Modeling meiotic chromosome pairing: a tug of war between telomere forces and a pairing-based Brownian ratchet leads to increased pairing fidelity

We modeled meiotic chromosome dynamics using a previously described Brownian dynamics model. Each chromosome is represented as a list of nodes representing beads connected by springs, with each node subjected to Langevin random forces. The frictional force acting on each node is proportional to its velocity. These forces yield the following equation of motion, which is used to update the velocity and position of each node at each time step:

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \xi {{\stackrel{\rightharpoonup}{v}}_{i}}=&~{{k}_{s}}\left(\left\Vert{{{\stackrel{\rightharpoonup}{x}}}_{i}}-{{{\stackrel{\rightharpoonup}{x}}}_{i-1}}\right\Vert-{{L}_{eq}}\right){{\hat{u}}_{i}}+{{k}_{{\rm s}}}\left(\left\Vert{{{\stackrel{\rightharpoonup}{x}}}_{i}}-{{{\stackrel{\rightharpoonup}{x}}}_{i+1}}\right\Vert\right.\nonumber \\ & \left.-{{L}_{{\rm eq}}}\right){{\hat{u}}_{i+1}}+\sigma {{\hat{n}}_{{\rm r}}}+\phi {{\hat{n}}_{{\rm t}}}\left({{\delta}_{i,0}}+{{\delta}_{i,N}}\right)\nonumber \end{align} \tag{ 1 }$$

where x_i and v_i are the position and velocity vectors at node i, ξ is the friction coefficient, k_s and L_eq are the spring constant and equilibrium length of the links, σ is the average magnitude of the Langevin (thermal) random force, ϕ is the average magnitude of the cytoskeletally generated random force acting at the telomeres, n_r and n_t are randomly oriented unit vectors, and u_i represents a unit vector directed from node i  −  1 to node i. In the current version of the model, successive randomly oriented unit vectors n_r and n_t are completely independent of each other, such that both the Langevin and telomere forces fluctuate in direction. At each time point, a new orientation for each of the random unit vectors is chosen from a uniform distribution of angles in spherical coordinates. The magnitude of each random force is assumed to be constant at every time step, so σ and ϕ are constants. We note that the term describing the Langevin random force actually represents the average net resultant of all random collisions with thermally excited solvent molecules during a single time-step. In our simulations the random unit vectors describing the motion of the telomeres are taken to always be tangential to the surface of the nucleus. We note that our representation of telomere motion, as a set of randomly oriented forces acting tangential to the nuclear surface and uncorrelated with the telomere forces on other chromosomes, would not be applicable in fission yeast, which differs from other systems in that the telomeres remain closely juxtaposed during the active movements and active forces drive large motions of the whole nucleus.

This bead-spring chain model represents a Rouse model of polymer dynamics (Grosberg and Khokhlov 1994) and thus assumes freely rotating links and lack of a barrier to strand passage. Such Rouse models of polymer dynamics have been shown to be consistent with experimental measurements of interphase chromatin motion (Hajjoul et al 2013). The Rouse model is a 'phantom polymer' model that lacks topological constraints. Order of magnitude scaling arguments based on the relative density of topoisomerase strand-passage sites compared to the density of chromatin interlocks have shown that there is effectively no barrier to strand passage because whenever an interlock would form, it is resolved by topoisomerase activity on a sufficiently rapid timescale (Sikorav and Jannink 1994). Experimental studies have confirmed that topoisomerase activity can convert an entangled DNA melt into a viscous fluid, indicating that strand passage can readily occur on a times-scale relevant to chromosome motion (Kundukad and van der Maarel 2010). Topoisomerase II activity is observed during meiosis, and defects in topoisomerase II activity lead to rapid arrest of meiosis (Rose and Holm 1993, Zhang et al 2014), confirming that this enzyme is active in strand-passage during the stage we are simulating. Obviously an interesting area for future computational analysis will be the effect of limitations on strand-passage rates. We note that one advantage of using the Rouse model is that our results could in principle be linked to analytical theory of polymers in a straightforward manner.

