MATHEMATICAL ANALYSIS AND MODELING OF DNA SEGREGATION MECHANISMS

. 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 chal- lenging 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 com- plex biomedical systems.


1.
Introduction. DNA segregation is a complicated process that is critical for cell proliferation and survival [16,30]. Failures during segregation can result in aberrant DNA contents (aneuploidy), a phenomenon which is prevalent in human cancers [24,28]. The fidelity of chromosome segregation during cell division is monitored by control mechanisms called checkpoints, which ensure that particular criteria are met before moving on irreversibly to the next phase [30].
In mitosis, the Spindle Assembly Checkpoint (SAC; [33]) ensures that all chromosomes are properly attached to spindle microtubules via their kinetochores. Even a single unattached or misattached chromosome is sufficient to keep the checkpoint active and engaged [32,31]. (A human mitotic cell has 46 chromosomes and 92 kinetochores.) In budding yeast (Saccharomyces cerevisiae and Drosophila male), an additional control checkpoint exists to place the correct DNA into the right cell during asymmetric cell division. This regulation is known as the Spindle Position loops, and two ODEs. It is structurally fully distinct to the known smallest biochemical model [41] and structurally comparable to the yeast mitotic model [39,17]. Second, the same model structure was applied to the SPOC, with both SPBs included. Subsequently, a one-parameter bifurcation was computed for these models, in order to demonstrate the realistic biochemical switches. Eventually, numerical simulations were carried out for the system as a system of ordinary differential equations (ODEs), and also as partial differential equations (PDEs; reaction-diffusion systems) with various parameters. 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.

Methods.
2.1. Reaction equations. The reaction rules governing the Spindle Assembly Checkpoint (SAC) system are (cf. Fig. 2A): The Spindle Position Checkpoint mechanism (SPOC) has very similar reaction rules (see below), but differs significantly in the initial concentrations, rate contacts, and signals of exactly two SPBs (cf. Fig. 4A): 2.2. Ordinary differential equations model. By applying the law of massaction kinetics, the reaction rules (Eqs.(1-3) and Eqs.4-6) can be translated into sets of time-dependent nonlinear ordinary differential equations (ODEs). The translation is done by computing dS/dt = N v(S) with state vector S, flux vector v(S) and stoichiometric matrix N . The initial amounts for reaction species and rate constants are taken from the literature (cf. Table 1).
2.3. Partial differential equations model. Adding a diffusion term as a second spatial derivative transforms the system into one of coupled partial differential equations (PDEs), which is known as a reaction-diffusion system and has the following general form: where [C i ] refers to the concentration of species i = {1, . . . , 4}. The first term on the right hand side represents the diffusion and the second one represents the biochemical reactions R j = {R 1 , ..., R 4 } where species i is involved. The constant D i denotes per-species diffusion for species i. t specifies time and P symbolizes phenomenological parameters. The operator ∇ refers to the spatial gradient in spherical coordinates ∇ (r, θ, ϕ) = ∂ ∂r e r + 1 r ∂ ∂θ e θ + 1 r sin(θ) ∂ ∂ϕ e ϕ . Recent studies have shown that spherical symmetry is suitable and sufficient for use when modeling the SAC mechanism [21,12,13]; thus, we use the spherically symmetric form. Eq. (7) reduces to the following system of PDEs, which depend on t and r: 2.4. Numerical simulations of the ODEs model. The systems of ODEs were implemented in the freely-available software package XPPAUT [8], and integrated using the Rosenbrock method (stiff solver). The bifurcation analyzes and the related numerical integrations were conducted with AUTO [6] via an XPPAUT interface.

