Limited Resources Induce Bistability in Microtubule Length Regulation

Open Access

Limited Resources Induce Bistability in Microtubule Length Regulation

Matthias Rank, Aniruddha Mitra, Louis Reese, Stefan Diez, and Erwin Frey

Phys. Rev. Lett. 120, 148101 – Published 5 April 2018

Abstract

The availability of protein is an important factor for the determination of the size of the mitotic spindle. Involved in spindle-size regulation is kinesin-8, a molecular motor and microtubule (MT) depolymerase, which is known to tightly control MT length. Here, we propose and analyze a theoretical model in which kinesin-induced MT depolymerization competes with spontaneous polymerization while supplies of both tubulin and kinesin are limited. In contrast to previous studies where resources were unconstrained, we find that, for a wide range of concentrations, MT length regulation is bistable. We test our predictions by conducting in vitro experiments and find that the bistable behavior manifests in a bimodal MT length distribution.

Received 17 August 2017
Revised 29 January 2018

DOI:https://doi.org/10.1103/PhysRevLett.120.148101

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.

Published by the American Physical Society

Physics Subject Headings (PhySH)

Intracellular transport Stochastic processes

Cytoskeleton Filaments Motor system Proteins

Physics of Living SystemsStatistical Physics & Thermodynamics

Authors & Affiliations

Matthias Rank¹, Aniruddha Mitra^2,3, Louis Reese⁴, Stefan Diez^2,3,*, and Erwin Frey^1,†

¹Arnold-Sommerfeld-Center for Theoretical Physics and Center for NanoScience, Ludwig-Maximilians-Universität München, Theresienstraße 37, 80333 München, Germany
²B CUBE—Center for Molecular Bioengineering and Center for Advancing Electronics Dresden (cfaed), Technische Universität Dresden, Arnoldstraße 18, 01307 Dresden, Germany
³Max Planck Institute of Molecular Cell Biology and Genetics, Pfotenhauerstraße 108, 01307 Dresden, Germany
⁴Department of Bionanoscience, Kavli Institute of Nanoscience, Faculty of Applied Sciences, Delft University of Technology, Van der Maasweg 9, 2629 HZ Delft, The Netherlands

^*stefan.diez@tu-dresden.de
^†frey@lmu.de

Article Text

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Supplemental Material

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References

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Issue

Vol. 120, Iss. 14 — 6 April 2018

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Images

Figure 1
Sketch of the model. (a) A MT in a closed volume interacts with molecular motors. (b) Motors attach to the MT lattice at rate $ω_{A}$ and detach at rate $ω_{D}$ . Motors proceed stepwise toward the plus end at rate $ν$ , provided the next site is unoccupied. At the tip, motor-induced lattice depolymerization (rate $δ$ ) competes with spontaneous polymerization (rate $γ$ ). $ω_{A}$ and $γ$ and depend on the concentrations of the proteins available in the closed volume [Eq. (1)].
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Figure 2
Basic phenomenology. Differently colored traces depict different simulation runs under the indicated starting conditions. Dotted lines show results of the full mean-field theory. (a),(b) MTs evolve toward a stationary length, which depends on the concentrations of Kip3 and tubulin. (c),(d) The corresponding motor density $ρ$ on the MT is shown; see Fig. S2(e) in the Supplemental Material: $ρ$ increases with distance from the minus end and peaks at the plus end. (e) The MT length is bistable for a range of concentrations.
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Figure 3
Theoretical results. (a),(b) In silico scans of the stationary length of MTs, shown in color, as a function of $c_{m}$ and $c_{T}$ . Simulations start from a fully depolymerized lattice (short) in (a); in (b), the MT is initially fully polymerized (i.e., long). In the region bounded by the red lines (obtained from the full MF theory; see Supplemental Material Sec. S. II), the stationary length differs for these two cases: here, MT dynamics is bistable. (c) Rate of change of the MT length $\partial_{t} L$ as a function of $L$ at $c_{T} = 1.5 μ M$ for three different motor concentrations, as obtained from the approximate MF theory. For low and high motor concentrations, MT length is monostable, while for intermediate concentrations, two stable stationary states are separated by an unstable state (bistability). (d) Comparison of the steady-state length obtained from simulations (blue) and the full MF theory (orange) at $c_{T} = 2 μ M$ . (e) Stability diagram as obtained from the full MF theory.
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Figure 4
Experimental results. (a) Length distribution of MTs grown for 3 h and subsequently incubated with various concentrations of Kip3 for 1 h. The distribution is unimodal for 0, 4, and 400 nM Kip3; it is bimodal for 20 nM Kip3, indicating that length regulation is bistable. (b) Box plots for the MT length show that the median MT length decreases as the Kip3 concentration is increased (left axis). The dashed line (right axis) indicates the average number of MTs per field of view (FOV). (c) Length distributions of MTs initially grown for 3 h and subsequently incubated with 20 nM Kip3 for various amounts of time. Within an hour, a bistable distribution is established, and its shape is conserved as the incubation time is increased.
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