Reverse engineering of force integration during mitosis in the Drosophila embryo

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Roy Wollman^1,2, Gul Civelekoglu-Scholey^1,2, Jonathan M Scholey^1,2 & Alex Mogilner^1,3,4

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The mitotic spindle is a complex macromolecular machine that coordinates accurate chromosome segregation (Karsenti and Vernos, 2001; Pines and Rieder, 2001; Scholey et al, 2003) (Figure 1A). In the Drosophila syncytial embryo, hundreds of mitotic spindles progress synchronously through a sequence of transitions, in which periods of rapid pole–pole separation are interspersed with quiescent pauses (Brust-Mascher and Scholey, 2007). One of the hypotheses is that this spindle elongation is driven by a balance of forces generated by microtubules and multiple molecular motors, but how these forces are integrated remains unclear, since the temporal activation profiles and the mechanical characteristics of the relevant motors are largely unknown. This hypothesis is supported by the fact that comparison of changes in spindle length time series in five mutant and biochemically inhibited spindles reveals characteristic defects in pole–pole separation compared to wild-type spindles (Figure 1C). Each defect can be most naturally explained by a shifting force balance in the spindle resulting from the inhibition of specific molecular motors.

Figure 1

The spindle protein machinery. (A) A cartoon that shows all major components of the spindle. Four MT populations (astral (as), kinetochore (kt), chromosome arm (chr) and inter-polar (ip) MTs) extend from the poles creating the spindle. Molecular motors bind to MTs and either regulate their ends' kinetics, or slide them, or exert forces on the chromosomes and centrosomes. (B) Eight possible MT–motor combinations, with the respective velocities and forces acting on a single MT. asMT: cortical dynein pulling the MT generates an outward force F₁ on the spindle pole. chrMT are anchored at the pole, while MT polymerization and chromokinesins generate a pushing force F₂; a force F₃ is associated with such an MT if depolymerases are activated at the MT's minus end in addition to motor activity at the MT's plus end. ktMT: an inward force, F₄ is generated on an MT anchored at the pole while kt motors act on the MT plus end; modified force F₅ acts on an MT depolymerized at the minus end in addition to the plus end motor activity; force F₆ is exerted if ktMT is depolymerized at its minus end and anchored at its plus end. ipMT: an outward force, F₇ results from the combination of kinesin-5 and kinesin-14 actions on the MT anchored at the pole, while a force F₈ is exerted by these motors on an MT being depolymerized at its minus end. (C) The experimentally measured time series for spindle length (pole–pole distance) in wild type (WT) and inhibited spindles used in the optimization process (details in Supplementary Figure 1, referenced in Supplementary Table 2). Colors correspond to motor colors in the legend. Previous studies revealed that the double inhibition of kinesin-5 and kinesin-14 fully rescues metaphase spindle assembly (dashed blue line, not used here) (Sharp et al, 2000b). However, recent studies suggest that this effect results from the partial inhibition of kinesin-5, whereas a more complete inhibition leads to prometaphase spindle collapse even in kinesin-14 null mutants (green line, used here), suggesting that an additional unknown inward force also opposes kinesin-5 and contributes to the collapse (Brust-Mascher and Scholey, in preparation).

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In this paper, we attempt to 'reverse engineer' the spindle—i.e. to use experimental data on the spindle elongation to understand the temporal activation sequence and the mechanical characteristics of the force generators acting during mitosis—and in a sense to reconstitute the spindle in silico. This presents a challenge that seems prohibitive: with more than 10 types of molecular motors being involved, each characterized by unknown mechanical, kinetic and regulatory parameters, and the structural complexity of the spindle, it is impossible to use intuition and traditional modeling to explain the dynamics associated with the sequence of transitions characteristic of mitotic progression. To address this challenge, we develop and utilize a novel computational algorithm that, first, automatically and randomly constructs a 'virtual spindle' from eight 'building blocks' (different microtubule–motor configurations that generate critical spindle forces). Each such virtual spindle is a dynamic combination of these building blocks composition of which changes over time based on motors' switching on and off at random times. Each virtual spindle is characterized, in addition to these random switching times, by a random choice of 30 kinetic and mechanical parameters characterizing the motors and microtubules. Thus, we have a very high (39-) dimensional 'model parameter space'—a vast set of plausible different models characterized by various possible combinations of molecular motors' and microtubules' force–velocity relations, with variations in the kinetics and timing of on/off regulatory switches. Second, our algorithm solves force balance equation for each of hundreds of thousands of models resulting in predictions for spindle forces, velocities and length. Third, the predicted time series for the spindle length are automatically compared to the data, and the models are 'screened' so that those whose predictions compare poorly with the data are discarded. Fourth, a genetic algorithm automatically modifies or 'mutates' the successful 'selected' models, and repeated stochastic optimization results in the evolution of increasingly adequate multiple models, each producing excellent fits to all the available experimental data. Finally, cluster analysis identifies minimal sets of successful strategies for force integration in mitosis.

