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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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PEF variation over the metaphase plate impacts oscillation amplitude and deviation from alignment.(A) Average absolute PEF (on sister 1) binned by distance r from the centre of the metaphase plate (bins of approximately equal number; alternating grey and white boxes). (B) Average profile of the proportion of the opposing force (spring + PEF) to F− on lead sister during a LIDS for trajectories with r ≥ 4 μm (cyan) and r ≤ 3 μm (blue). Switch events aligned as described under Fig 8. Solid and dashed lines indicate mean and ±s.e.m., respectively. (C-F) Trajectories binned by distance r as in (A) with bin mean of (C) alignment deviation; (D) oscillation amplitude; (E) metaphase plate thickness; (F) inter-sister distance shown. Lines in (D,F) indicate s.d., n = 843, see text for definitions.
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pcbi.1004607.g009: PEF variation over the metaphase plate impacts oscillation amplitude and deviation from alignment.(A) Average absolute PEF (on sister 1) binned by distance r from the centre of the metaphase plate (bins of approximately equal number; alternating grey and white boxes). (B) Average profile of the proportion of the opposing force (spring + PEF) to F− on lead sister during a LIDS for trajectories with r ≥ 4 μm (cyan) and r ≤ 3 μm (blue). Switch events aligned as described under Fig 8. Solid and dashed lines indicate mean and ±s.e.m., respectively. (C-F) Trajectories binned by distance r as in (A) with bin mean of (C) alignment deviation; (D) oscillation amplitude; (E) metaphase plate thickness; (F) inter-sister distance shown. Lines in (D,F) indicate s.d., n = 843, see text for definitions.

Mentions: Our analysis revealed that the PEF strength coefficient α increased with distance r from the centre of the metaphase plate (Fig 4I). We expect this to have an impact on the oscillatory dynamics, i.e., the average PEF force should be higher at the spindle periphery although it also depends on the degree to which the oscillation amplitude decreases in response to a more constraining PEF potential. In fact, a higher PEF is expected to both decrease the oscillation amplitude and improve the alignment of the sisters at the metaphase plate. We measure the latter with the alignment deviation (defined as the standard deviation of the distribution of time averaged sister mid-points 〈(X1 + X2)/2〉) and quantify the oscillation amplitude with the standard deviation of the sister mid-point trajectory; in essence this is a between and within variance analysis of the sister mid-point (ANOVA). We show in Fig 9A that the average PEF does indeed increase with r (ρ = 0.13, p < 10−4), although less than the PEF coefficient (Fig 4I) with only a 50% increase towards the periphery. The dependence on r was also evident in profiles of the opposing force (PEF + spring force) to the leading sister; however, even at the periphery of the plate stalling is atypical (Fig 9B). Excursions away from the plate decreased under the higher PEF potential at increased r with both a decrease in the alignment deviation and oscillation standard deviation (Fig 9C and 9D). The finite length of the trajectories inflates both these measures because of incomplete periods within each trajectory. We also confirmed that the higher PEF at the periphery caused the metaphase plate to thin using the plate thickness statistic of Jaqaman et al. [8] (the standard deviation of the pooled sister mid-point positions over time and trajectories; Fig 9E). In part the reduced oscillation amplitude contributes to the increased trajectory stochasticity since the oscillatory signal relative to noise is reduced, potentially explaining the reduction in τ towards the periphery (inversely proportional to noise; Fig 4J).


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

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

PEF variation over the metaphase plate impacts oscillation amplitude and deviation from alignment.(A) Average absolute PEF (on sister 1) binned by distance r from the centre of the metaphase plate (bins of approximately equal number; alternating grey and white boxes). (B) Average profile of the proportion of the opposing force (spring + PEF) to F− on lead sister during a LIDS for trajectories with r ≥ 4 μm (cyan) and r ≤ 3 μm (blue). Switch events aligned as described under Fig 8. Solid and dashed lines indicate mean and ±s.e.m., respectively. (C-F) Trajectories binned by distance r as in (A) with bin mean of (C) alignment deviation; (D) oscillation amplitude; (E) metaphase plate thickness; (F) inter-sister distance shown. Lines in (D,F) indicate s.d., n = 843, see text for definitions.
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Related In: Results  -  Collection

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pcbi.1004607.g009: PEF variation over the metaphase plate impacts oscillation amplitude and deviation from alignment.(A) Average absolute PEF (on sister 1) binned by distance r from the centre of the metaphase plate (bins of approximately equal number; alternating grey and white boxes). (B) Average profile of the proportion of the opposing force (spring + PEF) to F− on lead sister during a LIDS for trajectories with r ≥ 4 μm (cyan) and r ≤ 3 μm (blue). Switch events aligned as described under Fig 8. Solid and dashed lines indicate mean and ±s.e.m., respectively. (C-F) Trajectories binned by distance r as in (A) with bin mean of (C) alignment deviation; (D) oscillation amplitude; (E) metaphase plate thickness; (F) inter-sister distance shown. Lines in (D,F) indicate s.d., n = 843, see text for definitions.
Mentions: Our analysis revealed that the PEF strength coefficient α increased with distance r from the centre of the metaphase plate (Fig 4I). We expect this to have an impact on the oscillatory dynamics, i.e., the average PEF force should be higher at the spindle periphery although it also depends on the degree to which the oscillation amplitude decreases in response to a more constraining PEF potential. In fact, a higher PEF is expected to both decrease the oscillation amplitude and improve the alignment of the sisters at the metaphase plate. We measure the latter with the alignment deviation (defined as the standard deviation of the distribution of time averaged sister mid-points 〈(X1 + X2)/2〉) and quantify the oscillation amplitude with the standard deviation of the sister mid-point trajectory; in essence this is a between and within variance analysis of the sister mid-point (ANOVA). We show in Fig 9A that the average PEF does indeed increase with r (ρ = 0.13, p < 10−4), although less than the PEF coefficient (Fig 4I) with only a 50% increase towards the periphery. The dependence on r was also evident in profiles of the opposing force (PEF + spring force) to the leading sister; however, even at the periphery of the plate stalling is atypical (Fig 9B). Excursions away from the plate decreased under the higher PEF potential at increased r with both a decrease in the alignment deviation and oscillation standard deviation (Fig 9C and 9D). The finite length of the trajectories inflates both these measures because of incomplete periods within each trajectory. We also confirmed that the higher PEF at the periphery caused the metaphase plate to thin using the plate thickness statistic of Jaqaman et al. [8] (the standard deviation of the pooled sister mid-point positions over time and trajectories; Fig 9E). In part the reduced oscillation amplitude contributes to the increased trajectory stochasticity since the oscillatory signal relative to noise is reduced, potentially explaining the reduction in τ towards the periphery (inversely proportional to noise; Fig 4J).

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