Identifying the motive force is central to explaining chromosome motions during mitosis. Presently, there is no consensus on what it is. The author has proposed a minimal assumptions model for the dynamics of post-attachment chromosome motions based on nanoscale electrostatics. Given the electrical properties of tubulin and the dynamic instability of microtubules, it is possible to account for prometaphase post-attachment, metaphase, and anaphase chromosome motions within a comprehensive model. The model addresses all of the following in a unified manner:

• Efficiency of aster and spindle assembly and the motive force for the motion of asters and forming half-spindles.
• Chromatid pair attachment.
• Motion of monovalently attached chromatid pairs.
• Motion of bivalently attached chromatid pairs and chromosome congression.
• Metaphase chromatid pair oscillations.
• Chromatid separation and anaphase-A chromosome motion.
• Anaphase-B pole separation.
• An ab-initio calculation of the maximum tension force exerted by a microtubule during mitosis that falls within the experimental range.
• Poleward force generation of chromosomes at poles with associated microtubule flux.

Electrostatic Considerations in Mitosis

By L. John Gagliardi

iUniverse, Inc.

Copyright © 2009 L. John Gagliardi
All right reserved.
ISBN: 978-1-4401-7378-3

Contents

1 Introduction......................................................................71.1 Preliminary considerations......................................................71.2 Some cellular electrostatics....................................................101.3 Spindle assembly and dynamics...................................................151.4 Summary.........................................................................192 Electrostatics in Nuclear Envelope Breakdown......................................212.1 Mechanical equilibrium of a nuclear envelope....................................212.2 Equilibrium of membrane charge clusters.........................................252.3 Electrostatics in nuclear envelope disassembly..................................282.4 Energy considerations in nuclear envelope disassembly...........................302.5 Electrostatics in nuclear envelope reassembly...................................322.6 Summary.........................................................................343 Electrostatic Force in Poleward Chromosome Motions................................363.1 Introduction....................................................................363.2 Electrostatic microtubule disassembly force at cell poles.......................373.3 Electrostatic microtubule disassembly force at kinetochores.....................433.4 Penetrating microtubules at a centrosome........................................493.5 Summary.........................................................................504 Induced Charge in Poleward Motions................................................524.1 Introduction....................................................................524.2 Induced charge on centrosomes and kinetochores..................................534.3 Electrostatic microtubule disassembly force at cell poles.......................574.4 Electrostatic microtubule disassembly force at kinetochores.....................604.5 Summary.........................................................................635 Electrostatics in Mitotic Chromosome Motions......................................655.1 Introduction....................................................................655.2 Antipoleward nanoscale electrostatic assembly force.............................685.3 Prometaphase and metaphase chromosome motions...................................715.4 Anaphase-A chromosome motion....................................................765.5 Anaphase-B chromosome motion....................................................795.6 Summary.........................................................................82A Some results from electrostatics..................................................85B Debye theory of counterion screening..............................................87C Electrostatic stress..............................................................92D Equation 2.1......................................................................94E Equation 2.10.....................................................................96F Bibliography......................................................................99

Chapter One

Introduction

1.1 Preliminary considerations

Although the current paradigm in cell biology has been quite successful in solving a number of problems in the understanding of cell structure and function, important problems remain open. Some of the most notable of these are in the area of cell division. Given the constraints wittingly or unwittingly imposed by the molecular biology paradigm, it would seem that models of mitotic motions and events have become more and more complex, and therefore-to this observer at least-more and more unsatisfactory.

It may be helpful at this point to give an example. It is a common parlor trick to attract small bits of paper by a charged hard rubber hair comb. This motion is simply explained in terms of the known negative charge on the comb attracting the nearer induced positive charge on the bits of paper with the paper bits remaining overall electrically neutral and the displaced negative charges farther from the comb. There is no need to identify the specific molecules in the paper (or in the comb, for that matter) that are responsible for the attractive force and the subsequent motion. In fact, since the charged comb will attract many different kinds of paper as well as many other substances, attempting to identify the molecules in the various substances is counterproductive to explaining what is happening. Similarly, as will be discussed later, chromosome and other mitotic motions can be described in terms of electric charge distributions. However before this can be done, one must first establish that electrostatic force is significant within cells in spite of counterion screening.

