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Mathematical analysis and modeling of DNA segregation mechanisms

  • * Corresponding author: Bashar Ibrahim.

    * Corresponding author: Bashar Ibrahim.
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  • 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.

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


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  • 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.

    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).

    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.

    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).

    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]
    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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