We discovered that there is a tremendous variety of plausible models, many of which are complex and counterintuitive, that can explain behavior of the wild-type spindle, so the wild-type data set, on its own, is not sufficient to discriminate between the multiple potential mechanisms of mitosis. Then, further search using data for mutant and biochemically perturbed spindles eliminated thousands of possible models and identified just six distinct strategies (model groups) for microtubule–motor integration that agree with all available data. Though there are not enough data to definitively propose a single spindle model, this result is valuable for guiding future experimental work.

Indeed, several model properties, such as timing of activity of crucial bipolar kinesin, dynein and depolymerase kinesin motors, are highly conserved among the six models, hinting that these properties are required for proper spindle design and mitotic progression (Figure 3). Also, all six models predict that the spindle is largely balanced by the outward forces generated by motors on the inter-polar microtubules, opposed by inward forces generated by motors on the kinetochore microtubules, and that characteristic forces' magnitudes are in the range of hundreds of picoNewtons. In addition to illuminating properties of the entire spindle, the modeling suggests that specific biophysical properties of the participating molecular motors are conserved. Prediction for two of the kinesin motors are in agreement with recent in vitro biochemical studies (Tao et al, 2006). While many of the model features are conserved and uniform, the six identified model groups have interesting biological differences among them, and we suggest specific experiments that in the future can help to eliminate five plausible models and narrow down on a definitive quantitative model of the Drosophila spindle.

Figure 3

Cluster analysis of all identified models. (A) Results of the cluster analysis for models that fit all available experimental data. Dendrogram shows the hierarchical tree of all 1000 models. Each imaginary vertical line across the panel corresponds to a specific model fitting all available experimental data. Time series (represented on the y axis of each panel running from top (early) to bottom (late)) for the forces on the four MT populations and cohesion forces from prometaphase (t=0) till the end of anaphase B (t=278 s) follow immediately below the dendrogram. Six identified clusters within the tree are color-coded. The forces (in picoNewtons) are color coded according to the bar shown at the upper left corner (extreme red (blue) corresponds to 1500 pN (- 1500 pN)). The forces are abbreviated as follows: F_as—total force on astral MTs; F_ip—total force on inter-polar MTs; F_chr—total force on MTs reaching to the chromosome arms; F_kt—total force on the kinetochore fiber and F_coh—forces induced by the cohesion complex that hold sister chromatids together. For the reference, the time series for the pole–pole distance are shown immediately below the force bar. Time series for 10 motor switches' activity follow immediately below the force time series. White and black correspond to active and inactive motors, respectively. The switches are: P_dep—pole depolymerizer; P_chr—chromokinesin; P_dyn—dynein; P_k5—kinesin-5 sliding motor; P_k14—kinesin-14; P_kt—combined kt motors; P_mt—MT plus end depolymerization activity at the kinetochore; P_poly—MT plus end polymerization activity at the kinetochore; P_as—switch regulating the number of astral MTs; P_ovrlp—switch regulating the number of MTs at the overlap zone at the spindle equator. (B) The time series for the forces and switches predicted by a single selected model from group 1 as an illustrative example. The vertical bars (left) in the upper half of the panel represent the color-coded forces for each MT population, whereas the corresponding graphs (blue line, right) depict the change in magnitude of the corresponding force with time. The vertical bars in the lower half of the panel (left) represent the black–white switching 'off' or 'on' of the specific motors at the times indicated on the graphs (green line, right) for this particular model.

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