The electromagnetic interaction is primarily responsible for the structure of matter from atoms to objects. Much of physics, all of chemistry, and most of biology are in this realm of sizes. Primitive eukaryotic cells had to divide prior to the evolution of very many biological mechanisms, and it is reasonable to assume that basic physics and chemistry played dominant roles in both mitosis (nuclear division) and cytokinesis (cytoplasmic division). It is proposed that nanoscale electrostatic interactions played a major role in the dynamics of cell division in primitive cells, and that the fundamental solutions to the problem of cell division that were found by primitive cells may largely persist in modern eukaryotic cells.

The mitotic spindle is responsible for the segregation of sister chromatids during cell division. Chromosomes are attached to the spindle with their kinetochores [Euteneuer and McIntosh, 1981] attached to the plus ends of microtubules [Rieder, 1982; Bergen et al., 1980]. Chromosome movement is dependent on kinetochore-microtubule dynamics: a chromosome can move toward a pole only when its kinetochore is connected to microtubules emanating from that pole [Nicklas and Kubai, 1985]. A number of experimental studies have been undertaken to obtain information regarding microtubule dynamics, force production, and kinetochore function in mitotic cells. These experiments have revealed that the spindle can produce more force than is actually required to move a chromosome at the observed speeds for post-attachment movements, and that the force for the poleward motion of chromosomes can be localized at or near kinetochores [Nicklas, 1983; Mitchison et al., 1986; Gorbsky et al., 1987; Nicklas, 1989; Mitchison, 1989; Alexander and Rieder, 1991; Inoue and Salmon, 1995] or at spindle poles [Mitchison and Salmon, 1992; Maddox et al., 2002; Zhang and Chen, 2003]. Quite some time ago, M. S. Cooper addressed a possible link between endogenous electrostatic fields and the eukaryotic cell cycle [Cooper, 1979]. An early review by Jaffe and Nuccitelli [Jaffe and Nuccitelli, 1977] focused on the possible influence of relatively steady electric fields on the control of growth and development in cells and tissues.

Microtubules continually assemble and disassemble, so the turnover of tubulin is ongoing. The characteristics of microtubule lengthening (polymerization) and shortening (depolymerization) follow a pattern known as "dynamic instability": that is, at any given instant some of the microtubules are growing, while others are undergoing rapid breakdown. In general, the rate at which microtubules undergo net assembly-or disassembly-varies with mitotic stage [Alberts et al., 1994a].

Changes in microtubule dynamics are integral to changes in the motions of chromosomes during the stages of mitosis. Poleward and antipoleward chromosome motions occur intermittently during prometaphase and metaphase. Antipoleward motions dominate during the congressional movement of chromosomes to the cell equator, and poleward motion prevails during anaphase-A.

Chromosome motion during anaphase has two major components, designated as anaphase-A and anaphase-B. The poleward movement of anaphase-A is accompanied by the shortening of kinetochore microtubules at kinetochores and/or spindle poles. The second component, referred to as anaphase-B, involves the separation of the poles. Both components contribute to the increased separation of chromosomes during mitosis. It is proposed in this book that these changes in chromosome motions during mitosis can be attributed to changes in microtubule dynamics based on electrostatics. It is further proposed that the influence of intracellular pH changes on kinetochore microtubule dynamics is primarily responsible for post-attachment prometaphase and metaphase chromosome motions.

1.2 Some cellular electrostatics

In the cytoplasmic medium (cytosol) within biological cells, it has been generally thought that electrostatic fields are subject to strong attenuation by screening with 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 [Benedek and Villars, 2000a], and since cells of interest in the present work (i.e. eukaryotic) can be taken to have much larger dimensions, one would be tempted to conclude that electrostatic force could not be a major factor in providing the cause for mitotic chromosome movements in biological cells. However, the presence of microtubules, as well as other factors to be discussed shortly, change the picture completely.

