Mathematical analysis and modeling of DNA segregation mechanisms

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April 2018, 15(2): 429-440. doi: 10.3934/mbe.2018019

Mathematical analysis and modeling of DNA segregation mechanisms

Bashar Ibrahim ^,

Department of Mathematics and Computer Science, University of Jena, Ernst-Abbe-Platz 2,07743 Jena, Germany

* Corresponding author: Bashar Ibrahim.

Received January 02, 2017 Accepted May 04, 2017 Published June 2017

The precise regulation of cell life division is indispensable to the reliable inheritance of genetic material, i.e. DNA, in successive generations of cells. This is governed by dedicated biochemical networks which ensure that all requirements are met before transition from one phase to the next. The Spindle Assembly Checkpoint (SAC) is an evolutionarily mechanism that delays mitotic progression until all chromosomes are properly linked to the mitotic spindle. During some asymmetric cell divisions, such as those observed in budding yeast, an additional mechanism, the Spindle Position Checkpoint (SPOC), is required to delay exit from mitosis until the mitotic spindle is correctly aligned. These checkpoints are complex and their elaborate spatiotemporal dynamics are challenging to understand intuitively. In this study, bistable mathematical models for both activation and silencing of mitotic checkpoints were constructed and analyzed. A one-parameter bifurcation was computed to show the realistic biochemical switches considering all signals. Numerical simulations involving systems of ODEs and PDEs were performed over various parameters, to investigate the effect of the diffusion coefficient. The results provide systems-level insights into mitotic transition and demonstrate that mathematical analysis constitutes a powerful tool for investigation of the dynamic properties of complex biomedical systems.

Keywords: Mathematical biology, modeling and simulation, mitotic control, spindle assembly checkpoint, spindle position checkpoint.

Mathematics Subject Classification: Primary: 46N60, 34A09, 35K57, 32W50; Secondary: 46N20.

Citation: Bashar Ibrahim. Mathematical analysis and modeling of DNA segregation mechanisms. Mathematical Biosciences & Engineering, 2018, 15 (2) : 429-440. doi: 10.3934/mbe.2018019

References:

