Hargreaves D et al. (2024), Relaxation and Noise-Driven Oscillations in a M...

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Bull Math Biol 2024 Aug 03;869:113. doi: 10.1007/s11538-024-01341-w.

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Relaxation and Noise-Driven Oscillations in a Model of Mitotic Spindle Dynamics.

Hargreaves D , Woolner S , Jensen OE .

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During cell division, the mitotic spindle moves dynamically through the cell to position the chromosomes and determine the ultimate spatial position of the two daughter cells. These movements have been attributed to the action of cortical force generators which pull on the astral microtubules to position the spindle, as well as pushing events by these same microtubules against the cell cortex and plasma membrane. Attachment and detachment of cortical force generators working antagonistically against centring forces of microtubules have been modelled previously (Grill et al. in Phys Rev Lett 94:108104, 2005) via stochastic simulations and mean-field Fokker-Planck equations (describing random motion of force generators) to predict oscillations of a spindle pole in one spatial dimension. Using systematic asymptotic methods, we reduce the Fokker-Planck system to a set of ordinary differential equations (ODEs), consistent with a set proposed by Grill et al., which can provide accurate predictions of the conditions for the Fokker-Planck system to exhibit oscillations. In the limit of small restoring forces, we derive an algebraic prediction of the amplitude of spindle-pole oscillations and demonstrate the relaxation structure of nonlinear oscillations. We also show how noise-induced oscillations can arise in stochastic simulations for conditions in which the mean-field Fokker-Planck system predicts stability, but for which the period can be estimated directly by the ODE model and the amplitude by a related stochastic differential equation that incorporates random binding kinetics.

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Species referenced: Xenopus laevis
Genes referenced: pdf
GO keywords: mitotic spindle