Microtubules can be thought of as intermediaries that extend the reach of the electrostatic interaction over cellular distances, making this second most potent force in the universe available to cells in spite of their ionic nature. A presentation of some background material in electrostatics is summarized in Appendix A and a brief introduction to Debye lengths and counterion screening is given in Appendix B.

Microtubules are 25 nm diameter cylindrical structures comprised of protofilaments, each consisting of tubulin dimer subunits, 8 nm in length, aligned lengthwise parallel to the microtubule axis. The protofilaments are bound laterally to form a sheet that closes to form a cylindrical microtubule. The structure of microtubules is similar in all eukaryotic cells. Cross sections reveal that the wall of a microtubule consists of a circle of 4 to 5 nm diameter subunits. In most cases, the circle contains 13 subunits; however, 11, 12, 14, or 16 have also been observed. Neighboring dimers along protofilaments exhibit a small (B-lattice) offset of 0.92 nm from protofilament to protofilament, as depicted in Figure 1.1.

This offset will be approximated as 1 nm in the calculations in subsequent chapters since protofilament curling distributions for disassembling microtubules are more significant in determining the distances of protofilament free ends from various cellular structures. Protofilament curling of a disassembling microtubule is depicted in Figure 1.2.

Experiments have shown that the intracellular pH (p[H.sub.i]) of many cells rises to a maximum at the onset of mitosis, subsequently falling during the later stages [see, for example, Steinhardt and Morisawa, 1982; Amirand et al., 2000].

Although it is experimentally difficult to resolve the exact starting time for the beginning of the decrease in p[H.sub.i] during the cell cycle, it appears to decrease 0.3 to 0.5 pH units from the typical peak values of 7.3 to 7.5 measured earlier during prophase [Steinhardt and Morisawa, 1982].

Studies [Schatten et al., 1985] have shown that in vivo microtubule growth (polymerization) is favored by higher pH values. It should be noted that in vitro studies of the role of pH in regulating microtubule assembly indicate a pH optimum for assembly in the range of 6.3 to 6.4. The disagreement between in vitro and in vivo studies has been analyzed in relation to the nucleation potential of microtubule organizing centers like centrosomes [Schatten et al., 1985], and it has been suggested that pHi regulates the nucleation potential of microtubule organizing centers [Kirschner, 1980; De Brabander et al., 1982; Deery and Brinkley, 1983]. This favors the more complex physiology characteristic of in vivo studies to resolve this question. It will therefore be assumed in this book that in vivo experimental design is more appropriate for experiments relating to pH conditions affecting microtubule assembly.

A number of investigations have focused on the electrostatic properties of microtubule tubulin subunits [Sataric et al., 1993; Brown and Tuszynski, 1997; Baker et al., 2001; Tuszynski et al., 1998]. Large scale calculations of the tubulin molecule have been carried out using molecular dynamics programs along with protein parameter sets. The dipole moment of tubulin has been calculated to be as large as 1800 Debye (D) [Brown and Tuszynski, 1997; Tuszynski et al., 1995]. Experiments [Tuszynski et al., 1995; Sackett, 1997] have shown that tubulin net charge depends strongly on pH, varying quite linearly from -12 to -28 (electron charges) between pH 5.5 and 8.0. This could be significant for microtubule dynamics during mitosis because, as noted above, many cell types exhibit a decrease of 0.3 to 0.5 pH units from a peak at prophase during mitosis.

It has been determined that tubulin has a large overall negative charge of 20 at pH 7, and that as much as 40 % of the charge resides on C-termini. The C-termini can extend perpendicularly outward from the microtubule axis as a function of p[H.sub.i]. It would seem reasonable to assume that an increased tubulin charge and the resulting greater extension of C-termini may be integral to an increased probability for microtubule assembly during prophase when p[H.sub.i] is highest. A higher p[H.sub.i] during prophase is consistent with increased interaction between the highly extended C-termini of tubulin dimers with appropriate regions of other nearest neighbor dimers. Given a decrease in p[H.sub.i] during mitosis, changes in microtubule assembly probabilities-in conjunction with nanoscale electrostatic interactions-could be responsible for the observed changes in chromosome motions during mitosis. In particular, a decrease in p[H.sub.i] during mitosis may act as a master clock controlling microtubule disassembly to assembly (disassembly/assembly) probability ratios during the phases of mitosis, thereby controlling the timing and dynamics of mitotic chromosome movements through metaphase. This will be discussed in more detail in Chapter 5 after the necessary groundwork has been developed.