[1]	S. F. Bakhoum, G. Genovese and D. A. Compton, Deviant kinetochore microtubule dynamics underlie chromosomal instability, Curr Biol, 19 (2009), 1937-1942. doi: 10.1016/j.cub.2009.09.055.
[2]	A. K. Caydasi, B. Ibrahim and G. Pereira, Monitoring spindle orientation: Spindle position checkpoint in charge Cell Div 5 (2010), p28. doi: 10.1186/1747-1028-5-28.
[3]	A. K. Caydasi, B. Kurtulmus, M. I. L. Orrico, A. Hofmann, B. Ibrahim and G. Pereira, Elm1 kinase activates the spindle position checkpoint kinase Kin4, J Cell Biol, 190 (2010), 975-989. doi: 10.1083/jcb.201006151.
[4]	A. K. Caydasi, M. Lohel, G. Grünert, P. Dittrich, G. Pereira and B. Ibrahim, A dynamical model of the spindle position checkpoint Mol Syst Biol 8 (2012), p582. doi: 10.1038/msb.2012.15.
[5]	L. M. Cherry, A. J. Faulkner, L. A. Grossberg and R. Balczon, Kinetochore size variation in mammalian chromosomes: An image analysis study with evolutionary implications, J Cell Sci, 92 (1989), 281-289.
[6]	E. J. Doedel, AUTO: A program for the automatic bifurcation analysis of autonomous systems, Congr Numer, 30 (1981), 265-284.
[7]	A. Doncic, E. Ben-Jacob and N. Barkai, Evaluating putative mechanisms of the mitotic spindle checkpoint, Proc Natl Acad Sci U S A, 102 (2005), 6332-6337. doi: 10.1073/pnas.0409142102.
[8]	B. Ermentrout, Simulating, Analyzing, and Animating Dynamical Systems: A Guide to Xppaut for Researchers and Students (society for industrial and applied mathematics, philadelphia), 2002. doi: 10.1137/1.9780898718195.
[9]	D. Görlich, G. Escuela, G. Gruenert, P. Dittrich and B. Ibrahim, Molecular codes through complex formation in a model of the human inner kinetochore Biosemiotics 7 (2014), p223. doi: 10.1007/s12304-013-9193-5.
[10]	G. Gruenert, B. Ibrahim, T. Lenser, M. Lohel, T. Hinze and P. Dittrich, Rule-based spatial modeling with diffusing, geometrically constrained molecules BMC Bioinf 11 (2010), p307. doi: 10.1186/1471-2105-11-307.
[11]	G. Gruenert, J. Szymanski, J. Holley, G. Escuela, A. Diem, B. Ibrahim, A. Adamatzky, J. Gorecki and P. Dittrich, Multi-scale modelling of computers made from excitable chemical droplets, IJUC, 9 (2013), 237-266.
[12]	R. Henze, J. Huwald, N. Mostajo, P. Dittrich and B. Ibrahim, Structural analysis of in silico mutant experiments of human inner-kinetochore structure, Bio Systems, 127 (2015), 47-59. doi: 10.1016/j.biosystems.2014.11.004.
[13]	B. Ibrahim, In silico spatial simulations reveal that MCC formation and excess BubR1 are required for tight inhibition of the anaphase-promoting complex, Mol Biosyst, 11 (2015), 2867-2877. doi: 10.1039/C5MB00395D.
[14]	B. Ibrahim, Spindle assembly checkpoint is sufficient for complete Cdc20 sequestering in mitotic control, Comput Struct Biotechnol J, 13 (2015), 320-328. doi: 10.1016/j.csbj.2015.03.006.
[15]	B. Ibrahim, Systems biology modeling of five pathways for regulation and potent inhibition of the anaphase-promoting complex (APC/C): Pivotal roles for MCC and BubR1, Omics, 19 (2015), 294-305. doi: 10.1089/omi.2015.0027.
[16]	B. Ibrahim, Toward a systems-level view of mitotic checkpoints, Prog Biophys Mol Biol, 117 (2015), 217-224. doi: 10.1016/j.pbiomolbio.2015.02.005.
[17]	B. Ibrahim, A mathematical framework for kinetochore-driven activation feedback in the mitotic checkpoint, Bull Math Biol, 79 (2017), 1183-1200. doi: 10.1007/s11538-017-0278-1.
[18]	B. Ibrahim, S. Diekmann, E. Schmitt and P. Dittrich, In-silico modeling of the mitotic spindle assembly checkpoint PLoS One 3 (2008), e1555. doi: 10.1371/journal.pone.0001555.
[19]	B. Ibrahim, P. Dittrich, S. Diekmann and E. Schmitt, Stochastic effects in a compartmental model for mitotic checkpoint regulation, J Integr Bioinform, 4 (2007), 77-88. doi: 10.2390/biecoll-jib-2007-66.
[20]	B. Ibrahim, P. Dittrich, S. Diekmann and E. Schmitt, Mad2 binding is not sufficient for complete Cdc20 sequestering in mitotic transition control (an in silico study), Biophys Chem, 134 (2008), 93-100. doi: 10.1016/j.bpc.2008.01.007.
[21]	B. Ibrahim and R. Henze, Active transport can greatly enhance Cdc20:Mad2 formation, Int J Mol Sci, 15 (2014), 19074-19091. doi: 10.3390/ijms151019074.
