Fig 1.
Components of Stochastic Reaction-Diffusion-Dynamics model implemented in CellDynaMo package.
A) Schematic of the Stochastic Reaction-Diffusion-Dynamics model. 1) Spherical centrosomes (CS) with the microtubules’ (MTs) plus-ends on the surface (400-nm radius, 750 MTs per centrosome); 2) Microtubules (MTs) described by 12.0-nm spherical beads, each representing ~40 4.0-nm αβ-tubulin dimers; 3) Kinetochores’ corona regions (on spherical surface) in which the complexes of Ndc80 bound to MTs with their kinetochore-associated domains, spherical 8.0-nm beads, are labelled; 4) Kinetochore pairs (KTs) and sister Chromatids (Chs) represented by spherical beads with 362.5-nm radius (number of beads depends on the size of the chromatid arms, see B); 5) Ndc80 protein complexes with KT via the spherical kinetochore binding domains and the labelled 60.0-nm triple-helical coiled coil domain modeled as a harmonic spring connected to corona surface from one side and to MT plus-end from other side; 6) Blue space is Phosphatase (enzyme performing dephosphorylation of Ndc80), which is uniformly distributed in the interior space of the cell; 7) Aurora B kinase (AB; enzyme performing phosphorylation of Ndc80) is described by the spherical gradient of its concentration with the central maximum in the space between the kinetochores (red cloud); 8) Aurora A kinase (AA; enzyme performing phosphorylation of Ndc80) is described by the spherical gradient of its concentration with the twin maxima centered on the two centrosomes (yellow cloud). In the model, components 1–4 are described using the Langevin Dynamics in the Brownian diffusion limit (Table 1), whereas components 5–8 are modeled using the stochastic reaction-diffusion master equation (Table 2). 9) Cellular membrane is modeled parametrically as a smooth repulsive potential for all cell components. B) More detailed view of metaphase chromosomes (CHs) comprised of identical paired sister Chromatids (Chs). The entire length of each chromatid is represented by connected beads and neighboring beads on the adjacent sister chromatids are connected by a spring. C) Snapshot in 3D of a single trajectory from the CellDynaMo simulation that shows all the components, including cell membrane (black grid). D) Closer cell interior view of the Stochastic Reaction-Diffusion-Dynamics model showing all components: MTs (lime tubes), KTs and Chs (blue beads), corona with Ndc80 seeds (orange wall sections). CS pair are hidden behind point clouds of AA kinase (yellow). AB kinase (red point cloud) is located between KTs.
Table 1.
Components of CellDynaMo described with Langevin Dynamics approach.
Size, shape, and number of each component described using the Langevin Dynamics description of stochastic processes, including centrosomes (CS), microtubules (MTs), kinetochore (KT) corona, chromosomes (CH), and Ndc80 binding domains (Fig 1). Also provided is the information about force-generating properties, number of particles (spherical beads) used to describe each component, and some additional information such as MT persistence length lp, surface curvature of the KT corona χ, number of beads per chromatid n, and contour length of a chromatid L.
Table 2.
Components of CellDynaMo described with Reaction-Diffusion Master Equation formalism.
Functional role and the number of copies of each component described with RDME, including MT-Ndc80 complex, Phosphatase enzyme (PH), Aurora A enzyme (AA), Aurora B enzyme (AB), and cellular membrane. Also provided is the information about the force-generating properties and the distribution of each component inside the cell.
Fig 2.
Kinetochore-microtubule attachments in Stochastic Reaction-Diffusion-Dynamics Model.
A) A regular kinetochore (KT) representation. Blue bead represents the main body of the KT and orange wall section is the outer KT, which is covered by grid of Ndc80 seeds. B) Schematic of the KT model. Grid of beads within corona are connected to each other inside some cut-off radius (200 nm); all of them are also attached to the center of mass of the corresponding KT. This set of features was made to keep the KT shape constant. C) In the Stochastic Reaction-Diffusion-Dynamics model, we can vary the curvature of the outer KT from the most curved (beads cover the KT surface) to almost flat. D) Another option of the model is to cover a KT with molecular “armor” blocking the MT access to KTs, thus mimicking the role of CH arms. E) A more detailed look at the KT-MT interface. Here MTs are lime tubes. MTs are attached to the corona with the help of Ndc80-complexes (cyan lines). F) Types of KT-MT attachments are illustrated by examples coming from snapshots taken from the simulations.
Fig 3.
Design of Stochastic Reaction-Diffusion-Dynamics model-based implementation of CellDynaMo package.
CellDynaMo requires the initial input (reaction rate constants, copy numbers of biomolecules), the force field parameters (stretching and bending rigidities, KT-MT attachment strength), and cell morphology (number of chromosomes or kinetochore pairs, membrane shape). These specify cell morphology and geometry of spatial arrangements of different cell components (Fig 1), biochemical kinetics (Tables 2, 3 and 4), molecular transport (Table 3), and force-generating properties (Table 1). A list of parameters used in CellDynaMo package is provided in Table A in S1 Text. The RDME is solved numerically for all subcells at each time point. When changes to the mechanical state occur, the RDME switches off and the LD switches on. When a new state of mechanical equilibrium is reached, CellDynaMo writes an output for a particular time point, which includes coordinate file, force file, and file with subcell specific content. These can then be used to analyze and visualize the simulation data, and to compare with experiments and with theoretical predictions.
Table 3.
Microtubule dynamic processes and transport properties of Aurora A and Aurora B enzymes.