It is generally accepted that the charge on the plus free ends of microtubules proximal to kinetochores is negative. (According to existing convention, these ends are designated plus because of their more rapid growth, there being no reference to charge in the use of this nomenclature.) Because of the electric dipole nature of the tubulin dimer subunits comprising microtubules, the net charge at the slower growing minus ends of microtubules proximal to a centrosome matrix will be assumed positive. As will be noted in Section 1.3, the assumption of a net negative charge on centrosomes is consistent with experiment. However, positive charge at the minus ends of microtubules will induce negative charge on an electrically neutral centrosome matrix area adjacent to the free minus ends of the microtubules comprising the astral, polar, and kinetochore microtubule bundles, obviating the need to assume negative charge on centrosome matrices for some cell types. Calculations will be carried out in Chapters 3 and 4 for electrostatic force generation between positively charged microtubule minus ends and negatively charged centrosome matrices.

Similarly, kinetochores may manifest positive charge at pole-facing surfaces. Evidence for this comes from the presence within kinetochores of highly basic molecules in the Dam1 complex. In particular, the isoelectric points of Dam1p, Duo1p, and Spc34p are 9.97, 10.76, and 8.6, respectively. Significantly, experiments have revealed that the microtubule binding module of the Dam1 complex involves these three molecules; acidic proteins Ask1p, Spc19p, and Dad2p fail to bind [Westermann et al., 2005].

Kinetochore molecules self-assemble onto highly condensed, negatively charged DNA at centromeres [Alberts et al., 1994b], indicating that kinetochores may exhibit positive charge. This is an example of an important aspect of electrostatic interactions within cells: namely their longer range compared to other intracellular molecular interactions and the resulting capacity of electrostatic force to organize molecules and structures within cells. As will be discussed shortly, quite apart from the ability of microtubules to extend electrostatic interactions over cellular distances, the range of electrostatic fields within the cytosol itself is longer than ordinary counterion screening considerations would dictate. This will be seen throughout the present work to have important implications for a number of mitotic events.

Analogous to the situation for induced negative charge on a centrosome matrix, it may not be necessary to assume net positive charge on kinetochores since the negatively charged free plus ends of kinetochore microtubule bundles will induce positive charge on kinetochores. This possibility is considered in Chapter 4. The calculations in Chapters 3 and 4 demonstrate that the magnitude of the motive force for poleward motion of chromosomes is sufficient given either permanent or induced charge on centrosome matrices and kinetochores.

1.3 Spindle assembly and dynamics

It is reasonable to expect that the electric dipole nature of tubulin subunits greatly assists in their self-assembly into the microtubules of the asters and spindle. Thus we may envision that electrostatic fields organize and align the electric dipole dimer subunits, thereby facilitating their assembly into the microtubules that form the asters and mitotic spindle [Gagliardi, 2002b]. This self-assembly would be aided by significantly reduced counterion screening due to layered water adhering to the net charge of the dipolar subunits. Such water layering to charged proteins has long been theorized [Jordan-Lloyd and Shore, 1938; Pauling, 1945] and has been confirmed by experiment [Toney et al., 1994]. Additionally, as will be described in Chapter 3, layered water between sufficiently close charged proteins has a dielectric constant that is considerably reduced from the bulk value far from charged surfaces, further increasing the tendency for an electrostatic assist to aster and spindle self-assembly. The question of what is meant by "sufficiently close" charged protein surfaces as well as the reduction in the dielectric constant between such surfaces will be addressed in Chapter 3.

(Continues...)

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