J.A. Tuszynski, M.V. Sataric, S. Portet and J.M. Dixon: Physical Interpretation of Microtubule...

8/3/2019 J.A. Tuszynski, M.V. Sataric, S. Portet and J.M. Dixon: Physical Interpretation of Microtubule Self-organization in Grav

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Physical interpretation of microtubule self-organizationin gravitational fields

Jack Tuszynski a,, Miljko V. Sataric a, Stphanie Portet a, John M. Dixon b

a Department of Physics, University of Alberta, Edmonton, AB T6G 2J1, Canadab Department of Physics, University of Warwick, Coventry CV4 7AL, United Kingdom

Received 24 December 2003; received in revised form 21 November 2004; accepted 29 March 2005

Communicated by A.R. Bishop

Abstract

This Letter discusses the role of key physical and chemical effects in the emergence of pattern formation in microtubule

assemblies in vitro under the influence of gravity. The roles of chemical kinetics, diffusion, gravitational drift and electrostatic

interactions affected by thermal fluctuations are discussed. The subtle interplay of these factors leads to the observed self-

organization phenomena exhibiting the features of a typical polar liquid crystal phase.

2005 Published by Elsevier B.V.

Microtubules (MTs) are the key protein filaments

of the cytoskeleton in eukaryotic cells [1] that form

several m long hollow cylinders (typically 25 nm in

diameter) and exhibit dynamically unstable polymer-

ization kinetics combined with substantial structural

rigidity. In the past decade Tabony et al. (TEx) demon-

strated a decisive role of the gravitational force (GF)

in the formation of spatio-temporal patterns by assem-

blies of MTs in vitro [25]. In TEx, MTs were assem-

bled in rectangular spectrometer cells with dimensions

40101 mm3 by warming purified tubulin (TB) andGTP in an appropriate buffer solution from 7 to 37 C.

* Corresponding author.

E-mail address: [email protected] (J. Tuszynski).

The most striking result was obtained with the largest

dimension of the cell (40 mm) parallel to GF leading

to a pattern of stripes of highly oriented MT bundles

(Fig. 1(a)). With the smallest dimension of the cell

(1 mm) parallel to the GF, circular vortex-like mor-

phologies appeared. The critical period for this process

which lasted about 12 hours in [4] or 5 hours in [3,5]

before the equilibrium striped pattern was reached, is

however the first 6 minutes. If in this period the sam-

ple is subjected to the GF, the characteristic pattern

develops (Fig. 1), while microgravitational conditions

do not lead to pattern formation at all.

The process of MT polymerization from TB is

an example of non-linear chemical kinetics due to

the so-called dynamical instability [1]. The structural

anisotropy of the two MT ends results from an asym-

0375-9601/$ see front matter 2005 Published by Elsevier B.V.

doi:10.1016/j.physleta.2005.03.059
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Fig. 1. (a) During the first 6 minutes [35] after the onset of assembly, when the largest dimension of the cell (40 mm) is parallel to g, patterns

which we call curtains of highly oriented MTs are developed. The left half of the assembly is at an angle of 45 while the right half is at anangle of 135 with respect to the horizontal x-axis. (b) The characteristic periodicities developed after 12 hours in [4] or 5 hours in [3,5] withina single stripe for the samples exposed to GF ofg during the first 10 minutes.

metry of the growing and shrinking processes (the end+ grows faster than the end that tends to shrink)and this appears to be correlated with the electrostatic

charge distribution anisotropy [21]. While growth is

gradual, shrinking can take the form of a catastrophe

triggered stochastically by the loss of a GTP lateral

cap [6]. Importantly, in a single event thousands of TB

dimers may be released into solution.

TB dimers are negatively charged globular proteins

that are highly screened by positive counterions of

the solvent. The thermal fluctuations of the counter-charges on TBs cause a van der Waals attractive force

[7] that acts at a short distance and induces the ag-

gregation of TB dimers. We claim that these forces

correlate to initiate catastrophic events enabling TB

dimers to form avalanche correlated clusters (ACC)

which can reach significant sizes providing conditions

for symmetry breaking due to GF.

