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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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Determination of oscillatory trajectories using three quality statistics.(A) Histogram of explained variance (EV) of each trajectory. (B) Explained variance of each trajectory arranged by cell. (C) Histogram of log Bayes factor of the kinetochore model Mcoh against Brownian motion MBM (log B[Mcoh/MBM]) of each trajectory. (D) log B[Mcoh/MBM] of each trajectory arranged by cell. (E) Histogram of directional correlation statistic DΔt of each trajectory. (F) DΔt of each trajectory arranged by cell. (G) log B[Mcoh/MBM] plotted against EV for each trajectory. Trajectories are coloured according to DΔt; blue have significant directional correlations. (H) Mean trajectory positions within the metaphase plate viewed along the spindle axis (Y, Z) coloured by EV. All converged trajectories are shown in panels (A-G) (n = 1169); only those from grey area in (G) are shown in (H) (n = 843).
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pcbi.1004607.g003: Determination of oscillatory trajectories using three quality statistics.(A) Histogram of explained variance (EV) of each trajectory. (B) Explained variance of each trajectory arranged by cell. (C) Histogram of log Bayes factor of the kinetochore model Mcoh against Brownian motion MBM (log B[Mcoh/MBM]) of each trajectory. (D) log B[Mcoh/MBM] of each trajectory arranged by cell. (E) Histogram of directional correlation statistic DΔt of each trajectory. (F) DΔt of each trajectory arranged by cell. (G) log B[Mcoh/MBM] plotted against EV for each trajectory. Trajectories are coloured according to DΔt; blue have significant directional correlations. (H) Mean trajectory positions within the metaphase plate viewed along the spindle axis (Y, Z) coloured by EV. All converged trajectories are shown in panels (A-G) (n = 1169); only those from grey area in (G) are shown in (H) (n = 843).

Mentions: EV varied from around 0 to 66% (Fig 3A) with mean 26 ± 14% (± distribution s.d.). The variability in trajectories is apparent in the range of EV shown on a per cell basis (Fig 3B), indicating that all cells have a similar profile of near deterministic and highly stochastic trajectories. EV allowed us to rank trajectories by how well the model fitted—the trajectory shown in Fig 1C and reverse engineered in Fig 2 had the highest EV —but does not provide support for the model compared to any other since it is an intrinsic measure of fit. We therefore complemented the EV statistic with a comparative test using the Bayes factor B[Mcoh/MBM] between the coherence-incoherence model Mcoh and a Brownian motion (BM) model MBM (we also compared against BM models with an inter-sister spring and drift with almost identical results; see section 2.2 of S1 Text and S3A and S3B Fig). Surprisingly, B[Mcoh/MBM] showed significant preference for MBM; we found log B [Mcoh/MBM] < 0 for over 95% of the trajectories (Fig 3C and 3D; a log B[Mcoh/MBM] < 0 indicates preference for MBM). This is due to the lack of temporal structure in the MBM —displacements are treated as independent, identically Gaussian distributed so B[Mcoh/MBM] is predominantly a measure of whether the kinetochore displacements are Gaussian or not. Trajectories where Mcoh is preferred have very regular saw-tooth oscillations and thus an over-representation of large displacements; B[Mcoh/MBM] is thus a good discriminator of strong oscillating trajectories. Comparing EV with B[Mcoh/MBM] demonstrated a correlation as expected (overall ρ = 0.14, p = 0.0008; Fig 3G), but also revealed a group of outlier trajectories (green shading in figure) that had higher than average B[Mcoh/MBM] but low EV. These trajectories tended to have a few excessively large displacements indicative of tracking errors. We thus filtered these from the analysis by restricting to trajectories approximating the linear relationship between B[Mcoh/MBM] and EV (the selection region shown as a grey bar in Fig 3G; ρ = 0.90 for grey region) comprising 843 out of 1169 converged trajectories, 72%.


