Inertial force as a possible factor in mitosis

BioSystems, 17 (1985) 301-315 301 Elsevier Scientific Publishers Ireland Ltd.

I N E R T I A L F O R C E AS A P O S S I B L E F A C T O R IN M I T O S I S

JONATHAN WELLS

The Graduate Sch<~>l, Yale University, 417 Yale Station, New Haven, CT 06520 (U.S.A.)

(Received April 2nd, 1984) (Revision received January 11th, 1985)

Some previous studies of cell division have suggested that chromosome movements in mitosis involve two distinct forces: one which pulls chromosomes poleward by means of attached fibers, and another which tends to push chromosome arms away from the pole. The latter force may also be a factor in non-chromosomal spindle transport, by which objects other than chromosomes are transported toward or away from spindle poles.

Based on a survey of previous literature, this paper makes a prima facie case for describing this latter force as "inertial", since in some respects it can be simulated by contrifugation. A theoretical analysis demonstrates that an inertial force could arise in the spindle from postulated high-frequency, small-amplitude Oscillations, which could be caused by changes in coherently precessing electron spin alignments at the spindle poles. Some possible experimental approaches to the problem are briefly outlined.

Keywords: Mitosis; Spindle transport; Spindle oscillations; Inertial; Spin alignments.

I . I n t r o d u c t i o n

Although m a n y aspects of mitosis r e m a i n poorly unde r s t c ~ l , it has a t leas t become in- c reas ingly c lear t h a t cell division involves more t h a n one mechan i sm. For example , the ini t ial s epa ra t ion of s i s te r chromat ids , t he i r subsequen t poleward movemen t , and the con- t rac t ion of the c leavage fu r row in la te a n a p h a s e are all known to be funct iona l ly i n d e p e n d e n t of each other . (For reviews, see Schrader , 1953; Mazia, 1961; Luykx, 1970; Nicklas, 1971; Rappapor t , 1971; Ba je r and Mol~-Bajer, 1972; Inou~, 1981; P icke t t -Heaps et al., 1982).

In addi t ion to the force which r m o v e s chromosomes poleward du r ing anaphase , mitot ic spindles exhibi t a n o t h e r t r a n s p o r t p rope r ty which t ends to move chromosomes away f rom the poles, and which m a y be the same p rope r ty which causes non-chromosomal bodies wi th in spindles to move toward or away from the poles. This l a t t e r t r a n s p o r t p rope r ty depends in some way on the p resence of microtubule~; but , a t leas t as it affects non-chromosomal bodies, it has so fa r re-

ma ined "complete ly unexp la ined" (Bajer and Mol~-Bajer, 1982). A su rvey of previous l i tera- t u r e shows t h a t this l a t t e r aspect of spindle t r a n s p o r t m a y have the charac te r i s t ics of an iner t ia l force which, l ike cent r i fugal force, affects objects according to t he i r re la t ive densities. This p ap e r offers a theore t ica l account of how a spindle migh t gene ra t e an iner t ia l force of sufficient m a g n i t u d e to be a factor in mitot ic movements .

2. E v i d e n c e f o r a n i n e r t i a l f o r c e a s o n e f a c t o r i n m i t o s i s

2.1. C h r o m o s o m e m o v e m e n t s

I t is wel l -es tabl ished t h a t mitot ic chromosomes are a t t ached to the spindle pole by fibers which are neces sa ry for normal anaphase m o v em en t s (Cornman, 1944; Nicklas and Staehly , 1967; Begg and Ellis, 1979; Nicklas e t al., 1981; Ellis and Begg, 1981). When k ine tochore fibers are d i s rup ted by u l t rav io le t mic robeam i r rad ia t ion , chromosomes cease to move poleward (Bajer and Mol~-Bajer, 1961);

0303-2647/85/$03.30 © 1985 Elsevier Scientific Publishers Ireland Ltd. Published and Prir~ted in Ireland

302

and when chromosomes move poleward their kinetochores invariably lead the way, suggesting a pulling force from the pole which is t ransmit ted by kinetochore fibers (Nicklas, 1961; Nicklas and Koch, 1972).

Kinetochore fibers are also important in pre-anaphase movements. Uretz et al. (1954) and Bloom et al. (1955) showed tha t irradiation of kinetochores prevents chromosomes from forming a normal metaphase plate. More precisely, Izutsu (1959) demonstrated tha t irradiating the fiber of a single chromatid in prometaphase causes it to move with its sister chromatid toward the opposite pole (i.e. in the direction of the unirradiated fiber). Metaphase congression occurs only if (1) the fibers of both chromatids are irradiated, so tha t the poleward pulling force is zero; or (2) neither fiber is irradiated so tha t the poleward pulling force is exerted equally in opposite directions. Recent experiments by Seto et al. (1969) and McNeill and Berns (1981) tend to corroborate these results.

These data point to a poleward pulling force, active from prometaphase through anaphase, which is t ransmit ted by kinetochore fibers. Other evidence, however, in- dictates tha t chromosomes may be transported away from the pole without such fibers. For example, when kinetochore fibers are irradiated in anaphase, not only do chromosomes stop their poleward movement, but they also frequently return to the spindle equator (Bajer and Mol~-Bajer, 1961; Bajer, 1972). In some species, kinetochores gather at the metaphase plate while chromosome arms are pushed laterally outward from the spindle equator (0stergren et al., 1960; Bajer and 0stergren, 1963; Lambert and Bajer, 1972). In still other species, even the kinetochores of some chromosomes separate from the main body of the spindle as they are pushed laterally away from the equator (Schrader, 1946; Hughes-Schrader, 1948). Their lateral movement appears to be checked only by kinetochore fibers, which in these particular chromosomes form late or else grow to unusual lengths (Fig. 1). Schrader (1947) concluded tha t

i ?:;'.:.:...

D..~ ..:

e

Fig. I. Lateral exclusion of chromosomes from mitotic spindles. (a) and (b) Brachystethus rub~maculatus. The small, dark objects are chromosomes undergoing the usual mitotic movements within the spindle, while the large, white mass to the left is a chromosome which is being pushed laterally away from the spindle equator. The dotted lines represent both spindle fibers and kinetochore fibers (from Schrader, 1946) (c) HumbertieUa indica. The chromosome on the right is being pushed laterally away from the metaphase plate. Unlike the laterally excluded chromosome in (a) and (b), this one is connected to only one pole. Although Schrader thought that the lateral movement might be due merely to pushing by abnormally elongated kinetochore fibers, this example suggests that some other factor must be involved (from Hughes- Schrader, 1948).

this phenomenon "involves positive as well as negative forces, i.e. the give and take of chromosomal fibers as well as the tendency of the spindle body to evict the chromosomes."

In addition to the poleward pulling force, then, there seems to be a repelling force, originating at the spindle poles, which tends to move chromosomes toward the equator and then to push them laterally away from the spindle. This conclusion is not new: 0s tergren (1945) postulated "two antagonistic tenden- cies" influencing chromosomes at the metaphase plate: one striving to move the kinetochores "inward (centripetally)", and the other striving to move the chromatids themselves "outward (centrifugally)". The latter tendency may be equivalent to a "force" acting

303

to "repel" chromosome arms away from the spindle poles ((~stergren et al., 1960). More recently, Mol~-Bajer et al. (1975) distinguish- ed between a poleward "pulling force" on kinetochore fibers and an "aster eliminating property" which pushes chromosome arms away from the pole.

