A Mathematical Framework for Kinetochore-Driven Activation Feedback in the Mitotic Checkpoint
AA Mathematical Framework for Kinetochore-DrivenActivation Feedback in the Mitotic Checkpoint
Bashar Ibrahim
Department of Mathematics and Computer Science, University of Jena, Ernst-Abbe-Platz 2, 07743 Jena
Abstract
Proliferating cells properly divide into their daughter cells through a process thatis mediated by kinetochores, protein-complexes that assemble at the centromere ofeach sister chromatid. Each kinetochore has to establish a tight bipolar attachmentto the spindle apparatus before sister-chromatid separation is initiated. The SpindleAssembly Checkpoint (SAC) links the biophysical attachment status of the kineto-chores to mitotic progression, and ensures that even a single misaligned kinetochorekeeps the checkpoint active. The mechanism by which this is achieved is still elusive.Current computational models of the human SAC disregard important biochemicalproperties by omitting any kind of feedback loop, proper kinetochore signals, andother spatial properties such as the stability of the system and diffusion effects. Toallow for more realistic in silico study of the dynamics of the SAC model, a minimalmathematical framework for SAC activation and silencing is introduced. A nonlin-ear ordinary differential equation model successfully reproduces bifurcation signal-ing switches with attachment of all 92 kinetochores and activation of APC/C bykinetochore-driven feedback. A partial differential equation model and mathemat-ical linear stability analyses indicate the influence of diffusion and system stability.The conclusion is that quantitative models of the human SAC should account for thepositive feedback on APC/C activation driven by the kinetochores which is essentialfor SAC silencing. Experimental diffusion coefficients for MCC sub-complexes arefound to be insufficient for rapid APC/C inhibition. The presented analysis allowsfor systems-level understanding of mitotic control and the minimal new model canfunction as a basis for developing further quantitative-integrative models of the celldivision cycle. a r X i v : . [ q - b i o . S C ] J un Introduction
Monitoring the fidelity of chromosome segregation during the cell life cycle relies ontransition control mechanisms, called checkpoints, which ensure that all criteria are metbefore moving on irreversibly to the next phase [49, 23]. The major control mechanismin mitosis is called the Spindle Assembly Checkpoint (SAC; sometimes referred to as themitotic checkpoint [52]). The SAC ensures that all chromosomes are properly attachedto spindle microtubules through their kinetochores. Even a single unattached or mis-attached kinetochore (out of 92 in a human cell) is sufficient to keep the SAC engaged,yet the mechanism by which this is achieved is still elusive [51, 50]. A malfunction inthe SAC process can cause aneuploidy and lead to tumorigenesis [55, 18, 6].Biochemically, it is thought that unattached or misaligned kinetochores catalyzethe formation and broadcasting of a ‘wait’ signal to the environment (cf. Fig. 1, R1-R5). This counters the activation of the ubiquitin ligase anaphase promoting com-plex/cyclosome (APC/C) by Cdc20. APC/C activity is thought to be inhibited in mul-tiple ways. It can be directly prevented by a potent inhibitor, the Mitotic CheckpointComplex (MCC), which consists of the four checkpoint proteins Mad2, BubR1, Bub3,and Cdc20 (cf. Fig. 1, R7-R8). Furthermore, it has been suggested that MCC sub-complexes (such as BubR1 and Mad2:Cdc20) interact with APC/C [12, 17]. Anotherinhibitor, called the mitotic checkpoint factor 2 (MCF2), is associated with APC/C inthe checkpoint arrested state but its composition is unknown [10] (cf. Fig. 1, R7).Recent compelling data shows that the MCC itself can bind to Cdc20 that has alreadybound to APC/C or to a free Cdc20 [31] (cf. Fig. 1, R6). With the exception ofMCF2, all complexes inhibiting APC/C rely on the presence of unattached kinetochoresfor sufficiently rapid formation [23]. Immediately following proper attachment of thefinal kinetochore to the microtubules, inhibitors rapidly dissolve, ultimately resulting inactive APC/C (cf. Fig. 1, R9). APC/C that has been activated by Cdc20 contributesto the degradation