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	Fig. 1 The mitotic spindle and metaphase plate oscillate during the metaphase stage of mitotic cell division. a Time-lapse images of a mitotic spindle (GFP- -tubulin, green) and metaphase plate (mCherry-Histone 2B, magenta) during metaphase of a cell dividing in a Xenopus laevis embryo, at stage 10–11. The metaphase plate lies perpendicular to the fusiform shape of the mitotic spindle. b Blue and orange circles indicate the measured termini of the metaphase plate. c, f Tracked positions of metaphase-plate termini over a full course of metaphase, for two cells in an excised Xenopus animal cap at stage 10–11. d The x-components of the termini tracked in (c), showing oscillatory motion as a function of time. e The x-components of the termini of the metaphase plate tracked in (f), showing non-oscillatory motion as function of time. Arrows indicate the relevant measured terminus. Data from Hargreaves (2023), obtained using methods described in Appendix A (Color figure online)
	Fig. 2 Diagram of a spindle pole in three states. a The spindle pole (green) lies between the upper and lower cortex, displaced a distance z(t) from the mid-point. Force generators (orange) at each cortex comprise a motor protein head and an elastic linker which produce pulling forces . b Movement of the spindle pole affects the linker extensions of the motor proteins: movement away from the upper cortex lengthens the linkers of the upper force generators while compressing the linkers of the lower force generators. c Force generators with more extended linkers have an increased unbinding rate. Unbound generators cannot produce a pulling force (indicated by a grey force generator) (Color figure online)
	Fig. 3 Stochastic simulations can predict spontaneous oscillations of the spindle pole position. a Evolution of the non-dimensionalised spindle pole position through time. Dots correspond to moments in the cycle of interest and correspond colour-wise with the dots and diamonds plotted in (c). For later reference, the red bar identifies the oscillation period predicted by (18b) below. b The number of bound force generators (green) in the i) upper (+) and ii) lower (−) cortex (left y-axis) through time. The average extensions of the bound (magenta) and unbound (blue) force generators in the i) upper (+) and ii) lower (−) cortex are also shown (right y-axis). Coloured arrows correspond temporally to coloured symbols in (a). c Average extension of the bound generators in the upper (blue) and lower (orange) cortices as a function of pole position. Parameters in (a,b,c) are as in Table 2 with (see article for mathematical equation) A simulation when the unbinding of the force generator is no longer tension-sensitive, with . (e) A simulation when the restoring force is increased by a factor of 100 to (see article for mathematical equation) (Color figure online)
	Fig. 4 The effect of varying the magnitude of diffusion in the Fokker–Planck description. a, e Example solution to Eqs. (7a, 7b, 10), showing the pole position, z versus time t. Diffusion parameters (see article for mathematical equation) are a factor of 10 smaller in the right column than in the left column. b, f Heat map of (see article for mathematical equation) . c, g Heat map of (see article for mathematical equation) . d, h Probability density functions in the upper cortex at two instances of time. Solid lines: (see article for mathematical equation) , when the spindle pole is at and moving toward its minimum value (see article for mathematical equation) . Dotted lines: (see article for mathematical equation) , when the spindle pole is at (see article for mathematical equation) and moving toward its maximum value (see article for mathematical equation) . The peak widths scale with (see article for mathematical equation) and (see article for mathematical equation) as indicated. h The three regions used to reduce the system of PDEs to ODEs are indicated by roman numerals I, II, and III. The behaviour of the pdfs in the lower cortex are in antiphase to the behaviour seen here. Solutions were obtained using parameters as in Table 2 plus: (see article for mathematical equation) ; a–d baseline diffusivities (see article for mathematical equation) , (see article for mathematical equation) ; e–h (see article for mathematical equation) , (see article for mathematical equation) (Color figure online)
	Fig. 5 The stability boundary between oscillatory and non-oscillatory solutions is affected by the magnitude of diffusive terms. a Numerically solving the Fokker–Planck system (circles) reveals a boundary in (see article for mathematical equation) space which separates oscillatory from non-oscillatory solutions. Each circle represents a numerical solution, labelled magenta if the spindle pole has sustained oscillations and blue if the spindle pole position decays to (see article for mathematical equation) for large t. The point with the green boundary is the location in parameter space at which the solutions (c) and (d) sit. Other parameters are as in Table 2 except that (see article for mathematical equation) and (see article for mathematical equation) . For later reference, the shaded magenta area represents the region where oscillatory solutions exist as determined by stability analysis of the ODEs (19b) using equivalent parameters. The dashed curve (black) shows the same threshold in the limit of weak restoring force (see article for mathematical equation) , see (14)) determined by (21). The dotted magenta curve shows the asymptote of the lower boundary for (see article for mathematical equation) and (see article for mathematical equation) , as in (24). The dashed green curve shows the stability boundary (G1) predicted by (20b) from Grill et al. (2005). b The relationship between the period of oscillation and the binding rate (see article for mathematical equation) using (19a), along the neutral stability curve (19b). The period is unbounded as (see article for mathematical equation) . Points denote the periods taken from PDE solutions along the approximate neutral curve identified in (a). The magenta curves represent the approximations to the period for small (see article for mathematical equation) as in (25). The blue curves represent the approximations to the period as (see article for mathematical equation) as in (23). The dashed green curve shows the period (G2) predicted by (20a) from Grill et al. (2005). c Spindle pole position z in time t at the example point (green in (a)), from a PDE solution. d A PDE solution replicating (c) except that (see article for mathematical equation) and (see article for mathematical equation) (Color figure online)
	Fig. 6 Comparison of PDE (a, c) and ODE (b, d, e) solutions for equivalent parameters. PDE and ODE solutions for equivalent parameters are presented, with non-equivalent solutions separated by a dotted line. a, c Solutions of the PDEs; (b, d, e) solutions of the ODEs. First column: spindle pole position z. Second column: centre of the bound pdf as a function of pole position (see article for mathematical equation) . Third column: amplitude of the bound pdf as a function of the location of its peak (see article for mathematical equation) for PDE solutions (a, c); (see article for mathematical equation) for ODE solutions (b, d, e). PDE solutions were obtained using parameters are as in Table 2 except (see article for mathematical equation) , (see article for mathematical equation) , a and c (see article for mathematical equation) and . ODE solutions obtained using b equivalent parameters to (a); d equivalent parameters to (c); and e Equivalent parameters to (a) with (see article for mathematical equation) . Line colours correspond to solutions in each cortex (blue = upper, orange = lower). The black curves in the centre column represent the predicted limit cycle as (see article for mathematical equation) , as determined by the inversion of (30). Scatterpoints denote the positions of: maximum amplitude (blue (see article for mathematical equation) , yellow ), maximum spindle pole velocity (cyan (see article for mathematical equation) , magenta (see article for mathematical equation) (Color figure online)
	Fig. 8 Amplitude estimation of noise-induced oscillations. a The (3, 3) component of spectrum matrix using (43), for parameters as in Table 2 with (see article for mathematical equation) (black) and (see article for mathematical equation) (red). b Amplitude estimation from (see article for mathematical equation) versus distance to the neutral curve (see article for mathematical equation) for (see article for mathematical equation) (black), (see article for mathematical equation) (blue) and (see article for mathematical equation) (green). c, d Example pole dynamics from stochastic simulations and e, f the corresponding histograms weighted by time spent at each z-position. Shaded regions denote the interquartile range. Parameters as in Table 2 with (c, e) (see article for mathematical equation) , (d, f) (see article for mathematical equation) (Color figure online)
	Supplementary file 1 (avi 753 KB)
	Supplementary file 2 (mp4 3733 KB)
	Fig. 9 Graphical map of extension states for unbound and bound force generators. a Unbound generators in state (see article for mathematical equation) may extend or retract with probabilities (see article for mathematical equation) and (see article for mathematical equation) . Bound generators in state (see article for mathematical equation) may extend or retract with probabilities (see article for mathematical equation) and (see article for mathematical equation) . Bound generators may unbind or vice-versa with rate constants (see article for mathematical equation) and (see article for mathematical equation) respectively. Diagrams of force generators show corresponding extension and binding states. Each individual force generator n exists within these states. b Concatenated list of rate triplets to show numbering scheme. Probabilities from (see article for mathematical equation) to (see article for mathematical equation) correspond to force generators (see article for mathematical equation) which exist in the upper cortex. Probabilities (see article for mathematical equation) to (see article for mathematical equation) correspond to force generators (see article for mathematical equation) which exist in the lower cortex
	Fig. 7 Increasing (see article for mathematical equation) results in a neutral curve which underestimates the threshold number of N. Numerically solving the Fokker–Planck system (circles) reveals a boundary in (see article for mathematical equation) space which separates oscillatory from non-oscillatory solutions. Each circle represents a numerical solution, labelled magenta if the spindle pole has sustained oscillations and blue if the spindle pole position decays to (see article for mathematical equation) for large t. The black line represents the neutral curve separating oscillatory and decaying solutions as determined by stability analysis of the ODEs (19b) using equivalent parameters. The green curve shows the stability boundary (G1) predicted by (20b) from Grill et al. (2005). All parameters are as in Table 2 except that (see article for mathematical equation) , (see article for mathematical equation) , and (see article for mathematical equation) (Color figure online)