Even more striking evidence for the ex- istence of two such forces is provided by monopolar division in Sciara spermatocytes, in which some chromosomes move poleward in normal fashion during anaphase while others (the paternal homologs) move backward away from the pole with their arms leading and their kinetochores trailing (Fig. 2). According to Metz (1933, 1938), the arms-first orienta- tion of the retlreating paternal chromosomes shows that they are not being pushed away from the pole by their kinetochores. Further- more, although such backward movement may depend in some way on the presence of '!poorly defined astral rays" (Bajer and Mole- Bajer, 1981), i1~ does not seem to be due to pulling by fibers (Metz, 1933, 1938; Abbott et al., 1981). Metz concluded that monopolar division demonstrates the presence of two distinct mitotic forces: one which pulls a chromosome poleward by means of its kinetochore fiber, and another which tends to move the chromosome as a whole away from the pole.

These conclusions suggest that chromosomes should move poleward only when pulled by kinetochore fibers. There are three situa-

Fig. 2. Three successive stages in anaphase during monopolar division in a Sciara coprophila spermatocyte. After congregating semi-spherically near the pole, all of the chromosomes eKcept the paternal homologs move toward the pole in normal fashion, while the paternal homologs move backward away from the pole with their arms leading and their kinetochores (with attached fibers) trailing (from Metz, 1933).

tions, however, in which this seems not to be the case. These apparent anomalies warrant individual attention.

(1) Chromosomes with no detectable kinetochores can be produced experimentally by irradiating cells with X-rays (Carlson, 1938) or fl-rays (Bajer, 1958a; Bajer and Mol~- Bajer, 1963). During prometaphase, many of these acentric fragments move towards the spindle poles, where some of them are then eliminated into the cytoplasm. Others move to the equator, where they tend either to remain until late anaphase or to move laterally out of the spindle. Even though such fragments lack kinetochores, however, Carlson (1938) and Bajer (1958a) were unable to establish that they lacked connections to spindle fibers altogether. Bajer and Mol~-Bajer (1963) suggested that some fragment movements might be due to "neo-centric" activity, in which spindle fibers become attached to parts of a chromosome other than its kinetochore. Although the suggestion was tentative with respect to radiation-induced fragments, neo-centric activity is well-established in other cases (0ster- gren and Prakken, 1946; Rhoades, 1952; Bajer and Ostergren, 1961); and spindle fiber attachments to chromosomes without detectable kinetochores are common in some organisms (Ris and Kubai, 1970; Friedl~inder and Wahrman, 1970; Rieder, 1982). Given the like- lihood that chromosome fragments are sometimes moved by neo-centric at tachments, the question is whether such movement tends to be toward the spindle poles or away from them. Since t h e evidence for whole chromosomes indicates that poleward movement depends on fibers, and that chromosomes without any functional fibers at all move to the equator, it seems reasonable to suppose that neo-centric activity produces poleward movement in prometaphase. This supposition is s trengthened by recent evidence that, in at least one case, poleward movement of acentric fragments depends on pulling by fibers (Fuge, 1975). Finally, the reportedly t ransient nature of neocentric a t tachments (Bajer and Ostergren, 1961; Bajer and Mol~-Bajer, 1963) could result

304

in detachment of some chromosome fragments after they have moved to the spindle poles, thus accounting for their subsequent elimination into the cytoplasm.

(2) In most cases of mitosis, metaphase chromosomes lie at the spindle equator in a plate perpendicular to the spindle axis. In some cases of plant mitosis, however, the kinetochores of long, slender chromosomes arrange themselves at the metaphase plate while the arms lie parallel to the spindle axis, pointing toward the poles (Bajer and Mol~- Bajer, 1972). Ostergren et al. (1960) postulated a "pumping action" of the spindle to account for this phenomenon (and for poleward transport in general). However, 0ster- gren (1949) had earlier noted a "tendency of the spindle to arrange rod-shaped bodies in parallel to the spindle fibers". Since spindle fibers are most numerous and most parallel at the metaphase plate (Nicklas, 1971; Bajer and Mol~-Bajer, 1975), the arrangement of long chromosome arms parallel to the spindle axis could be due merely to crowding by closely packed spindle fibers. Alternatively, or perhaps in addition to this tendency, some chromosome arms might be moved poleward by neo-centric activity similar to that discus- sed above.

(3) Experimentally-induced chromosome fragments which collect at the spindle equator from prometaphase through early anaphase are often observed to move poleward in late anaphase (Carlson, 1938; Bajer, 1958a; Mol~- Bajer, 1965). It seems likely that cytoplasmic currents are responsible for this movement. Although mitotic movements before mid- anaphase could not be due simply to currents in the spindle (Bajer and Mol~-Bajer, 1956; 0stergren et al., 1960), cytoplasmic currents have been observed to flow inward at the equator in late anaphase both in animal cells (Carlson, 1938; Hiramoto, 1968) and in plant cells (Bajer and Mol~-Bajer, 1956), with the result that cytoplasmic inclusions are trans= ported poleward.

Therefore, none of these apparent anomalies is inconsistent with the conclusion that two

distinct forces affect mitotic chromosomes from prometaphase through mid-anaphase: a poleward pulling force transmitted to the chromosomes by attached fibers, and another force which tends to move chromosomes away from the pole.

In some respects, this latter force can be simulated by centrifugation. When cells are centrifuged during mitosis, chromosomes are displaced in the centrifugal direction while spindle poles are displaced centripetally (Luyet, 1935; Beams and King, 1938; Shimamura, 1940). As a result, each chromosome is subjected to a force which tends to move it away from the pole, its movement being hindered only by its kinetochore fiber (Fig. 3). Of course, this similarity between the effects of centrifugal force and the chromosome-repelling mechanism in spindle transport could be coincidental, and it does not rule out other possible explanations. The similarity

' : " ~ " : ~ i i : ' " ..:~',

@ b

Fig. 3. Effects of centrifugation on dividing cells. (a) Cen- trifugation tends to force chromosomes centrifugally away from the spindle pole, to which they remain attached only by their kinetochore fibers (from Luyet, 1935). (b) An uncentrifuged cell in which the ends of a bipolar spindle are anchored to the cell periphery. (c) The same cell after centrifugation. As in (a), the chromosomes are displaced centrifugally, remaining "suspended" from the spindle poles by their kinetochore fibers. The dots at the centripetal pole are fat globules (from Shimamura, 1940).

305

seems sufficient, however, to justify a tentative working ]hypothesis that the repelling mechanism has the characteristics of an inertial force originating at spindle poles. Such a force would cause chromosomes unrestrained by kinetochore fibers to move backward away from a single pole (as in monopolar division), or to move to the plane midway between two poles and then proceed laterally outward (as in bipolar division). Before speculating about the possible nature of the force, however, it is relevant to inquire whether the effects of spindle transport on non-chromosomal bodies are consistent with this working hypothesis.