of Cyclin B, which is an essential requirement for mitotic exit [23](cf. Fig. 1, R10).Simultaneously, APC/C:Cdc20 tags Securin for degradation by proteasome. Securinbinds and inhibits Separase, a protease required to cleave Cohesin, which is the glueconnecting the two sister-chromatids of each chromosome (cf. Fig. 1, R11). Thus,activation of APC/C by Cdc20 initiates sister-chromatid separation, which marks thetransition to mitotic exit. The APC/C activation mechanism is known as SAC silencing,in which APC/C itself plays a role, driven by kinetochores, in the disassembly of its owninhibitors [64, 63]. Additionally, many proteins are also involved in the silencing process,such as P31 comet , UbcH10, and Dynein (see the review [23]).Both spindle assembly checkpoint activation and silencing are complex and it is dif-ficult to observe their spatial features experimentally. Each SAC component has variouslocalizations and states upon which the interactions depend. And a single small hu-man kinetochore (radius ∼ . µ m) has to inhibit all APC/Cs in the cell (average radius ∼ µ m), and after the last attachment, the inhibitors have to be switched off rapidly.Also, kinetochores have dynamical complex structure which change over the cell-cycle2hases. Mitotic regulation is also challenging theoretically as computational methodscan be hindered by the combinatorial explosion in the number of intermediate compo-nents (complexes) and explicit representations. Also, the different components ofteninteract nonlinearly in space and time, and in the presence of various feedback loops,which lead to remarkable phenomena that are difficult to predict [15, 28, 36, 16, 14, 61].Mathematical models have helped to illuminate fundamental modules of the SAC con-trols [8, 53, 21, 40, 20, 22, 27, 45, 37, 25, 13, 24, 26, 29, 3]. These models yet conceivedSAC activation on details molecular level in order to distinguish between different path-ways. However, none of these models have considered SAC silencing, included importantproperties such as feedback loops (e.g. the APC/C positive feedback activation loop) orhave been subjected to stability analyses. In this study, a concise model for both SACactivation and silencing in humans was engineered, accounting for all kinetochore signalsand also the APC/C activation loop. Ordinary differential equation (ODE) model sim-ulations were computed, along with a single parameter bifurcation analysis. The effectsof important parameters on the dynamics of the system were then studied, followed by apartial differential equation (PDE) model and its linear stability with various parametervalues. The SAC model shown in Fig. 1 was manually reduced to a core model that containsthree biochemical reaction equations (Eqs. (1-3)) describing the dynamics of the follow-ing four species: I (Inhibitor production like Mad2 or Cdc20 or BubR1), I* (cf. MCC),I*:APC/C (or MCC:APC/C), and APC/C. Four additional additional reactions (Eqs.(4-7)) were incorporated to represent the production and degradation of both CyclinB and Securin. Mathematically, these additional reactions do not affect the overalldynamics. The full biochemical reaction rules describing the SAC system are:[I] k , Un.-Kin −−−−−−−−−−−−− (cid:42) [I*] (1)[I*] + [APC/C] k −−−−−−−−−−−−− (cid:42)(cid:41) −−−−−−−−−−−−− k − [I*:APC/C] (2)[I*:APC/C] k , Att.-Kin , [APC/C] −−−−−−−−−−−−−− (cid:42) [I] + [APC/C] (3) k −−−−−−−−−−−−− (cid:42) [Cyclin B] (4) k −−−−−−−−−−−−− (cid:42) [Securin] (5)[Cyclin B] k , [APC/C] −−−−−−−−−−−−− (cid:42) (6)[Securin] k , [APC/C] −−−−−−−−−−−−− (cid:42) (7)Where Att.-Kin and Un.-Kin refer to attached and unattached kinetochores, respec-tively. The APC/C positive feedback-loop that is driven by the kinetochore is incorpo-3 ad1 C-Mad2 O-Mad2*Mad1 C-Mad2 O-Mad2* Cdc20C-Mad2
BubR1Bub3
Cdc20C-Mad2 BubR1Bub3
MCC
APC/C
Inhibitors production (SAC On) APC/C activation (SAC Off)
BubR1Bub3
R2 R1R3 R5R4 R9 U n a tt ac h e d K i n e t o c ho r e APC/C inhibition (SAC On) Mitotic Exit SAC Off
APC/C R8 Cdc20C-Mad2 BubR1Bub3 R6 Cyclin B
Cyclin B
Securin
Separase
Securin
Separase
R10 R11 I m APC/C I m I I I I APC/C R7 APC/C I n I n APC/C
Figure 1: Schematics illustrating the intracellular signaling of spindle assembly check-point activation and silencing.
