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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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Posterior distribution from a trajectory with strong oscillations.Inferred posterior distributions and model parameters for the single trajectory shown in Fig 1C for (A) the K-fibre (de)polymerisation velocities v+ (blue) and v− (red); (B) spring constant κ; (C) anti-poleward force gradient α; (D) noise parameter τ = s−2; (E) natural length L; (F) probabilities of not switching state per time-point when sisters are coherent (blue) and incoherent (red). The informed Gaussian prior for L determined through nocodazole treatment (see S1 Fig) is shown in red in (E). Posterior distributions consist of over 5,000 samples (see section 1.2 of S1 Text for convergence protocols). (G) Trajectories for sisters of trajectory in Fig 1C with switching points marked by dashed lines. (H) Probability of switching per frame shown for each sister. (I) Posterior probabilities of sister state over the course of the trajectory. Incoherent state –/– does not occur during this particular trajectory.
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pcbi.1004607.g002: Posterior distribution from a trajectory with strong oscillations.Inferred posterior distributions and model parameters for the single trajectory shown in Fig 1C for (A) the K-fibre (de)polymerisation velocities v+ (blue) and v− (red); (B) spring constant κ; (C) anti-poleward force gradient α; (D) noise parameter τ = s−2; (E) natural length L; (F) probabilities of not switching state per time-point when sisters are coherent (blue) and incoherent (red). The informed Gaussian prior for L determined through nocodazole treatment (see S1 Fig) is shown in red in (E). Posterior distributions consist of over 5,000 samples (see section 1.2 of S1 Text for convergence protocols). (G) Trajectories for sisters of trajectory in Fig 1C with switching points marked by dashed lines. (H) Probability of switching per frame shown for each sister. (I) Posterior probabilities of sister state over the course of the trajectory. Incoherent state –/– does not occur during this particular trajectory.

Mentions: Parameter estimation and hidden state determination for the trajectory in Fig 1C is illustrated in Fig 2, a trajectory with good oscillatory behaviour. Parameter posteriors appear Gaussian and all show low variance, i.e., low parameter uncertainty (Fig 2A–2F). The natural length L posterior is close to the prior; the shift is probably due to the approximate nature of the symmetry, i.e., the twist carries some information. Switching events were confidently identified for both sisters (Fig 2G and 2H). The inferred hidden state shows strong regions of coherence (sisters moving in the same direction) interspersed with short periods of incoherence (sisters moving in opposing directions) that correspond to contraction (+/+) in this trajectory (Fig 2I). There is high confidence in assignment of the sister polymerisation state (Fig 2I) indicative of a highly deterministic behaviour (strong clear oscillations). This particular trajectory shows the previously reported [7] switching choreography wherein the lead sister switches first at every directional switching event. This is also evident directly from the trajectory time-series Fig 1C. This lead sister driven dynamics is responsible for the contracting incoherent state observed between coherent runs and gives the ‘standard’ choreography with the inter-sister distance relaxing at a switching event and increasing over the following half-period as the lead sister moves away [12].


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

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

Posterior distribution from a trajectory with strong oscillations.Inferred posterior distributions and model parameters for the single trajectory shown in Fig 1C for (A) the K-fibre (de)polymerisation velocities v+ (blue) and v− (red); (B) spring constant κ; (C) anti-poleward force gradient α; (D) noise parameter τ = s−2; (E) natural length L; (F) probabilities of not switching state per time-point when sisters are coherent (blue) and incoherent (red). The informed Gaussian prior for L determined through nocodazole treatment (see S1 Fig) is shown in red in (E). Posterior distributions consist of over 5,000 samples (see section 1.2 of S1 Text for convergence protocols). (G) Trajectories for sisters of trajectory in Fig 1C with switching points marked by dashed lines. (H) Probability of switching per frame shown for each sister. (I) Posterior probabilities of sister state over the course of the trajectory. Incoherent state –/– does not occur during this particular trajectory.
© Copyright Policy
Related In: Results  -  Collection

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

pcbi.1004607.g002: Posterior distribution from a trajectory with strong oscillations.Inferred posterior distributions and model parameters for the single trajectory shown in Fig 1C for (A) the K-fibre (de)polymerisation velocities v+ (blue) and v− (red); (B) spring constant κ; (C) anti-poleward force gradient α; (D) noise parameter τ = s−2; (E) natural length L; (F) probabilities of not switching state per time-point when sisters are coherent (blue) and incoherent (red). The informed Gaussian prior for L determined through nocodazole treatment (see S1 Fig) is shown in red in (E). Posterior distributions consist of over 5,000 samples (see section 1.2 of S1 Text for convergence protocols). (G) Trajectories for sisters of trajectory in Fig 1C with switching points marked by dashed lines. (H) Probability of switching per frame shown for each sister. (I) Posterior probabilities of sister state over the course of the trajectory. Incoherent state –/– does not occur during this particular trajectory.
Mentions: Parameter estimation and hidden state determination for the trajectory in Fig 1C is illustrated in Fig 2, a trajectory with good oscillatory behaviour. Parameter posteriors appear Gaussian and all show low variance, i.e., low parameter uncertainty (Fig 2A–2F). The natural length L posterior is close to the prior; the shift is probably due to the approximate nature of the symmetry, i.e., the twist carries some information. Switching events were confidently identified for both sisters (Fig 2G and 2H). The inferred hidden state shows strong regions of coherence (sisters moving in the same direction) interspersed with short periods of incoherence (sisters moving in opposing directions) that correspond to contraction (+/+) in this trajectory (Fig 2I). There is high confidence in assignment of the sister polymerisation state (Fig 2I) indicative of a highly deterministic behaviour (strong clear oscillations). This particular trajectory shows the previously reported [7] switching choreography wherein the lead sister switches first at every directional switching event. This is also evident directly from the trajectory time-series Fig 1C. This lead sister driven dynamics is responsible for the contracting incoherent state observed between coherent runs and gives the ‘standard’ choreography with the inter-sister distance relaxing at a switching event and increasing over the following half-period as the lead sister moves away [12].

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