Dynamic processes involving microtubules (MT) and the values of kinetic rate constants and characteristic timescales associated with the MT growth, shortening, catastrophe and rescue; Δl = 24 nm is the amount by which MT length increases or decreases when MT grows or shortens, respectively. Also shown are the diffusion constants and diffusion timescales for Aurora B and Aurora A enzymes.
Table 4.
Biochemical reactions at kinetochore-microtubule interface.
Enzymatic reactions (e.g., phosphorylation and dephosphorylation) and association-dissociation reactions, which involve the MT associated protein Ndc80 linking MTs with KTs, and the reaction rate constants and characteristic timescales. The subscript p = 0, 1,…,6 denotes the number of phosphate groups attached to Ndc80 and changes in the rate constants and reaction propensities for MT-Ndc80 complex dissociation.
Fig 4.
Exploring the role of Phosphatase and Aurora B kinase on dynamics of kinetochore-microtubule attachments.
A) Initial position and orientation of the KT pair inside the cell. The KT pair is placed close to the equatorial plate, shifted by 1.5 μm below and along the axis perpendicular to the axis of the spindle. Orientation of the KT pair axis is along the axis perpendicular to the spindle axis. The AB particles are spatially distributed around the center between the KT pair as shown in the blowout. B) Time dependent evolution of KT-KT orientation angle (solid lines; left y-axis) and distance between KT-pair and equatorial plate (dash-dotted lines, right y-axis) for the Phosphatase to Aurora B (P:AB) ratio = 1:100 (blue), 1:10 (red), and 1:1 (black). Each curve shows results from a single simulation run. C) Probability to find each type of attachment for different P:AB ratio = 1:100 (blue bars), 1:10 (red bars), and 1:1 (black bars). Statistics was collected from n = 8 independent runs for each of these three cases. D) The P:AB ratio influences the average bond lifetime and the frequency of attachment-detachment switches. Shown is evolution of the maximum number of microtubules (MTs) attached to a single KT from a single centrosome for P:AB ratio = 1:100 (blue lines), 1:10 (red) and 1:1 (black) during 30 min of simulation. Changes in the KT-MT attachment status are shown in the inset. E) Snapshot of the final state of the system (red and cyan spheres are particles of Aurora B kinase and Phosphatase).
Fig 5.
Exploring the effect of the Aurora A presence.
A) Snapshot of the final KT pair position and orientation (amphitelic attachment) for a simulation with AA. B) Probability to find each type of attachment for the system without AA (blue bars) and for the system with AA (red bars). Statistics were collected from n = 20 simulation runs for both cases. C) Addition of AA kinases to the system changes the Phosphatase to Aurora B ratio close to centrosomes (CSs). This influences the average bond lifetime and the frequencies of attachment/detachment switches. Blue line shows the evolution of total number of microtubules (MTs) attached to a single KT pair from both CSs for the case study without AA, red line shows the number of MTs for the case study with AA. D) An example of how adding AA kinases to the system corrects the trajectory of the KT pair. Lines show the 2D projection (xz-plane) of the KT pair trajectory during 30 min simulations. For the case study without AA (blue line), KT pair reaches one of the CSs and for the case study with AA (red line), KT pair stops right before the cloud of AA kinases.
Fig 6.
Describing chromosome arms and flexible corona surface in Stochastic Reaction-Diffusion-Dynamics Model.
A) Probability to find each type of attachment for a single chromosome (CH) with CH arms. Blue and red bars represent statistics for kinetochore pairs (KTs) without and with CH arms, respectively. The statistics are based on n = 20 simulation runs for each case. B) Snapshot after 30 min simulation of biological time shows a representative example of merotelic attachment. C) Changes in KT-KT orientation angle (solid lines; left y-axis) and distance between KT-pair and equatorial plate (dash-dotted lines, right y-axis) over time for the most representative simulation run for the case study with CH arms present. D) Number of MTs vs. time profiles for the same simulation run as in panel C showing proper amphitelic attachment. Abbreviations used: L1 and L2 denote the numbers of MTs from the left CS attached to the first KT and second KT, respectively; R1 and R2 are the numbers of MTs from the right CS attached to the first KT and second KTs, respectively. E) Probability to find each type of attachment for a rigid corona surface (blue) and for a flexible corona surface (red). The statistics are based on n = 20 simulation runs for each case study. F) KT-MT interface and corona flexing for a case of amphitelic attachment. G) KT-MT interface and corona flexing for a case of merotelic attachment.
Fig 7.
Influence of stochastic noise on dynamics of KT-MT attachments.
Comparison of the types and numbers of KT-MT attachment from the simulations with the random force component (shaded bars) and without random force component (blank bars) for: A) A single KT pair with the Phosphatase to Aurora B (P:AB) ratio = 1:10, B) A single CH with CH arms (see also Fig J in S1 Text). The KT-KT orientation angle vs. time (solid lines) and distance to the equatorial plate vs. time profiles (dashed lines) for the case of a single KT pair with P:AB ratio = 1:10 (panel C) and for a single CH with CH arms (panel D). Black lines in panel C corresponds to simulation run without thermal fluctuations and demonstrate amphitelic attachment. Red and blue lines correspond to simulation runs with thermal fluctuations and demonstrate monotelic and merotelic attachments, respectively. In panel D, black and green lines represent simulation runs without and with thermal fluctuations, respectively. Panels E and F demonstrate the profiles for the number of MTs attached to KTs from both CS for a single KT pair with P:AB ratio = 1:10 and for a single CH with CH arms, respectively. In panels E and F, the assignment of curve color is same as in panels C and D. Abbreviation L1, L2, R1, and R2 are same as in Fig 6D.