2.2. Non-chromosomal spindle transport

Bodies other than chromosomes which are transported by mitotic spindles (and by the rays of individual asters) include fat or oil droplets, mit~'hondria, persistent nucleoli, yolk particles and various other types of granules.

When Nicklas and Koch (1972) used micromanipulation to place cytoplasmic granules within normally granule-free spindles, they observed that some of the granules moved poleward while others moved laterally out of the spindles. The granules "probably include[d] a variety of materials such as lipid droplets and mitochondria", but Nicklas and Koch did not report whether there was any difference between the behavior of lipid droplets and that of mitochondria. Other evidence, however, indicates that oil droplets generally move toward the pole while mitochondria move away from it. Chambers (1917) used a microneedle to push "oil-like droplets" from the cytoplasm of an egg into the periphery of an aster, and observed that the droplets moved along the astral rays toward the center. Bloom et al. (1!}55) reported that oil droplets in the spindle are "attracted" by the pole. On the other hand, mitchondria caught in the spindle are often pushed to the equatorial region a n d - t h e n forced laterally outward (Hughes-Schrader, 1948; Bajer and ()stergren, 1963). Since centrifugation experiments have

demonstrated that oil droplets are less dense than mitochondria.(Luyet and Ernst, 1934; Beams, 1951; Tilney and Marsland, 1969), it appears that spindle transport of these two cell components resembles an inertial force.

Persistent nucleoli in spindles are sometimes transported poleward and sometimes eliminated laterally at the equator (()stergren et al., 1960). Nucleoli in centrifuged cells are sometimes unaffected and sometimes displaced centrifugally (Kostoff, 1930; Luyet and Ernst, 1934), suggesting variations in their densities. Nevertheless, in the absence of at- tempts to correlate density variations with spindle transport behavior, it cannot be clearly established that these data support the working hypothesis.

Yolk is the densest cytoplasmic component in many egg cells, and is usually excluded from mitotic spindles altogether; but asters sometimes contain yolk particles. Wilson (1925) reported that such asters are typically surrounded by several concentric zones, with the yolk particles in the outermost zone and mitochondria just inside it. Since centrifugation concentrates mitochondria in a layer just centripetal to the yolk (Tilney and Marsland, 1969), this seems to be consistent with the working hypothesis. Rebhun (1963) reported that small yolk particles in Cistenides eggs migrate into asters during mitosis, but the density of these particles relative to other astrally located particles was not reported.

By staining and centrifuging certain animal eggs, Pasteels (1958) and Mulnard (1958) were able to distinguish two types of meta- chromatic granules, a and fl, with different densities. The less dense fl granules, but not the denser a granules, were commonly found in close association with mitotic spindles, and were observed to "invade" asters (Mulnard et al., 1959), suggesting a possible correlation between density and aster transport behavior. On the other hand, Rebhun (1959, 1960) reported that sometimes a as well as fl granules "show.. . an astral localization," though Rebhun's observations did not exclude the possibility that the fl granules accumulated closer to

306

the astral center than the denser a granules. Feulgen-positive granules in Cyclops spin-

dles, according to Stich (1954), move poleward in prometaphase, then toward the equator in metaphase and early anaphase, then poleward again in late anaphase. Stich observed that the granules coalesced and increased in size before moving from the pole to the equator, but he did not determine whether any density change accompanied this size increase. No observed change in the granules accompanied their return from the equator to the pole, which may have been due (as Stich thought) to late anaphase cytoplasmic currents such as those described above.

In addition to such cytoplasmic currents, another factor complicating the picture of granule transport in mitotic spindles was described by Nicklas and Koch (1972). During anaphase chromosome movements, some granules were observed to move poleward ahead of the chromosomes, and at a somewhat slower velocity, until the chromosomes by- passed them and their movement ceased. Such movements seem to be a case of pushing by chromosomes ra ther than of true non- chromosomal transport.

Although the net movement in non- chromosomal spindle t ransport is toward or away from the pole, the movement is generally intermittent and may even include tempor- ary reversals of direction (Rebhun, 1972). It would be implausible to at t r ibute such reversals to transient density changes (Rebhun, 1964), but such irregularities may indicate that the force acts intermittently and/or un- evenly, producing scattered, occasional dead zones and cytoplasmic eddies in the spindle.

In summary, it must be said that the evidence from non-chromosomal t ransport is more ambiguous than the evidence from chromosome movements. Although the behavior of oil droplets and mitochondria seems to support the hypothesis that one mechanism in spindle t ransport has the characteristics of an inertial force, reports of the behavior of other non-chromosomal bodies are incomplete or equivocal. It can be said, however, that the

evidence from non-chromosomal transport does not falsify the hypothesis, while the evidence from chromosome movements tends to corroborate it. Therefore, it seems worthwhile to inquire how a spindle might possibly generate an inertial force of sufficient magnitude to be a factor in mitotic movements.

3. A poss ible inertial force in mitos is

3.1. Nature of the force and its possible molecular origin

Inertial forces, technically speaking, are not fundamental physical forces but "pseudo- forces" arising from accelerated motion. Accel- eration can be due either to a change in the magnitude of a velocity (as in linear acceleration), or to a change in its direction (as in rotational motion), or to a combination of the two. Therefore, at least in principle, an inertial force in a mitotic spindle could arise from rotational movements of the spindle itself.

Mitotic spindles in some plant and animal cells have been observed to rotate slowly about their long axis, through an angle any- where from 10 ° to 270 °. Frequently, a spindle then reverses the direction of its rotation, and oscillates in this manner from prometaphase through late anaphase. Such oscillations may exhibit irregular periodicity of the order of minutes (Bajer, 1958b; McGrath, 1959; Abe and Nishikawa, 1981). Short of actual rotation, some spindles exhibit a "trembling" (Izutsu et al., 1979) which could be due to rotation with a much shorter period of oscillation. Extrapolating from these cases, it is possible that even spindles which appear to be stat ionary may actually have some angular velocity which is so small, or is reversing its direction so frequently, that it would be diffi- cult to detect. In any case, it is clear that some mechanism (which is, as yet, unknown) is capable of producing rotation and oscillation in mitotic spindles. The same mechanism might also cause other, less easily detectable spindle movements which would be capable of

307

generating significant inertial forces. One physical mechanism which could,

theoretically, ~jve rise to the spindle movements described above is the Einstein-de Haas effect. If electron spins in a ferromagnetic substance are suddenly aligned, the accompanying change in angular momentum can produce macroscopic rotation. Richardson (1908) suggested and Einstein and de Haas (1915) demonstrated that an iron cylinder suspended by a thin fiber will rotate slightly when an external magnetic field is suddenly turned on. Although a spindle pole is not an iron cylinder in an external magnetic field, the same principle could apply here if the pole were capable of producing changes in a substantial number of electron spin alignments.

