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Proc Natl Acad Sci U S A. Nov 7, 2006; 103(45): 16758–16763.
Published online Oct 25, 2006. doi:  10.1073/pnas.0603174103
PMCID: PMC1636528
Biophysics

Modeling dual pathways for the metazoan spindle assembly checkpoint

Abstract

Using computational modeling, we investigate mechanisms of signal transduction. We focus on the spindle assembly checkpoint, where a single unattached kinetochore is able to signal to prevent cell cycle progression. The inhibitory signal switches off rapidly once spindle microtubules have attached to all kinetochores. This requirement tightly constrains the possible mechanisms. Here we investigate two possible mechanisms for spindle checkpoint operation in metazoan cells, both supported by recent experiments. The first involves the free diffusion and sequestration of cell cycle regulators. This mechanism is severely constrained both by experimental fluorescence recovery data and by the large volumes involved in open mitosis in metazoan cells. By using a simple mathematical analysis and computer simulation, we find that this mechanism can generate the inhibition found in experiment but likely requires a two-stage signal amplification cascade. The second mechanism involves spatial gradients of a short-lived inhibitory signal that propagates first by diffusion but then primarily by active transport along spindle microtubules. We propose that both mechanisms may be operative in the metazoan spindle assembly checkpoint, with either able to trigger anaphase onset even without support from the other pathway.

Keywords: kinetochore, mathematical modeling, signal transduction, concentration gradients

The question of how a signal emanating from a small, compact structure in a cell can be amplified and propagated to an entire cell is fundamental to cell biology (1). An excellent example is provided by the spindle assembly checkpoint (SAC) (2), which regulates cell cycle progression from metaphase to anaphase during mitosis. The segregation of sister chromatids that occurs during anaphase is permitted only after all of the kinetochores are attached by microtubules to the mitotic spindle. Even a single unattached kinetochore can signal to the rest of the cell and prevent cell cycle progression (3, 4). A fundamental issue is how a relatively small structure, such as a kinetochore, can generate sufficient signal to robustly communicate with distant subcellular locations (1). Moreover, this signal must switch off rapidly, within a period of minutes, after complete kinetochore attachment to spindle microtubules (4). These requirements strongly constrain the possible signal transduction mechanisms. In this paper, we focus particularly on the SAC in cases where the nuclear envelope breaks down before SAC activity (open mitosis), as in metazoan cells. In this context, we examine two distinct models: a diffusive sequestration model and a model involving active signal transport along spindle microtubules. We believe that both of these pathways may be in simultaneous operation in the metazoan SAC.

The metaphase/anaphase transition is triggered by an intricate sequence of events centered around the proteins securin, cyclin B, and separase. The first step is the ubiquitination of securin and cyclin B by the anaphase-promoting complex/cyclosome (APC/C) (5), a process that tags securin/cyclin B for destruction via the 26S proteasome. This degradation allows separase to cleave the cohesin complex that tethers sister chromatids together. Once the cohesin complex has been cleaved, the sister chromatids are pulled apart to opposite poles by the mitotic spindle. To prevent premature entry into anaphase, the SAC must prevent securin/cyclin B ubiquitination by the APC/C until proper attachment of all chromosomes to the spindle. Evidence has accumulated for a number of overlapping and therefore possibly redundant mechanisms for SAC operation. The APC/C is known to be stimulated by Cdc20 binding; hence, a plausible way to achieve APC/C inhibition is to inhibit the ability of Cdc20 to bind to the APC/C. One possibility is that Cdc20 is held and sequestered in an inactive form via binding to Mad2, with the production of Cdc20–Mad2 being promoted by unattached kinetochores (68). Key proteins identified at unattached kinetochores include Bub1, Mad1, Mad2, BubR1 (Mad3 in budding yeast), Bub3, and Cdc20. Moreover, fluorescence recovery after photobleaching (FRAP) experiments have revealed that some of these proteins, including Mad2, BubR1, and Cdc20, turnover rapidly at unattached kinetochores (6, 9). Furthermore, the available evidence suggests that Mad2 exists in two forms: open (O-Mad2) and closed (C-Mad2) (10, 11), with the closed form adopted when bound to Cdc20. Production of C-Mad2–Cdc20 may be catalyzed by the kinetochore-bound C-Mad2–Mad1 complex. Intriguingly, refs. 7 and 8 propose that C-Mad2–Cdc20 away from the unattached kinetochore can convert further cytosolic O-Mad2 and Cdc20 into their bound C-Mad2–Cdc20 state. In this way, the relatively weak signal coming from an unattached kinetochore can be amplified, leading to comprehensive Cdc20 sequestration throughout the cell. Recent experiments have further implicated a protein called p31comet in switching off this signal after complete kinetochore attachment (7, 12, 13).