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

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

Determination of oscillatory trajectories using three quality statistics.(A) Histogram of explained variance (EV) of each trajectory. (B) Explained variance of each trajectory arranged by cell. (C) Histogram of log Bayes factor of the kinetochore model Mcoh against Brownian motion MBM (log B[Mcoh/MBM]) of each trajectory. (D) log B[Mcoh/MBM] of each trajectory arranged by cell. (E) Histogram of directional correlation statistic DΔt of each trajectory. (F) DΔt of each trajectory arranged by cell. (G) log B[Mcoh/MBM] plotted against EV for each trajectory. Trajectories are coloured according to DΔt; blue have significant directional correlations. (H) Mean trajectory positions within the metaphase plate viewed along the spindle axis (Y, Z) coloured by EV. All converged trajectories are shown in panels (A-G) (n = 1169); only those from grey area in (G) are shown in (H) (n = 843).
© Copyright Policy
Related In: Results  -  Collection

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

pcbi.1004607.g003: Determination of oscillatory trajectories using three quality statistics.(A) Histogram of explained variance (EV) of each trajectory. (B) Explained variance of each trajectory arranged by cell. (C) Histogram of log Bayes factor of the kinetochore model Mcoh against Brownian motion MBM (log B[Mcoh/MBM]) of each trajectory. (D) log B[Mcoh/MBM] of each trajectory arranged by cell. (E) Histogram of directional correlation statistic DΔt of each trajectory. (F) DΔt of each trajectory arranged by cell. (G) log B[Mcoh/MBM] plotted against EV for each trajectory. Trajectories are coloured according to DΔt; blue have significant directional correlations. (H) Mean trajectory positions within the metaphase plate viewed along the spindle axis (Y, Z) coloured by EV. All converged trajectories are shown in panels (A-G) (n = 1169); only those from grey area in (G) are shown in (H) (n = 843).
Mentions: EV varied from around 0 to 66% (Fig 3A) with mean 26 ± 14% (± distribution s.d.). The variability in trajectories is apparent in the range of EV shown on a per cell basis (Fig 3B), indicating that all cells have a similar profile of near deterministic and highly stochastic trajectories. EV allowed us to rank trajectories by how well the model fitted—the trajectory shown in Fig 1C and reverse engineered in Fig 2 had the highest EV —but does not provide support for the model compared to any other since it is an intrinsic measure of fit. We therefore complemented the EV statistic with a comparative test using the Bayes factor B[Mcoh/MBM] between the coherence-incoherence model Mcoh and a Brownian motion (BM) model MBM (we also compared against BM models with an inter-sister spring and drift with almost identical results; see section 2.2 of S1 Text and S3A and S3B Fig). Surprisingly, B[Mcoh/MBM] showed significant preference for MBM; we found log B [Mcoh/MBM] < 0 for over 95% of the trajectories (Fig 3C and 3D; a log B[Mcoh/MBM] < 0 indicates preference for MBM). This is due to the lack of temporal structure in the MBM —displacements are treated as independent, identically Gaussian distributed so B[Mcoh/MBM] is predominantly a measure of whether the kinetochore displacements are Gaussian or not. Trajectories where Mcoh is preferred have very regular saw-tooth oscillations and thus an over-representation of large displacements; B[Mcoh/MBM] is thus a good discriminator of strong oscillating trajectories. Comparing EV with B[Mcoh/MBM] demonstrated a correlation as expected (overall ρ = 0.14, p = 0.0008; Fig 3G), but also revealed a group of outlier trajectories (green shading in figure) that had higher than average B[Mcoh/MBM] but low EV. These trajectories tended to have a few excessively large displacements indicative of tracking errors. We thus filtered these from the analysis by restricting to trajectories approximating the linear relationship between B[Mcoh/MBM] and EV (the selection region shown as a grey bar in Fig 3G; ρ = 0.90 for grey region) comprising 843 out of 1169 converged trajectories, 72%.

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
Related in: MedlinePlus