Admittedly, the postulate that spindle poles could contain a substantial number of aligned electron spins is speculative, and may even seem far-fetched, since such highly ordered electron behavior has previously been thought to be limited to inorganic systems. Recent discoveries, however, have shown that organic molecules are capable of certain other types of highly ordered electron behavior, including piezoelectricity, pyroelectricity and ferroelec- tricity (Athenstaedt, 1974; Sessler, 1982; Lov- inger, 1983), as well as superconductivity (Parkin et al., 1983; Maugh, 1984). These recent discoveries make it reasonable to con- sider the possibility that an organic system might be capable of the magnetic interactions postulated here.

Such interactions, if present, would presumably be due to a lattice of organometallic molecules which contain magnetic ions. Since the spin state of such ions is determined by factors such as their oxidation state and the nature and arrangement of ligands surrounding them (Reed, 1981; Scheidt, 1981), redox reactions and/or stereochemical changes at the spindle pole could produce significant alt- erations in the net spin, and therefore in the net angular momentum. As a result, a spindle which is free to rotate (for example, about its long axis) cou]ld acquire a small angular velocity

o5- a~ I (1)

in which zlSis the change in the angular momentum, I is the moment of inertia and cytoplasmic damping effects are (for now) ignored.

Assuming that the average ferromagnetic moment per magnetic ion at the spindle pole is only 1% that of iron, a spindle pole in an average cell (with a diameter of, say, 5 x 1 0 - 5 m) could contain the equivalent of roughly 101° aligned moments (with their accompanying angular momenta). Such a pole could then generate a AS of the order of 10 25 kg m ~ s - 1, and thus an ~ (before damping) of the order of 10-3s-1. This could account for (a) a slow rotation around the long axis of the spindle (corresponding to the observations of Bajer (1958b) and Abe and Nish- ikawa (1981)); (b) a relatively slow oscillation due to periodic reversals in the sign of ~IS and therefore in the direction of rotation (corresponding to the observations of McGrath (1959), Abe and Nishikawa (1981) and perhaps Izutsu et al. (1979)).

Spindle movements (a) and (b) are, however, insufficient to generate a significant inertial force. Therefore, if such a force were present it would have to be due to some additional movement, perhaps (c) a superimposed oscillation of very high frequency and very small amplitude (which, if present, has not yet been directly observed). This fast oscillation (c) could, theoretically, arise from electron spin precession associated with aligned moments at the spindle pole. In a magnetic field B, electron spins precess with an angular velocity

4 ~ = B ~ (2)

in which ~ is the Bohr magneton and g is the gyromagnetic ratio (Eisberg and Resnick, 1974). Even in the absence of an external magnetic field, there exist interactions among magnetic ions which act as effective internal magnetic fields of the order of 1 T (Ashcroft and Mermin, 1976). Consequently, if spindle

308

pole molecules possessed the magne t ic in- te rac t ions pos tu la ted here , t hey could be accompanied by coheren t precess ion of the order of 1011s- 1. As the following model demon- s t ra tes , it is possible to der ive a s ignif icant iner t ia l force from the fast oscil lation (c) which this coheren t precess ion would super impose upon spindle m o v e m e n t s (a) and (b). Such a force could account for those previously unexp la ined aspects of mitot ic movemen t s described above.

3.2. A mathemat ica l model

The following model t akes as i ts s t a r t i ng point a single as ter . Since the t r a n s p o r t prop- e r ty which is of in te res t he re is found in as te r s (Chambers , 1917), monopola r spindles (Metz, 1933), and each ha l f of a b ipolar spindle (Bajer, 1982), it should be possible to move from such a s t a r t i ng point to a t heo ry which is applicable to mitot ic spindles in genera l , both as t ra l and anas t ra l .

Assuming t h a t an idealized, spher ica l ly symmet r ica l a s t e r is f ree to ro ta t e in a n y direct ion while i ts cen te r r ema ins fixed, the cen te r of the a s t e r can serve as the origin of bo th a s t a t i ona ry coordinate sys t em xyz (fixed re la t ive to the cell m e m b r a n e and the lab) and a ro t a t ing coordinate sys tem x ' y ' z ' (fixed rela- t ive to the as t ra l rays). I f t he r e is an in t e rna l magne t ic field a t t he cen te r of the a s t e r which has a fixed direct ion re la t ive to the a s t r a l rays, t h e n t h a t field defines the z ' axis; and e lect ron spins a t the cen t e r of the a s t e r will m a k e some cons tan t angle A wi th the z ' axis and precess about it wi th some angu l a r velocity ¢. Al ter ing the a l i g n m e n t of those spins will t h e n produce a change in angu l a r momen- t u m AS which will also m a k e a cons tan t angle

wi th the z ' axis and precess about i t wi th angu la r velocity ~. Assuming (for the t ime being) t h a t the damping effects of the cyto- p lasm m a y be ignored, the a s t e r will conserve angu l a r m o m e n t u m by acqui r ing a small angu l a r velocity o3 which, l ike AS, will also m a k e a cons tan t angle A wi th the z ' axis and precess about it wi th angu l a r veloci ty ~. To

simplify calculations, the rotating coordinate system x'y'z' is then defined so that the position vector F (designating the point at which acceleration is to be calculated) is fixed in the y'z' plane and makes a constant angle 0 with the z' axis (in other words, F is fixed in the aster); and the stationary axes xyz are chosen so that the two coordinate systems exactly coincide at a time t = 0 when o5 is in the x'z' plane (Fig. 4).

Calculations may be further simplified by

Z,~Z" _.1

A S

J

X~

Fig. 4. Stationary coordinate system xyz and rotating coordinate system x'y'z' at time t = 0. The position vector is fixed in the y'z' plane and makes a constant angle e with the z' axis. The rotation vector o~, which is always an- tiparallel to the angular momentum vector AS, makes a constant angle A with the z' axis but precesses around it with angular velocity 4- At t = 0, o5 is in the x'z' plane. The precession vector 4) lies constantly on the z' axis. The vectors are not drawn to scale: actually, ~ >> (5. Looking at the coordinate systems from above, (5 is precessing very rapidly counterclockwise about the z' axis. The slow rotation (a) corresponds to a slow clockwise rotation ofx'y' (and thus f) about the z' axis, with angular velocity -oJ cos A. The slow oscillation (b) simply corresponds to reversals in the direction of this movement. The fast oscillation (c) can be visualized by ignoring these slow movements and imagining that the point designated by ~ is rotating rapidly (about 1011 times/s) in a very small (about 10-19m) elliptical orbit which is perpendicular to the vector ~.

. y , y

309

noting that 4; is many orders of magnitude faster than 05. The difference is so great that during the first, millisecond following t = 0, (5 will have caused the rotating coordinate system to make ~nly about a millionth of a re- volution relative to the stat ionary coordinate system, but 4; will have caused 05 to make about a hundred million revolutions around the z' axis. This means that, on a time scale of the order of minutes, the only net non-zero component of ~; will be - to cos A ~', which will cause the aster to rotate slowly about the z' axis. This corresponds to the slow rotation (a), and reversals in the direction of this rotation (presumably due to changes in the sign of A~) would correspond to the slow oscillation (b). These two movements are so slow, relative to the fast oscillation (c), that they may be ignored for the purpose of calculating the acceleration at the point designated by ?.