Inhibitors production (red box):
The protein Mad2 ispresent in two stable conformations differing in the spatial arrangement of a ‘safety-belt’that is either open (O-Mad2) or closed (C-Mad2) [5, 42]. O-Mad2 is able to tran-siently bind Cdc20 (R1). Meanwhile, O-Mad2 is recruited to unattached kinetochoresby Mad1-C-Mad2 to form the ternary complex Mad1:C-Mad2:O-Mad2* [5, 42], whichcan bind Cdc20 efficiently, and switches to a closed conformation upon Cdc20-binding(R2-R3). The resulting complex, Cdc20:C-Mad2, together with BubR1:Bub3, formsthe tetrameric mitotic checkpoint complex (MCC, R4), which is a potent inhibitor ofAPC/C. The Cdc20 also binds the BubR1:Bub3 complex independently (R5). MCC,which contains Cdc20 as a subunit, can bind an additional free Cdc20 (R6).
APC/Cinhibition (yellow box):
The symbol I n refers to APC/C potential inhibitors (cf. MCC,Mad2:Cdc20, bubR1:bub3, and MCF2; see R7). The I m symbol denotes the ability ofthe MCC complex to additionally inhibit Cdc20 that is already bound to APC/C (R8). APC/C activation (green box):
When all 92 kinetochores are correctly attached fromopposite poles to the mitotic spindle, the SAC is switch off and APC/C is activated byCdc20 via the APC/C:Cdc20 complex (R9).
Mitotic Exit (blue box):
Active APC/Ccontributes to the degradation of mitotic Cyclin B (R10), and causes Securin (buddingyeast Pds1) to be tagged for degradation by the proteasome, making Seprase fully active(R11). 4ated in Eq. (3.
The reaction rules (Eqs. (1-7)) can be translated into sets of time-dependent nonlinearordinary differential equations (ODEs). The translation is done by applying the generalprinciples of mass-action kinetics, computing dS/dt = N v ( S ) with state vector S , fluxvector v ( S ) and stoichiometric matrix N . This results in one time-dependent ODE foreach species such as Cyclin B or Securin. For example the equation for Cyclin B wouldbe: d[Cyclin B] / dt = k − k [APC/C][Cyclin B]. Adding a diffusion term to each differential equation transforms the system into a set ofcoupled partial differential equations (PDEs) known as the
Reaction-Diffusion system ,that are functions of space and time and the following general form: ∂ [ C i ] ∂t = D i ∇ [ C i ] (cid:124) (cid:123)(cid:122) (cid:125) Diffusion + R j ( { [ C i ] } ; P) (cid:124) (cid:123)(cid:122) (cid:125) Reaction . (8)Where [ C i ] denote concentrations for species i = { , ..., } . On the right hand side,the first term refers to the diffusion and the second term represents the biochemicalreactions R j = { R , ..., R } for species i . The constant coefficient D i represents itsspecies diffusion. t is time and P represents phenomenological parameters. The operator ∇ refers to the spatial gradient in spherical coordinates ( (cid:126) ∇ ( r, θ, ϕ ) = ∂∂r (cid:126)e r + r ∂∂θ (cid:126)e θ + rsin ( θ ) ∂∂ϕ (cid:126)e ϕ ). Recent mathematical models of the SAC mechanism show that sphericalsymmetry can be used without loss of generality [27, 20], and this form is used here.The system in Eq. (8) reduces to the following PDEs depending on t and r : ∂ [ C i ] ∂t = Dr ∂∂r ( r ∂ [ C i ] ∂r ) (cid:124) (cid:123)(cid:122) (cid:125) Diffusion + R j ( { [ C i ] } ; P) (cid:124) (cid:123)(cid:122) (cid:125) Reaction . (9) The ODE models were implemented in the freely-available software package XPPAUT [9]and integrated using a stiff solver. The bifurcation analyses were computed with AUTO[7] using the XPPAUT interface. The PDE reaction-diffusion systems were implementedin matlab (MathWorks) and simulated using the pdepe function. The pdepe solver canhandle systems of parabolic and elliptic PDEs in one space variable and time, perfectlymatching the presented SAC model. This function uses the method of lines, spatiallydiscretizing the problem in space and converting it to a system of ordinary differentialequations that can be solved using the numerical stiff solver ode15s in Matlab [56]. Theode15s ODEs solver can solve the differential-algebraic systems that frequently arise inPDE systems. 