However, the above “Mad template” model is not the only proposed mechanism for APC/C repression. BubR1 and Bub3 also are known to bind Cdc20 and, thus, repress the APC/C (14). Indeed, BubR1 appears to be a more potent inhibitor of Cdc20 than Mad2, and both BubR1 and Mad2 may mutually promote each other's binding to Cdc20 (15). Furthermore, Bub1 is believed to phosphorylate Cdc20, possibly also repressing the APC/C (16). In addition, the overall copy number of Cdc20 is down-regulated until all of the kinetochores are attached (17). Clearly, reducing the overall number of Cdc20 will impair the effectiveness of the APC/C before anaphase. Moreover, not only does Cdc20 form complexes with Mad2 and BubR1, but it also is believed to form a separate complex called the mitotic checkpoint complex, consisting of Mad2, BubR1, Bub3 and Cdc20 (18). This complex also appears to be a potent inhibitor of the APC/C (18). In addition, the APC/C is regulated by phosphorylation through the kinase Cdk1 (19). Lastly, the APC/C itself has intriguing localization properties, localizing, for example, to unattached kinetochores (20). This positioning may also have implications for the mechanism of APC/C inhibition.

Recently, a pioneering paper by Doncic et al. (21) introduced a careful mathematical analysis to the question of how the SAC functions. Focusing on yeast, they analyzed various simplified models of how an unattached kinetochore can signal to the rest of the cell. Yeast undergoes closed mitosis, during which, even at the onset of anaphase, the nuclear envelope is still complete. As a consequence, a diffusive signal from an unattached kinetochore is required only to propagate within the nuclear volume of a few cubic micrometers. However, one important difference between yeast and metazoan cells is that the latter undergo open mitosis, during which the nuclear envelope has broken down by the time the SAC is active; hence, any freely diffusing signal from an unattached kinetochore must propagate throughout the cytoplasm, a volume a thousand times larger than the nucleus of a yeast cell. Furthermore a considerable amount of data are available concerning the metazoan SAC in the form of fluorescent bleaching data and estimates for the copy numbers of the relevant molecules. In this paper, we therefore use computational modeling to analyze the SAC in metazoan cells. We develop a model that incorporates two-step signal amplification, a mechanism with close similarities to previously investigated multistep signal cascades (22, 23). This similarity points toward a close connection between the SAC and other cell signaling systems, such as MAPK cascades. Importantly, we find that our model generates robust inhibition in metazoan cells, unlike models without an amplification step.

As discussed above, the most popular models of APC/C inhibition involve transport solely by diffusion. However, these models are hard to reconcile with the experiments of Rieder et al. (24). By using cells with two independent spindles, Rieder et al. (24) found that unattached kinetochores on one spindle did not block anaphase onset in a neighboring complete spindle. This experiment appears to indicate that any “wait anaphase” signal would have to be limited to a single spindle, and not diffuse throughout the cytosol. This finding is in clear contradiction with the mechanisms introduced above. For this reason, we develop a second model in which a short-lived signal from an unattached kinetochore first diffuses to a nearby microtubule before being actively transported along spindle microtubules. In this way the signal is restricted to the spindle. Moreover, we propose that both the active transport and diffusive mechanisms may be in simultaneous use in metazoan cells. We suggest that either of the signaling mechanisms may individually be able to initiate anaphase, even without support from the other pathway. With this assumption, our models are then fully consistent with both the experiments of Rieder et al. (24) and with a diffusive sequestration model (7).

Results

Mechanisms with Diffusive Transport.

We analyze the case for which the kinetochores control the concentration of a freely diffusing species c. When a kinetochore is unattached and therefore is signaling, a large majority of the species c is in the c* state, but when the last kinetochore itself switches off, the c* species rapidly decays to the c state, thus communicating the switch-off (attachment) to the rest of the cell. We will sometimes refer to the c* species as being in the inhibiting state because it is this state that prevents securin/cyclin B ubiquitination by the APC/C.

The concentrations of the freely diffusing species are almost uniform.

The time scale for diffusion across the cell is τD = rc2/D, whereas the mean time for a molecule to collide with the kinetochore is τC = rc3/(Drk). Here, D is the diffusion constant for the protein; rc is the distance across the cell (typically a few tens of micrometers); and rk [similar, equals] 0.2 μm (25) is the radius of the kinetochore. If we assume that the lifetime for the inhibiting c* species is much longer than τD, then any gradient in its concentration will clearly be small. Furthermore, the time scale for diffusion across the cell is much smaller than that for collisions with the kinetochore, τDC = rk/rc [double less-than sign] 1. Hence, each molecule of the c species crisscrosses the cell many times between kinetochore reactions, so gradients in its concentration also are small in most of the cell. If, on the other hand, the lifetime of the inhibiting species is short with respect to τD, then its concentration will no longer be uniform and, instead, density gradients will form. This scenario will arise later on when we consider models with active transport. However, for longer lifetimes, we can model the inhibition by using simple ordinary differential equations.

