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 1), and is a target of the Cdc20 molecule, although there are some conflicting evidences that APC/C is located on the kinetochores 21 . This assumption can affect the binding rate, but does not impact the construction of our model. For a cell containing N chromosomes, the targets of the Cdc20 molecules are the N associated kinetochores, containing the APC/C complexes. When a Cdc20 molecule reaches a kinetochore, it activates the APC/C complex and this can trigger a cascade of reactions (detailed in the background section) leading to the separation of the sister chromatids. The goal of the spindle assembly checkpoint signal is to prevent this activation of APC/C by Cdc20, when at least one of the chromosomes is not properly attached to the microtubules responsible for the chromatids migration. The spindle assembly checkpoint signal consists in the production of proteins such as Mad2, BubR1 and Mad3 4 , generated by unattached kinetochores. These proteins diffuse in large quantity in the cell to inhibit the APC/C binding by Cdc20 molecules. Indeed, these proteins form with Cdc20 a complex called mitotic checkpoint complex (MCC). Similar complexes can be found in yeast, in which a BubR1-related Mad3 protein might inhibit Cdc20 as a pseudo-substrate 22 23 24 . The formation of this complex results in Cdc20 ubiquitylation, which prevents APC/C activation. In our model, MCC will represent the complex of inhibitory proteins before it binds Cdc20, in contrast with the usual terminology where MCC includes Cdc20. Cdc20 molecules are produced from the dissociation of a complex 25 , which could be a subcomplex of Mad2 and Cdc20 resulting from MCC:Cdc20 disassembly, promoted by p31 26 . In summary, the above chemical reactions can be summarized as

C o m p l e x : C d c 20 → λ C o m p l e x + C d c 20 ( production ) C d c 20 + M C C → k - 1 M C C : C d c 20 ( ubiquitylation ) C d c 20 + A P C / C → μ A P C / C : C d c 20 ( activation ) .

where the rate λ measures the production of Cdc20 and k _-1 the degradation, while μ is the arrival rate for a CdC20 to an APC/C site. We shall compute in the next paragraphs the joint probability

p k ( t ) = P r ( | C d c 20 | ( t ) = k , no activation occurred before time t )

that the APC/C is not activated by any free Cdc20 molecule at time t. To compute this probability, we first derive a Markov equation. The difficulty is that this joint probability contains a discrete variable counting the number of Cdc20 molecules and a binary one, which monitors whether or not an activation of APC/C by Cdc20 has occurred before time t. The state space of this Markov process is completed by adding a state describing that activation occurred before time t. It is modeled as an absorbing state of probability

p * ( t ) = P r ( activation occurred before time t ) ,

which accounts for all the activations which have happened before time t from all the states k. Starting with k active Cdc20 molecules and no activation, there are three possible transitions (figure 2): 1) one Cdc20 molecule is inhibited, so that k-1 active molecules are left 2) one Cdc20 molecule activates the APC/C 3) one Cdc20 molecule is generated, leading to the transition from k to k+1 active molecules. Thus, the probabilities pk satisfy the chemical master equations 18 20

ṗ 0 = - λ S p 0 + k - 1 p 1 ṗ k = - ( λ ( S - k ) + ( μ N + k - 1 ) k ) p k + λ ( S - k + 1 ) p k - 1 + k - 1 ( k + 1 ) p k + 1 , for 1 ≤ k ṗ * = ∑ k μ N k p k .

The production rate of Cdc20 molecules is proportional to the number of remaining available complex molecules given by λ(S - k). Indeed the Cdc20 molecules are produced during metaphase by dissociation from a pool of complexes, which limits the level of Cdc20 to a maximum of S molecules 25 . In addition, we shall emphasize that there is another interpretation of equation (4): indeed, dissociation of the MCC:Cdc20 complex, producing Cdc20 molecule, leads also to equation (4). When none of the N target kinetochores have been activated, the arrival rate for a Cdc20 molecule to an APC/C is μN, where μ = 1 τ and τ is the mean time for a Cdc20 molecule to reach the APC/C site. This mean time can be approximated by 27 28 29

μ = 3 r D π R 3 ,

where D is the diffusion coefficient of a Cdc20 molecule, a is the radius of the APC/C complex, R the radius of the cell. As the mechanisms underlying the production and the regulation of MCC are still unclear, we consider that the MCC concentration is homogeneous and remains constant over time to guarantee a robust inhibition of Cdc20. Thus, k_-1 is given by the Smoluchovski formula for the binding rate of a Brownian particle

k - 1 = 2 π b D [ M C C ] ,

where [MCC] is the concentration of MCC, uniform over the cell and b is the radius of the Cdc20 binding site. When the SAC starts, no free Cdc20 molecules are present in the cell, thus we choose for the initial conditions pk(0) = δ _k,0. Because there can only be S Cdc20 molecules, we have for all time t, pk(t) = 0 and k > S.