5he same initial concentrations were applied to all models for comparability and con-sistency (Table 1). The specific values were chosen according to data from the literature([60, 57, 11, 19, 34, 48, 58, 38, 59]). The kinetic rate constants were also taken from lit-erature when available (e.g. [33, 41, 5, 43, 24]; Table 1). For the rate constants ( k , and k , see Table 1), several computations with values representing the whole physiologicallyreasonable parameter range were compared (see Fig. 2D and Fig. 3). The trajectoriesare discussed in corresponding figures. In the PDE simulations, the cell is taken to be a sphere with radius R . The lastunattached kinetochore is modeled as a sub-sphere (radius r ) located in the center ofthe cell. Boundary conditions are assumed to be reflective and the numbers of all inter-acting elements are assumed to be conserved. The PDEs are assumed to be sphericallysymmetric. Reactions are considered to follow the mass-action-kinetics law and all rateconstants are listed in (Table 1). The initial conditions for the PDEs are correspond touniform distribution. The reason is that APC/C is found experimentally to be localizedat kinetochores, spindles and poles [62, 35] while MCC is found to be localized in boththe kinetochores and the cytosol [29]. Thus it is most appropriate to assume a uniformdistribution. 6 Results and Discussions
The SAC mechanism (Fig. 1) consists of four modules: Inhibitors production (Fig. 1red box), APC/C inhibition (Fig. 1 yellow box), APC/C activation (Fig. 1 green box),and mitotic exit (Fig. 1 blue box). This mechanism was reduced to a minimal coremodel (shaded area shown in Fig. 2A) comprising three reaction equations (R1-R3)that describe the dynamics of four main species: I (represents either Mad2 or Cdc20or BubR1), I* (cf. MCC), I*:APC/C (or MCC:APC/C), and APC/C. These reactionswere translated into a set of coupled nonlinear ordinary differential equations under theassumption of mass action kinetics for all reactions. In order to further reduce the model,the following biochemical assumptions were assumed. The total concentration of APC/Cin the system is constant and can be calculated using [APC/C] tot = [
AP C/C ] + [ I ∗ : AP C/C ]. The same is true for I , using [I] tot = [I] + [I*] + [I*:APC/C]. For simplicity,the total amount of I* was taken to be [I*] tot =[I*]+[I*:APC/C]. The core SAC systemcan be written as the following pair of nonlinear ODEs:d[I*] tot d t = k .U ([I] tot − [I*] tot ) − k .A [APC/C][I*:APC/C]) (10)d[APC/C]d t = − k [I*][APC/C] + ( k − + A [APC/C])[I*:APC/C] (11)Finally, an extended model is generated by adding Cyclin B, Securin, and also presumingthat APC/C is in steady state, so that putting the total concentration of [APC/C] tot into the system gives the following full ODE-based model:d[I*] tot d t = k .U ([I] tot − [I*] tot ) − k .A ([APC/C] tot − [I*:APC/C])[I*:APC/C])(12)[I*:APC/C] = − b ± √ b − ac a (13)d[Cyclin B] tot d t = k − k [APC/C][Cyclin B] (14)d[Securin] tot d t = k − k [APC/C][Securin] (15)where, a = − k − Ab = k [I*] + k [APC/C] tot + k − + A. [APC/C] tot c = − k tot And U refers to the number of unattached kinetochores that have an additional ODEequation: d U/ dt = − α.U . A refers to the number of attached kinetochores and is equal7o A = 92 − U . It is clear that the addition of Cyclin B and/or Securin does not affectthe system’s behavior because these species depend on the active APC/C level.The results of numerical simulations (see Section 2.4) are shown in Fig. 2B. Theyare consistent with experimental findings in terms of APC/C level and the timing ofanaphase [1, 40]. The concentration of APC/C (Fig. 2B brown line) is