[22]	B. Ibrahim, R. Henze, G. Gruenert, M. Egbert, J. Huwald and P. Dittrich, Spatial rule-based modeling: A method and its application to the human mitotic kinetochore, Cells, 2 (2013), 506-544. doi: 10.3390/cells2030506.
[23]	B. Ibrahim, E. Schmitt, P. Dittrich and S. Diekmann, In silico study of kinetochore control, amplification, and inhibition effects in MCC assembly, Bio Systems, 95 (2009), 35-50. doi: 10.1016/j.biosystems.2008.06.007.
[24]	G. J. Kops, B. A. Weaver and D. W. Cleveland, On the road to cancer: Aneuploidy and the mitotic checkpoint, Nat Rev Cancer, 5 (2005), 773-785. doi: 10.1038/nrc1714.
[25]	P. Kreyssig, G. Escuela, B. Reynaert, T. Veloz, B. Ibrahim and P. Dittrich, Cycles and the qualitative evolution of chemical systems PLoS One 7 (2012), e45772. doi: 10.1371/journal.pone.0045772.
[26]	P. Kreyssig, C. Wozar, S. Peter, T. Veloz, B. Ibrahim and P. Dittrich, Effects of small particle numbers on long-term behaviour in discrete biochemical systems, Bioinformatics, 30 (2014), i475-i481. doi: 10.1093/bioinformatics/btu453.
[27]	M. Lohel, B. Ibrahim, S. Diekmann and P. Dittrich, The role of localization in the operation of the mitotic spindle assembly checkpoint, Cell Cycle, 8 (2009), 2650-2660. doi: 10.4161/cc.8.16.9383.
[28]	S. Marques, J. Fonseca, P. MA Silva and H. Bousbaa, Targeting the spindle assembly checkpoint for breast cancer treatment, Curr Cancer Drug Targets, 15 (2015), 272-281. doi: 10.2174/1568009615666150302130010.
[29]	H. B. Mistry, D. E. MacCallum, R. C. Jackson, M. A. J. Chaplain and F. A. Davidson, Modeling the temporal evolution of the spindle assembly checkpoint and role of Aurora B kinase, Proc Natl Acad Sci U S A, 105 (2008), 20215-20220. doi: 10.1073/pnas.0810706106.
[30]	A. Musacchio and E. D. Salmon, The spindle-assembly checkpoint in space and time, Nat Rev Mol Cell Biol, 8 (2007), 379-393. doi: 10.1038/nrm2163.
[31]	C. L. Rieder, R. W. Cole, A. Khodjakov and G. Sluder, The checkpoint delaying anaphase in response to chromosome monoorientation is mediated by an inhibitory signal produced by unattached kinetochores, J Cell Biol, 130 (1995), 941-948. doi: 10.1083/jcb.130.4.941.
[32]	C. L. Rieder, A. Schultz, R. Cole and G. Sluder, Anaphase onset in vertebrate somatic cells is controlled by a checkpoint that monitors sister kinetochore attachment to the spindle, J Cell Biol, 127 (1994), 1301-1310. doi: 10.1083/jcb.127.5.1301.
[33]	A. D. Rudner and A. W. Murray, The spindle assembly checkpoint, Curr Opin Cell Biol, 8 (1996), 773-780. doi: 10.1016/S0955-0674(96)80077-9.
[34]	R. P. Sear and M. Howard, Modeling dual pathways for the metazoan spindle assembly checkpoint, Proc Natl Acad Sci U S A, 103 (2006), 16758-16763. doi: 10.1073/pnas.0603174103.
[35]	L. F. Shampine and M. W. Reichelt, The matlab ode suite, SIAM J Sci Comput, 18 (1997), 1-22. doi: 10.1137/S1064827594276424.
[36]	R. D. Skeel and M. Berzins, A method for the spatial discretization of parabolic equations in one space variable, SIAM J Sci Comput, 11 (1990), 1-32. doi: 10.1137/0911001.
[37]	F. Stegmeier, M. Rape, V. M. Draviam, G. Nalepa, M. E. Sowa, X. L. Ang, E. R. McDonald, M. Z. Li, G. J. Hannon, P. K. Sorger, M. W. Kirschner, J. W. Harper and S. J. Elledge, Anaphase initiation is regulated by antagonistic ubiquitination and deubiquitination activities, Nature, 446 (2007), 876-881. doi: 10.1038/nature05694.
[38]	S. Tschernyschkow, S. Herda, G. Gruenert, V. Döring, D. Görlich, A. Hofmeister, C. Hoischen, P. Dittrich, S. Diekmann and B. Ibrahim, Rule-based modeling and simulations of the inner kinetochore structure, Prog Biophys Mol Biol, 113 (2013), 33-45. doi: 10.1016/j.pbiomolbio.2013.03.010.
[39]	A. Verdugo, P. K. Vinod, J. J. Tyson and B. Novak, Molecular mechanisms creating bistable switches at cell cycle transitions Open Biol 3 (2013), 120179. doi: 10.1098/rsob.120179.
[40]	Z. Wang, J. V. Shah, M. W. Berns and D. W. Cleveland, In vivo quantitative studies of dynamic intracellular processes using fluorescence correlation spectroscopy, Biophys J, 91 (2006), 343-351. doi: 10.1529/biophysj.105.077891.
[41]	T. Wilhelm, The smallest chemical reaction system with bistability BMC Syst Biol 3 (2009), p90. doi: 10.1186/1752-0509-3-90.