Inhibitor production only at a kinetochore.

In the simplest possible model, the inhibiting c* species is produced only at unattached kinetochores. We denote the steady-state rate of production of c* at the final unattached kinetochore by Joff. To allow the inhibition to be switched off at the beginning of anaphase, the inhibiting c* species must be unstable. We model this instability by a first-order decay c* → c, with rate constant α. Cdc20 is known to activate the APC/C, thereby triggering anaphase. Because Mad2 and BubR1 are known to bind to Cdc20, we can tentatively identify c as Cdc20, whereas c* is a Mad2–Cdc20 or BubR1–Cdc20 complex. Because the kinetochore is such a small structure, we now address the question of whether the flux of c* produced only at a single unattached kinetochore is sufficient to maintain inhibition. At steady state, the fluxes on and off a kinetochore must be the same, i.e., Jon = Joff. Furthermore, at steady state, the rate of production of c* at the last unattached kinetochore must equal its rate of decay, i.e., Joff = αN*c, where N*c is the number of inhibiting c* molecules. Experimental data, largely on marsupial PtK1 and PtK2 cells, gives estimates for many of the model parameters (6, 26). From this data we will be able to estimate the value of N*c and see whether good inhibition can be obtained.

Experimental evidence for the flux Joff is available from FRAP data for the recovery of fluorescence of checkpoint proteins at unattached kinetochores. These data provide a direct measure of dissociation rates. Together with an estimate of the copy number of each kinetochore-bound protein, we can then derive the off flux Joff. For BubR1 (fast phase, see ref. 6) the FRAP half-life (t1/2) is 3 s, whereas for Mad2 the half-life is 10–20 s (6, 9), with a kinetochore occupancy Nk of ≈1,000 molecules for both (6). These measurements give an off flux of approximately Joff = Nk ln 2/t1/2 ≈ 1,000 × ln 2/3 ≈ 200 s−1 for BubR1 and Joff ≈ 50 s−1 for Mad2. Note that these fluxes are actually the maximum possible, given that not all of the released molecules will be in the inhibiting form. The lifetime of the inhibiting species α−1 has not been directly measured. However, anaphase begins ≈20 min after the last kinetochore becomes attached (4). Clearly, the number of inhibiting c* molecules must decrease dramatically over a period of no more than ≈10 min, giving an upper bound of α−1 ≈ 10 min, a value that is also consistent with the data of ref. 27. If we sum the fluxes of Mad2 and BubR1 and set α−1 = 10 min, the number of molecules Nc* = Joff α−1 [similar, equals] 150, 000. The total number of Cdc20 molecules is ≈800, 000 (6). Thus, with the experimental fluxes and assuming a decay rate as slow as is reasonable we find that, at steady state, a single kinetochore can only sequester ≈20% of the Cdc20. Furthermore, if the lifetime of the inhibiting c* molecules is shorter than supposed by our upper bound α−1 ≈ 10 min or if the flux Joff is lower, then an even smaller fraction will be in the inhibiting state. Hence, the formation of Mad2/Cdc20 and BubR1/Cdc20 complexes solely at an unattached kinetochore cannot maintain good inhibition.

The experimental FRAP data used in the above analysis was not specifically derived from the final unattached kinetochore. Potentially the final unattached kinetochore might have a higher turnover rate, thereby generating greater inhibition. We therefore develop a complementary approach to determine whether a reaction only at a single unattached kinetochore is in principle sufficient to sequester most of the Cdc20. We estimate the inhibition that would be generated if the on flux Jon at the kinetochore were maximized, i.e., diffusion limited with a diffusion constant D at the top of the range of the values measured in vivo. Diffusion constants in the cytoplasm of metazoan cells have typically yielded values in the range 1–30 μm2·s−1 (2831). Also, Howell et al. (6) found that a bleached spot of cytoplasmic GFP-Cdc20 0.8 μm across recovered its fluorescence in <0.2 s, implying a diffusion constant [greater, similar]5 μm2·s−1. If the on-flux to the unattached kinetochore is diffusion limited, then using a plate geometry appropriate for a kinetochore, the on-flux at steady state is given by Jon ≈ 4DrkNc/vc (32). Here, Nc is the number of c molecules, vc = 6,000 μm3 is the cytoplasmic volume (6), and we use a fast diffusion constant D = 30 μm2·s−1. At steady state the flux Jon must be balanced by the c* → c flux, Jc*→c = αNc*. Equating these two fluxes and defining the constant total number of molecules as N = Nc + Nc* (neglecting a negligible number at the kinetochore), we have Nc*/N = (1 + αvc/4Drk)−1 ≈ 3/4. Hence, even in an optimistic scenario, only approximately three-quarters of the Cdc20 can be sequestered. Furthermore, to reach this limit, high flux rates are required, much higher than found experimentally for BubR1/Mad2 (see above). For these reasons, it is unlikely (although still theoretically possible) that the metazoan SAC could function through this method.

