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Experiments implicating bound volume positive charge at kinetochores interacting with negative charge at microtubule free ends have prompted our calculation of the force at kinetochores for chromosome poleward motility during mitosis. We present here a corroborating force calculation between positively charged Hec1 tails in kinetochores and negatively charged C-termini at microtubule free ends. Based on experimentally-known charge magnitudes on Hec1 tails and C-termini at microtubule free ends, an
*ab initio* calculation of poleward (tension) force per microtubule that falls within the experimental range is demonstrated. Due to the locations of C-termini charges on concave sides of splaying microtubules, this attractive force between subsets of low curvature splaying microtubule protofilaments C-termini eventually fails for subsets of protofilaments with more pronounced curvature, thus generating poleward force as microtubules depolymerize in a dynamic coupling, as observed experimentally. The mechanism by which kinetochores establish and maintain a dynamic coupling to microtubules for force production during the complex motions of mitosis remains elusive, and force generation at kinetochores has emerged as a signature problem in chromosome motility. In agreement with experiment, two separate calculations show that attractive electrostatic interactions over nanometer distances account for poleward chromosome forces at kinetochores.

Force generation at kinetochores has emerged as one of the signature problems in mitotic movements. Consistent with theoretical predictions made over a decade ago [

Our purpose here is to mathematically corroborate that calculation with one that is based on direct interactions between positively charged unstructured Ndc80Hec1 tails in kinetochores and negatively charged C-termini at the free ends of microtubules, supporting an electrostatic-based model that explains poleward force generation.

Miller et al. [

Chromosomes can move toward a proximal pole only when their kinetochores are connected to microtubules coming from that pole [

The electrostatic properties of tubulin have been well-studied [

In the context of force generation for chromosome motility at kinetochores, Miller et al. [

Since the lengths of Hec1 tails are much longer than the location volumes of C-termini charge distributions, Hec1 tails will be modeled as very long linear charges, with a linear charge density λ C/m (Coulombs/meter). A simple application of Gauss’s law [

E = λ / 2 π ε r (1)

where ε (=kε_{0}) is the kinetochore permittivity, ε_{0} = 8.85 pF/m (picoFarads/meter), and k is the kinetochore dielectric constant. Note that the relatively small contributions from edge effects near the ends of the Hec1 tails are neglected in this calculation.

The N-terminal tail of Hec1 contains an equivalent positive charge Q of 10 (electron charges, e) [

For the force per protofilament, we have:

F p f = q E = n e λ / 2 π ε r (2)

where q = ne is the charge of n electrons on C-termini of a protofilament interacting with a Hec1 tail. Consistent with their open structures, a cytosol-saturated kinetochore is expected to have a dielectric constant midway between the kinetochore dry value and cytoplasmic water [_{pf} = 0.6n pN/pf (picoNewtons per protofilament).

Kinetochores generally number at least 8 Hec1 proteins per microtubule [

Since microtubule C-termini are on the concave sides of progressively splaying microtubules, increasing protofilament curvature will lead to a separation of the charges on Hec1 tails and C-termini. Subsets of low curvature splaying protofilaments produce poleward force, while other subsets of protofilaments with more pronounced curvature in later stages of depolymerization fail to bind. Accordingly, poleward forces are generated as microtubules depolymerize, in agreement with observation.

Electrostatic fields within the cytosol are subject to strong attenuation due to screening by oppositely charged ions (counterion screening), decreasing exponentially to much smaller values over a distance of several Debye lengths. The Debye length within cells is typically given to be of order 1 nm [

Cellular electrostatics is also strongly influenced by reduced counterion screening due to layered water adhering to charged molecules. Such water layering – with consequent reduction or elimination of Debye screening – at charged proteins has long been theorized [

Given positive charge at kinetochores and negative charge on plus ends of microtubules, it is difficult to conceptualize there not being an attractive electrostatic poleward-directed force between these structures. A direct calculation of the electrostatic force between positively charged Hec1 tails and negatively charged C-termini at and near the free ends of microtubules supports an electrostatic force generating mechanism for poleward chromosome motions during mitosis. A singular strength of the present calculation is that the disassembly rate of microtubules at kinetochores is explicitly shown to be correlated with force production at kinetochores. In a broader context, understanding the underlying forces and mechanisms that dictate chromosome movements through mitosis will be critical to the development of approaches to circumvent anomalous cell divisions (e.g., cancer).

LJG conceptualized the theoretical aspects of this article and DHS provided intellectual contributions. Both authors read and approved the final manuscript.

The authors declare they have no competing interests.

Gagliardi, L.J. and Shain, D.H. (2019) Electrostatic Mechanism for Depolymerization-Based Poleward Force Generation at Kinetochores. Open Journal of Biophysics, 9, 198-203. https://doi.org/10.4236/ojbiphy.2019.93014