show all references

References:

[1]	S. F. Bakhoum, G. Genovese and D. A. Compton, Deviant kinetochore microtubule dynamics underlie chromosomal instability, Curr Biol, 19 (2009), 1937-1942. doi: 10.1016/j.cub.2009.09.055.
[2]	A. K. Caydasi, B. Ibrahim and G. Pereira, Monitoring spindle orientation: Spindle position checkpoint in charge Cell Div 5 (2010), p28. doi: 10.1186/1747-1028-5-28.
[3]	A. K. Caydasi, B. Kurtulmus, M. I. L. Orrico, A. Hofmann, B. Ibrahim and G. Pereira, Elm1 kinase activates the spindle position checkpoint kinase Kin4, J Cell Biol, 190 (2010), 975-989. doi: 10.1083/jcb.201006151.
[4]	A. K. Caydasi, M. Lohel, G. Grünert, P. Dittrich, G. Pereira and B. Ibrahim, A dynamical model of the spindle position checkpoint Mol Syst Biol 8 (2012), p582. doi: 10.1038/msb.2012.15.
[5]	L. M. Cherry, A. J. Faulkner, L. A. Grossberg and R. Balczon, Kinetochore size variation in mammalian chromosomes: An image analysis study with evolutionary implications, J Cell Sci, 92 (1989), 281-289.
[6]	E. J. Doedel, AUTO: A program for the automatic bifurcation analysis of autonomous systems, Congr Numer, 30 (1981), 265-284.
[7]	A. Doncic, E. Ben-Jacob and N. Barkai, Evaluating putative mechanisms of the mitotic spindle checkpoint, Proc Natl Acad Sci U S A, 102 (2005), 6332-6337. doi: 10.1073/pnas.0409142102.
[8]	B. Ermentrout, Simulating, Analyzing, and Animating Dynamical Systems: A Guide to Xppaut for Researchers and Students (society for industrial and applied mathematics, philadelphia), 2002. doi: 10.1137/1.9780898718195.
[9]	D. Görlich, G. Escuela, G. Gruenert, P. Dittrich and B. Ibrahim, Molecular codes through complex formation in a model of the human inner kinetochore Biosemiotics 7 (2014), p223. doi: 10.1007/s12304-013-9193-5.
[10]	G. Gruenert, B. Ibrahim, T. Lenser, M. Lohel, T. Hinze and P. Dittrich, Rule-based spatial modeling with diffusing, geometrically constrained molecules BMC Bioinf 11 (2010), p307. doi: 10.1186/1471-2105-11-307.
[11]	G. Gruenert, J. Szymanski, J. Holley, G. Escuela, A. Diem, B. Ibrahim, A. Adamatzky, J. Gorecki and P. Dittrich, Multi-scale modelling of computers made from excitable chemical droplets, IJUC, 9 (2013), 237-266.
[12]	R. Henze, J. Huwald, N. Mostajo, P. Dittrich and B. Ibrahim, Structural analysis of in silico mutant experiments of human inner-kinetochore structure, Bio Systems, 127 (2015), 47-59. doi: 10.1016/j.biosystems.2014.11.004.
[13]	B. Ibrahim, In silico spatial simulations reveal that MCC formation and excess BubR1 are required for tight inhibition of the anaphase-promoting complex, Mol Biosyst, 11 (2015), 2867-2877. doi: 10.1039/C5MB00395D.
[14]	B. Ibrahim, Spindle assembly checkpoint is sufficient for complete Cdc20 sequestering in mitotic control, Comput Struct Biotechnol J, 13 (2015), 320-328. doi: 10.1016/j.csbj.2015.03.006.
[15]	B. Ibrahim, Systems biology modeling of five pathways for regulation and potent inhibition of the anaphase-promoting complex (APC/C): Pivotal roles for MCC and BubR1, Omics, 19 (2015), 294-305. doi: 10.1089/omi.2015.0027.
[16]	B. Ibrahim, Toward a systems-level view of mitotic checkpoints, Prog Biophys Mol Biol, 117 (2015), 217-224. doi: 10.1016/j.pbiomolbio.2015.02.005.
[17]	B. Ibrahim, A mathematical framework for kinetochore-driven activation feedback in the mitotic checkpoint, Bull Math Biol, 79 (2017), 1183-1200. doi: 10.1007/s11538-017-0278-1.
[18]	B. Ibrahim, S. Diekmann, E. Schmitt and P. Dittrich, In-silico modeling of the mitotic spindle assembly checkpoint PLoS One 3 (2008), e1555. doi: 10.1371/journal.pone.0001555.
[19]	B. Ibrahim, P. Dittrich, S. Diekmann and E. Schmitt, Stochastic effects in a compartmental model for mitotic checkpoint regulation, J Integr Bioinform, 4 (2007), 77-88. doi: 10.2390/biecoll-jib-2007-66.
[20]	B. Ibrahim, P. Dittrich, S. Diekmann and E. Schmitt, Mad2 binding is not sufficient for complete Cdc20 sequestering in mitotic transition control (an in silico study), Biophys Chem, 134 (2008), 93-100. doi: 10.1016/j.bpc.2008.01.007.
[21]	B. Ibrahim and R. Henze, Active transport can greatly enhance Cdc20:Mad2 formation, Int J Mol Sci, 15 (2014), 19074-19091. doi: 10.3390/ijms151019074.