A series of experiments on charged biopolymers,

including DNA, F-actin fibers and MTs indicate that

in the presence of small amounts of polyvalent salts,

the electrostatic interaction is attractive contradicting

the basic PoissonBoltzmann theory. The attraction

originates from charge fluctuations along the rods due

to the exchange between condensed and free counte-

rions. As inter-filament separation decreases, charge

fluctuations on one filament couple more strongly with

those on its neighbor. Such is the case in TEx due to

the presence of Mg2+ ions. In the first six minutes ofthe MT growth process in TEx, the growth rate may

reach V = 1.17 107 m s1 and is isotropic withthe lengths of MTs reaching up to 50 m [8]. MTsthat are this long can affect polymerization kinetics of

their neighbors [9] especially the catastrophe events

which could also initiate polarizational kinks [1012]

propagating along MTs and eventually triggering the

catastrophic shrinking of the filament (Fig. 2). Thus

within a minute several MTs could be depolymerized

into free dimers, packed more tightly than in solution

but less so than in stable MTs.

A single TB dimer has about 40e negative charges

[13]. If only 10% of this charge is unscreened by coun-


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Fig. 2. Massive depolymerization caused by attraction of MTs and

mediated by moving kinks.

terions, then qeff= 4e and the Bjerrum length lB =q2eff(40kB T )

1, with = 80 for water, at roomtemperature, can reach about 15 nm. This indicates

that two TBs separated by lB can be electrostatically

correlated. Thus, if there is sufficient concentration of

TBs from colocalized collapses of MTs, this could

provide the required condition for the appearance of

an ACC.

The Peclet number Pe [14] relevant for the sedi-

mentation of clusters in a solution is defined as the

ratio of the buoyant energy in the GF and the thermal

energy

(1)Pe =m0(1 /0)gd

kB T,

where m0 is the cluster mass, g the gravitational ac-

celeration, d the cluster diameter, the density of

solution, 0 the density of the cluster and kB T the

thermal energy. For a single TB dimer with a mass

of m0 = 1.84 1022 kg and a diameter of roughlyd= 6 nm, the Peclet number at room temperature isof the order of 1010 indicating that the gravitational

drift is negligible. For a minimal TB concentration inwater of approximately 10% necessary for ACC with

a diameter of 1 m, we estimate the corresponding

Peclet number as follows. Since ACC is a sponge-

like material with 90% water content, from Eq. (1)

using m0 = 5.4 1016 kg and /0 = 0.95, we findPe = 0.07 101. This means that drift-led sedimen-tation of ACCs becomes competitive with an other-

wise dominantly diffusive behavior of TB in solution.

Smaller ACCs are less competitive but still contribute

to the pattern formation in TEx. Drifting along GF

Fig. 3. Left: avalanche-like correlated cluster (ACC) of TB

(d = 1 m) created by catastrophic disassembly events. Right:showered with rephosphorylated TB, nucleation centers have grown

into highly oriented parallel MTs.

lines, ACCs encounter randomly oriented MT nucle-

ation centers. We propose that sweeping through the

solution ACCs orient their centers so as to initiate anew MT growth predominantly in the direction of GF

(Fig. 3).

The reactiondiffusiondrift equation for the TB

concentration C(z,t) around the growing tip of the

MT with the GF directed along the z-axis is derived

from the models of Odde [8] and Prigogine [15] that

account for the specificity of the TEx system

(2)D2C +g Cz

V Cz

L (z z0)=C

t.

The first term above is a diffusion term. The convec-tion term proportional to g, describes the drift of a

tubulin dimer induced by the GF. The convection term,

proportional to V, is due to the fact that the tip of a

MT, that is fixed at the origin, continues to grow at a

relative velocity V. Finally, L (z z0), where z0 isthe length of MT at t= 0, describes the initial randomACCs represented by a spatial step function with asso-

ciated delta-function time correlations.

Extrapolating from the TB diffusion coefficient

for in vivo sea urchin eggs at 25 C as D = 5.9


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1012 m2 s1 [16], we find D = 70 1012 m2 s1under the TEx conditions (T = 37 and the viscosity = 102 Poise). The GF drift mobility = m0D(1

0 )(kB T )1 is quite sensitive to the ACC size. Therate of MT growth is estimated about V = 1.9 107 m s1 at TEx [6,17].

After a Fourier transformation with respect to z and

t, Eq. (2) leads to the following correlation function

(3)|Cqw |2= |Lqw |

2D2q4 + [(g V )q w]2 ,

where q is the wave-number and w is the correspond-

ing frequency. The characteristic relaxation time of the

waves of TB concentration spreading throughout the

TEx cell in the z-direction is given from the disper-

sion relation as

(4)= 2|q(g V )| +Dq 2 .

From the basic periodicity (A) in the TEx pattern

(Fig. 1(b)), we take the wavelength to be = 3 mmand estimate = 9 103 s = 2.5 h. This is consistentwith the order of magnitude of the reported time of

5 hours in [3,5] for the development of self-organized

MT patterns.

