Implications of alternative routes to APC/C inhibition by the mitotic checkpoint complex

The mitotic checkpoint (also called spindle assembly checkpoint) is a signaling pathway that ensures faithful chromosome segregation. Mitotic checkpoint proteins inhibit the anaphase-promoting complex (APC/C) and its activator Cdc20 to prevent precocious anaphase. Checkpoint signaling leads to a complex of APC/C, Cdc20, and checkpoint proteins, in which the APC/C is inactive. In principle, this final product of the mitotic checkpoint can be obtained via different pathways, whose relevance still needs to be fully ascertained experimentally. Here, we use mathematical models to compare the implications on checkpoint response of the possible pathways leading to APC/C inhibition. We identify a previously unrecognized funneling effect for Cdc20, which favors Cdc20 incorporation into the inhibitory complex and therefore promotes checkpoint activity. Furthermore, we find that the presence or absence of one specific assembly reaction determines whether the checkpoint remains functional at elevated levels of Cdc20, which can occur in cancer cells. Our results reveal the inhibitory logics behind checkpoint activity, predict checkpoint efficiency in perturbed situations, and could inform molecular strategies to treat malignancies that exhibit Cdc20 overexpression.

Furthermore, we use the following notation to refer to the net association and dissociation reaction of two species X and Y that can form a complex Z: where [X] stands for the concentration of species X. To indicate the steady state level of X, we use the notation [X].
The dissociation constant is defined as For some derivations we assume that the rate is approximately the same for all reactions. In that case we will simply refer to it as K D . We write down equations only for the concentrations of complexes. The concentrations of free A, M , and C can then be obtained from conservation relations that are justified by the observation that all checkpoint proteins and APC/C are stable and that total Cdc20 is at steady state. All numerical simulations were carried out using the Python package "SloppyCell" [S1, S2] and custom written Python functions. Table S1 lists experimental measurements of the relevant species in different organisms derived from a comprehensive survey of the scientific literature. Relevant for our model are the concentrations of free Mad2 (not bound to Mad1), Mad3, APC/C, and Cdc20. With very few exceptions, these concentrations do not differ by more than a factor of five. Different studies come to very different conclusions, illustrating the difficulty of accurately determining absolute values. As a general scheme, however, total levels of APC/C are typically lower than Mad2/3 and Cdc20 (see also S1 Fig). For the simulations we further assumed that all dissociation constants are small (i.e. strong binding) and have the same value. Specifically, we used:

Model Parameters
in dimensionless units (i.e. normalized to the total amount of Mad levels in wild type).

Sequential Inhibition Model Equations
The wiring diagram in Fig 2 (i) corresponds to the following set of equations: Furthermore, we have the following conservation relations: Approximation for small Cdc20 In the following we will often exploit the fact that the models we consider are detailed balanced. This means that at steady state all forward reactions are individually balanced against their corresponding reverse reactions, or which directly leads to For the sequential inhibition model, detailed balancing can easily be shown. Setting (9) to zero, we immediately get But then it follows from (7) and (8) that both other reactions are also detailed balanced.
Therefore, we obtain For small levels of C total we can assume that most species are in their free form and therefore use the approximation Then from (16) and (17) we immediately get Thus, in this regime the relative amounts of AC and M CC1 are directly related to the total amounts and the corresponding binding reactions. Furthermore, using (20) we can rewrite (18) as or The quadratic dependence means that AM CC2 will dominate for [AC] larger than k. But k is a small number if binding is generally strong and/or [M total ] > [A total ]. Note that k is small even if all reactions have the same binding strength and and [M total ] = [A total ]. This is because there are three binding reactions leading to the inhibited, but only one reaction leading to the active species. Mathematically this is reflected by the product of two K D s in the numerator of (22). Together this explains the "funneling" effect that we observe for small levels of Cdc20. We can find approximate solutions for the species concentrations as a function of [C total ] under the assumption of strong binding (i.e. K D A total , M total ). In this case [ÂM CC2] dominates the sum in (12). In particular, we have which explains the initially approximately linear behavior of APC/C MCC2 in S2 Fig. Based on this first order approximation, we can use Equations (16)-(18) to derive expressions for the other species as well: These expressions indicate how the steady state concentrations for small levels of Cdc20 depend on parameters and concentrations. In particular, they explain the approximately linear dependence of APC/C MCC2 and the square root dependence of the other species. The approximations are in good quantitative agreement with the simulations for small values of K D . But even for larger values they provide a qualitative understanding of the behavior of the model (S2 Fig).