Initial generation of a random-walk chain, and relaxation according to equation (1), have been previously described (Marshall and Fung 2016). Unless otherwise specified, all simulations in this paper run for 100 000 time steps. The chromosomes are confined inside a spherical nuclear envelope by a repulsive force applied to any node moving outside the spherical shell in a direction normal to the surface with a spring constant k_nuc as previously described (Marshall and Fung 2016). Telomeres are constrained to be located on the nuclear surface by calculating the radial distance from each telomeric node and then translating the telomere along the radius by this distance, so that after each step the telomere is always returned to a position on the nuclear surface (Marshall and Fung 2016).

Pairing is modeled on an individual node basis, such that node 1 on a chromosome is only allowed to pair with node 1 on the homolog, node 2 with node 2, and so on. The simulations in this paper assume that all nodes are able to undergo pairing, although this is an adjustable parameter. As the homologous chains undergo their random movement, whenever corresponding loci move within a defined capture radius of each other, and if the local orientation of the two chains is closer to parallel than to antiparallel as judged by the dot product of a local orientation vector, they are set to a paired state and thereafter forced to move together until such time as they become unpaired. Unpairing of paired loci occurs with a uniform probability given by the unpairing rate constant k_unpair per iteration. The rate constant gives the probability of unpairing at any given timepoint because the duration of the time step is one time unit. Affinity between a given pair of loci is reflected by the value of k_unpair, such that tightly associating loci (representing correct homologous associations) will have a low value for k_unpair, while incorrect associations would have a higher value for k_unpair. Thus, in our model, discrimination between correct and incorrect associations is reflected entirely by a difference in off-rates.

Table 1 gives the parameter values used for the simulations presented here, following order of magnitude estimates previously described (Marshall and Fung 2016). Our model is not intended to precisely represent the actual chromosomes of any particular species, but rather is designed to be an abstract model that captures essential features of meiotic chromosome pairing. For this reason, our primary concern in choosing parameters is only to ensure that they are of the correct relative orders of magnitude.

Briefly, we employ a length scale in which 1 unit of length in the simulation corresponds with 100 nm of length in an actual cell. The equilibrium spring length for the links between nodes was chosen to be 1 unit (100 nm), consistent with the reported persistence length of yeast chromatin (Dekker 2008). With this length scale, a nuclear radius of 5 units corresponds to a nuclear diameter of 1 micron, as is appropriate for yeast nuclei. Assuming that each link of 1 unit (100 nm) corresponds to 10 kb, based on prior estimates setting the packing density of chromatin at 110–150 bp nm⁻¹ (Bystricky et al 2004), a 50 node chain would correspond to a fully stretched chromosome length of 500 kb, which is in the range of actual yeast chromosome size. Capture distance was chosen based on the picture of a chromosome chain as a series of random polymer globules, and assuming that the diameter of each such globule equals the statistical link length. Based on the physical model of a polymer as a chain of random globules each corresponding to a statistical segment with diameter equal to the link length, this choice of capture distance corresponds to pairing taking place when the distance between the centers of the random globules of two homologous loci are separated by their radii, thus indicating a substantial degree of overlap between the two globules.

We set a timescale of 100 ms per simulation time step. With this assumption, our simulation of 100 000 time steps corresponds to a pairing time on the order of 5 h, consistent with observations in yeast (note here we refer specifically to pairing and not to subsequent events of meiosis such as synapsis or crossing-over). Based on measurements of the diffusion constant (500 nm² s⁻¹) for yeast chromatin (Marshall et al 1997) using the relation δ  =  sqrt(6ΔtD), we obtain a root mean-squared (rms) displacement during a single time step of δ  =  17 nm. This displacement is produced by thermal energy of a magnitude k_BT, which we take as approximately 4 pN nm inside a yeast cell grown at 30 C. Hence the rms force σ acting to drive this displacement is k_BT/δ or 0.24 pN.