[22]	B. Ibrahim, R. Henze, G. Gruenert, M. Egbert, J. Huwald and P. Dittrich, Spatial rule-based modeling: A method and its application to the human mitotic kinetochore, Cells, 2 (2013), 506-544. doi: 10.3390/cells2030506.
[23]	B. Ibrahim, E. Schmitt, P. Dittrich and S. Diekmann, In silico study of kinetochore control, amplification, and inhibition effects in MCC assembly, Bio Systems, 95 (2009), 35-50. doi: 10.1016/j.biosystems.2008.06.007.
[24]	G. J. Kops, B. A. Weaver and D. W. Cleveland, On the road to cancer: Aneuploidy and the mitotic checkpoint, Nat Rev Cancer, 5 (2005), 773-785. doi: 10.1038/nrc1714.
[25]	P. Kreyssig, G. Escuela, B. Reynaert, T. Veloz, B. Ibrahim and P. Dittrich, Cycles and the qualitative evolution of chemical systems PLoS One 7 (2012), e45772. doi: 10.1371/journal.pone.0045772.
[26]	P. Kreyssig, C. Wozar, S. Peter, T. Veloz, B. Ibrahim and P. Dittrich, Effects of small particle numbers on long-term behaviour in discrete biochemical systems, Bioinformatics, 30 (2014), i475-i481. doi: 10.1093/bioinformatics/btu453.
[27]	M. Lohel, B. Ibrahim, S. Diekmann and P. Dittrich, The role of localization in the operation of the mitotic spindle assembly checkpoint, Cell Cycle, 8 (2009), 2650-2660. doi: 10.4161/cc.8.16.9383.
[28]	S. Marques, J. Fonseca, P. MA Silva and H. Bousbaa, Targeting the spindle assembly checkpoint for breast cancer treatment, Curr Cancer Drug Targets, 15 (2015), 272-281. doi: 10.2174/1568009615666150302130010.
[29]	H. B. Mistry, D. E. MacCallum, R. C. Jackson, M. A. J. Chaplain and F. A. Davidson, Modeling the temporal evolution of the spindle assembly checkpoint and role of Aurora B kinase, Proc Natl Acad Sci U S A, 105 (2008), 20215-20220. doi: 10.1073/pnas.0810706106.
[30]	A. Musacchio and E. D. Salmon, The spindle-assembly checkpoint in space and time, Nat Rev Mol Cell Biol, 8 (2007), 379-393. doi: 10.1038/nrm2163.
[31]	C. L. Rieder, R. W. Cole, A. Khodjakov and G. Sluder, The checkpoint delaying anaphase in response to chromosome monoorientation is mediated by an inhibitory signal produced by unattached kinetochores, J Cell Biol, 130 (1995), 941-948. doi: 10.1083/jcb.130.4.941.
[32]	C. L. Rieder, A. Schultz, R. Cole and G. Sluder, Anaphase onset in vertebrate somatic cells is controlled by a checkpoint that monitors sister kinetochore attachment to the spindle, J Cell Biol, 127 (1994), 1301-1310. doi: 10.1083/jcb.127.5.1301.
[33]	A. D. Rudner and A. W. Murray, The spindle assembly checkpoint, Curr Opin Cell Biol, 8 (1996), 773-780. doi: 10.1016/S0955-0674(96)80077-9.
[34]	R. P. Sear and M. Howard, Modeling dual pathways for the metazoan spindle assembly checkpoint, Proc Natl Acad Sci U S A, 103 (2006), 16758-16763. doi: 10.1073/pnas.0603174103.
[35]	L. F. Shampine and M. W. Reichelt, The matlab ode suite, SIAM J Sci Comput, 18 (1997), 1-22. doi: 10.1137/S1064827594276424.
[36]	R. D. Skeel and M. Berzins, A method for the spatial discretization of parabolic equations in one space variable, SIAM J Sci Comput, 11 (1990), 1-32. doi: 10.1137/0911001.
[37]	F. Stegmeier, M. Rape, V. M. Draviam, G. Nalepa, M. E. Sowa, X. L. Ang, E. R. McDonald, M. Z. Li, G. J. Hannon, P. K. Sorger, M. W. Kirschner, J. W. Harper and S. J. Elledge, Anaphase initiation is regulated by antagonistic ubiquitination and deubiquitination activities, Nature, 446 (2007), 876-881. doi: 10.1038/nature05694.
[38]	S. Tschernyschkow, S. Herda, G. Gruenert, V. Döring, D. Görlich, A. Hofmeister, C. Hoischen, P. Dittrich, S. Diekmann and B. Ibrahim, Rule-based modeling and simulations of the inner kinetochore structure, Prog Biophys Mol Biol, 113 (2013), 33-45. doi: 10.1016/j.pbiomolbio.2013.03.010.
[39]	A. Verdugo, P. K. Vinod, J. J. Tyson and B. Novak, Molecular mechanisms creating bistable switches at cell cycle transitions Open Biol 3 (2013), 120179. doi: 10.1098/rsob.120179.
[40]	Z. Wang, J. V. Shah, M. W. Berns and D. W. Cleveland, In vivo quantitative studies of dynamic intracellular processes using fluorescence correlation spectroscopy, Biophys J, 91 (2006), 343-351. doi: 10.1529/biophysj.105.077891.
[41]	T. Wilhelm, The smallest chemical reaction system with bistability BMC Syst Biol 3 (2009), p90. doi: 10.1186/1752-0509-3-90.