The different forms of rotational acceleration, each of which can give rise to a pseudoforce, are given by the right-hand side of the Coriolis :Equation (Symon, 1971):

d2F d*2F d*F d05 d t 2 = - ~ + 05X(05XF) + 205X ~ + - ~ XF

(3)

in which F is a :position vector designating the point at which acceleration is to be calculated, 05 is the angular velocity, and the asterisks indicate measurements taken in the rotating coordinate system. Since ~ is assumed to be fixed in the aster, the first term on the right would be zero; since (5 is very small, the second term (centrifugal acceleration) would be negligible; and the third term (coriolis acceleration), like the first, would be zero. A significant acceleration could arise only from the fourth term, which can now be calculated.

During the fi:rst millisecond following t = 0, the axes x 'y 'z ' will have moved so little relative to xyz that to a very close approximation the motion of 05 can be wri t ten as

05 ~ sin A cos ~bt

+ to sin A sin ~)t :y - to cos A ~ (4)

During this same millisecond, the position of point ~ (again, to a very close approximation) remains

g ~ r sin 0 )

+ r c o s 0 (5)

Then, if the slow rotation term - to cos A ~ in Eqn. 4 is disregarded, the velocity of point during this millisecond is

05XF ~- mr sin A cos 0 sin Ct

- wr sin A cos 0 cos ~bt

+ wr sinA sin 0 cos ~bt ~ (6)

This means that, shortly after t = 0, the motion of~ due to the fast oscillation (c) may be closely approximated by

f ( 0 5 X f ) d ' t ~ - [ - ~ g i n A c o s O ( c o s ~ b t - 1 ) ] ~

[ tor ] + r sin 0 - -~- sin A cos 0 sin ~bt 3;

[ (or ] + r cos 0 + -~- sin A sin 0 sin ~bt

(7)

in which the integration constants have been determined by conditions at t = 0. According to Eqn. 7, • would move in an elliptical orbit which is perpendicular to the astral rays, and which has a maximum size of approximately (2wr sin A)/~b. Given the values used above, this would be of the order of 10-19 m, or several orders of magnitude less than an atomic diameter. It is not surprising, therefore, that the fast oscillation (c), if present, has so far gone undetected.

Despite this extremely small displacement, the acceleration at ~ would be far from negligible. From Eqns. 4 and 5, the lateral acceleration would be

t X F ~ ¢oor sin A cos 0 cos ~bt

+ ~bwr sin A cos 0 sin ~bt )

- ~btor sin A sin 0 sin ~bt (8)

310

Assuming, for the purpose of this model, that the fast oscillation (c) is t ransmit ted by astral rays (microtubules) to every point in the aster, including every point in the fluid within the aster, a movable body in the fluid bounded by a cone of microtubules would then experience the acceleration given by Eqn. 8. The lateral components of this acceleration would average zero; but the cone-shaped arrangement of the microtubules would produce a radial component of acceleration, much as a centrifuge with tilted sides would produce a vertical component of acceleration. In other words, aster geometry would convert the lateral acceleration in Eqn. 8 to a radial acceleration

a = 1 sin v do5 -3-7 z~

1 ~bwr sin ~ sin A ~/cos 2 0 + sin 2 0 sin s ~bt

(9)

in which r is the angle at the apex of the cone of microtubules (Fig. 5). If ~ were, say, 2 °, then a movable body in an aster in the average cell described above would experience a radial acceleration of the order of the acceleration due to gravity. Therefore, the fast oscillation (c) could give rise to an inertial force which is presumably of sufficient magnitude to be a factor in mitotic movements

3.3. Extension of the model to mitosis

Since the displacement of ~ in Eqn. 7 is so extremely small, the angles 0 and ~ would remain essentially constant. This means that the z' axis, for practical purposes, may be regarded as identical to the z axis with respect to all three of the spindle movements described above. The model presented here would thus be applicable not only to asters and monopolar spindles, but also to bipolar spindles, provided that the long axis of the spindle coincides with the z/z' axis. It would also be applicable to anastral as well as astral spindles, assuming that the poles of the former contain the postulated spin alignments.

Fig. 5. Hypothetical cross-section through a cone of spindle microtubules radia t ing from the pole (with the angle v exaggerated for clarity). Assuming t ha t the ent i re cone, its contents included, is undergoing the fast oscillation (c), a movable body within the cone, a t a distance r from the pole, would experience the acceleration given by Eqn. 8. From the geometry of the situation, a t any given in s t an t this acceleration (i) can be resolved into a component which is normal to a side of the cone (ii), and another component which is tangent ia l (iii). This la t ter component can be fur ther resolved to yield a component (iv) which is directed along the axis of the cone and which is less t han the original acceleration (i) by a factor sin .] cos ~, or ~ sin ~. This is the radial acceleration a given by Eqn. 9.

If the AS generated by one pole of a bipolar spindle were in the same direction as that generated by the other, the entire spindle would tend to rotate about its long axis. This would be the slow rotation (a) described above. If the AS generated by one pole were opposite in direction to that generated by the other, this slow rotation would be reduced, perhaps even to zero. Nevertheless, since the displacement caused by the fast oscillation (c) is several orders of magnitude less than an atomic diameter, there would presumably be no significant interference between the oscillating microtubules of one half-spindle and those of the other. Therefore, counter-rotation of opposite poles would not prevent the two half-spindles from undergoing the fast oscillation (c).

Spindles in which the AS of one pole is not

311

opposite to that of the other would acquire some slow rotation (a). Due to damping by cytoplasmic viscosity, the observed rotation would be significantly less than the oJ cos predicted by the model. For the small-amplitude oscillatiion (c), however, viscosity would be negligible. In fact, since viscosity results from large numbers of molecular collisions on a scale many orders of magnitude larger than an atomic diameter, whereas the amplitude of the oscillation is many orders of magnitude 'smal- ler than an atomic diameter, it is doubtful whether viSCOSiLty could even be defined in this, case. Therefore, it is unlikely that cytoplasmic viscosity would prevent a spindle from oscillating in such a way as to produce a significant inertial force.

Perhaps the most serious theoretical difficul- ty with the mcdel is the assumption that the fast oscillatiorL (c) is t ransmit ted by microtubules throughout the entire half-spindle. Although spindle fibers and related structures are reported to be relatively rigid (Wilson, 1925; Pease, 1!)63; Bajer, 1968), such reports are based on phenomena which are so much larger and slower than the fast oscillation (c) that it is doubtful whether they are relevant here. A similar reservation would apply to existing theore~ical t rea tments of fluid effects in flagellar motion (Chwang and Wu, 1971; Winet and Keller, 1976). It may eventually be possible to define the problem in quantum- mechanical t e ~ s , perhaps as a phonon prop- agation in organic polymers; but it may be necessary to await the development of new conceptual tools, such as the "biological mechanics" suggested by Thornburg (1967), before phenomena like this can be dealt with ade- quately.

