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Inferring the Forces Controlling Metaphase Kinetochore Oscillations by Reverse Engineering System Dynamics.

Armond JW, Harry EF, McAinsh AD, Burroughs NJ - PLoS Comput. Biol. (2015)

Bottom Line: We found the K-fibre force to be the dominant force throughout oscillations, and the centromeric spring the smallest although it has the strongest directional switching signature.There is also structure throughout the metaphase plate, with a steeper PEF potential well towards the periphery and a concomitant reduction in plate thickness and oscillation amplitude.Future work will now be able to map out how individual proteins contribute to kinetochore-based force generation and sensing.

View Article: PubMed Central - PubMed

Affiliation: Warwick Systems Biology Centre and Mathematics Institute, University of Warwick, Coventry, United Kingdom.

ABSTRACT
Kinetochores are multi-protein complexes that mediate the physical coupling of sister chromatids to spindle microtubule bundles (called kinetochore (K)-fibres) from respective poles. These kinetochore-attached K-fibres generate pushing and pulling forces, which combine with polar ejection forces (PEF) and elastic inter-sister chromatin to govern chromosome movements. Classic experiments in meiotic cells using calibrated micro-needles measured an approximate stall force for a chromosome, but methods that allow the systematic determination of forces acting on a kinetochore in living cells are lacking. Here we report the development of mathematical models that can be fitted (reverse engineered) to high-resolution kinetochore tracking data, thereby estimating the model parameters and allowing us to indirectly compute the (relative) force components (K-fibre, spring force and PEF) acting on individual sister kinetochores in vivo. We applied our methodology to thousands of human kinetochore pair trajectories and report distinct signatures in temporal force profiles during directional switches. We found the K-fibre force to be the dominant force throughout oscillations, and the centromeric spring the smallest although it has the strongest directional switching signature. There is also structure throughout the metaphase plate, with a steeper PEF potential well towards the periphery and a concomitant reduction in plate thickness and oscillation amplitude. This data driven reverse engineering approach is sufficiently flexible to allow fitting of more complex mechanistic models; mathematical models of kinetochore dynamics can therefore be thoroughly tested on experimental data for the first time. Future work will now be able to map out how individual proteins contribute to kinetochore-based force generation and sensing.

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Oscillating stochastic kinetochore model.(A) Model schematic showing the orientation of the two sisters and the respective forces (arrows). Sister kinetochores are connected by a linear spring (producing force Fspring, green) and are attached to a K-fibre in either a polymerising (F+, blue) or depolymerising (F−, red) state. The direction of polar ejection forces (PEF, FPEF) is indicated by orange arrows. Sisters may be off-axis (twist angle θ); the spring force (and natural length) are then projected onto the metaphase plate normal. A directional switch is shown with the trailing sister (left) switching first (K-fibre catastrophe), followed by the originally leading sister (K-fibre rescue). See main text for details. (B) Simulated trajectory from model Eq (5) showing the (normal) distance from the metaphase plate of the two sisters. Parameters used were pc = 0.94, pic = 0.61, v+ = 0.05 μm s−1, v− = −0.03 μm s−1, L = 0.8 μm, κ = 0.05 s−1, α = 0.03 s−1 and τ = 1000 s2 μm−2, (Δt = 2s). See Methods for parameter explanation. (C) Example sister pair trajectory, from kinetochore tracking of live-cell imaging data, showing the (normal) distance from the metaphase plate of the two sisters. (D) The hidden Markov chain transition network for K-fibres switching between polymerising (+) and depolymerising (–) states used in the MCMC algorithm model. pc, pic are the probabilities of remaining coherent and incoherent, respectively, and qc = 1−pc and qic = 1−pic. (E) Time-lapse sequence of kinetochores. Kinetochores are marked by eGFP-CENP-A. Yellow bordered panel at right shows single kinetochore pair indicated by yellow box in first image.
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pcbi.1004607.g001: Oscillating stochastic kinetochore model.(A) Model schematic showing the orientation of the two sisters and the respective forces (arrows). Sister kinetochores are connected by a linear spring (producing force Fspring, green) and are attached to a K-fibre in either a polymerising (F+, blue) or depolymerising (F−, red) state. The direction of polar ejection forces (PEF, FPEF) is indicated by orange arrows. Sisters may be off-axis (twist angle θ); the spring force (and natural length) are then projected onto the metaphase plate normal. A directional switch is shown with the trailing sister (left) switching first (K-fibre catastrophe), followed by the originally leading sister (K-fibre rescue). See main text for details. (B) Simulated trajectory from model Eq (5) showing the (normal) distance from the metaphase plate of the two sisters. Parameters used were pc = 0.94, pic = 0.61, v+ = 0.05 μm s−1, v− = −0.03 μm s−1, L = 0.8 μm, κ = 0.05 s−1, α = 0.03 s−1 and τ = 1000 s2 μm−2, (Δt = 2s). See Methods for parameter explanation. (C) Example sister pair trajectory, from kinetochore tracking of live-cell imaging data, showing the (normal) distance from the metaphase plate of the two sisters. (D) The hidden Markov chain transition network for K-fibres switching between polymerising (+) and depolymerising (–) states used in the MCMC algorithm model. pc, pic are the probabilities of remaining coherent and incoherent, respectively, and qc = 1−pc and qic = 1−pic. (E) Time-lapse sequence of kinetochores. Kinetochores are marked by eGFP-CENP-A. Yellow bordered panel at right shows single kinetochore pair indicated by yellow box in first image.