Figure 1. Schematics illustrating the intracellular signaling of mitotic transition control mechanisms. (A) Dependency of spindle assembly checkpoint signaling on microtubule attachment. Kinetochores which are not attached to the spindle apparatus generate a 'wait'-signal. Chemically SAC precludes the mitotic progression by inhibiting the activation of APC/C, presumably through sequestration of the APC/C-activator Cdc20. Even a single unattached or misattached kinetochore can maintain the spindle checkpoint. If all kinetochores are correctly attached from opposite poles to the mitotic spindle, SAC is turned off and APC/C:Cdc20 formation is turned on. Paired chromosomes are held together by protein complexes called cohesin, which is depredated by active APC/C:Cdc20 complexes. $n$ refers to the number of chromosomes, as SAC is conserved from yeast to human ($n=46$ in human cells). (B) Dependency of spindle position checkpoint signaling on the correct alignment of spindle pole bodies. If the cell progresses into anaphase with a misaligned spindle, SPOC delays mitotic exit to provide the cell with extra time to make the correction. The roles of Mad2 and Cdc20 in the SAC are similar to the respective roles of Bfa1 and Tem1 in the SPOC. SPOC prevents exit from mitosis through inhibition of the MEN-activator Tem1 by Bfa1 until the spindle is properly aligned. Tem1 'inactive' indicates the GDP-bound form; 'active' refers to the GTP-bound form.