To derive the frictional coefficient ξ, we note that a node of the chain moves by δ  =  17 nm in a single timestep corresponding to 0.1 s, creating an apparent velocity of 170 nm s⁻¹. From the relation ξ  =  σ/v we obtain a value for ξ of 1.4  ×  10⁻³ pNs nm⁻¹ which is equivalent to 1.4  ×  10⁻⁶ Kg s⁻¹. To see if this is reasonable, we can calculate the viscosity corresponding to this frictional coefficient assuming Stoke's law for a perfect sphere, such that η  =  ξ/6πR. Assuming the de Gennes model of spherical globules whose diameter matches the link length, we take R  =  50 nm, which yields a value for viscosity of 1.4 Pa s. This is approximately 1000 times the viscosity of pure water. The viscosity for the fluid phase of nucleoplasm, experienced by protein-sized objects, has been measured to be at least 10  ×  that of water (Erdel 2015, Speil 2010), but for chromosome sized objects diffusing through a dense polymer melt of other chromatin, we would certainly expect the viscosity to be several orders of magnitude higher.

Our simulations employ a spring constant for the link between nodes of magnitude 0.25 force units per distance unit which, given that one distance unit is equivalent to 100 nm and 0.15 force units (σ) are equivalent to 0.24 pN, gives a spring constant of 0.004 pN nm⁻¹. To see if this is reasonable, we calculate the Young's modulus assuming that each link is a cylinder of length 100 nm and diameter 100 nm. With this geometry, a spring constant of 0.004 pN nm⁻¹ for extension of the cylinder by an external force acting on the ends corresponds to a Young's modulus of 60 Pa. This is somewhat less than the Young's modulus measured experimentally for interphase chromatin, for which a Young modulus of approximately 200 Pa has been reported (de Vries et al 2007). However it is of the same order of magnitude and in any case the actual Young's modulus for meiotic chromosomes appears not to have been measured.

Our simulations used a range of unpairing rate constants (k_unpair) of 0.001–0.1. The frequency of unpairing in units of s⁻¹ can be obtained from the unpairing rate constant by noting that k_unpair is the frequency of unpairing events per timestep and then multiplying by a conversion factor of 10 timesteps per second (based on the assumption of 0.1 s per timestep discussed above). This gives us a range of unpairing frequencies of 0.01–1 s⁻¹. In vitro kinetic measurements have found that the dissociation rate for RecA mediated DNA pairing is on the order of 0.1 s⁻¹ (Bazemore et al 1997), consistent with the range of unpairing rate constant values we have employed.

We summarize the following conversion factors that relate parameters of the model to real units. One time step in the simulation corresponds to 0.1 s in actual time. One distance unit in the simulation corresponds to 100 nm in the physical cell. One force unit in the simulation corresponds to 1.6 pN.

Although parameter values were chosen based on order of magnitude estimates from yeast, it should be noted that the simulations reported here are not intended to accurately represent molecular details of an actual biological system, but rather to test the concept of telomere force-driven discrimination of pairing using a highly simplified model.

To simulate competition between homologous and homeologous regions, we initialize the simulation with a total of four chromosomes, numbered 1–4. Chromosomes 1 and 2 are homologous to each other and homeologous to chromosomes 3 and 4. Chromosomes 3 and 4 are homologous to each other and homeologous to chromosomes 1 and 2. We will refer to pairing of a homologous locus with its corresponding locus on the homologous chromosome as correct pairing. If a locus pairs with the corresponding locus on either of the two homeologs, this is referred to as incorrect pairing. Within the simulation, unpaired nodes at corresponding positions on homologous or homeologous chromosomes are switched to the paired state whenever they approach each other within a pre-set capture radius. The probability of pairing is assumed to be the same regardless of whether the pairing is correct or incorrect. At each time point, paired loci have the possibility to switch to an unpaired state, with a probability that is different for correct versus incorrect pairings. The probability of switching from the paired to unpaired states in any given iteration are specified by the variables k_unpair_correct and k_unpair_incorrect. Unless specified otherwise, k_unpair_correct is always 0.001.