2.5.
Numerical simulations of the reaction-diffusion PDEs model. For the spatial simulations, the mitotic cell is considered as a 3-sphere with radius R. The last unattached kinetochore is a small 3-sphere with radius r, which is located in the center of the cell. All boundary conditions are assumed to be reflective and the numbers of all interacting elements to be conserved. Additionally, all PDEs are assumed to be spherically symmetric (solely as a function of t and r, cf. Eq. 8).
All reactions are assumed to follow the mass-action-kinetics law. The kinetic rate constants are taken from the literature, as given in Table 1.
The reaction-diffusion system of PDEs were solved numerically using MATLAB (MathWorks), and integrated using its predefined function called pdepe-solver, which solves systems of parabolic and elliptic PDEs in one space variable r and time t.
The pdepe-solver converts the PDEs to ODEs using a second-order accurate spatial discretization based on a fixed set of user-specified nodes [36]. This is done using piecewise non-linear Galerkin (regular case) and implicit Petrov-Galerkin (singular case, second-order accurate). The ordinary differential equations resulting from discretization in space are integrated via the multistep solver ode15s which is a variable order solver based on the numerical differentiation formulas (NDFs)using the numerical Gear method [35]. To check that our results were not influenced by the spatial discretization method used in pdepe, we repeated all simulations for 50, 100 and 1000 grid cells.
3. Results. Under these assumptions, the reduced system can be easily written as the following nonlinear ODEs: The parameter U refers to the number of unattached kinetochores, which vary from 92 to 0, while A references the number of attached kinetochores and is defined by A = 92 − U . APC/C can be presumed in steady state to analyze the bifurcation for the kinetochore signal versus the total MCC. Also, plugging the constant concentration [APC/CT] into the system 9-10 produces where the parameters a, b, and c are defined as follows: First, one-parameter bifurcation analysis was performed for the nonlinear system (Eq. 11 and 16). The aim is to demonstrate the bistable switch states influencing total MCC, while kinetochores are gradually attached. The simulations were conducted using AUTO software (see Methods). The results in Fig. 2B  Additionally, the dynamics as the change of concentrations over time were simulated and plotted (Fig. 2C) using XPPAUT (see Methods). The concentration of the APC/C component (Fig. 2C pink line) remains very low, as long as no kinetochores are attached. After approximately 25 minutes, the APC/C activity increases quickly to reach its maximum. This result is consistent with the experimental findings reported in the literature [1,27]. The inhibitor complexes MCC:APC/C that sequester APC/C display exactly the opposite behaviors compared to APC/C (Fig.  2C, brown line). MCC concentration behaves similarly to MCC:APC/C with respect to the difference in the initial amount (Fig. 2C, blue line).
Also interesting is the effect of the diffusion coefficient on the SAC system, particularly since the MCC diffusion constant is unknown. To investigate this, a second-derivative diffusion term was added to the original system (Eq.10-Eq.9). The resulting system of coupled PDEs is known as a reaction-diffusion system: This system was subjected to the initial conditions given in Table 1, with reflective (Neumann) boundary conditions and equivalent geometry details for the cell and kinetochore as specified in Methods. The reaction-diffusion system (Eq.13-Eq.14) was implemented in MATLAB and simulated with various diffusion constant values (Table 1). No qualitative changes were recorded using a wide range of diffusion coefficients for MCC from the literature and higher; all behave similarly as shown in the typical curves in Fig. 3A. To ensure that the assumptions used to reduce 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. the model had no influence of this result, a full system including three PDEs was re-simulated using various diffusion values for MCC. Again, no effects were observed (Fig. 3B). We conclude that diffusion has no major influence on the SAC model, and that the use of ODEs is in principle sufficient. This is certainly not applicable to other model structures, and cannot be generalized, particularly to high-dimensional systems or using low diffusion constants. 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).

3.2.
Mathematical framework of SPOC. Following the same steps as for SAC analysis as in the previous section, the SPOC wiring diagram (Fig. 4A)   Again, AUTO software was used to find the bifurcation curve (see Methods for details). The bifurcation curve (Fig. 4B) is shown for the number of misaligned SPBs versus the total concentration of the Bfa1:Bub2:Tem1 inhibitor. The system switches its bistable state at the value 1.99; subsequently, SPOC is turned off, and eventually Tem1 is rapidly activated.
The numerical simulations of the ODEs is depicted in Fig. 2C, using a stiff solver in XPPAUT. Tem1 (Fig. 2C pink line) is inactive until both SPBs are This study k −2 0.008 − 0.08s −1 0.001−0.08s −1 This study k 3 0.005 − 0.5s −1 0.001 − 0.5s −1 This study that each chromosome has established its attachment to the spindle apparatus before commencing sister-chromatid separation, while the SPOC mechanism assures correct spindle alignment in some asymmetric cell divisions. The complexity of the mitotic control system arises from its fundamental spatial feature. In SAC, a single unattached or incorrectly attached kinetochore (out of 92) has to inhibit all APC/Cs of the cell and solely after last proper attachment the inhibitor has to be switched off rapidly. This behavior likely implies of a feedback loop contribution. The same is applied for SPOC mechanism with the distinguish in the signal that represents two SPBs. Mathematical modeling can help to improve our molecular-level understanding of the interplay of SAC as well as SPOC components, allowing an understanding of the requirements that the system has to meet. To that end, a mathematical framework containing all signals in SAC (or SPOC) was studied. The network models were constructed based on biochemical reaction rules, and spatial characteristics, such as diffusion coefficients or cell size. Kinetochores and SPBs act as sensory-driven signals in SAC, and SPOC regulations, respectively. The simulation results (as ODEs or PDEs) were able to capture the desired behavior of both SAC and SPOC. Additionally to the simulations, a crucial feedback loop was built around the core components APC/C and MCC (or Tem1 and Bfa1 in SPOC) and their interplay. A one-parameter bifurcation diagram clearly depicts the bistability of the system and the realistic switch from active to inactive control states.
The presented mathematical models can be extended in future work to include various cell cycle checkpoints. Additionally, this approach will serve as a basis for designing experiments and evaluating novel hypotheses related to mitosis and cell cycle. The results provide systems-level details into the DNA segregation control mechanism and demonstrate that the combination of mathematical analysis with experimental data constitutes a powerful tool for investigation of complex biomedical systems.