[Paper Review] Diffusible crosslinkers cause superexponential friction forces

Harmen Wierenga, Pieter Rein ten Wolde|arXiv (Cornell University)|Jun 6, 2019

Microtubule and mitosis dynamics4 citations

TL;DR

This study reveals that diffusible passive crosslinkers generate superexponential friction forces between sliding microtubules in the mitotic spindle, arising from rare, discrete jumps driven by free-energy barrier crossings during crosslinker hopping. The friction coefficient increases superexponentially with crosslinker density due to steric exclusion effects, a mechanism that stabilizes the spindle's dynamic structure beyond what motor proteins alone can achieve.

ABSTRACT

The mitotic spindle lies at the heart of the spatio-temporal control over cellular components during cell division. The spindle consists of microtubules, which are not only crosslinked by motor proteins but also by passive binding proteins. These passive crosslinkers stabilize the highly dynamic mitotic spindle by generating friction forces between sliding filaments. However, it remains unclear how the friction coefficient depends on the number of crosslinkers and the size of the overlap between the microtubules. Here, we use theory and computer simulations to study the friction between two filaments that are crosslinked by passive proteins, which can hop between neighboring binding sites while physically excluding each other. The simulations reveal that the movement of one microtubule relative to the other is limited by free-energy barrier crossings, causing rare and discrete jumps of the microtubule that span the distance between adjacent crosslinker binding sites. We derive an exact analytical expression for the free-energy landscape and identify the reaction coordinate that governs the relative movement, which allows us to determine the effective barrier height for the microtubule jumps. Both through simulations and reaction rate theory, we make the experimentally testable prediction that the friction between the microtubules increases superexponentially with the density of crosslinkers.

Motivation & Objective

• To understand the origin of friction forces between sliding microtubules in the mitotic spindle, particularly those generated by passive, diffusible crosslinkers.
• To resolve the lack of a theoretical description for how the friction coefficient depends on crosslinker density and overlap length.
• To determine whether friction increases exponentially or superexponentially with crosslinker density, especially under steric exclusion constraints.
• To derive an analytical expression for the free-energy landscape governing microtubule sliding, identifying the reaction coordinate and effective barrier height.

Proposed method

• Modeling microtubule sliding as a process governed by discrete jumps of length one tubulin dimer, driven by crosslinker hopping between binding sites.
• Using reaction rate theory to describe the relative movement as free-energy barrier crossings, with the barrier height determined by the statistical mechanics of crosslinker configurations.
• Deriving an exact analytical expression for the free-energy landscape using combinatorial statistics of crosslinker distributions on overlapping microtubules.
• Applying a Taylor expansion and exponential approximation to the free-energy barrier height, capturing both low- and high-density regimes.
• Validating results through computer simulations and comparing analytical approximations to exact numerical calculations of the barrier height.
• Incorporating steric exclusion between crosslinkers and their thermal diffusion to model realistic biological conditions.

Experimental results

Research questions

• RQ1 How does the friction coefficient between two sliding microtubules depend on the density of diffusible passive crosslinkers?
• RQ2 What is the origin of the observed exponential increase in friction, and does it transition to superexponential behavior at higher densities?
• RQ3 How do steric exclusion effects between crosslinkers influence the free-energy landscape and the effective barrier height for microtubule sliding?
• RQ4 Can the relative movement of microtubules be accurately described as rare, discrete jumps, and what determines the jump size?
• RQ5 What analytical expression captures the friction coefficient across both low and high crosslinker densities?

Key findings

• The friction coefficient increases superexponentially with crosslinker density due to steric exclusion effects, contradicting the previously assumed exponential dependence.
• The relative sliding of microtubules occurs via rare, discrete jumps of exactly one tubulin dimer length, corresponding to the distance between adjacent crosslinker binding sites.
• The free-energy barrier height is analytically derived using combinatorial statistics of crosslinker configurations, with the reaction coordinate being the number of crosslinkers on the mobile microtubule.
• At low densities, the barrier height increases linearly with the number of crosslinkers, consistent with independent hopping; at high densities, exclusion effects dominate, leading to superexponential growth.
• The analytical approximation in Eq. S.44 captures the exact barrier height over a wide range of densities, outperforming second-order Taylor expansions.
• The model predicts a measurable superexponential increase in friction, providing an experimentally testable signature of passive crosslinker function in spindle stability.