Simulations for figures 4–6 started from a half-paired (amphitene) state in order to avoid having to simulate the initial pairing process. A chain of length 50 was generated, after which a second copy of the same chain was added to the system in which all nodes are in the same position in space as the corresponding nodes in the original chain. The first 25 nodes were then initialized to the paired state, and the second 25 nodes initialized to the unpaired state. The Brownian dynamics simulation was run for 2000 iterations with the probability of pairing and unpairing both set to zero, allowing the unpaired segments to relax subject to nuclear volume constraint. At this point, the probability of pairing was set to 1 for loci within the capture distance, and the telomere force was activated. In contrast to the pairing simulations, for this analysis the telomere force was only applied to the last node in each chain, and the forces were directed in opposite directions that are constant over time. This non-fluctuating force was used in order to directly analyze the ability of zippering to do work against an applied force.

Unless otherwise stated, these tug-of-war simulations all employed a nuclear radius of 50, which is roughly five times higher than our estimated nuclear radius for yeast, but which we use to ensure that nuclear radius did not limit the extent to which the telomeres could be pulled apart. This point is discussed in the text and analyzed in figure 6. Also, we note that for these simulations, the telomere force is always exerted constantly in one direction, and not with a randomly fluctuating orientation. The persistence of this applied force means that the magnitude of the force is not directly comparable with that used in previous simulations of pairing selectivity described above.

For each value of k_unpair, an initial range of values for the telomere force magnitude ϕ were tested, with results assessed in terms of how many nodes were paired at the end of the simulation, until values were found that bracketed the stall force, as judged by the fact that one value allows zippering and one causes unzippering. Once these bracketing values were found, binary search was implemented by testing a value of ϕ halfway between the current bracketing values, and retaining whichever pair of values continued to bracket the stall force. The process was concluded when a value was found in which the average number of paired nodes was in the range 24–26. Because the simulations began with half of the 50 nodes paired, final outcomes with half the nodes paired reflect a situation in which the zippered region neither elongated nor shortened, in other words, a stall of the zippering process.

In our previous simulation of meiotic homolog pairing, we found that even substantially high probability of unpairing still allowed processive zippering (Marshall and Fung 2016). Avidity effects create a situation in which even weak affinity is enough to get strong pairing, since many sites all act together. This result raised the possibility that weak or inexact homologous regions might be able to effectively compete with regions of perfect homology. We therefore augmented our simulation framework to represent a situation in which the cell contains four chromosomes, which we will denote a, b, c, and d (figure 1(A)). In the figure, chromosomes a and b are homologs of each other, whereas c and d are homologs of each other and homeologs of a and b. Each chromosome is modeled as a Rouse polymer represented by a bead-spring chain (figure 1(B)) of 50 loci, such that corresponding loci on homologous chromosomes will pair if they come within a specified capture radius (figure 1(C)), and will also unpair with a spontaneous unpairing rate constant k_unpair_correct, as in our previously published model. To simulate competition, as would occur with homeologous chromosomes, we also allow corresponding loci on a to pair with corresponding loci on c or d (the same is also true for chromosome b pairing with c or d) and vice versa. But in this case, if the pairing is between loci on a or b with loci on c or d, dissociation occurs with a different rate constant k_unpair_incorrect, which we assume is higher than k_unpair_correct. It is important to note from the outset that each locus has one correct locus that it can pair with, but two incorrect loci. With these assumptions in place, chromosome movement and pairing is simulated as described in materials and methods (figure 1(D)).

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The competition scenario outlined in figure 1(A) was simulated under a range of unpairing rate constants, with a typical result depicted in figure 2(A). We depict the results by drawing each chromosome as a color bar, and then showing a second color bar alongside it that indicates which of the other three chromosomes it is paired with. Both simulations were run using a ten fold difference in unpairing rate between the correct and incorrect homologs. If all loci paired and unpaired independently, one would expect that roughly one sixth of the loci would be incorrectly paired, given that for each correct pairing partner there are two incorrect pairing partners. Clearly, however, large stretches of incorrect pairing can occur. To gain a more systematic view of the relation between selectivity at the level of unpairing rates versus at the level of overall pairing, simulations of pairing competition were run with a range of different unpairing rates for the incorrect associations. As plotted in figure 2(B), the result is that even with a more than ten-fold difference in affinity, there is predicted to be substantial degrees of mispairing on average.