Mentions: Chromosomes are attached to, and their movements powered by, kinetochores, multi-protein machines that assemble on each sister chromatid and form dynamic attachments to bundles of kinetochore-microtubules (K-MTs) called K-fibres [1] (see Fig 1A). A long-standing challenge in the mitosis field is to measure the magnitude of forces that kinetochores can generate and identify the molecular components and mechanisms responsible. Nicklas and colleagues addressed this question by using calibrated micro-needles to pull on chromosomes in grasshopper spermatocytes, measuring the force needed to stall chromosome movement [2]. These classic experiments found that > 20 pN was necessary to slow, and 700 pN to stall, chromosome-to-pole movement in anaphase, while there was a much lower stall force (50 pN) for chromosome movement during congression. These measured values are considerably higher than the 0.1 pN that was calculated (based on Stokes law; force = viscosity × chromosome size × velocity) to be required for moving a chromosome under normal conditions [3, 4].


Inferring the Forces Controlling Metaphase Kinetochore Oscillations by Reverse Engineering System Dynamics.

Armond JW, Harry EF, McAinsh AD, Burroughs NJ - PLoS Comput. Biol. (2015)

Oscillating stochastic kinetochore model.(A) Model schematic showing the orientation of the two sisters and the respective forces (arrows). Sister kinetochores are connected by a linear spring (producing force Fspring, green) and are attached to a K-fibre in either a polymerising (F+, blue) or depolymerising (F−, red) state. The direction of polar ejection forces (PEF, FPEF) is indicated by orange arrows. Sisters may be off-axis (twist angle θ); the spring force (and natural length) are then projected onto the metaphase plate normal. A directional switch is shown with the trailing sister (left) switching first (K-fibre catastrophe), followed by the originally leading sister (K-fibre rescue). See main text for details. (B) Simulated trajectory from model Eq (5) showing the (normal) distance from the metaphase plate of the two sisters. Parameters used were pc = 0.94, pic = 0.61, v+ = 0.05 μm s−1, v− = −0.03 μm s−1, L = 0.8 μm, κ = 0.05 s−1, α = 0.03 s−1 and τ = 1000 s2 μm−2, (Δt = 2s). See Methods for parameter explanation. (C) Example sister pair trajectory, from kinetochore tracking of live-cell imaging data, showing the (normal) distance from the metaphase plate of the two sisters. (D) The hidden Markov chain transition network for K-fibres switching between polymerising (+) and depolymerising (–) states used in the MCMC algorithm model. pc, pic are the probabilities of remaining coherent and incoherent, respectively, and qc = 1−pc and qic = 1−pic. (E) Time-lapse sequence of kinetochores. Kinetochores are marked by eGFP-CENP-A. Yellow bordered panel at right shows single kinetochore pair indicated by yellow box in first image.
© Copyright Policy
Related In: Results  -  Collection