The fundamental difficulty with this mechanism can be presented in an alternative way that brings out the essential role played by the cell volume. The time each molecule spends in the c* form is on the order of α−1. The time taken for each molecule in the c state to find the unattached kinetochore is approximately vc/kk, where kk is the unattached kinetochore binding rate. This time scale is proportional to the cell volume and, hence, becomes longer as the cell size increases. As the volume increases, for fixed lifetime α−1, each molecule spends less and less time in the inhibiting state and eventually the mechanism fails.

Autocatalysis.

One way to try to increase the number of inhibiting molecules would be for the reaction that sequesters Cdc20 to occur not just on the kinetochore but also off the kinetochore. If c* itself catalyzes an off kinetochore cc* reaction, then the reaction is autocatalytic. Autocatalysis was considered by Doncic et al. (21), who found it to be unsatisfactory for the case of closed mitosis in yeast. The reason is that, for good inhibition, the off-kinetochore autocatalysis will likely have a large reaction rate, because the on-kinetochore reaction is on its own insufficient to maintain good inhibition (see above). Consequently, when the kinetochore reaction is switched off, the result is only a weak perturbation of an autocatalytic reaction. In other words, inhibition of a “go anaphase” signal either cannot be switched off or switches off only very slowly (see Fig. 1). Interestingly, the Mad2 template model proposed in ref. 7 is essentially identical to this mechanism, where the closed form of Mad2 catalyzes the conversion of O-Mad2 to C-Mad2 both on and off the kinetochore. Thus an unattached kinetochore inhibits anaphase by generating C-Mad2–Cdc20, which in turn generates more C-Mad2–Cdc20 off the kinetochore via an autocatalytic reaction. However, by itself this autocatalytic mechanism does not allow switch-off of the metazoan SAC for the same reason it does not for the SAC in yeast.

Fig. 1.
Plot, for the model of Eq. 1 showing the fraction of molecules Ne/N (solid line), Nc*/N (dashed line), and Ne*/N (dotted line) as a function of time, with the final kinetochore attaching at time t = 0, i.e., initial steady-state concentrations, but k ...

Autocatalysis could play a role in the inhibition, provided an additional process switches off the autocatalytic reaction once the last kinetochore attaches to a microtubule. It has been suggested that the protein p31comet could play a role in this context (7, 12, 13). Potentially p31comet could be up-regulated to abruptly switch off the SAC. However, the question is then how this signal could be turned on so rapidly after microtubule attachment to the final kinetochore. If p31comet competes with O-Mad2 for C-Mad2 binding (13), then this is no easier than regulating the concentration of free Cdc20, our original problem. Potentially, p31comet could be up-regulated by using a two-step reaction process; however, such a scheme would have to be conceptually similar to that discussed in Model with an amplification step. We emphasize that p31comet also could play other important roles, for example, in switching off kinetochore signaling after microtubule attachment (33).

Model with an amplification step.

We now turn to a model with an off-kinetochore, but nonautocatalytic, reaction. This involves the species e, e*, and c*. We first assume that an e species cycles through the unattached kinetochore, where it is converted to the inhibiting e* form. We further assume that e* is able to convert e to a second inhibiting c* species, giving a second step to the signal amplification. Importantly the c* form cannot convert further e into the c* or e* forms. This form of amplification ensures that only molecules that have passed through the unattached kinetochore participate in amplifying the inhibitory signal. Because the inhibitory signal is not produced autocatalytically, it can switch off rapidly after the final kinetochore–microtubule attachment. The reaction processes are

equation image

where K is an unattached kinetochore. It is natural to associate the e species with free Cdc20, and the e* and c* species with complexes of Cdc20, such as C-Mad2–Cdc20 and BubR1–Cdc20. We are assuming that one form of the complex (e*) can catalyze the production of further complexes (c*), which differ in that c* cannot participate in manufacturing further complexes. This distinction between the two forms is a clear prediction of our modeling. However, we cannot definitively identify the difference between the c* and e* species, which could, for example, involve phosphorylation or a conformational change. Clearly, our reaction scheme is schematic; more complex schemes based on the same principles are certainly feasible. One possibility is that there could be more than two steps to the amplification process. Alternatively, p31comet could be rapidly up-regulated by using a two-step reaction, as discussed previously. Another possibility is using a two-step reaction to modulate the decay rate α. Both possibilities could, in principle, lead to both good inhibition and rapid checkpoint switch-off but are fundamentally similar to the scheme of Eq. 1. The key prediction of the model of Eq. 1 is for at least two species allowing nonautocatalytic amplification off the kinetochore and hence strong sequestration.