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Figure 2. Spindle Assembly Checkpoint (SAC) model set-up and analysis of all 92 kinetochores. (A) Schematic representation of the biochemical reaction network of the SAC mechanism. SAC proteins/complexes are shown as nodes, and the interactions between them as edges. Abbreviations are APC/C, anaphase promoting complex/cyclosome; Cdc20, cell division cycle 20 homolog and MCC, Mitotic Checkpoint Complex. Unattached kinetochores enhance the production of the mitotic checkpoint complex (MCC) consisting of BubR1, Bub3, Mad2, and Cdc20. Eventually, MCC binds tightly to and inhibits the APC/C in a manner preventing Cdc20 from interacting with mitotic APC/C. Immediately after the last kinetochore attachment to microtubules, the inhibitors dissolve, eventually resulting in active APC/C. This reactivation process is known as SAC silencing, where APC/C plays a role in its feedback loop. (B) Single parameter bifurcation curve. Shown is the number of attached kinetochores versus the total concentration of the MCC inhibitor. Stable node points and stable steady states are indicated by solid lines, while unstable saddle points are shown by dashed lines. Stable and unstable steady states meet at saddle-node bifurcation points, indicated by solid circles. When the number of attached kinetochore raises above approximately 91.98, the SAC checkpoint disengages and APC/C is activated. As the cell enters anaphase, MCC lowers back to zero. The black dashed line explains how the switch flips from the SAC-active state to the SAC-inactive state as the number of attached kinetochores increase. The number of attached kinetochores in the cell determines whether MCC activity is high (SAC active state) or low (SAC silence state). The bifurcation curve can be shifted to left or right depending on the values of $k_3$ and $k_2$. The earliest shift (to the left) can take place at high values of both $k_3$ as well as $k_1$ (1.1 and 0.5 respectively). (C) Numerical simulations using ODEs, SAC model. Dynamical behavior of core SAC component concentration is plotted versus time. All parameter settings are according to Table 1 (see text for more details).

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Figure 4. Spindle Position Checkpoint model with both SPBs signals. (A) The biochemical reactions governing the SPOC mechanism. Shown are SPOC proteins/complexes and the interactions between them. Abbreviations are (see for details [2,3]): BUB2, budding uninhibited by benomyl; BFA1, byrfour-alike-1; SPB, spindle pole body; and Tem1, Ras-like GTPase. (B) SPOC bifurcation curve, parameterized as the number of aligned SPBs versus the total concentration of the Bfa1:Bub2:Tem1. Unstable saddle points are shown by dashed lines, and stable node point steady states are indicated by solid lines. As the number of aligned SPBs increases above approximately 1.99 (about to be both correctly aligned), the SPOC checkpoint switches off. As the cell finalizes anaphase with both SPBs aligned, Tem1 activity becomes high and Bfa1:Bub2 becomes inactive. The black dashed line represents how the switch flips from the SPOC-active state to the SPOC-inactive state as the number of aligned SPBs increase. The bifurcation curve is sensitive to some parameters, and it can be very slightly shifted to left or right. (C) Numerical simulations of SPOC model. Dynamical behavior is very similar qualitatively but not quantitatively to that of the SAC model. The concentrations are drawn versus time. Tem1 concentration is kept low, as long as the SPBs are not aligned. After about 7 minutes, SPOC switches off and Tem1 is rapidly activated. Meanwhile, Bfa1 curve displays a steep downwards trend. All parameter settings are according to Table 1.