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Show All Figures
getmorefigures.php?uid=PMC4664287&req=5

pcbi.1004607.g001: Oscillating stochastic kinetochore model.(A) Model schematic showing the orientation of the two sisters and the respective forces (arrows). Sister kinetochores are connected by a linear spring (producing force Fspring, green) and are attached to a K-fibre in either a polymerising (F+, blue) or depolymerising (F−, red) state. The direction of polar ejection forces (PEF, FPEF) is indicated by orange arrows. Sisters may be off-axis (twist angle θ); the spring force (and natural length) are then projected onto the metaphase plate normal. A directional switch is shown with the trailing sister (left) switching first (K-fibre catastrophe), followed by the originally leading sister (K-fibre rescue). See main text for details. (B) Simulated trajectory from model Eq (5) showing the (normal) distance from the metaphase plate of the two sisters. Parameters used were pc = 0.94, pic = 0.61, v+ = 0.05 μm s−1, v− = −0.03 μm s−1, L = 0.8 μm, κ = 0.05 s−1, α = 0.03 s−1 and τ = 1000 s2 μm−2, (Δt = 2s). See Methods for parameter explanation. (C) Example sister pair trajectory, from kinetochore tracking of live-cell imaging data, showing the (normal) distance from the metaphase plate of the two sisters. (D) The hidden Markov chain transition network for K-fibres switching between polymerising (+) and depolymerising (–) states used in the MCMC algorithm model. pc, pic are the probabilities of remaining coherent and incoherent, respectively, and qc = 1−pc and qic = 1−pic. (E) Time-lapse sequence of kinetochores. Kinetochores are marked by eGFP-CENP-A. Yellow bordered panel at right shows single kinetochore pair indicated by yellow box in first image.
Mentions: Chromosomes are attached to, and their movements powered by, kinetochores, multi-protein machines that assemble on each sister chromatid and form dynamic attachments to bundles of kinetochore-microtubules (K-MTs) called K-fibres [1] (see Fig 1A). A long-standing challenge in the mitosis field is to measure the magnitude of forces that kinetochores can generate and identify the molecular components and mechanisms responsible. Nicklas and colleagues addressed this question by using calibrated micro-needles to pull on chromosomes in grasshopper spermatocytes, measuring the force needed to stall chromosome movement [2]. These classic experiments found that > 20 pN was necessary to slow, and 700 pN to stall, chromosome-to-pole movement in anaphase, while there was a much lower stall force (50 pN) for chromosome movement during congression. These measured values are considerably higher than the 0.1 pN that was calculated (based on Stokes law; force = viscosity × chromosome size × velocity) to be required for moving a chromosome under normal conditions [3, 4].

Bottom Line: We found the K-fibre force to be the dominant force throughout oscillations, and the centromeric spring the smallest although it has the strongest directional switching signature.There is also structure throughout the metaphase plate, with a steeper PEF potential well towards the periphery and a concomitant reduction in plate thickness and oscillation amplitude.Future work will now be able to map out how individual proteins contribute to kinetochore-based force generation and sensing.

View Article: PubMed Central - PubMed

Affiliation: Warwick Systems Biology Centre and Mathematics Institute, University of Warwick, Coventry, United Kingdom.

ABSTRACT
Kinetochores are multi-protein complexes that mediate the physical coupling of sister chromatids to spindle microtubule bundles (called kinetochore (K)-fibres) from respective poles. These kinetochore-attached K-fibres generate pushing and pulling forces, which combine with polar ejection forces (PEF) and elastic inter-sister chromatin to govern chromosome movements. Classic experiments in meiotic cells using calibrated micro-needles measured an approximate stall force for a chromosome, but methods that allow the systematic determination of forces acting on a kinetochore in living cells are lacking. Here we report the development of mathematical models that can be fitted (reverse engineered) to high-resolution kinetochore tracking data, thereby estimating the model parameters and allowing us to indirectly compute the (relative) force components (K-fibre, spring force and PEF) acting on individual sister kinetochores in vivo. We applied our methodology to thousands of human kinetochore pair trajectories and report distinct signatures in temporal force profiles during directional switches. We found the K-fibre force to be the dominant force throughout oscillations, and the centromeric spring the smallest although it has the strongest directional switching signature. There is also structure throughout the metaphase plate, with a steeper PEF potential well towards the periphery and a concomitant reduction in plate thickness and oscillation amplitude. This data driven reverse engineering approach is sufficiently flexible to allow fitting of more complex mechanistic models; mathematical models of kinetochore dynamics can therefore be thoroughly tested on experimental data for the first time. Future work will now be able to map out how individual proteins contribute to kinetochore-based force generation and sensing.

Show MeSH