very low evenwhen a single kinetochore is not attached. After around 18 minutes, the APC/C ac-tivity increases rapidly to reach its maximum value. The complexes I*:APC/C (orMCC:APC/C) exhibit behavior that is opposite to that of APC/C (Fig. 2B, red line).I* concentration follows I*:APC/C with a difference only in the initial concentration(Fig. 2B, blue line).A one parameter bifurcation analysis was performed for the nonlinear ODE-basedsystem (Eqs. 12 and 13). Simulations of the bifurcation curves were performed usingthe AUTO software (Section 2.4). The desired behavior is a bistable switch influencingthe total I* as kinetochores are attached one-by-one. Fig. 2C shows the result, whichis a typical S-shape in the number of attached kinetochores as a function of the totalconcentration of the inhibitor, I*. Solid lines refer to stable nodes while dashed linesrefers to the unstable saddles. Stable and unstable states meet at saddle node bifurcationpoints that are indicated by solid circles. When nearly all kinetochores are attached(91.85 kinetochores), the SAC checkpoint turns off and APC/C is activated rapidly.Total I* falls back to zero as the cell exits from mitosis. The black line indicates howthe switch alternates between the SAC-active and SAC-inactive states as the number ofattached kinetochores increases. All parameters were taken from Table 1.In order to determine the sensitivity to parameter values, various ranges of param-eter values were examined (Fig. 2D and Fig. 3). The bifurcation diagram is mostsensitive to k values. The working range is between 0.01 s − and 30 s − . As k in-creases, the bifurcation point and curve are shifted to the right (Fig. 2D). Low valuesof k (c.f. 0.01 s − ) initiate the SAC switch before 83 kinetochores are attached, whichcould cause grave risk to the cell. All other k values higher than 0.5 s − generate a safeand appropriately-timed switch. k values have some minor effects on the system (Fig.3). Its working values are between 0.01 s − ad 100 s − . Increasing k value to 100 canshift the bifurcation point higher to meet at higher amount of total I*.8 S pe c i e s C on c en t r a t i on u M T i m e i n m i nu t e s
0 10 20 30 40 50
I*T
I*:APC/C
APC/C
Cyclin BSecurin A N u m be r o f a tt a c hed k i ne t o c ho r e s
91 920.20.1 [I * ] t o t C on c en t r a t i on u M SA C on SA C o ff B N u m be r o f a tt a c hed k i ne t o c ho r e s
83 88 90 91 92 [I * ] t o t C on c en t r a t i on u M k values APC/C I * APC/C I * I Cyclin B
Cyclin B
SecurinSecurin
R6 R7R4 R5
C D k k k1= Figure 2: Spindle Assembly Checkpoint ODE model. (A) Biochemical reaction networkfor SAC activation and silencing. I is the inhibitor producing, for example, BubR1, Mad2and Cdc20. I* is a potent APC/C inhibitor which mainly refers to the MCC complexin this model. Production of I* is enhanced by signals from unattached kinetochores.This is SAC activation (or APC/C inhibition). Directly following attachment of the finalkinetochore to the spindle microtubules, the inhibitor dissolves and APC/C is activated.This is SAC silencing (or APC/C activation). A positive feedback loop takes place viaAPC/C itself, which is driven by kinetochores. Nodes represent core SAC proteins orcomplexes, edges refer to their interactions. (B) Numerical solution of the ODE modelshowing the concentration of each SAC component versus time. Once all kinetochoresare attached (at 17 minutes), APC/C is activated. Active APC/C tags both Securin andCyclin B for degradation. Cyclin B is degraded in two phases, first by APC/C:Cdc20 andsecond via APC/C:Cdh1 during mitotic exit. Thus Cyclin B level does not reach zero inour simulation, which is consistent with the literature [30]. All parameter values are setaccording to Table 1 (see the text for more details). (C) Single parameter bifurcationcurve showing kinetochore signals versus