A mechanomolecular model for the movement of chromosomes during mitosis driven by a minimal kinetochore bicyclic cascade

CellDynaMo–stochastic reaction-diffusion-dynamics model: Application to search-and-capture process of mitotic spindle assembly

We introduce a Stochastic Reaction-Diffusion-Dynamics Model (SRDDM) for simulations of cellular mechanochemical processes with high spatial and temporal resolution. The SRDDM is mapped into the CellDynaMo package, which couples the spatially inhomogeneous reaction-diffusion master equation to account for biochemical reactions and molecular transport within the Langevin Dynamics (LD) framework to describe dynamic mechanical processes. This computational infrastructure allows the simulation of hours of molecular machine dynamics in reasonable wall-clock time. We apply SRDDM to test performance of the Search-and-Capture of mitotic spindle assembly by simulating, in three spatial dimensions, dynamic instability of elastic microtubules anchored in two centrosomes, movement and deformations of geometrically realistic centromeres with flexible kinetochores and chromosome arms. Furthermore, the SRDDM describes the mechanics and kinetics of Ndc80 linkers mediating transient attachments of microtubules to the chromosomal kinetochores. The rates of these attachments and detachments depend upon phosphorylation states of the Ndc80 linkers, which are regulated in the model by explicitly accounting for the reactions of Aurora A and B kinase enzymes undergoing restricted diffusion. We find that there is an optimal rate of microtubule-kinetochore detachments which maximizes the accuracy of the chromosome connections, that adding chromosome arms to kinetochores improve the accuracy by slowing down chromosome movements, that Aurora A and kinetochore deformations have a small positive effect on the attachment accuracy, and that thermal fluctuations of the microtubules increase the rates of kinetochore capture and also improve the accuracy of spindle assembly.

The CellDynaMo package models, in 3D, any cellular subsystem where sufficient detail of the macromolecular players and the kinetics of relevant reactions are available. The package is based on the Stochastic Reaction-Diffusion-Dynamics model that combines the stochastic description of chemical kinetics, Brownian diffusion-based description of molecular transport, and Langevin dynamics-based representation of mechanical processes most pertinent to the system. We apply the model to test the Search-and-Capture mechanism of mitotic spindle assembly. We find that there is an optimal rate of microtubule-kinetochore detachments which maximizes the accuracy of chromosome connections, that chromosome arms improve the attachment accuracy by slowing down chromosome movements, that Aurora A kinase and kinetochore deformations have small positive effects on the accuracy, and that thermal fluctuations of the microtubules increase the rates of kinetochore capture and also improve the accuracy.

Regulating positioning and orientation of mitotic spindles via cell size and shape

Proper location of the mitotic spindle is critical for chromosome segregation and the selection of the cell division plane. However, how mitotic spindles sense cell size and shape to regulate their own position and orientation is still largely unclear. To investigate this question systematically, we used a general model by considering chromosomes, microtubule dynamics, and forces of various molecular motors. Our results show that in cells of various sizes and shapes, spindles can always be centered and oriented along the long axis robustly in the absence of other specified mechanisms. We found that the characteristic time of positioning and orientation processes increases with cell size. Spindles sense the cell size mainly by the cortical force in small cells and by the cytoplasmic force in large cells. In addition to the cell size, the cell shape mainly influences the orientation process. We found that more slender cells have a faster orientation process, and the final orientation is not necessarily along the longest axis but is determined by the radial profile and the symmetry of the cell shape. Finally, our model also reproduces the separation and repositioning of the spindle poles during the anaphase. Therefore, our work provides a general tool for studying the mitotic spindle across the whole mitotic phase.

Accurate phosphoregulation of kinetochore-microtubule affinity requires unconstrained molecular interactions

Accurate regulation of kinetochore–microtubule affinity is driven by incremental phosphorylation of an NDC80 molecular “lawn,” in which NDC80–microtubule bonds reorganize dynamically in response to the number and stability of microtubule attachments.