To quantify the inhibition capacity of the SAC, we estimate the probability P(t) that at time t, no APC/C has been activated, so that no chromosomal migration could have been initiated. This probability is given by

P ( t ) = ∑ k = 0 + ∞ p k ( t ) .

We shall compute P(t) using the generating function

f ( t , x ) = ∑ k = 0 + ∞ p k ( t ) x k .

Using equation (4), f satisfies a first order PDE

∂ f ∂ t = λ S ( x - 1 ) f + ( - λ x 2 + ( λ - μ N - k - 1 ) x + k - 1 ) × ∂ f ∂ x .

Using the characteristics method, we look for a solution of

Ẋ = λ X 2 - ( λ - μ N - k - 1 ) X - k - 1 ,

which is of Riccati type. From the classical substitution x = - 1 λ u ′ u we obtain the linear second order differential equation

u ″ + ( λ - μ N - k - 1 ) u ′ - k - 1 λ u = 0 .

Thus, the solution for characteristics is

x C ( t ) = - 1 λ r 1 e r 1 t + r 2 C e r 2 t e r 1 t + C e r 2 t ,

where C is a constant and r ₁ and r ₂ are the two roots of the quadratic polynomial associated with (11)

r 1 = 1 2 - λ + μ N + k - 1 + ( - λ + μ N + k - 1 ) 2 + 4 k - 1 λ

r 2 = 1 2 - λ + μ N + k - 1 - ( - λ + μ N + k - 1 ) 2 + 4 k - 1 λ .

Along one of these characteristics, f satisfies the linear first order ODE

d f ( t , x C ( t ) ) d t = λ S ( x C ( t ) - 1 ) f ( t , x C ( t ) ) .

The general solution for equation (15) is

f ( t , x C ( t ) , K ) = K exp λ S ∫ 0 t ( x C ( u ) - 1 ) d u .

At time t = 0, we have p _k (0) = δ _k0, initial conditions set K = 1. To find p(t), we shall select the characteristic for which at time t, x _C (t) = 1. Solving this, yields to

C ( t ) = - e ( r 1 - r 2 ) t λ + r 1 λ + r 2 ,

and we obtain the characteristic

x C ( t ) ( u ) = - 1 λ r 1 - r 2 λ + r 1 λ + r 2 exp [ ( r 1 - r 2 ) ( t - u ) ] 1 - λ + r 1 λ + r 2 exp [ ( r 1 - r 2 ) ( t - u ) ] .

Finally, the probability P(t) of no activation is

P ( t ) = exp - λ S t + 1 λ … × ∫ 0 t r 1 - r 2 λ + r 1 λ + r 2 e ( r 1 - r 2 ) u 1 - λ + r 1 λ + r 2 e ( r 1 - r 2 ) u d u

= exp - λ S t + 1 λ r 1 u - ln 1 - λ + r 1 λ + r 2 × e ( r 1 - r 2 ) u 0 t

= e - λ S t - r 1 S t - λ - r 2 + ( λ + r 1 ) e ( r 1 - r 2 ) t ( r 1 - r 2 ) S

= e - λ t S ( λ + r 1 ) e - r 2 t - ( λ + r 2 ) e - r 1 t λ ( r 1 - r 2 ) S

Finally,

P ( t ) = ( λ + r 1 ) e - ( λ + r 2 ) t - ( λ + r 2 ) e - ( λ + r 1 ) t r 1 - r 2 S

P is a decreasing function of time, and remains constant for λ = 0 and μ = 0. In figure 3, we plot P as a function of time for different values of λ and k_-1, and as a function of λ and k _-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).

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 k Cdc20 molecules, the production rate is given by λ(S-k) and the destruction rate k _-1(k+1). Thus, the probability p _k (t|NA) that k Cdc20 molecules are inside the cell, conditioned that no activation has occurred, satisfies the Master equations

ṗ 0 ( t | N A ) = - λ S p 0 ( t | N A ) + k - 1 p 1 ( t | N A ) ṗ k ( t | N A ) = - ( λ ( S - k ) + k - 1 k ) p k ( t | N A ) + λ ( S - k + 1 ) p k - 1 ( t | N A ) + k - 1 ( k + 1 ) p k + 1 ( t | N A ) .

The equilibrium probabilities p _k (∞) and the mean number N ̄ for such a system is 30

p k ( ∞ ) = S k λ k - 1 k 1 + λ k - 1 S

N ̄ = S λ / k - 1 1 + λ / k - 1 .

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.