total I*. Unstable saddle points are shown bydashed lines and stable node points by solid lines. Both stable and unstable states meetat saddle node bifurcation points shown by solid circles. The SAC checkpoint is releasedand APC/C activated only when almost all kinetochores are attached (approximately91.98). As the cell enters anaphase, I* falls back to zero. The black line indicates how theswitch flips from the SAC-active state to the SAC-inactive state as number of attachedkinetochores increases. (D)The bifurcation curves are sensitive to k values (see text fordetails). 9 u m be r o f a tt a c hed k i ne t o c ho r e s
89 90 91 92 00.20.1 [I * ] t o t C on c en t r a t i on u M {0.005, 0.01, 0.05, 0.1, 0.5) k values k3= Figure 3: Sensitivity to k values. Bifurcation curve for the unattached kinetochoresignals versus the total I ∗ level. Different curves shown the sensitivity of the analysisto k values. Unstable saddle points are shown by dashed lines and stable nodes pointsby solid lines. Both stable and unstable states meet at saddle-node bifurcation pointsshown by solid circles. Several studies have indicated the importance of diffusion in the SAC mechanism (e.g.[8], [20]). However, these models represented either smaller budding yeast cells usingPDEs or detailed SAC activation using ODEs. Here, a minimal core SAC model repre-sented by PDEs is examined.Cyclin B and Securin reactions are eliminated because these reactions are the outputof active APC/C and cannot affect APC/C regulation. To this end, a second derivativediffusion term was added to the SAC system (Eq.11-Eq.10), which leads to a set of cou-pled partial differential equations known as a
Reaction-Diffusion system with sphericalsymmetry (Section 2.3):d[I*] tot d t = Dr ∂∂r ( r ∂ [I*] tot ∂r ) + k .U ([I] tot − [I*] tot ) − k .A [APC/C][I*:APC/C]) (16)d[APC/C]d t = Dr ∂∂r ( r ∂ [ AP C/C ] ∂r ) − k [I*][APC/C] + ( k − + k .A [APC/C])[I*:APC/C](17)These are supplemented by reflective (Neumann) boundary conditions and assumptionsof spherical geometry for the cell and kinetochores (Section 2.5). Numerical simulationsof the PDE system (Eq.16-Eq.17) were conducted in matlab. The diffusion coefficientfor APC/C is known from the literature (Table 1) while that of MCC (or I ∗ ) is not.Therefore the simulations were repeated using four different values for the MCC diffusion10oefficient, (Table 1 and Fig. 4A-D). One of these values (referred to here as ‘realistic’) ischosen with reference to experimental work on MCC subunits (cf. Mad2, BubR1, Bub2,and Cdc20; see Table 1). The other values span a wide range to allow any effects ofdiffusion to be observed. Following the idea by Doncic et. al [8], only the last unattachedkinetochore was considered. Hence, the parameter U was assumed to be constant andto refer only to the last unattached kinetochore (cf. U = 1, and A = 92 − U = 91). Theaim was to reveal, if the last kinetochore is not yet attached, how fast MCC is able toinhibit high levels of APC/C in the cell.The simulations show that using a realistic diffusion coefficient for MCC (3 µm s − ),APC/C is inhibited very slowly after an hour (Fig. 4C). Using lower diffusion coefficients(1 µm s − ), only 10% of the APC/C level is inhibited (Fig. 4B) while APC/C is notinhibited at all (Fig. 4A) when using an even lower value (0.1 µm s − ) . Using a diffu-sion coefficient higher than the realistic value (10 µm s − ) achieves proper inhibition ofAPC/C (Fig. 4D). The conclusions are that a realistic diffusion coefficient is sufficientfor inhibiting APC/C, albeit slowly, but a coefficient 2-3 times larger is required forproper SAC functioning. Thus, it is recommended to use a SAC model based on a PDEmodel, rather than solely on ODEs, though it is not essential.11 AP C / C C on c en t r a t i on u M T i m e i n S e c ond D i s t a n c e i n u m AP C / C