decided to regenerate the underlying noise time series whenever the trajectory collides with the periphery to negate the effects of prior movement. This adaptation succeeded in creating a uniform distribution of positions across the nucleus. However, we acknowledge that our theoretical particle no longer satisfies the fluctuation dissipation theorem inherent to all Brownian motion, including FBM. More careful investigation of the behavior of chromatin at the boundary in silica and in vivo will help clarify the validity of this modification.
Binding of chromatin to NPCs was modeled using a simple two-state Markov model wherein a locus within the peripheral region (an annulus extending 50 nanometers from the nuclear boundary), can assume a bound state in the next step with a defined probability.
Particles bound to the NPC remain bound at a second defined probability for every step until it becomes unbound. A particle bound to the NPC is assumed to be interacting strongly with an NPC, their motion is inhibited, but not entirely arrested. We therefore scaled the step sizes of particles in the bound state with G and a parameters derived from non-linear regression of the MSD for the spindle pole body (Figure 2). In this way, we simulate the effective "pausing" of chromatin motion due to NPC interaction.

Source Code
Our simulation data and source code are openly available. Our simulations were implemented in Python, with routine algorithms like random noise generation or the fast Fourier transform from the NumPy library (Harris et al., 2020) and all other code implemented using custom libraries is available on GitHub (https://github.com/MCnu/YGRW). Analytical pipeline of two-dimensional tracking data is also available. All analyses were implemented in R and scripts are available on GitHub (https://github.com/MCnu/R_sim_scripts).  Brickner & Walter, 2004;Brickner et al., 2019) and scored as either nucleoplasmic (left) Table S1) and two telomeres (red) at different time intervals (t). The ribbon around the mean         paths were initiated at random positions within a 1µm diameter sphere in the center of the 2µm diameter nucleus and followed for 5 minutes. Paths that did not make contact with the nuclear periphery were excluded.  The fraction of all time points that d ≤ 0.55µm for each strain and media condition. For panels B-D, mean distances, lifetimes and fraction of timepoints clustered were also determined for pairs of randomly selected simulated paths (with or without zip code; red).  Cells in which the mean distance between the loci was > 0.55µm appear in orange, while cells in which the mean distance between the loci was ≤ 0.55µm appear in purple. For each plot, the slope and R 2 for a linear relationship between cor(d) and ∆θ are indicated. 40-50 cells were analyzed per strain and condition. Simulations are the 50 pairs of paths generated for Figure 6. having the LacO array integrated at both alleles of URA3 were grown in SDC ± histidine, the loci were tracked and analyzed for correlated step size and step angle as described for Figure 7.
The slope and R 2 for each plot is for a linear relationship between cor(d) and ∆θ.