Accurate chromosome segregation relies on dynamic interactions between microtubules (MTs) and the NDC80 complex, a major kinetochore MT-binding component. Phosphorylation at multiple residues of its Hec1 subunit may tune kinetochore–MT binding affinity for diverse mitotic functions, but molecular details of such phosphoregulation remain elusive. Using quantitative analyses of mitotic progression in mammalian cells, we show that Hec1 phosphorylation provides graded control of kinetochore–MT affinity. In contrast, modeling the kinetochore interface with repetitive MT binding sites predicts a switchlike response. To reconcile these findings, we hypothesize that interactions between NDC80 complexes and MTs are not constrained, i.e., the NDC80 complexes can alternate their binding between adjacent kinetochore MTs. Experiments using cells with phosphomimetic Hec1 mutants corroborate predictions of such a model but not of the repetitive sites model. We propose that accurate regulation of kinetochore–MT affinity is driven by incremental phosphorylation of an NDC80 molecular “lawn,” in which the NDC80–MT bonds reorganize dynamically in response to the number and stability of MT attachments.

Highly Transient Molecular Interactions Underlie the Stability of Kinetochore-Microtubule Attachment During Cell Division

Chromosome segregation during mitosis is mediated by spindle microtubules that attach to chromosomal kinetochores with strong yet labile links. The exact molecular composition of the kinetochore-microtubule interface is not known but microtubules are thought to bind to kinetochores via the specialized microtubule-binding sites, which contain multiple microtubule-binding proteins. During prometaphase the lifetime of microtubule attachments is short but in metaphase it increases 3-fold, presumably owing to dephosphorylation of the microtubule-binding proteins that increases their affinity. Here, we use mathematical modeling to examine in quantitative and systematic manner the general relationships between the molecular properties of microtubule-binding proteins and the resulting stability of microtubule attachment to the protein-containing kinetochore site. We show that when the protein connections are stochastic, the physiological rate of microtubule turnover is achieved only if these molecular interactions are very transient, each lasting fraction of a second. This "microscopic" time is almost four orders of magnitude shorter than the characteristic time of kinetochore-microtubule attachment. Cooperativity of the microtubule-binding events further increases the disparity of these time scales. Furthermore, for all values of kinetic parameters the microtubule stability is very sensitive to the minor changes in the molecular constants. Such sensitivity of the lifetime of microtubule attachment to the kinetics and cooperativity of molecular interactions at the microtubule-binding site may hinder the accurate regulation of kinetochore-microtubule stability during mitotic progression, and it necessitates detailed experimental examination of the microtubule-binding properties of kinetochore-localized proteins.

A mathematical model of ParA filament-mediated chromosome movement in Caulobacter crescentus

Caulobacter crescentus uses the dynamic interactions between ParA and ParB proteins to segregate copies of its circular chromosome. In this paper, we develop two mathematical models of the movement of the circular chromosome of this bacterium during division. In the first model, posed as a set of stochastic differential equations (SDE), we propose that a simple biased diffusion mechanism for ParB/ParA interactions can reproduce the observed patterns of ParB and ParA localization in the cell. The second model, posed as a set of nonlinear partial differential equations, is a continuous treatment of the problem where we use results from the SDE model to describe ParB/ParA interactions and we also track ParA monomer dynamics in the cytoplasm. For both models, we show that if ParB complexes bind weakly and nonspecifically to ParA filaments, then they can closely track and move with the edge of a shrinking ParA filament bundle. Unidirectional chromosome movement occurs when ParB complexes have a passive role in depolymerizing ParA filaments. Finally, we show that tight control of ParA filament dynamics is essential for proper segregation.

Springs, clutches and motors: driving forward kinetochore mechanism by modelling

As a mechanical system, the kinetochore can be viewed as a set of interacting springs, clutches and motors; the problem of kinetochore mechanism is now one of understanding how these functional modules assemble, disassemble and interact with one another to give rise to the emergent properties of the system. The sheer complexity of the kinetochore system points to a future requirement for data-driven mathematical modelling and statistical analysis based on quantitative empirical measurement of sister kinetochore trajectories. Here, we review existing models of chromosome motion in the context of recent advances in our understanding of kinetochore molecular biology.