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The results of figure 2(B) indicate that Brownian motion (in the form of thermally driven chromosome movement) is insufficient to overcome the avidity effect even for weak interactions. We can understand this effect by considering what happens when a pair of loci become unpaired. If they remain within the capture radius of each other during one time-step, they will immediately re-pair. Thus, unpairing can only happen if the just-unpaired loci diffuse apart from each other by a distance of more than the capture radius.

Perhaps active forces might be used to pull apart weakly bound regions to improve fidelity. In the framework just discussed, the extra force applied by the cytoskeleton would be able to drive recently unpaired loci far enough apart that they cannot immediately re-bind. On the other hand, it is not immediately obvious that application of such forces only to the telomeres would have an effect on the rest of the chromosome. To test whether active forces at the telomeres could enhance fidelity, we repeated the simulations of competition between chromosomes with high and low affinity in the presence of active randomly-directed forces applied to the telomeres. In these simulations, the magnitude of the active force ϕ was set to be 3.3 times the magnitude of the thermal random force, and switched direction with every timestep. The result of this simulation is shown in figure 3(A). For comparison, a line is also included which indicates the predicted discrimination if each locus paired and unpaired independently of the others in the absence of applied forces. When the ratio of unpairing rates is sufficiently close to 1, discrimination is similar in the presence of absence of the active random force. However, as the rate of unpairing for the incorrect loci increases, the discrimination becomes markedly better in the presence of active forces. Indeed for sufficiently large unpairing rate constant ratios, the active forces increase discrimination by two orders of magnitude. In vitro measurements of unpairing rate constants for RecA mediated DNA pairing between perfect homologs and homologs with mismatches showed that there was little discrimination for less than 10% mismatches but after that, increasing mismatches led to a difference in affinity of roughly five-fold (Bazemore et al 1997). Such a five-fold difference in unpairing rate constants is where we begin to observe an effect caused by active forces in figure 3(A). We emphasize, as noted already, that the molecular mechanisms that mediate meiotic homology pairing remain controversial, hence RecA mediated DNA pairing may or may not be a relevant comparison, but at least it suggests that the range of differences in affinity that our model employs are at least biologically plausible.

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We next asked how the increase in discrimination depends on the magnitude of the telomere random force (figure 3(B)). Simulations were performed in which the unpairing probability was ten fold higher for incorrect versus correct loci, and the magnitude of the telomere random force ϕ was swept through a range of values. When the magnitude of the telomere force was equal to the magnitude of the thermal random force, discrimination was the same as seen in the absence of any active force, as expected. Discrimination did not increase with small increases in the random force magnitude, but once the magnitude of the random telomere force exceeded the thermal force by a factor of two or more, discrimination improved as a monotonically increasing function of the magnitude of the force. Taken together, these analyses support the idea that random forces can, at least in principle, increase fidelity of meiotic pairing.

How does a mechanical property like the magnitude of a random force have an effect on a molecular property like discrimination of correct from incorrect loci? The processivity of zippering is suggestive of a mechanical system capable of force generation. Specifically, as the two homologous chromosomes zip together, the unpaired telomeres will be pulled closer. This large-scale mechanical behavior, which has the potential to perform work if the telomeres are pulled against an opposing force, is ultimately driven by the molecular-level interactions between homologous nodes in the chain. Zippering thus potentially creates a relationship between molecular recognition and chromosome mechanics.

We characterized the ability of zippering to generate force by simulating a condition in which the chromosomes start out fully paired along half their length, and then simulating pairing and unpairing in the presence of continuous force applied at the telomeres at one end of each chromosome (figure 4(A)). In these simulations, forces are applied in a uniform direction at each telomere, so that the telomeres would tend to be pulled apart. There are two possible outcomes—either the force will drive unzippering of the region as the chromosome ends are pulled apart, or the zippering will pull the chromosome ends together despite the applied force.

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Outcomes of these simulation results can be visualized in the form of pairing kymographs (Marshall and Fung 2016), in which different colors are used to represent the distance between homologous loci, with dark blue representing zero distance, and red representing the average distance between completely unpaired loci (figure 4(B)). Shades of yellow indicate regions at intermediate distances. To generate a kymograph, the inter-homolog distance map at any point in time is plotted as a vertical color bar, with the top and bottom of the bar representing the two ends of the chromosomes. These color bars are then stacked left to right in temporal order. Pairing at any given locus is indicated by a switch from red to blue as one traverses the plot from left to right, while unpairing is represented as a switch from blue to red. Processive zippering is indicated by upwardly diagonal edges, while unzippering is indicated by downwardly directed diagonal edges. Each of these pairing kymographs illustrates a simulation of 8000 timepoints.