Solving Eq. (2) for stationary patterns ( Ct= 0) and

for a single growing MT (L = 0), we obtain the z-dependence of the TB concentration profile

(5)C(z)= C C0 expVg

Dz

,

where C is the asymptotic concentration of C asz , and C0 is the TB concentration at the tip ofa growing MT. We estimate the critical radius of an

ACC for which the gradient of concentration tends to

zero, i.e., Vg = 0, to find

rcr

= 3V kB T

40D(1

0 )

1/3

=0.4 m,

(6)d= 0.8 m 1 m.At this size, the growth rate of an ACC equals the rate

of TB coagulation within a sweeping avalanche. Note

that this length scale corresponds to the smaller peri-

odicity (F) of MT bundles (Fig. 1(b)).

Importantly, the correlation length for this reaction

diffusiondrift effect, for Pe = 101 is

(7)lcor =

D(1+ Pe)Kcat

1/2= 3.3 mm,

where Kcat is the rate of catastrophes taken to be 7 106 s1 extrapolating from the estimate of Ref. [18]and experimental results of Ref. [19]. While this value

is close to the observed periodicity (A) of 3 mm inTEx (Fig. 1(b)), the situation would change drastically

if the conditions were close to the bifurcation point

(6 minutes after the onset of reaction), where Kcattends to zero and hence lcor diverges. Close to the bi-

furcation point this correlation effect becomes macro-

scopic giving rise to a globally self-organized state of

MT curtain domains (Fig. 1(a)). Thus both physi-

cal and chemical aspects of the process are needed in

order to create spatial and temporal self-organization

since they have comparable characteristic time scales.

The characteristic time for transport effects, tr,is related to the spread of TBs along the z-direction

through the length L = trD(1+ Pe). Substitutingtr = 5 h together with D and Pe used before one ob-tains L = 1.2 mm, which compares reasonably wellwith the width of one stripe (B) in TEx (Fig. 1(b)).

The chemical kinetics in TEx shows an over-shoot.

The MT assembly rate is at a maximum between 6 and

10 minutes after the onset of chemical instability af-

fecting the relative ratio of free TB and MTs. This co-

incides with the bifurcation time at which the system is

gravity-dependent due to local massive depolymeriza-

tions and the formation of ACCs. At a later time, theg-oriented MTs grow to be shorter (5 m on average).

Hence, further dynamical processes do not favor the

formation of isotropically oriented MTs. Highly ori-

ented MTs are on average separated by 0.3 m which

is beyond the Bjerrum length so that small repulsive

forces between MTs prevail. This is consistent with

liquid-crystal behavior of MTs within oriented cur-

tains. This also explains the reason why MTs do not

experience sedimentation effects in this phase of pat-

tern formation in TEx.

It has been conjectured in the literature that sin-gle MTs possess ferroelectric and piezoelectric prop-

erties [1012]. This has been supported by molecu-

lar dynamics calculations [20] made possible by the

electron crystallography determination of the atomic-

resolution structure of TB. Recently, it was reported

[21] that a large negative charge ( 20e per TBmonomer) is distributed on the surface of MTs with

a clear asymmetry between the + and ends. A fur-ther calculation of the permanent dipole moment per

TB dimer [22] (approximately 1.8 103 debye), indi-


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cated its orientation to be almost perpendicular to the

MT cylinder axis. This means that MTs are anisotropic

also with respect to their electric susceptibility. We as-

sume that M = 4, M = 2, for TEx conditions wherethe concentration of TB is 10 mg ml1 with 70% ofit being polymerized in MTs. Thus the relative vol-

ume ratio of MTs in solution is v = 0.007. It wasshown [7] that if the condition |v| 1 holds, with =

M s

M +s, the correlated thermal fluctuations of

counterions along MTs ensure that the forces between

two oriented bundles of MTs become attractive. The

corresponding free energy density for two bundles

separated by the distance R, containing N MTs per

m2 of area normal to the MTs axis has the form

F

vol= N2 kB T

64A2

3

2R3

2 +

1

4

+ 128

2cos2 + 1

(8)+ R2

+

1

8

+R 4 ln(2R)

,

where A = d24 is the cross section of a MT, =(

4 n2c e2

s kB T)1/2 is the inverse screening length with nc

denoting the average concentration of the condensed

counterions along MTs. The MTs dielectric anisotropy

is highly important and expressed through and

= M (

M +s )3M ss (

M +s )

. Finally, represents the angle

between the orientations of two neighboring bundles.