Approximation for large Cdc20
If C total is large, then every free molecule of M or A will quickly bind to a free molecule of C. With the approximation that this binding is instantaneous and M ≈ A ≈ 0, we can directly calculate the steady state values for the remaining species. For this it is sufficient to look at the three species M CC1, AC, and AM CC2, and the only reaction left to be considered is because the other two species are then determined by the conservation relations. At steady state we have From this we get which is a quadratic equation in [ÂM CC2] whose solutions are Only the "−" solution ensures that . Expressions for the other species can be directly inferred, using The

Competitive Inhibition Model
Equations: The competitive inhibition model includes the additional species M CC2 which is formed when M CC1 binds an additional molecule of C. The inhibited species AM CC2 is in this case formed by M CC2 binding to a free molecule of A. We therefore have to consider four equations: together with the following conservation relations: Approximation for small Cdc20 In the same way as before we can show that detailed balancing holds, and we get at steady state Equations (39) and (40) are identical to (16) and (17), and the only difference between (42) and (18) is one of the dissociation constants. Moreover, for strong binding [M CC2] is very small compared to the other species because, for example, from (41) and (42) So the model effectively reduces to the same equations as the sequential inhibition model. In particular we get the equivalent of (22) by rewriting (42) using (39): This explains why for small levels of Cdc20 the steady state behavior is basically the same for sequential and competitive inhibition given equivalent choice of parameters. Again, k is a small number provided that binding is strong and A total does not exceed M total . Moreover, we see that the funneling effect is indifferent to the order in which the complexes are formed. This is a straightforward consequence of mass action kinetics and detailed balancing. Analogously to the case of sequential inhibition, [AM CC2] dominates (38) for K D → 0, so we can derive the following approximations: [ÂC] ≈ 1 2 Again we reproduce the approximately linear behavior of APC/C AMCC2 and the square root behavior of the other species (with the exception for free M CC2, which increases linearly, but with a very small slope).

Approximation for large Cdc20
For high levels of C total , we can assume that meaning that all species that bind to free C (i.e. A, M , and M CC) are approximately zero. With the help of the conservation relation (38) we can then immediately derive This means that the system effectively reduces to a simple competition model where the inhibitor M CC2 competes with C for free A. It can be easily shown that the active species will always outcompete the inactive species if levels of Cdc20 are high. First of all, note that if [C total ] → ∞, then also [C] → ∞. From detailed balancing, we then get which entails or From Eq. (52) we can derive an analogous expression for [Ĉ]: or Furthermore, rewriting (51) and afterwards substituting (56) and (58), we get from which we obtain Combining (56), (58), and (60), we finally get and Furthermore, from (50) and (61) we get From (61) and (62) we obtain the simple expression Thus, the ratio of active to inactive APC/C increases linearly with the level of Cdc20. Note that these expressions depend only on the total amounts and not on the association/dissociation parameters (in particular, we did not use the assumption of strong binding). Most importantly, and as already shown, we will always get AC → A total and AM CC2 → 0 for C total → ∞. In other words, the competitive inhibition model always becomes checkpoint deficient for sufficiently high levels of Cdc20.
The approximations for the competitive inhibition model are shown in S2 Fig (ii).

Combined Model Equations
The combined model includes both ways of producing the inhibited species AM CC2. The corresponding set of equations is The conservation relations are the same as (36), (37), and (38). For this network the detailed balancing property does not follow directly from the equations, but requires certain restricting conditions on the rate constants. It can be shown that the condition for detailed balancing for this model is (a procedure for deriving this condition can be found for instance in [S3]). For our analysis we assume that detailed balancing holds. The condition is obviously fulfilled in the special case that all K D s are the same.

Approximation for small Cdc20
Given detailed balancing, the derivation of steady state expressions can be carried out in the same way as in the case of the competitive inhibition model in Section 3.2. The relationship between active and inhibited species is described both by (22) and (44) because the two equations coincide when condition (69) holds. This explains why the behavior of the combined model for small Cdc20 is the same as in the other two models.

Approximation for large Cdc20
As in Section 3.2, we can assume that all reactions involving free C are saturated. In particular, we have M CC1 ≈ 0, which means that the sequential production of the inhibitor R AC:M CC1 AM CC2 , that is added with respect to the competitive inhibition model, is negligible. As a consequence, the behavior of the combined model at saturating levels of Cdc20 is the same as the competitive inhibition model. The approximations for the combined model are shown in S3 Fig (iii).

Model with Mad3 as a separate species
Equations: To incorporate Mad3 as a separate species, we assume that first Mad2 (M 2) binds to Cdc20 (C) to form Mad2:Cdc20 (M 2C). Afterwards this complex binds to free Mad3 (M 3) to form M CC1. This translates to the following equations for the competitive inhibition case:   reactions to build the inhibited species. This means that the funneling effect is even more pronounced in a model with Mad3 as a separate species. This can be seen in S3 Fig B. Approximation for large Cdc20 For large levels of Cdc20 the model including Mad3 behaves very similarly to the simpler models, provided that [M 3 total ] = [M total ], i.e. M 3 is limiting. For the competitive inhibition scenario, the model is approximated by exactly the same reduced network as the model in 3.2. This is because M CC1 ≈ 0, and therefore also M 3, M 2C ≈ 0 (using again detailed balancing).
In the case of sequential inhibition we are left with two remaining reactions: R M 2C:M 3 M CC1 and R AC:M CC1 AM CC2 . Thus there is again an equilibrium between AC and AM CC2, but the levels are slightly shifted because [M CC1] < [M 3 total ]. Under the assumption that binding of M 2C to M 3 is strong, the asymptotic levels of AC and AM CC2 are very close to those in 3.1, as can be seen in S3 Fig B (i).