To perform the simulations, the two homologs were initially paired along their entire length, and then the first 25 nodes are set to the unpaired state while the second 25 nodes are locked in the paired state (k_unpair  =  0). The simulation was run for 2000 time points (indicated by black bars under each kymograph), giving the telomeres time to separate under the influence of the active forces. At t  =  2000, the unpairing rate for the second 25 nodes was set to a value of k_unpair  =  0.01. The simulation was then run for an additional 6000 time points. Representative results are illustrated in figure 4(B) for three different values of the telomere force magnitude ϕ. Because we did not allow the telomere forces to vary in direction, but rather applied a constant force at each telomere in opposing directions, the values of ϕ in these simulations should not be directly compared to those in the prior simulations, in which the direction of the forces fluctuated randomly between time points. The purpose of this simulation was not to simulate actual events during meiosis, but to test whether avidity-driven zippering could mechanically oppose a constant force.

As illustrated by the examples in figure 4(B), when the telomere force is weak, zippering still occurs. When the telomere force is sufficiently strong, the chromosomes become completely unzipped. At an intermediate force, a stall condition is reached in which neither zippering nor unzippering occur. Under these stall conditions, the degree of pairing fluctuates such that several nodes in a row may pair but then unpair, such that overall there is no processive change in pairing.

We can understand the results of figure 4 by imagining a tug of war between constant active force applied at the telomeres by the cytoskeleton, and an opposing force generated by processive zippering, which can be viewed as a Brownian ratchet (figure 5(A)). To see how this ratchet might work, we note that each step of zippering requires the two unpaired nodes following the last paired nodes to pair. Once this happens, the paired region extends by one node, and the unpaired telomeres will be drawn slightly closer together. Pairing requires unpaired loci to move within a specified capture radius. In the Brownian ratchet model, once the unpaired segments are maximally stretched by the applied telomere forces, pairing of the next set of unpaired nodes happens when the end-to-end length of one or both of the unpaired segments undergoes a thermally-driven positive length fluctuation such that the distance Δx between the two unpaired nodes becomes less than the capture distance. The unpaired segments can be viewed as springs in such a picture. The spring constant and equilibrium end-to-end length for a freely jointed chain are both functions of the chain length, such that as the chain length increases, the spring constant decreases as the inverse of the chain length (Grosberg and Khokhlov 1994). This spring-like behavior of the chains means that as the chain is stretched more and more by the telomere forces, the probability of thermal forces driving enough length fluctuation to pair the next set of nodes decreases, eventually making further pairing impossible.

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In this tug of war scheme, the probability of zippering versus unzippering depends on the telomere force and the unpairing rate constant. A forward zippering step will take place if the first-passage time for the end-to-end lengths increasing is less than the dwell time between successive unpairing events. The smaller k_unpair is, the more chance there is that the unpaired segments will undergo a large enough end-to-end extension to allow capture of the next unpaired nodes. In contrast, if k_unpair is large, then the paired nodes at the end of the paired regions are expected to unpair before the next set of nodes are able to fluctuate within one capture radius of each other. Because the first passage time for the spring endpoints moving within a capture radius depends on the telomere force, we expect that for any given k_unpair, there should be a corresponding value of the telomere force ϕ at which the probability of a forward step equals the probability of a backward step. This represents a stall force for the zippering Brownian ratchet. We explored the stall force of this system by carrying out simulations at a range of k_unpair (figure 5(B)). The stall force is seen to be a decreasing function of k_unpair. This result indicates that chromosomal regions that differ in their pairing affinity will have different stall forces. Given two different values of k_unpair for correct versus incorrect associations, it is thus possible to find a range of telomere forces which are able to stall or even reverse the zippering of weakly paired regions, while still permitting zippering of strongly pairing regions.