The above two-step process has close similarities to other multistep signaling cascades, such as those for MAPK (22, 23). However, there are differences. For example, the e species participates in both steps of the above amplification process. In MAPK cascades, on the other hand, a separate c species is converted to the c* form in the second amplification reaction. Nevertheless, the principle of using more than one step to provide robust amplification but with rapid response times is similar and is likely conserved across many different signaling systems. However, the difficulty of robust signaling in the SAC is particularly acute, because the initial signal emerges from such a small region (a single unattached kinetochore).

For the parameter values of the reactions in Eq. 1, we use α−1 = αe−1 = 5 min. These lifetimes are shorter than used previously: The two step nature of the reaction mechanism now dictates that shorter lifetimes are needed for switch-off within the appropriate time frame of <20 min. Despite these short lifetimes, a two-step reaction cascade ensures that robust signal amplification is still achieved. In dilute solution in vitro, rate constants for diffusion-limited protein–protein association are ≈10−3 to 10−2 μm3·s−1 (34), with the exception of some large rate constants, for which there is significant electrostatic attraction between the proteins. We take an in vivo rate constant at the top end of these values, with k = 10−2 μm3·s−1. For the rate constant kk, we use the diffusion limited value kk ≈ 4Dr (32). Assuming D = 20 μm2·s−1, i.e., fairly fast cytoplasmic diffusion, we find kk ≈ 20 μm3·s−1.

Because the gradients in the concentrations are small (see above), we can determine the time-dependence and steady-state values of the three species from ordinary differential equations:

equation image

equation image

where Nx is the number of molecules of species x = e, e*, and c*. At steady state (ss), we have

equation image

Defining the (constant) total number of molecules as N = Ne + Ne* + Nc*, we find

equation image

For N = 800,000, and using the above parameters, we find Ness ≈ 40,000, Ne*ss ≈ 40,000, and Nc*ss ≈ 720,000. Hence, ≈95% of the molecules are in the inhibiting state. For the single unattached kinetochore, we find that the diffusion limited on rate onto the kinetochore is approximately Jon ≈ 4DrNess/vc ≈ 100 s−1. Assuming a kinetochore population of ≈1,000, as found experimentally, gives a half-life for the kinetochore bound population of 5–10 s, which is roughly consistent with the observed Mad2 and BubR1 kinetochore half-lives (6, 9). As shown in Fig. 1, we also find that the signal switches off quite quickly after microtubule attachment to the final kinetochore. After 10 min, the fraction of the e molecules has increased from 5% to 24%, and, after 20 min, 60% is in the e form. Moreover the switch-on of the checkpoint is even quicker, with good inhibition being reestablished within 1 min of even a single kinetochore detachment (see Fig. 1 Inset). These results are in good agreement with cyclin B1 data from ref. 27, which suggests that switch-on is essentially an order of magnitude faster than SAC switch-off.

We therefore conclude that the above model is compatible with experiments. However, we did use a diffusion-limited value for kk and a value of k near the top end of the range of reaction-rate constants for typical proteins in a dilute solution (34). Reducing k by an order of magnitude weakens the level of inhibition, although >85% of the molecules are still in the inhibiting state. However, the model is not consistent with experiments if the reaction rates are further reduced or if diffusion is substantially slowed. Thus, we predict that if the mechanism of the SAC is diffusive-amplified sequestration, then measurements of the reaction rates will reveal rather fast reactions or possibly more than two steps to the amplification process.

For completeness we also analyze an alternative two-species model previously proposed by Doncic et al. (21) for the smaller volumes involved in the yeast SAC. The reaction scheme for their model is

equation image

Note that this scheme does not catalytically amplify the inhibitory signal. Here, one e* molecule can interact with only one c molecule, whereas in our previous model a single e* molecule can convert many molecules into the inhibiting form, thereby producing amplification. For good inhibition we require Ne*ss + Nc*ss ≈ 800,000. By using α−1 = αe−1 = 5 min, we find that the flux off the unattached kinetochore must be well over 1,000 s−1, faster than the diffusion limited maximum, even for high levels (105 copies) of the e species. This finding is, of course, not unexpected. The lack of amplification means that the flux of inhibitory molecules off the unattached kinetochore must be higher than in the catalytic model proposed above. We therefore believe that this model is probably not able to account for the SAC in metazoan cells.

Mechanisms Involving Active Transport.

As shown above, our model with an amplification step is able to explain many features of the metazoan SAC. However, it is not consistent with the experiments of Rieder et al. (24). They observed that an incomplete spindle did not inhibit another complete spindle 20 μm away. Furthermore, an unattached kinetochore was found to inhibit anaphase onset everywhere within its local spindle, even those parts >20 μm away. These findings are clearly incompatible with models for which inhibition propagates purely diffusively away from incomplete kinetochores into the cytoplasm.

