Institute for Biology (IBENS), Group of Computational Biology and Applied Mathematics, Ecole Normale Supérieure, 46 rue d'Ulm 75005 Paris, France

Background

Default activation of the spindle assembly checkpoint provides severe constraints on the underlying biochemical activation rates: on one hand, the cell cannot divide before all chromosomes are aligned, but on the other hand, when they are ready, the separation is quite fast, lasting a few minutes. Our purpose is to use these opposed constraints to estimate the associated chemical rates.

Results

To analyze the above constraints, we develop a markovian model to describe the dynamics of Cdc20 molecules. We compute the probability for no APC/C activation before time t, the distribution of Cdc20 at equilibrium and the mean time to complete APC/C activation after all chromosomes are attached.

Conclusions

By studying Cdc20 inhibition and the activation time, we obtain a range for the main chemical reaction rates regulating the spindle assembly checkpoint and transition to anaphase.

I. Background

A fundamental step in cell division consists in the alignment of each pair of chromosomes. This process occurs during metaphase, where centrosome nucleated microtubules interact with the chromosomes kinetochores to build the mitotic spindle. Only after all chromosomes have become aligned at the metaphase plate and when every kinetochore is properly attached to a bundle of microtubules, the cell enters anaphase. To prevent premature progression to anaphase, even if all-but-one of the kinetochores have been attached and the chromosomes are aligned, unattached or improperly attached kinetochores generate a signal inhibiting the anaphase activators. This process is called the spindle assembly checkpoint (SAC).

Although the exact mechanisms of the SAC and anaphase processes are still unclear, several key steps have been identified. Sister chromatids are initially bound by proteins such as cohesin. During anaphase onset, separase protein cleaves cohesin, thus allowing the sister chromatids to separate

The SAC has been modeled at a molecular level, however the parameters used

In the present article, our purpose is to study the inhibition followed by its fast activation of Cdc20, which is the key activator of the anaphase promoting complex. As the number of kinetochores implied in the SAC is small, the forward binding rate of a chemical reaction as it is classically computed in the continuously concentrated limit cannot be applied. To adequately describe chemical reactions in microdomains

II. Methods

Markovian modeling of APC/C activation and Cdc20 inhibition

We describe here the time evolution of the joint probability distribution of Cdc20 molecules and of APC/C complex activation, the later being responsible for the chromosome separation. In this model, APC/C is located on the chromosomes (figure

A schematic view of the spindle assembly checkpoint and anaphase

**A schematic view of the spindle assembly checkpoint and anaphase**. **A**: Before all chromosomes are all attached, the mitotic checkpoint complex inhibits the Cdc20 molecules binding with APC/C to prevent premature separation of sister chromatids. This signal ubiquitylates Cdc20. **B**: When all chromosomes are properly attached, the inhibiting signal is shut down. **C **and **D**: Activation of APC/C triggers the separation of the chromatids and ultimately the anaphase.

where the rate λ measures the production of Cdc20 and _{-1 }the degradation, while

that the APC/C is not activated by any free Cdc20 molecule at time

which accounts for all the activations which have happened before time

Markov diagram for the probability of number of Cdc20 molecules

**Markov diagram for the probability of number of Cdc20 molecules**. Cdc20 at state _{-1}

The production rate of Cdc20 molecules is proportional to the number of remaining available complex molecules given by λ(

where _{-1 }is given by the Smoluchovski formula for the binding rate of a Brownian particle

where [_{
k,0}. Because there can only be

The probability for no activation

To quantify the inhibition capacity of the SAC, we estimate the probability

We shall compute

Using equation (4),

Using the characteristics method, we look for a solution of

which is of Riccati type. From the classical substitution

Thus, the solution for characteristics is

where _{1 }and _{2 }are the two roots of the quadratic polynomial associated with (11)

Along one of these characteristics,

The general solution for equation (15) is

At time _{
k
}(0) = _{
k0}, initial conditions set _{
C
}(

and we obtain the characteristic

Finally, the probability

Finally,

_{-1}, and as a function of λ and _{-1 }at a given time. It is a decreasing function of λ (increasing the Cdc20 production rate decreases the probability of activation) and a decreasing function of k_{-1 }(increasing the inhibition of Cdc20 increases the probability for no activation).

The probability _{-1}

**The probability P for no activation during the SAC is represented as a function of the time and the rates λ and k**.

The distribution of Cdc20 at equilibrium

After the last chromosome attached and thus all kinetochores are properly positioned, the inhibition of APC/C:Cdc20 binding is suppressed and anaphase can start. The initial condition for the number of Cdc20 molecules for this new phase is the one obtained at equilibrium from the previous phase, in which Cdc20 is produced and destroyed by the SAC. When there are _{-1}(_{
k
}(

The equilibrium probabilities _{
k
}(∞) and the mean number

When the SAC is suppressed, Cdc20 is no longer inhibited and can activate APC/C to trigger anaphase. Using the distribution computed here we compute in the next section the mean time for complete separation of sister chromatids during anaphase.