The free energy thus consists of three terms reflecting

the monopolemonopole, dipoledipole and van der

Waals interactions, respectively.

Minimizing Eq. (8) with respect to the angle , we

obtain = /2 which agrees with the angle betweentwo halves of the curtain pattern (Fig. 1(a)). This

orientation is even more clearly expressed in the finalstriped pattern of TEx (Fig. 1(b)). It is evident that the

electrostatic interactions with counterion charge fluc-

tuations lead to the result that MTs polymerized in the

gravitational field during TEx form a polar liquid crys-

tal (LQ) phase [23]. Moreover, it is known [24] that

long cylinders generally tend to align parallel to each

other when their density exceeds a critical number

cr L2d1. For MTs (L = 5 m and d= 25 nm),this concentration is MTcr 1.5 1018 m3. For theconcentration and length of MTs in TEx, MTTEx = 3

1018 m3 > MTcr , easily satisfies the conditions forthe onset of a LQ phase. Consequently, the LQ fea-

tures dominate in the pattern formation behavior after

the critical bifurcation state is reached under the actionof GF.

LQ properties of MTs affect their alignment with

the surface of the cell. For long MT cylinders (L d)in TEx, condensing many MT ends at the surface car-

ries a high entropy cost and is therefore unlikely to

occur. This is consistent with the experimental fact

that MTs are aligned parallel to the cell wall corre-

sponding to the largest area 4010 mm2 and pointingtoward the smallest surface area. Moreover, due to an

oblique alignment of MTs with the smaller surface ar-

eas ( = /4) a further reduction in the number ofMTs in contact is accomplished. Since the position ofa MT end with respect to the wall is determined up

to its diameter d, this significantly reduces the config-

urational entropy of one MT in contact by Sconf=kB ln(

Ld

) 5.3kB . This is why MTs in TEx are notoriented vertically as expected from gravitational ACC

effects. This orientation forms a close-packed layer of

MTs perpendicular to the surface and creates a new

surface a distance L into the bulk, at which location

the next layer of MTs must be properly arranged to

further reduce the entropy. Oblique orientations in-

duce splay effects especially near the corners of theexperimental cell where gaps between MTs open up,

into which other MTs can penetrate, restoring the ran-

dom distribution of MT ends. However, in the bulk,

the highly parallel oriented MTs still have to satisfy

the cooperative entropy condition.

The rotational relaxation of a single cylinder of

fixed length is hindered by its disordered neighbors

when cylinder concentrations exceed L3. WithL = 5 m one obtains = 1016 m3, which is muchmore dilute than in TEx explaining why microgravity

conditions prevent pattern formation.MTs that grow in a disorderly fashion are prevented

from establishing LQ order. If the system already pos-

sesses highly parallel ordering, as is the case of gravi-

tationally directed MTs within curtains, the entropy-

frustrated state should be relaxed by the spread of slow

rotational waves starting from the walls and lasting ap-

proximately 12 hours [4] or 5 hours [3,5] depending

on experimental conditions. These rotations are facil-

itated by the existing dynamical instability of MTs

but now are less dramatic than at the moment of the


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chemical instability (6 minutes after the onset of the

reaction).

Implications of the microgravity effects and their

explanation presented here extend to cell behavior aswas seen in another space flight experiment [25]. Fur-

thermore, it was recently reported [26] that in climbing

plants competing for light energy, MTs create a twist

obliquely oriented toward the cells long axis. This

suggests that symmetry breaking by the GF together

with structural changes within TB subunits related to

the rigidity of MTs, is the mechanism responsible for

pattern formation. Patterns in [26] clearly reveal the

LQ phase of MTs in vivo.

In conclusion, our theoretical model allows us to

make qualitative predictions regarding the emergenceof MT patterns on a function of several controllable

parameters such as the GF strength, pH, TB concen-

tration, and temperature. A comprehensive analysis of

the effect of these parameters will be published else-

where.

Acknowledgements

The authors thank NSERC, MITACS-MMPD and

the ITP at the University of Alberta for financial

support. S. Portet has been supported by a Bhatiapost-doctoral fellowship. J.A. Tuszynski thanks Dr.

J. Tabony for his hospitality during his visit to Greno-

ble. Valuable discussions with Profs. A. Ramani of

Ecole Polytechnique Palaiseau and B. Grammaticos

of Universit Paris VII are gratefully acknowledged.

J.A. Tuszynski wishes to thank Dr. R. Binot for the in-

vitation to the microgravity workshop at the ESA at

Noordwijk, Holland.

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