The model becomes somewhat more com- plicated when the motion of bodies within the spindle is taken into account, since the first and third terms on the right-hand side of Eqn. 3 would not be zero for those bodies. Such motion is sufficiently slow, however, to guaran- tee that those terms would still be negligible in comparison wit]h the fourth term. A somewhat more significant complication would be intro-

duced by the asymmetrical distribution of chromosomes, since the center of mass of a half-spindle would no longer coincide exactly with the spindle pole. As a result, the origins of the two coordinate systems would be displaced relative to each other. The z' axis, however, would still essentially coincide with the z axis; and the resulting fast oscillation (c), though more complex than the oscillation described in Eqn. 7, would still be roughly of the same order of magnitude.

4. D i s c u s s i o n

The preceding analysis, though speculative, demonstrates at least a theoretical possibility: if there were a substantial number of aligned and coherently precessing electron spins at a spindle pole, then the pole could biochemically alter its net spin so as to produce a high- frequency, small-amplitude oscillation in the spindle which could give rise to a significant inertial force. Since the force would arise only during changes in spin alignments, its opera- tion would be intermittent. Its transmission would depend upon microtubules, though it would not be reducible to a simple pushing or pulling by them. Although the mathematical model presented above took a single aster as its start ing point, the postulated spin alignments could just as easily occur at the poles of an anastral spindle. Consequently, the theory would be applicable to mitosis in plants as well as animals.

The possibility that some aspects of spindle tl 'ansport might be explained by "oscillation or vibration of microtubules" was suggested by Bajer (1967), who made reference to the theory of Jarosch (1964). Jarosch's theory attr ibuted a variety of biological movements to screwlike rotations of protein helices, a mechanism quite different from the oscillation postulated here. Jarosch's theory, as applied to mitosis, has not yet been corroborated.

An older oscillatory model of mitosis, proposed by Lamb (1907) and revived by Cannon (1923) and Faberg~ (1942), relied on a hydro-

312

dynamic effect described by Bjerknes (1902). Bjerknes and his father had shown that two bodies pulsating or oscillating in a fluid at tract or repel each other depending on whether they pulsate or oscillate in the same or opposite

• . ~ . .

phase. These hydrodynamm theorms of mitosis, however, failed to account for the fact (now well established) that mitotic movements depend on the presence of microtubules.

More recently, the theory of Bornens (1979) attr ibutes certain aspects of mitosis to oscillations originating in rapidly rotating centrioles. According to Bornens, rotating centrioles t ransmit rhythmic signals along spindle microtubules which interfere with similar signals originating in kinetochores. Mitotic movements are at tr ibuted to the resulting interference patterns. A major objection to Bornens' theory, however, is that mitosis occurs in many organisms which lack centrioles.

A non-oscillatory mechanism which has been suggested to account for some chromosome movements away from spindle poles is pushing by microtubules. According to Hill (1981), it is theoretically possible that microtubule assembly could produce sufficient pushing force to deform a cell membrane; and Mol~-Bajer and Bajer (1983) have found evidence that some chromosome fragments in taxol-treated anastral spindles are pushed away from the polar regions by elongating microtubules. Since some aspects of spindle transport, however, seem not to be reducible to a simple pushing or pulling by microtubules, it is probable that one or more additional factors are involved, one of which could be the inertial force described in this paper.

Recently, Hayden and Allen (1984) observed particles translocating along microtubules in living keratocytes. Occasionally, a single particle was observed to move first in one direction and then in the other along the same microtubule. Such translocations may, of course, involve a mechanism distinct from the one proposed here; but since the direction of movement due to the fast oscillation (c) depends on relative density, such changes of direction

could be caused by slight fluctuations in the density of the cytoplasm surrounding a microtubule. The possibility tha t an inertial force could be a factor in intracellular t ransport in general is enhanced by the fact that microtubules in vivo tend to radiate from one or more organizing sites (Mitchison and Kirschner, 1984). If such microtubule organizing sites contain the spin alignments required by the theory presented here, the small angles between microtubules radiating from a common center would be sufficient to produce a significant inertial force.

There are at least three experimental approaches which could be used to test the theory: (1) micromanipulation studies which insert artificial particles of known densities into mitotic spindles could establish more clearly whether the transport mechanism does, in fact, behave like an inertial force; (2) ESR techniques could be helpful in determining whether spindle poles contain a substantial number of aligned electron spins; (3) it might be possible to detect the faint microwave radiation which would presumably be emitted by spindle poles if they contain aligned spins which are precessing coherently.

Acknowledgements

The author gratefully acknowledges valu- able discussions with Dr. Andrew Bajer. The author also wishes to thank Ted Clark, Jack Hettema, John Hissong, Timothy Jackson, David LeMaster, Mark Rit ter and Keith Thom- son for reading portions of the manuscript and offering helpful comments. Lucy Wells assisted in compiling the references.

References

Abe, S. and A. Nishikawa, 1981, Periodic rotation of chromosomes during the mitotic divisions in secondary spermatogonia of newt, Cynops pyrrhogaster. Dev., Growth Differ. 23, 165-173.

Abbott, A.G., J.E. Hess and S.A. Gerbi, 1981, Sper- matogenesis in Sciara coprophila. Chromosoma 83, 1-18.

Ashcroft, N.W. and N.D. Mermin, 1976, Solid State Physics (Holt, Rinehart a.nd Winston, New York) pp. 694--699.

Athenstaedt, H., 1974, Pyroelectric and piezoelectric properties of vertebr~Ltes. Ann. N.Y. Acad. Sci. 238, 68--93.

Bajer, A., 1958a, Cine-micrographic studies on chromosome movements in E-irradiated cells. Chromosema 9, 319-331.

Bajer, A., 1958b, C:ine-micrographic studies on mitosis in endosperm. Exp. Cell Res. 15, 370--383.

Bajer, A., 1967, Notes on ultrastructure and some properties of transport within the living mitotic spindle. J. Cell Biol. 33, 713-720.

Bajer, A., 1968, Chromosome movement and fine structure of the mitotic spirLdle. Syrup. Soc. Exp. Biol. 22, 285~-309.

Bajer, A., 1972, Influence of UV microbeam on spindle fine structure and anaphase chromosome movements, in: Chromosomes Today, Vol. 3, C.D. Darlington and K.R. Lewis (eds.) (Hal~aer, New York) pp. 63-69.

Bajer, A., 1982, Functional autonomy of monopolar spindle and evidence for c,scillatory movement in mitosis. J. Cell Biol. 93, 33-48.

Bajer, A. and J. Mol~-Bajer, 1956, Cine-micrographic studies on mitosis in endosperm. Chromosoma 7, 558--607.

Bajer, A. and J. Mo[~-Bajer, 1961, UV microbeam irradiation of chromosomes during mitosis in endosperm. Exp. Cell Res. 25, 251--267.

Bajer, A. and J. Mol~-Bajer, 1963, Cine analysis of some aspects of mitosis in endosperm, in." Cinemicrography in Cell Biology, G.G Rose (ed.) (Academic, New York) pp. 357-409.

Bajer, A. and J. Mcl~-Bajer, 1972, Spindle dynamics and chromosome movements. Int. Rev. Cytol. Suppl. 3.