C on c en t r a t i on u M T i m e i n S e c ond D i s t a n c e i n u m AP C / C C on c en t r a t i on u M T i m e i n S e c ond D i s t a n c e i n u m AP C / C C on c en t r a t i on u M T i m e i n S e c ond D i s t a n c e i n u m AC D = 3 uM2s-1 D = 10 uM2s-1D = 0.1 uM2s-1 D = 1 uM2s-1 BD Figure 4: Partial differential equation model simulations. Simulation diagrams (A-B)are based on low diffusion coefficient values (0.1 µm s − and 1 µm s − ), while (C-D) arebased on realistic and higher values (3 µm s − and 10 µm s − ).12 .3 Linear Stability Analysis As shown in the previous section, a realistic value for the diffusion coefficient does nothave a strong effect on the SAC model. However, this does not exclude the possibilitythat diffusion influences properties of the system such as its stability. To study this, alinear stability analysis was performed on the SAC PDE model (Eqs.16-17).Taking [APC/C] tot = [APC/C] + [I*:APC/C], and [I*] tot =[I*]+[I*:APC/C] (or equiv-alently [I*]=[I*] tot − [APC/C] tot - [APC/C]), the system will be:d[I*] tot d t = Dr ∂∂r ( r ∂ [I*] tot ∂r )+ k .U ([I] tot − [I*] tot ) − k .A [APC/C]([APC/C] tot − [APC/C])(18)d[APC/C]d t = Dr ∂∂r ( r ∂ [ AP C/C ] ∂r ) − k ([I*] tot − [APC/C] tot − [ AP C/C ])[APC/C]+( k − + k .A [APC/C])([APC/C] tot − [APC/C]) (19)Assuming small perturbations on I* and APC/C in standard form (see for example[39]): [I*] tot ( r, t ) = [I*] tot ( r ) + [I*] tot ( r, t ) (20)[APC/C]( r, t ) = [APC/C] ( r ) + [APC/C] ( r, t ) (21)where [I*] and [APC/C] denote the steady state, and [I*] and [APC/C] denote theunsteady-state (or disturbance). Substituting Eqs.20-21 into the PDE system (Eqs.16-17) leads to two separate systems. The linearized equations governing the disturbancesystem ared[I*] tot d t = Dr ∂∂r ( r ∂ [I*] tot ∂r ) − β ([I*] tot ) − k .A ([APC/C] [APC/C] tot +2[APC/C] [APC/C] + [APC/C] ) (22)d[APC/C] d t = Dr ∂∂r ( r ∂ [APC/C] ∂r ) + ( β [APC/C] − k [I*] tot )[APC/C] + β − ( k + k .A )[APC/C] (23)where, β = k .U , [APC/C] tot is constant, β = ( k [ I ∗ ] tot + ( k + k .A )[APC/C] tot ),and β = k − [APC/C] tot . Note that for hydrodynamic stability any second order termsin the disturbance system should be neglected [39]. Since the diffusion coefficient for I*(MCC complex) is unknown, the focus is on I*.The solution is assumed to be a propagating wave, dependent on both space andtime, and given by [ I ∗ ] tot = I e iγr − ct , and [ AP C/C ] =
AP C/C e iγr − ct . γ is the wave13umber and C is the complex speed of the wave C = C r + iC i where C i is the importantparameter for stability analysis. The disturbance is damped if C i is negative, whereasfor C i >
0, the disturbance will grow and lead to instability. After simplification, therelation becomes: C i = − k .U − k .A ∗ (( AP C/C ) /I ) − D.γ < The spindle assembly checkpoint (SAC) regulates the timing of chromosome segregationto prevent the creation of aneuploidy and tumorigenesis [55, 18, 6]. SAC is highlysensitive to kinetochore-microtubule attachment, such that even a single unattachedor mis-attached kinetochore can maintain checkpoint activation and keep the APC/Ccomplex inactive. The checkpoint is only silenced when all chromosomes are properlyattached to microtubules, after which APC/C is activated and anaphase onset occurs.The mechanism by which SAC activation and silencing is achieved is still largely elusive.Despite several systems-level mathematical efforts in recent years to model SAC ac-tivation, no model has been built with the level of detail in its mathematical abstractionthat is crucial for further developments [23]. For example, no model has consideredall kinetochore signals. In addition, to the best of our knowledge, there is no modelof the SAC that combines both SAC activation and silence, or one that includes anykind of feedback. This is the first work presenting