These observations motivate us to consider mechanisms in which the spindle itself plays an active role. If the signal is propagated within the spindle itself and not spread throughout the cytoplasm, then the obvious transport mechanism is via molecular motors. Propagation via active transport along microtubules is fast; motors can move at speeds of micrometers per second (35). Because the typical spindle length scale is ≈10 μm, transport across the spindle takes only seconds at that speed, consistent with rapid checkpoint switch-on/off. Of course, by definition, an unattached kinetochore is not connected by microtubules to a spindle pole. Hence, that kinetochore must first produce a freely diffusing species to carry the signal as before. This inhibitory species, which we denote by g*, initially diffuses through the cytoplasm but only until it either encounters a minus-end directed microtubule-bound molecular motor or decays to an inactive form, g. If the molecule encounters a microtubule-bound motor, this binding then stabilizes the active g* form and transports the inhibitory signal to a spindle pole.

However, before we can conclude that this model with active transport is consistent with the experimental data of Rieder et al. (24), we need to demonstrate that it is possible to find a lifetime for g* that is long enough to allow it to encounter a motor but short enough to prevent more than a small fraction diffusing 20 μm away. If g* is manufactured at a rate Jg* at unattached kinetochores and decays at a rate αg*, then the concentration cg*(r, t) satisfies the partial differential equation

equation image

We solve Eq. 7 at steady state after assuming spherical symmetry around the source (the kinetochore) and negligible concentration of g* at large distances from the source. The solution is

equation image

where λ = D/αg* and r is the distance from the kinetochore. Of course, the assumption of spherical symmetry is a gross approximation, especially because the kinetochore is a plate-shaped structure. However, we are only interested in qualitative results for which this approximation will be reasonable at large distances. If we set the cytoplasmic g* lifetime to be αg*−1 = 0.5 s, then even with a large diffusion constant of D = 20 μm2·s−1, λ ≈ 3 μm. Hence, the signal 20 μm away will be greatly attenuated. Because of the short lifetime of the g* form, the inhibitor forms a steep gradient inside the cell. Hence, we predict that subcellular concentration gradients, already believed to be important for microtubule growth and kinetochore capture, also play an important role in checkpoint function (36).

When we considered diffusive sequestration, we found kinetochore fluxes on the order of 100 molecules per second. If we assume a similar flux Jg* = 100 per second, then the concentration of g* 1 μm away from the source is ≈20 μm−3. At this concentration the reaction rate per motor is 20k, where k is the reaction rate between a pair of proteins. If we again take k = 10−2 μm3·s−1 (34), then we have a rate per motor complex of 0.2 s−1. Thus a motor will pick up a g* molecule within a few seconds if it is close to an unattached kinetochore. If we assume a motor density of 10 μm−1 moving at 1 μms−1, then we expect a rate on to a spindle pole of perhaps 5 s−1 per microtubule. Metazoan cells will have large numbers of spindle microtubules in the vicinity of an unattached kinetochore, increasing the on-rate still further. Even if some of the signal is lost in transit to the spindle pole, the flux is adequate to communicate the state of the kinetochore. Note that the localization of the inhibitory signal means that less amplification is needed: The flux off the kinetochore together with directed transport are by themselves sufficient to produce good inhibition. The next question is how the pole processes this information. One attractive possibility is that the g* molecules inhibit the spindle pole and are subsequently released back into the cytoplasm in the inactive g form. When the last kinetochore attaches, the active transport “wait anaphase” signal is switched off, releasing the inhibition and allowing the spindle pole to broadcast a “go anaphase” signal. This signal could be actively transported by plus-end-directed motors to communicate with connected kinetochores on the same spindle. However, in the experiments of ref. (24), once one spindle had entered anaphase, the other spindle also progressed to anaphase regardless of whether it contained unattached kinetochores. This finding suggests that a final “go anaphase” signal is transmitted via diffusion.

For active transport models there are few relevant experimental data. As a result, our modeling has inevitably been more speculative and less detailed than for diffusive sequestration models. In particular, it is not clear what the signaling molecule g* might be. Presumably it cannot involve BubR1–Cdc20 or Mad2–Cdc20, given that we require a short lifetime in the cytoplasm. Furthermore, these molecules are not known to bind to minus-end-directed motors. The active transport model nevertheless predicts that the inhibitory signal is propagated away to the spindle pole by a minus-end-directed motor. It is tempting to associate the motor protein dynein with this role; however, this assignment is problematic. If dynein were performing this function, then inhibition of dynein would effectively switch off the inhibitory active transport “wait anaphase” signal, resulting in anaphase progression. However, experiments have revealed precisely the opposite effect: Inhibition of dynein leads to inhibition of cell cycle progression (37). Furthermore, this block was not due to a more general effect of dynein inhibition, because injecting Mad2 antibodies in dynein-inhibited cells still led to rapid anaphase entry (37). Hence, either dynein is not sufficiently inhibited in this experiment, implying that some inhibitory signal can still leak through, or other motors are involved. Dynein is already known to transport kinetochore components (26, 37). However, this transport is associated with the removal of Mad2 binding sites at kinetochores once a microtubule has attached (37). Without these binding sites, Mad2 cannot cycle through a kinetochore. When this removal occurs at the final attached kinetochore, Cdc20 can no longer be sequestered by Mad2 and instead will be free to trigger anaphase progression.