Activation of APC/C

When all kinetochores are properly attached, the SAC is shut down and the activation of APC/C:Cdc20 complex triggers a cascade of reactions leading to cohesin ubiquitylation at the chromosome sites _{-1 }to decay to 0 and thus we take the equilibrium Cdc20 concentrations as the initial conditions for the activation of APC/C. For our analysis, we further consider that the time for all kinetochores to be attached is not too short compared to the degradation and production time scale, so that the CdC20 concentration is close to equilibrium. When there are

From the state (

Schematic representation of the Markov Chain associated with the joint probability _{k,m}(

**Schematic representation of the Markov Chain associated with the joint probability p _{k,m}(t) to have k bounds APC/C and m free Cdc20 molecules**.

with the initial condition

computed in equation (25). In that case, the mean time

where

leads to

In practice, we solve this linear system of equations numerically and in figure _{-1 }and λ. We find that _{-1 }(inhibition decreases the number of Cdc20 at equilibrium and thus the time to reach the threshold, after the source of inhibition is terminated). These variations go in the opposite direction compared to the probability of no activation during the SAC. Thus we expect that using the probability _{-1 }as we will describe now.

The time _{-1}

**The time τ is plotted as a function of the parameters λ and k**. The parameter valuers are given in table 1.

III. Results

Quantitative constraints on the rates λ and _{-1}

We now apply our previous modeling to determine the rates of production λ and the backward binding rate _{-1}. Indeed, during SAC, a strong inhibition signal imposes that the probability for no activation remains very high and thus, the degradation rate _{-1 }has to be high enough compared to the production rate λ. In contrast, a fast activation during anaphase forces the mean time to activate all the kinetochores to be short, thus the production rate λ has to be high. These opposite constraints allow us to determine a range for the parameters λ and _{-1}. We use the following quantitative constraints.

1. First, the probability _{1 }where all chro mosomes get properly attached in the metaphase plate. It has been estimated that _{1 }≈ 20 min _{1}) that no activation occurred before time _{1}

2. Second, during the anaphase onset, the time 〈_{
s
}〉 for all chromosomes to get separated is short. Since APC/C activation triggers the chromosome sep aration, we can consider that _{
S
}is the time for all APC/C to get activated. Indeed, biophysical data _{
s
}should be limited in time

Using formula (23) for the probability _{1}) and integrating numerically the time 〈_{
S
}〉 from the matrix equation (31), we determine a range of validity for these parameters by a geometrical domain Ω represented in figure _{1 }∩ Ω_{2}, where

Representation of the domain Ω (red)

**Representation of the domain Ω (red)**. **A**: The probability for no activation at time _{-1}. Other parameters are those of table I B : The mean time to threshold as a as a function of parameters λ and _{-1}. Other parameters are those of table I. C : The curve 1 is given as the level line associated

and

We tested the prediction of our model on PTK2 cells, originating from kangaroo rat kidney, used in studies on mitosis because there are only a few large chromosomes and the cells remain flattened during mitosis. For these cells, the concentration of bound complex from which Cdc20 is produced is approximately 50 nM ^{3 }(for a flat cell of size 10 μm and of height 1 ^{3}). For larger cells, the number of molecules can be multiplied by 10 or 100. Thus, during the SAC, it is tempting to think that the system escapes the stochastic limit. However, because the number of Cdc20 is small at early metaphase and limited by the inhibition of MCC, the stochastic regime is still controlling the behavior of the system and in addition, the inhibition is strong enough to maintain a low level of Cdc20. In figure _{1 }and Ω_{2}, the first is on the left of curve 1, while the second is on the right side of curve 2. Our analysis can be generalized by changing the two conditions C1 and C2 for specific cell types. The other parameters are summarized in table

**Parameter**

**Description**

**Value**

cell volume

≈ 100 ^{3 }

Diffusion coefficient of Cdc20

^{2}/

radius of APC/C complex

≈ 10^{-2 }

radius of MCC binding site

≈ 2

Number of chromosomes

13

Initial number of complexes

3000

binding rate

≈ 2.10^{-4}^{-1}

IV. Discussion and Conclusion

Based on the two main constraints _{-1}. Actually, this range approximatively depends on the ratio

The constraint _{-1}. Because _{-1 }is given by the Smolu-chovski formula 2_{-1}. For example, when the number of MCC is in the range of 10000 (which corresponds to the Mad2 concentration of 200nM found in _{-1 }≈ 24 and in that case, we approximatively get for the production rate

For example, fixing the value

Authors' contributions

KDD and DH designed research and wrote the paper. KDD and DH performed research. All authors read and approved the final manuscript.

Acknowledgements

We would like to thank J. Pines, Z. Xu and D. Peric for fruitful discussions. D. H research is supported by an ERC-Starting Grant.