The simulations of figures 4 and 5 were conducted using parameters in which the diameter of the nucleus is large relative to the relaxed length of the chromosomes, allowing the paired chromosomes to be fully pulled apart. The logic of choosing an initially large radius for the nucleus was that as the telomere force increases, the unpaired portion of the chromosome will stretch out more and more (figure 6(A)), but cannot stretch to a length greater than the radius of the nucleus. If the nucleus is too small, then even large telomere forces would not be able to complete unzippering, simply because there would not be enough space for the unpaired regions of the chromosomes to stretch out (figure 6(B)).

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We therefore explored the effect of nuclear radius on the tug of war simulations. We observed that the extent of unpairing achievable with strong telomere forces was not dependent on nuclear radius until the radius dropped below 25. Beyond this point, the number of unpaired nodes dropped proportionally with the radius (figure 6(C)). This is the expected result given that the link length of the chromosome chain is 1 unit per link, such that the length of the initially unpaired segments would, if fully stretched, be 25 for each homolog. At maximal stretch, the two unpaired segments would thus fully span a nucleus of radius 25, and it would not be physically possible to pull them any further apart. For smaller radii, the unpaired segments could not fully stretch out unless zippering proceeded, resulting in a smaller number of unpaired nodes at the end of the simulation.

The processivity of homolog pairing speeds up pairing, but here we have shown that it also creates a problem for fidelity of the process, in that avidity effects allow even weakly paired loci to zip up, leading to potentially incorrect associations. Figure 2 demonstrates that this effect causes the discrimination of correct from incorrect homologs to be less than expected if the individual loci were allowed to pair and unpair independently.

Our simulations (figure 3(A)) indicate that random active forces, even when they are only applied at the telomeres, can enhance the discrimination in meiotic homology pairing. Interestingly, we found that active telomere forces were able to increase discrimination above the level expected by independent pairing/unpairing of loci. We propose that this increased discrimination can be explained by a tug of war model, in which zippering is mechanically opposed by the tendency for telomeres to be driven apart from each other. The stall force of zippering is expected to depend on the off-rate of pairing, such that weakly paired regions would have a lower stall force. If the average force pulling the telomeres apart is greater than the stall force for incorrectly paired regions, but smaller than the stall force for correctly paired regions, the effect of the force would be to unzip any incorrect associations, while still allowing correct associations to proceed. In such a case, perfect discrimination should be possible.

The role of meiotic chromosome motion in unpairing was recently probed with chemical inhibitors to transiently arrest motion in fission yeast, followed by washout to restore motion, monitoring pairing during the process (Chacon et al 2016). In that study, it was found that loci normally experienced fluctuating pairing states, such that a given locus would pair and unpair many times during the course of prophase. However, loci that happened to be paired when the motion was stopped, failed to unpair once the motion was resumed. It was inferred that the prolonged pairing created a pairing-locked state that could not be readily unpaired. Based on our simulation results, we suggest an alternative interpretation. Transient arrest of motion would allow paired loci to initiate zippering, which would, if allowed to progress long enough, lock the sites together such that even when motion resumed, they could not be unpaired. This would explain the lack of unpairing after transient motion arrest. Moreover, our results suggest that processive zippering in the absence of motion would lead to erroneous pairing with heterologous or homeologous chromosomes, which might then produce recombination intermediates that could not be properly resolved.

We note that active forces are likely to play multiple roles, with active unzippering being just one of them. In organisms like C. elegans, in which pairing is determined by highly specific pairing centers whose interactions are mediated by specific proteins, the initial interactions presumably have sufficient fidelity that extensive zippering of incorrect regions is unlikely to be an issue. For species with abundant repeat sequences widely distributed in the genome, incorrect zippering would be a serious problem. We therefore suspect that different species will vary substantially in the degree to which active motion contributes to pairing fidelity. Such considerations lead us to expect that experimental outcomes of abrogating telomere motion may vary widely between species.