In summary, our active-transport model is consistent with the experimental data of Rieder et al. We can easily find a lifetime for the inhibitory species g* that is long enough for g* to reach an adjacent microtubule, thereby communicating the state of the kinetochore to the spindle pole, but short enough for strong attenuation at another spindle 20 μm away.

Discussion

In this paper, we have shown that two models with quite different mechanisms, the diffusive reaction cascade model and a model with active transport, are possible signaling mechanisms for the SAC. These two models are schematically illustrated in Fig. 2.

Fig. 2.
A schematic of the diffusive two-step reaction model (A) and the active-transport model (B). Kinetochores are shown in green; red dashed lines denote diffusion; and solid red arrows denote motion via active transport. Sister chromatids are shown in blue, ...

Dual Pathways for the Metazoan SAC?

One attractive possibility to reconcile the above models and the experimental data is that, in metazoan cells, both mechanisms are used. Interestingly, cells with unattached kinetochores microinjected with Mad2 antibodies prematurely entered anaphase (38). This procedure will flood the cell with unsequestered Cdc20, whereas unattached kinetochores will still be signaling a “wait anaphase” signal via the active transport mechanism. The fact that the cell then enters anaphase indicates that an active transport mechanism is probably not sufficient on its own for checkpoint function. On the other hand, the experiments of Rieder et al. (24) show that an inhibitory diffusive pathway can be overruled by a second pathway, which, as we have seen, is likely to rely on active transport. We therefore propose that either mechanism can trigger anaphase onset, even without support from the other pathway. The switch-off of an active-transport-based “wait anaphase” signal or the release of sufficient freely diffusible Cdc20, by switching off an efficient sequestration apparatus, are each separately capable of generating cell cycle progression. With this assumption, our modeling is then entirely consistent with experiment.

Future Work.

Although some of the key principles used by the metazoan SAC are starting to become clear, there is still much that remains to be understood. Even for models of signaling via diffusion, which have been more actively pursued, further quantitative measurements would be very useful.

For example, if the in vivo diffusion constants were found to be small, then some possible models could be eliminated. In addition, we predict that amplification without autocatalysis is likely to be required, which implies the existence of at least two inhibiting species (e* and c*). Of course, the in vivo checkpoint dynamics will likely be much more complicated than our simple outline model, but the requirement for amplification will likely remain. Better characterization of the BubR1/Mad1/Mad2/Cdc20 protein dynamics should therefore allow these predictions to be tested. We would also like to emphasize the close connection between our diffusive two-step reaction and other signal cascades, such as those for MAPK.

For the alternative active transport pathway, a first goal would be to directly observe and image its components. For example, it would instructive to search for the transport of checkpoint proteins along spindle microtubules prior to microtubule attachment. Furthermore, disruption of appropriate minus-end motors may be able to generate premature anaphase onset by disrupting the active transport inhibitory signal. As we have discussed previously, it is also important to examine how any such signal is processed by the spindle pole to provide inhibition.

In general, future experimental work will need to measure more of the model parameters before we can make reliable quantitative predictions for intracellular signaling. Nevertheless, as we have shown, computational models can play a useful role in discriminating between viable and inviable mechanisms of checkpoint function.

Acknowledgments

We thank Fred Chang, Alexey Khodjakov, Yinghui Mao, and Kim Nasmyth for useful discussions. M.H. was supported by The Royal Society.

Abbreviations

SAC
spindle assembly checkpoint
APC/C
anaphase-promoting complex/cyclosome
FRAP
fluorescence recovery after photobleaching
O-Mad2
open form of Mad2
C-Mad2
closed form of Mad2

Footnotes

The authors declare no conflict of interest.

This article is a PNAS direct submission.