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Figure 3. Numerical simulation of SAC Reaction-Diffusion system with spherical symmetry. (A) Simulation of two PDEs, one per components (i.e. MCC and APC/C). (B) Simulation of three PDEs, one per component (MCC, APC/C, and MCC:APC/C). Both panels were generated for various diffusion constants and showed no qualitative changes; therefore, shown are typical curves for MCC:APC/C (to the left) and APC/C (to the right).

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Table 1. Kinetic Parameters of the SAC and the SPOC Models

	$\mathbf{Symbol}$	$\mathbf{SAC value}$	$\mathbf{SPOC value}$	$\mathbf{Remark}$
Initial amount
	APC/C	0.09 $\mu M$		[37]
	MCC	0.15 $\mu M$		[15]
	Tem1		0.06 $\mu M$	[4]
	Bfa1		0.04 $\mu M$	[4]
	Bub2		0.04 $\mu M$	[4]
Diffusion constants
	MCC	1-20 ${\mu m^2}{s^{-1}}$		This study
	APC/C	1.8 ${\mu m^2}{s^{-1}}$		[40]
	Cdc20	19.5 ${\mu m^2}{s^{-1}}$		[40]
	Mad2	5 ${\mu m^2}{s^{-1}}$		[13]
Environment
	Radius of the kinetochore	0.1$\mu m$	0.01$\mu m$	[5]
	Radius of the cell	10 $\mu m$	4$\mu m$	[21]
Rate constants
	kinetochores or SPBs	0-92	0-2	[16]
	$k_1$	$1-100 s^{-1}$	$1-50 s^{-1}$	This study
	$k_2$	$50-100 \mu\text{M}^{-1}s^{-1}$	$10-50 s^{-1}$	This study
	$k_{-2}$	$0.008-0.08 s^{-1}$	$0.001-0.08 s^{-1}$	This study
	$k_{3}$	$0.005-0.5 s^{-1}$	$0.001-0.5 s^{-1}$	This study

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	$\mathbf{Symbol}$	$\mathbf{SAC value}$	$\mathbf{SPOC value}$	$\mathbf{Remark}$
Initial amount
	APC/C	0.09 $\mu M$		[37]
	MCC	0.15 $\mu M$		[15]
	Tem1		0.06 $\mu M$	[4]
	Bfa1		0.04 $\mu M$	[4]
	Bub2		0.04 $\mu M$	[4]
Diffusion constants
	MCC	1-20 ${\mu m^2}{s^{-1}}$		This study
	APC/C	1.8 ${\mu m^2}{s^{-1}}$		[40]
	Cdc20	19.5 ${\mu m^2}{s^{-1}}$		[40]
	Mad2	5 ${\mu m^2}{s^{-1}}$		[13]
Environment
	Radius of the kinetochore	0.1$\mu m$	0.01$\mu m$	[5]
	Radius of the cell	10 $\mu m$	4$\mu m$	[21]
Rate constants
	kinetochores or SPBs	0-92	0-2	[16]
	$k_1$	$1-100 s^{-1}$	$1-50 s^{-1}$	This study
	$k_2$	$50-100 \mu\text{M}^{-1}s^{-1}$	$10-50 s^{-1}$	This study
	$k_{-2}$	$0.008-0.08 s^{-1}$	$0.001-0.08 s^{-1}$	This study
	$k_{3}$	$0.005-0.5 s^{-1}$	$0.001-0.5 s^{-1}$	This study

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American Institute of Mathematical Sciences