a mathematical analysis that inte-grates both SAC activation and silencing. The model is minimal in that it containsonly four reactions and four species, considers all 92 kinetochore signals, and incorpo-rates kinetochore-driven feedback on APC/C activation (for SAC silencing). The modelincorporates differing levels of detail. The ODE-based model was simulated and gen-erated a bifurcation diagram realistic switch behavior over a wide range of parameters.The model was also compared with experimental findings and found to be consistent(Table 2).Parameters were selected carefully. All models have the same initial concentrationswhich were chosen according to data from the literature (Table 1). The reaction rateconstants were also taken from literature where they were known. For two rate constants k , and k , several computations were performed using values representing the wholephysiologically reasonable parameter range (see Fig. 2D and Fig. 3).It had been thought that diffusion is required for proper SAC functioning (e.g. [23]).To investigate the role of diffusion, a second level of detail was implemented in a PDE-based reaction-diffusion model. Active transport (or convection) was not consideredbecause there is no experimental evidence for transport of either APC/C or MCC. Ac-tive transport is essential for inhibitor formation, which has been studied extensively inthe literature and is therefore not considered here. It has been shown both experimen-tally and mathematically that active transport of O-Mad2 towards the spindle mid-zone14ncreases the efficiency of Mad2-activation in inhibiting Cdc20 [27, 19, 54]. The diffu-sion coefficient for APC/C is known [65]. The MCC diffusion coefficient is not knownbut was approximated based on its sub-components (Mad2, Cdc20, BubR1 and Bub3).This value of (3 µm s − ) was found to be insufficient for rapid APC/C inhibition. FastAPC/C inhibition, and therefore proper SAC functioning, was found to require a valuethat was at least twice as high. This may be due to noise in the experimental dataor because there is an additional mechanism that increase the rate independent of thekinetochores ([29]).The mathematical modeling and analysis presented here can serve as a basis formore sophisticated models be used to evaluate novel hypotheses related to mitosis andthe cell cycle. A combination of further experimental work and mathematical analysisbe necessary to fill in the gaps in our understanding of the cell life cycle.15 cknowledgements This work was supported by the European Commission HIERATIC Grant 062098/14.The author gratefully acknowledges the visiting fund of the Institute for NumericalSimulation (INS) at Bonn University.
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Initial amountAPC/C 0.09 µM µM µM µM [60, 57]I*(cf. MCC) 0.15 µM µM µM µM [22]I 0.15 µM µM µM µM [11, 19,34]I*:APC/C 0 0 0 0Cyclin B 0.2 µM µM µM µM [48, 58,38]Securin 0.18 µM µM µM µM [59]U 92 92 92 92 [24, 23]A 92-U 92-U 92-U 92-U [24, 23]Diffusion coefficientsAPC/C 1.8 µm s − [65]I* (cf. MCC) 0.1-10 µm s − ThisstudyI (Cdc20) 19.5 µm s − [65]I (Mad2) 5 µm s − [20]I(Mad2:Cdc20) 4 µm s − [20]I(BubR1:Bub3) 7.9 µm s − [60, 11]EnvironmentRadius of thekinetochore 0.1 µm [4]Radius of thecell 10 µm [27]Rate constants α . s − . s − . s − . s − k s − . − s − s − s − Thisstudy k µ M − s − µ M − s − µ M − s − µ M − s − [24] k − . s − . s − . s − . s − [24] k . s − . s − . − . s − . s − Thisstudy k . s − . s − . s − k . s − . s − . s − k . s − . s − . s − [33, 41] k . s − . s − . s − [33, 41]Diffusion for MCC was calculated using D A.B = D A .D B D A + D B , where D A and D B are thediffusion coefficients for A (Mad2:Cdc20) and B (BubR1:Bub3), respectively.22able 2: Comparison of the in vitro and in silico mutation experiments withrespect to APC/C activityTypeof mu-tation Experiment in-vitro in-silico