Bajer, A. and J. Mcle-Bajer, 1975, Lateral movements in the spindle and the mechanism of mitosis, in: Molecules and Cell Movement, S. Inou~ and R.E. Stephens (eds.) (Raven, New York) pp. 77-96.

Bajer, A. and J. Mol,~-Bajer, 1981, Mitosis: studies of living ce l l s -a revision of basic concepts, in: Mitosis/ Cytokinesis, A. Zimmerman and A. Forer (eds.) (Academic, New York) pp. 277-299.

Bajer, A. and J. Mol~-Bajer, 1982, Asters, poles, and transport properties within spindlelike microtubule arrays. Cold Spl~ng Harb. Syrup. Quant. Biol. 46, 263-283.

Bajer, A. and G. (~)stergren, 1961, Centromere-like behaviour of non-centromeric bodies. Hereditas 47, 563-598.

Bajer, A. and G. OsLergren, 1963, Observations on transverse movements within the phragmoplast. Hereditas 50, 179-195.

Beams, H.W., 1951, The effects of ultracentrifugal force on the cell with spe,:ial reference to division. Ann. N.Y. Acad. Sci. 51, 1349-1364.

313

Beams, H.W. and R.L. King, 1938, An experimental study of mitosis in the somatic cells of wheat. Biol. Bull. 75, 189-207.

Begg, D.A. and G.W. Ellis, 1979, Micromanipulation studies of chromosome movement. J. Cell Biol. 82, 528--541.

Bjerknes, F.V., 1902, Vorlesungen tiber hydrodynamische fernkrtifte nach C.A. Bjerknes theorie (Barth, Leipzig).

Bloom, W., R.E. Zirkle and R.B. Uretz, 1955, Irradiation of parts of individual cells. Ann. N.Y. Acad. Sci. 59, 503-513.

Bornens, M., 1979, The centriole as a gyroscopic oscillator. Biol. Cell. 35, 115-132.

Cannon, H.G., 1923, On the nature of the centrosomal force. J. Genet. 13, 47-78.

Carlson, J.G., 1938, Mitotic behavior of induced chromosomal fragments lacking spindle attachments in the neuroblasts of the grasshopper. Proc. Natl. Acad. Sci. U.S.A. 24, 500-507.

Chambers, R., 1917, Microdissection studies. J. Exp. Zool. 23, 483-505.

Chwang, A.T. and T.Y. Wu, 1971, A note on the helical movement of micro-organisms. Proc. R. Soc. London, ser. B 178, 327-346.

Cornman, I., 1944, A summary of evidence in favor of the traction fiber in mitosis. Am. Nat. 78, 410--422.

Einstein, A. and W.J. de Haas, 1915, Experimenteller nachweis der amp~reschen molekularstr~me. Verh. Dtsch. Phys. Ges. 17, 152-170.

Eisberg, R. and R. Resnick, 1974, Quantum Physics (John Wiley, New York) pp. 291-305.

Ellis, G.W. and D.A. Begg, 1981, Chromosome micromanipulation studies, in: Mitosis/Cytokinesis, A. Zim- merman and A. Forer (eds.) (Academic, New York) pp. 155-179.

Faberge, A.C., 1942, Homologous chromosome pairing: the physical problem. J. Genet. 43, 121-144.

Friedltinder, M. and J. Wahrman, 1970, The spindle as a basal body distributor. J. Cell Sci. 7, 65--89.

Fuge, H., 1975, Anaphase transport of akinetochoric fragments in tipulid spermatocytes. Chromosoma 52, 149-158.

Hayden, J.H. and R.D. Allen, 1984, Detection of single microtubules in living cells: particle transport can occur in both directions along the same microtubule. J. Cell Biol. 99, 1785-1793.

Hill, T.L., 1981, Microfilament or microtubule assembly or disassembly against a force. Proc. Natl. Acad. Sci. U.S.A. 78, 5613-5617.

Hiramoto, Y., 1968, The mechanics and mechanisms of cleavage in the sea-urchin egg. Symp. Soc. Exp. Biol. 22, 310-327.

Hughes-Schrader, S., 1948, Expulsion of the sex chromosome from the spindle in spermatocytes of a mantid. Chromosoma 3, 257-270.

Inou~, S., 1981, Cell division and the mitotic spindle. J. Cell Biol. 91, 131s-147s.

314

Izutsu, K., 1959, Irradiation of parts of single mitotic apparatus in grasshopper spermatocytes with an ultra- violet microbeam. Mie Med. J. 9, 15-29.

Izutsu, K., H. Sate and Y. Ohnuki, 1979, Behavior of mitotic spindles in dividing cells - description of a film, in: Cell Motility: Molecules and Organization, S. Hatano, H. Ishikawa, and H. Sato (eds.) (University Park, Balti- more) pp. 281-287.

Jarosch, R., 1964, Screw-mechanical basis of protoplasmic movement, in: Primitive Motile Systems in Cell Biology, R.D. Allen and N. Kamiya (eds.) (Academic, New York) pp. 599-620.

Kostoff, D., 1930, Protoplasmic viscosity in plants. Proto- plasma 11, 177-183.

Lamb, A.B., 1907, A new explanation of the mechanics of mitosis. J. Exp. Zool. 5, 27-33.

Lambert, A. and A. Bajer, 1972, Dynamics of spindle fibers and microtubules during anaphase and phragmoplast formation. Chromosoma 39, 101-144.

Lovinger, A.J., 1983, Ferroelectric polymers. Science 220, 1115-1121.

Luyet, B.J., 1935, Behavior of the spindle fibers in centrifuged cells. Proc. Soc. Exp. Biol. Med. 33, 163--165.

Luyet, B.J. and R.A. Ernst, 1934, On the comparative specific gravity of some cell components. Biodynamica 1, 1-14.

Luykx, P., 1970, Cellular mechanisms of chromosome distribution. Int. Rev. Cytol. Suppl. 2.

Maugh, T.H., 1984, New organic superconductors. Science 226, 37.

Mazia, D., 1961, Mitosis and the physiology of celt division, in: The Cell, Vol. 3, J. Brachet and A.E. Mirsky (eds.) (Academic, New York) pp. 77-412.

McGrath, R.A., 1959, Oscillation of the nucleus and mitotic figure in the neuroblast of the grasshopper Chortophaga viridifasciata (De Geer). Exp. Cell Res. 16, 459-462.

McNeill, P.A. and M.W. Berns, 1981, Chromosome behavior after laser micreirradiation of a single kinetochore in mitotic PtK 2 cells. J. Cell Biol. 88, 543-553.

Metz, C.W., 1933, Monocentric mitosis with segregation of chromosomes in Sciara and its bearing on the mechanism of mitosis. Biol. Bull. 64, 333-347.

Metz, C.W., 1938, Chromosome behavior, inheritance and sex determination in Sciara. Am. Nat. 72, 485-520.

Mitchison, T. and M. Kirschner, 1984, Micretubule assembly nucleated by isolated centresomes. Nature 312, 232-237.

Mol~-Bajer, J., 1965, Telophase segregation of chromosomes and amitosis. J. Cell Biol. 25, 79-93.