References

1. Nasmyth K. Cell. 2005;120:739–746. [PubMed]
2. Musacchio A, Hardwick KG. Nat Rev Mol Cell Biol. 2002;3:731–741. [PubMed]
3. Rieder CL, Cole RW, Khodjakov A, Sluder G. J Cell Biol. 1995;130:941–948. [PMC free article] [PubMed]
4. Rieder CL, Schultz A, Cole R, Sluder G. J Cell Biol. 1994;127:1301–1310. [PMC free article] [PubMed]
5. Castro A, Bernis C, Vigneron S, Labbé J-C, Lorca T. Oncogene. 2005;24:314–325. [PubMed]
6. Howell BJ, Moree B, Farrar EM, Stewart S, Fang G, Salmon ED. Curr Biol. 2004;14:953–964. [PubMed]
7. Antoni AD, Pearson CG, Cimini D, Canman JC, Sala V, Nezi L, Mapelli M, Sironi L, Faretta M, Salmon ED, Musacchio A. Curr Biol. 2005;15:214–225. [PubMed]
8. DeAntoni A, Sala V, Musacchio A. Philos Trans R Soc London B. 2005;360:637–648. [PMC free article] [PubMed]
9. Shah JV, Botvinick E, Bonday Z, Furnari F, Berns M, Cleveland DW. Curr Biol. 2004;14:942–952. [PubMed]
10. Sironi L, Mapelli M, Knapp S, Antoni AD, Jeang K-T, Musacchio A. EMBO J. 2002;21:2496–2506. [PMC free article] [PubMed]
11. Luo X, Tang Z, Rizo J, Yu H. Mol Cell. 2002;9:59–71. [PubMed]
12. Xia G, Luo X, Habu T, Rizo J, Yu H. EMBO J. 2004;23:3133–3143. [PMC free article] [PubMed]
13. Mapelli M, Filipp FV, Rancati G, Massimiliano L, Nezi L, Stier G, Hagan RS, Confalonieri S, Piatti S, Sattler M, Musacchio A. EMBO J. 2006;25:1273–1284. [PMC free article] [PubMed]
14. Tang Z, Bharadwaj R, Li B, Yu H. Dev Cell. 2001;1:227–237. [PubMed]
15. Fang G. Mol Biol Cell. 2002;13:755–766. [PMC free article] [PubMed]
16. Tang Z, Shu H, Oncel D, Chen S, Yu H. Mol Cell. 2004;16:387–397. [PubMed]
17. Pan J, Chen R-H. Genes Dev. 2004;18:1439–1451. [PMC free article] [PubMed]
18. Sudakin V, Chan GKT, Yen TJ. J Cell Biol. 2001;154:925–936. [PMC free article] [PubMed]
19. Kraft C, Herzog F, Gieffers C, Mechtler K, Hagting A, Pines J, Peters J-M. EMBO J. 2003;22:6598–6609. [PMC free article] [PubMed]
20. Acquaviva C, Herzog F, Kraft C, Pines J. Nat Cell Biol. 2004;6:892–898. [PubMed]
21. Doncic A, Ben-Jacob E, Barkai N. Proc Natl Acad Sci USA. 2005;102:6332–6337. [PMC free article] [PubMed]
22. Heinrich R, Neel BG, Rapoport TA. Mol Cell. 2002;9:957–970. [PubMed]
23. Kholodenko BN. Nat Rev Mol Cell Biol. 2006;7:165–176. [PMC free article] [PubMed]
24. Rieder CL, Khodjakov A, Paliulis LV, Fortier TM, Cole RW, Sluder G. Proc Natl Acad Sci USA. 1997;94:5107–5112. [PMC free article] [PubMed]
25. McEwen BF, Ding Y, Heagle AB. Chromosome Res. 1998;6:123–132. [PubMed]
26. Howell BJ, Hoffman DB, Fang G, Murray AW, Salmon ED. J Cell Biol. 2000;150:1233–1249. [PMC free article] [PubMed]
27. Clute P, Pines J. Nat Cell Biol. 1999;1:82–87. [PubMed]
28. Wojcieszyn JW, Schlegel RA, Wu ES, Jacobson KA. Proc Natl Acad Sci USA. 1981;78:4407–4410. [PMC free article] [PubMed]
29. Lang I, Scholz M, Peters R. J Cell Biol. 1986;102:1183–1190. [PMC free article] [PubMed]
30. Seksek O, Biwersi J, Verkman AS. J Cell Biol. 1997;138:131–142. [PMC free article] [PubMed]
31. Schmiedeberg L, Weisshart K, Diekmann S, Hoerste GMZ, Hemmerich P. Mol Biol Cell. 2004;15:2819–2833. [PMC free article] [PubMed]
32. Berg HC. Random Walks in Biology. Princeton: Princeton Univ Press; 1993. p. 28.
33. Vink M, Simonetta M, Transidico P, Ferrari K, Mapelli M, De Antoni A, Massimiliano L, Ciliberto A, Faretta M, Salmon ED, Musacchio A. Curr Biol. 2006;16:755–766. [PubMed]
34. Northrup SH, Erickson HP. Proc Natl Acad Sci USA. 1992;89:3338–3342. [PMC free article] [PubMed]
35. Howard J. Mechanics of Motor Proteins and the Cytoskeleton. Sunderland, MA: Sinauer; 2001.
36. Wollman R, Cytrynbaum EN, Jones JT, Meyer T, Scholey JM, Mogilner A. Curr Biol. 2005;15:828–832. [PubMed]
37. Howell BJ, McEwen BF, Canman JC, Hoffman DB, Farrar EM, Rieder CL, Salmon ED. J Cell Biol. 2001;155:1159–1172. [PMC free article] [PubMed]
38. Gorbsky GJ, Chen R-H, Murray AW. J Cell Biol. 1998;141:1193–1205. [PMC free article] [PubMed]

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