In the results above, we first showed that randomly fluctuating forces acting at the telomeres were able to enhance pairing selectivity (figure 3). We then argued that telomeric forces are engaged in a tug of war with chromosome zippering such that forces exist that can unpair low affinity interactions but not high affinity interactions (figures 4 and 5). However, the tug of war simulations employed forces acting in constant, opposite directions, so as to pull the chromosomes apart. This choice was made in order to directly probe the mechanical aspect of unpairing. But how then do the results of the tug of war analysis, in which directionally persistent forces are applied to push the telomeres apart, relate to the analysis of figure 3, in which randomly fluctuating forces were used, whose direction was unbiased with respect to the relative position of the two telomeres? If we consider two objects, each undergoing independent random walks, the distance between them will tend to increase proportionally to the square root of elapsed time. This tendency to constantly increase the distance can be viewed as creating an effective force pulling the objects apart, albeit a force that varies as a function of the distanced between the objects. Given that the two telomeres are constrained to lie on the nuclear surface, there will be an average distance between them after which there will no longer be a consistent tendency for them to pull further apart. But if the telomeres were to be come very close, as in the final stages of zippering, then most random fluctuations would in fact tend to increase the distance between the telomeres, just as would be the case with a persistent directed force.

The simplified model implemented here assumes that homologous loci dissociate with a constant probability that does not depend on the applied force. Because of this assumption, we explain the enhancement of discrimination in terms of telomeric force acting in opposition to zippering by slowing or stalling the continued pairing of unpaired sites, thus giving already-paired sites a chance to dissociate before they are locked in by zippering. However, an alternative hypothesis is that pulling forces exerted between chromosomes might increase the dissociation rate for weakly homologous loci, by providing enough energy to overcome the energetic barrier for dissociation. Such an alternative model would be equivalent to the idea that forces provide sufficient additional energy at the B cell receptor to overcome the energetic barrier for dissociation of lower affinity antigens (Natkanski et al 2013). Single molecule experiments in which DNA strands are pulled apart by optical tweezers (Danilowicz 2003) have shown that the rate constant of dissociation depends not only on the base pair composition and the degree of nucleotide mismatches between the two strands, but also on the applied force (Hatch 2007). In meiosis, we predict that physical pulling applied to the chromosomes should increase the rate of unpairing, with the applied pulling force more likely to pull apart incorrect regions that lack extensive homology and thus cannot anneal as strongly. In this case, active forces would increase meiotic fidelity by catalyzing reversals of transient pairing. Our current simulation, in which the unpairing probability is assumed constant, argues that such selective unpairing is not required, but we cannot rule out that it might also make an additional contribution to fidelity in vivo. It is intuitively obvious that if a force-dependent dissociation is introduced, this will enhance the ability of forces to promote pairing fidelity, thus our central conclusions would still hold.

As shown in figure 6, if the nucleus is not sufficiently large, active unzippering can be impeded simply by virtue of the fact that the telomeres cannot get farther apart than the diameter of the nucleus. In light of this observation, it is worth noting that nuclei tend to be significantly larger in meiotic cells than in mitotic cells of the same species. This enlargement is usually assumed to be required to support high levels of protein synthesis during egg formation. Based on the results of figure 6, however, we hypothesize that the large size of meiotic nuclei may, at least in part, contribute to pairing fidelity by allowing more extensive unzipping. There is, however, another solution to the constraint imposed by nuclear size. If the nuclear envelope is sufficiently stretchable, then it is possible, at least in theory, for active forces to pull telomeres apart by a distance greater than the equilibrium nuclear diameter by deforming the nuclear envelope. This would be manifest as chromosomes that extend beyond the primary mass of chromosomes inside the nucleus. Such 'maverick' chromosomes have been directly visualized in budding yeast meiosis (Scherthan et al 2007). The fact that telomere forces appear to act over such long distances that chromosomes move apart from the rest of the nuclear mass is consistent with our hypothesis that these forces are serving to unzip incorrectly paired regions. In contrast, the existence of maverick chromosomes is less consistent with the idea that the primary role of these forces is to promote pairing by increasing the frequency of chromosome collisions, since the maverick chromosomes are actually pulled away from the other chromosomes. In any case, it is clearly important to develop experimental strategies to explore the interplay of nuclear radius and maverick chromosome behavior on different aspects of meiotic pairing selectivity and kinetics.