A Dynamical Model for Activating and Silencing the Mitotic Checkpoint

The spindle assembly checkpoint (SAC) is an evolutionarily conserved mechanism, exclusively sensitive to the states of kinetochores attached to microtubules. During metaphase, the anaphase-promoting complex/cyclosome (APC/C) is inhibited by the SAC but it rapidly switches to its active form following proper attachment of the final spindle. It had been thought that APC/C activity is an all-or-nothing response, but recent findings have demonstrated that it switches steadily. In this study, we develop a detailed mathematical model that considers all 92 human kinetochores and all major proteins involved in SAC activation and silencing. We perform deterministic and spatially-stochastic simulations and find that certain spatial properties do not play significant roles. Furthermore, we show that our model is consistent with in-vitro mutation experiments of crucial proteins as well as the recently-suggested rheostat switch behavior, measured by Securin or CyclinB concentration. Considering an autocatalytic feedback loop leads to an all-or-nothing toggle switch in the underlying core components, while the output signal of the SAC still behaves like a rheostat switch. The results of this study support the hypothesis that the SAC signal varies with increasing number of attached kinetochores, even though it might still contain toggle switches in some of its components.

We can use this formula to calculate how long it takes for all O-Mad2 particles to diffuse to a kinetochore. In particular, we can calculate the average hitting time, which is the time needed for half the particles to diffuse to a kinetochore before that time. It is the half-time of the following reaction: (2) Using the relationship between the half-time and the reaction rate we can estimate the maximum turnover rate of C-Mad at the kinetochores. Assuming a nucleus radius of 6µm, a kinetochore radius of 0.1µm and the Mad2 diffusion coefficient of 16.61µm 2 s −1 , we find a half-time of 43 s, which is consistent with experimental findings and results in a turnover rate of 0.016s −1 at each kinetochore.

Text S2 Calculating diffusion coefficients
In addition to the reactions and their governing laws, a particle simulation also needs spatial information. It requires knowledge of the shape and size of every protein and their relative diffusion coefficients. For the sake of simplicity, we model all proteins as spheres with uniform densities. Given the mass, which is known from lab-experiments, and assuming a uniform spherical particle, we can calculate the radius of the sphere using in which m i refers to lab-measured mass of the molecule in Da and r i is the radius of the molecule in nm. Following the Stokes-Einstein equation we can calculate the diffusion coefficient using where r i = radius of the assumed spherical-particle in m η = viscosity of the medium in Ns/m 2 (0.891 × 10 −3 in water; 6.75 × 10 −3 in nucleus)

Text S3 Coarse-graining
Typical time-steps in particle simulations are 10 −9 s, which would imply 1.2 × 10 11 time-steps to realize the full metaphase (around 20 real-time minutes). Realistic particle numbers are around 1, 000, 000 per species, which leads to an infeasible large time requirement for simulating the full system. For reference, simulating 1, 751 particles for a total of 8 × 10 7 time-steps takes around one week. For this reason, we introduce two coarse-graining techniques: (1) pseudo particles and (2) scaling of the time.
One approach to reducing the amount of particles is to enlarge their reaction-surface and decrease their number. Every particle has a radius r i and a surface where it interacts with other particles. As particles are assumed to be spherical, this surface is 4 3 πr 2 i . A number of particles can be merged to one bigger one: where N i is the number of particles of type i,N i is the reduced number andr i is the increased radius of pseudo-particle i. Under the constraint that the reaction surface must be conserved, the formula for the new radius is given by It is also possible to reduce the time in the same manner. A process taking time T can be reduced toT using a factor f t so that To ensure that all reactions reach the steady state in timeT (if such a state exists), every reaction rate has to be multiplied by the factor f : In a reaction system with relevant spatial characteristics, the time-scaling means that a given particle may no longer be able to cover the distance d between point P = (p x , p y , p z ) and Q = (q x , q y , q z ). Thus, the diffusion coefficients for each type of particle must also be modified. In n-dimensional space the mean-square-displacement (MSD) is an estimate for the average distance a particle has moved and is calculated using where d = dimension of the simulation space The square root of the MSD is used as the mean distance that the particle has moved from its origin and this dist i must also conserved through the time-scaling process: This theoretical result is valid given the limited range of diffusion coefficients of ≈ 0.5 − 30µm 2 s −1 . If the compressionfactor f exceeds 100 (which is quite a low factor) then the diffusion coefficient can no longer be adapted in this way, because the displacements of particles in each time-step would become large and the system would become unstable.
An alternative way to make the particle move from P to Q in the scaled timeT is to decrease the distance between the points by scaling the space (the reaction volume). A space of dimensions X, Y, Z can be transformed using where every individual point is transformed according to This transformation represents a consistent spatial dilation of all axis by an as yet unknown factor f s . Given that dilatations are length-conserving, the distance in the new space is scaled directly by the factor f s according to One distance of interest is the MSD (described above), which gives the average displacement of a particle by diffusion. The relation between this distance and the time-compression-factor is Those two scalings in time and space ensure that a particle which took δ seconds to move a distance η in the original system, needs f t δ seconds to cover a distance from f s η in the scaled system.  Figure S2. Shown is the schematic reaction volume used for the spatially-stochastic simulation. The whole nuclear space with a radius of 6µm is modeled, but all species are hold in the green area. This space exclusion is necessary to guarantee fast turnover from O-Mad2 to C-Mad2. All our model only take place in the green area, which can be seen as a well-mixed soup.  Kinetochores initial amount is 92 and they do not diffuse. Other species particles start from zero. Their mass and diffusion coefficient combine from the basic blocks.