Mol~-Bajer, J. and A. Bajer, 1983, Action of taxol on mitosis: modification of microtubule arrangements and function of the mitotic spindle in Haemanthus endo- sPerm. J. Cell Biol. 96, 527-540.

Mol~-Bajer, J., A. Bajer and A. Owczarzak, 1975, Chromo- some movements in prometaphase and aster transport in the newt. Cytobios 13, 45--65.

Mulnard, J., 1958, La m~tachromasie in vivo et son

analyse cytochimique dans l'oeuf de l'annelide Chaetop- terus pergamentaceus. Arch. Biol. 69, 645-685.

Mulnard, J., W. Auclair and D. Marsland, 1959, Meta- chromasia observed in the living eggs of Arbacia punctulata and its cytochemical analysis. J. Embryol. Exp. Morphol. 7, 223-240.

Nicklas, R.B., 1961, Recurrent pole-to-pole movements of the sex chromosome during prometaphase I in Mclano- plus differentialis spermatocytes. Chromosoma 12, 97-115.

Nicklas, R.B., 1971, Mitosis, in: Advances in Cell Biology, Vol. 2, D.M. Prescott, L. Goldstein, and E. McConkey (eds.) (Appleton-Century-Crofts, New York) pp. 225-297.

Nicklas, R.B. and C.A. Koch, 1972, Chromosome micromanipulation. Chromosoma 39, 1-26.

Nicklas, R.B. and C.A. Staehly, 1967, Chromosome micromanipulation. Chromosoma 21, 1-16.

Nicklas, R.B., D.F. Kubai and T.S. Hays, 1981, Spindle structure after chromosome micromanipulation, in: Biological Functions of Microtubules and Related Struc- tures, H. Sakai, H. Mohri and G.G. Borisy (eds.) (Academic, Tokyo) pp. 227-231.

0stergren, G., 1945, Transverse equilibria on the spindle. Bot. Not., 467-468.

Ostergren, G., 1949, A survey of factors working at mitosis. Hereditas 35, 525-528.

Ostergren, G. and R. Prakken, 1946, Behaviour on the spindle of the actively mobile chromosome ends of rye. Hereditas 32, 473--494.

Ostergren, G., J. Mol~-Bajer and A. Bajer, 1960, An interpretation of transport phenomena at mitosis. Ann. N.Y. Acad. Sci. 90, 381-408.

Parkin, S.S.P., E.M. Engler, R.R. Schumaker, R. Lagier, V.Y. Lee, J.C. Scott and R.L. Greene, 1983, Superconduc- tivity in a new family of organic conductors. Phys. Rev. Lett. 50, 270--273.

Pasteels, J., 1958, Nouvelles recherches sur la meta- chromasie in vivo et sur l'histochimie de roeuf normal et centrifug~ de Paracentrotus lividus. Arch. Biol. 69, 591-619.

Pease, D.C., 1963, The ultrastructure of flagellar fibrils. J. Cell Biol. 18, 313--326.

Pickett-Heaps, J.D., D.H. Tippit and K.R. Porter, 1982, Rethinking mitosis. Cell 29, 729-744.

Rappaport, R., 1971, Cytokinesis in animal cells. Int. Rev. Cytol. 31, 169-213.

Rebhun, L.I., 1959, Studies of early cleavage in the surf clam, Spisula solidissima, using methylene blue and toluidine blue as vital stains. Biol. Bull. 117, 518--545.

Rebhun, L.I., 1960, Aster-associated particles in the cleavage of marine invertebrate eggs. Ann. N.Y. Acad. Sci. 90, 357-380.

Rebhun, L.I., 1963, Saltatory particle movements and their relation to the mitotic apparatus, in: The Cell in Mitosis, L. Levine (ed.) (Academic, New York~ pp. 67-103.

Rebhun, L.I., 1964, Saltatory particle movements in cells,

in: Primitive Motile Systems in Cell Biology, R.D. Allen and N. Kamiya (eds.) (Academic, New York) pp. 503-525.

Rebhun, L.I., 1972, Polarized intracellular particle transport: saltatory movements and cytoplasmic streaming. Int. Rev. Cytol. 32, 93-137.

Reed, C.A., 1981, Oxidation states, redox potentials and spin states, in: The Biological Chemistry of Iron, H.B. Dunford, D. Dolphin, K.N. Raymond and L. Sieker (eds.) (D. Reidel, DordI~cht) pp. 25-42.

Rhoades, M.M., 1952, Preferential segregation in maize, in: Heterosis, J.W. Gowen (ed.) (Iowa State College, Ames) pp. 66-80.

Richardson, O.W., 1908, A mechanical effect accompanying magnetization. Phys. Rev. 26, 248-253.

Rieder, C.L., 1982, The formation, structure, and composi- tion of the mammalian kinetochore and kinetochore fiber. Int. Rev. Cytol. 79, 1-58.

Ris, H. and D.F. Kubai, 1970, Chromosome structure. Annu. Rev. Genet. 4, 263-294.

Scheidt, W.R., 1981, Magnetic complexities in por- phinatoiron (III) complexes, in: The Biological Chemis- try of Iron, H.B. Dunford, D. Dolphin, K.R. Raymond and L. Sieker (eds.) (D. Reidel, Dordrecht) pp. 261-272.

Schrader, F., 1946, The elimination of chromosomes in the meiotic divisions ofBrachystethus rubrornaculatus. Dal- las. Biol. Bull. 90, 19-31.

Schrader, F., 1947, Data contributing to an analysis of

315

metaphase mechanics. Chromosoma 3, 22-47. Schrader, F., 1953, Mitosis, 2nd edn. (Columbia Universi-

ty, New York). Sessler, G.M., 1982, Polymeric electrets, in: Electrical

Properties of Polymers, D.A. Seanor (ed.) (Academic, New York) pp. 241-284.

Seto, T., J. Kezer and C.M. Pomerat, 1969, A cinemato- graphic study of meiosis in salamander spermatocytes in vitro. Z. Zellforsch. 94, 407-424.

Shimamura, T., 1940, Studies on the effect of the centrifugal force upon nuclear division. Cytologia 11, 186-216.

Stich, H., 1954, Stoffe und str6mungen in der spindel von Cyclops strenuus. Chromosoma 6, 199-236.

Symon, K.R., 1971, Mechanics, 3rd edn. (Addison-Wesley, Reading MA) p. 278.

Thornburg, W., 1967, Mechanisms of biological motility. Theor. Exp. Biophys. 1, 77-127.

Tilney, L.G. and D. Marsland, 1969, A fine structural analysis of cleavage induction and furrowing in the eggs ofArbacia punctulata. J. Cell Biol. 42, 170-184.

Uretz, R.B., W. Bloom and R.E. Zirkle, 1954, Irradiation of parts of individual cells. Science 120, 197-199.

Wilson, E.B., 1925, The Cell in Development and Heredity, 3rd edn. (Macmillan, New York).

Winet, H. and S.R. Keller, 1976, Spirillum swimming: theory and observations of propulsion by the flagellar bundle. J. Exp. Biol. 65, 577-602.

Inertial force as a possible factor in mitosis

Documents

Transcript of Inertial force as a possible factor in mitosis