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 study the time for such activation, we consider that production and degradation of MCC are fast enough so that the MCC concentration decreases rapidly once all the kinetochores are attached. In that case, we can neglect the transient time for the rate k _-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 k Cdc20 present in the cell and m of them are bound to APC/C, it results that the association rate is μ(N - m)(k - m) 18 . We shall now estimate the joint probabilities that there are k Cdc20 molecules, and m activated APC/C by Cdc20

p k , m ( t ) = P r ( | C D C 20 | = k , m   activated   APC / C   by   Cdc 20 )

From the state (k,m), the transition rate to activation of APC/C located on another kinetochore is then μ(k - m)(N - m). Thus, we get the following Markov chain (represented in figure 4)

ṗ 0 , 0 = - λ S p 0 , 0 ṗ k , 0 = - ( λ ( S - k ) + μ N k ) p k , 0 + λ ( S - k + 1 ) p k - 1 , 0 ṗ k , k = - λ ( S - k ) p k , k + μ ( N - k + 1 ) p k , k - 1 ṗ k , m = - ( λ ( S - k ) + μ ( N - m ) ( k - m ) ) p k , m + λ ( S - k + 1 ) p k - 1 , m + μ ( N - m + 1 ) ( k - m + 1 ) p k , m - 1 ,

with the initial condition

p k , m ( 0 ) = S k λ k - 1 k 1 + λ k - 1 S δ 0 , m ,

computed in equation (25). In that case, the mean time τ that all APC/C are activated is obtained by analyzing a continuous markov process that reaches a given threshold 20 . Using formula 12 of 20 , the mean time to threshold is expressed as a sum

τ = ∑ k = 0 N - 1 ∑ m = 0 S a k , m ,

where a k , m = ∫ 0 ∞ p k , m ( t ) d t . Integrating the system of equation (28) from 0 to + ∞ with the initial conditions

p k , m ( 0 ) = δ m , 0 S k λ k - 1 k 1 + λ k - 1 S ,

leads to

- p 0 , 0 ( 0 ) = - λ S a 0 , 0 - p k , 0 ( 0 ) = - ( λ + μ N ) ( S - k ) a k , 0 + λ ( S - k + 1 ) a k - 1 , 0 0 = - λ ( S - k ) a k , k + μ ( N - k + 1 ) a k , k - 1 0 = - ( λ ( S - k ) + μ ( N - m ) ( k - m ) ) a k , m + μ ( N - m + 1 ) ( S - k ) a k , m - 1 + λ ( S - k + 1 ) a k - 1 , m .

In practice, we solve this linear system of equations numerically and in figure 5, we plot the mean τ as a function of the parameters k _-1 and λ. We find that τ is a decreasing function of λ (the faster Cdc20 is produced, the faster the threshold of bindings is reached) and an increasing function of the rate k _-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 P and this mean time τ will lead to limit the range of the parameters λ and k _-1 as we will describe now.

We now apply our previous modeling to determine the rates of production λ and the backward binding rate k _-1. Indeed, during SAC, a strong inhibition signal imposes that the probability for no activation remains very high and thus, the degradation rate k _-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 k _-1. We use the following quantitative constraints.

1. First, the probability P of no activation remains high enough during the time τ ₁ where all chro mosomes get properly attached in the metaphase plate. It has been estimated that τ ₁ ≈ 20 min 31 . Thus by fixing a threshold of 0.95 for the probabil ity P(τ ₁) that no activation occurred before time τ ₁

P ( τ 1 > 20 min ) ≥ 0 . 95 ( C 1 ) .

2. Second, during the anaphase onset, the time 〈τ _s 〉 for all chromosomes to get separated is short. Since APC/C activation triggers the chromosome sep aration, we can consider that τ _S is the time for all APC/C to get activated. Indeed, biophysical data 32 suggest that τ _s should be limited in time τ' ≈ 10 min. Thus,

τ S ≤ τ ′ ( C 2 ) .

Using formula (23) for the probability P(τ ₁) and integrating numerically the time 〈τ _S 〉 from the matrix equation (31), we determine a range of validity for these parameters by a geometrical domain Ω represented in figure 6, as the intersection Ω = Ω₁ ∩ Ω₂, where

Ω 1 = { ( λ , k - 1 ) s . t . P ( τ 1 ( λ , k - 1 ) > 20 min ) ≥ 0 . 95 }

and

Ω 2 = { ( λ , k - 1 ) s . t . τ S ( λ , k - 1 ) ≤ τ ′ } .

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 25 during interphase. This concentration is of the same order as the one reported in 13 , and is equivalent to 3000 molecules restricted in a volume of 100 μm ³ (for a flat cell of size 10 μm and of height 1 μm, leading to a volume of 10 × 10 × 1 μm 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 6, we represented the two domains Ω₁ and Ω₂, 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 1. Surprisingly, Ω is not bounded, but it provides an interesting and new range for the parameters.