A simple formulation of microtubuie dynamics: quantitative implications of

the dynamic instability of microtubuie populations in vivo and in vitro

PETER M. BAYLEY*, MARIA J. SCHILSTRA and STEPHEN R. MARTIN

Division of Physical Biochemistry, National Institute for Medical Research, The Ridgeway, Mill Hill, London AT17 IAA, England

•Author for correspondence

Summary

A simple formulation of microtubuie dynamic in-stability is presented, which is based on the exper-imental observations by T. Horio and H. Hotani ofcoexisting, interconverting growing and shrinkingmicrotubules. Employing only three independent,experimentally determined parameters for a givenmicrotubuie end, this treatment accounts quantitat-ively for the principal features of the observeddynamic behaviour of steady-state tubulin micro-tubules in vitro. Experimental data are readilyreproduced for microtubuie length redistribution,and for the kinetics of tubulin exchange processes,including pulse-chase properties. The relative im-portance of dynamic incorporation and that due totreadmilling are assessed. Dynamic incorporationis found to dominate the overall exchange proper-ties; polarized incorporation due to treadmillinggenerally becomes significant only when the dy-namics are largely suppressed.

This treatment also permits simulation of certaincellular phenomena, showing how microtubuie re-nucleation can control microtubuie growth, bymeans of changes in microtubuie number concen-tration in a system at constant microtubuie mass.

A relatively simple extension of the formulationaccounts quantitatively for non-steady-state micro-tubuie properties, e.g. dilution-induced rapid disas-sembly and the oscillatory mode of microtubuieassembly. The principles relating dynamic insta-bility and oscillatory behaviour are clearly indi-cated. Possible mechanisms of the switching ofmicrotubules are briefly discussed.

Key words: microtubules, dynamics, simulation, dynamicinstability, model, regulation.

Introduction

The cytoskeleton of eukaryotic cells contains three majorfilamentous protein structures: actin filaments, inter-mediate filaments and microtubules (Schliwa, 1986). Ofthese, the microtubules undergo a major spatial redistri-bution during the cell cycle (Dustin, 1987; Amos &Eagles, 1987). They change from the dispersed cytoplas-mic microtubuie arrays of the interphase cell, to thespecific localized structures of the mitotic spindle, andare subsequently redistributed as the cytoplasmic micro-tubules of the daughter cells. Since the discovery ofconditions under which tubulin can undergo reversibleassembly-disassembly in vitro, much work has beendirected at understanding the kinetics of microtubuieassembly (Correia & Williams, 1983; Purich & Kristof-ferson, 1984; Engelborghs, 1988), and the factors thatcontrol the regulation of microtubuie arrays in vivo(Kirschner & Mitchison 1986a). It is a major outstandingquestion to evaluate how far the dynamic characteristicsof microtubuie arrays in vivo are the expression offundamental physical properties as observed in vitro, andhow far they are controlled by biochemical effects, suchas the modulation of microtubuie structure due toenzymic modification.

Journal of Cell Science 93, 241-254 (1989)Printed in Great Britain © The Company of Biologists Limited 1989

Observations of the fast kinetics of microtubuie disas-sembly in vitro (Farrell et al. 1983; Karr et al. 1980) andthe observation that the average length of a microtubuiepopulation increased upon limited dilution (Mitchison &Kirschner 1984a,b) led to the proposal of the 'dynamicinstability' mechanism, in which the microtubuie popu-lation consists of two sub-populations of growing andshrinking microtubules. Dark-field microscopy of tubu-lin microtubules in vitro showed that microtubules doindeed undergo large length excursions, with one endbeing more active than the other (Horio & Hotani, 1986).Transitions between growing and shrinking phases occurapparently at random, with no correlation between thebehaviour at the two ends. Similar results have beenreported (Cassimeris et al. 1987; Walker et al. 1988).

A number of observations have also been made of thetime-dependent redistribution of microtubuie lengths invivo (e.g. see Salmon et al. 1984; Salmon & Wadsworth,1986; Saxton et al. 1984; Schulze & Kirschner,1986,1987; Sammak et al. 1987; Wadsworth & Salmon,1986; Soltys & Borisy, 1985; Gorbsky et al. 1987;Sammak & Borisy, 1988; Kristofferson et al. 1986;Cassimeris <tf al. 1986; Walkers al. 1986; Koshland et al.1988). These studies show that rearrangements of micro-

241

tubules in vivo can occur on a short time scale. Althoughquestions of the irreversible ('catastrophic') nature of thedisassembly process have been controversial (Williams etal. 1986; Kirschner & Mitchison, 19866), and the ratesdetermined for growing and shrinking of microtubulesvary somewhat and may be modulated by the presence ofmicrotubule-associated proteins (MAPs) (Horio &Hotani, 1986; Farrell et al. 1987), it is clear that theimplications of dynamic instability must be evaluatedquantitatively in interpreting the dynamic properties ofmicrotubules both in vitro and in vivo.

These considerations have stimulated attempts to de-fine numerical models describing the behaviour of dy-namic microtubules. Direct analytical solutions are, ingeneral, precluded, although various approximations arepossible (Hill, 1984; Chen & Hill 1985a,6; Rubin, 1988).Hill and coworkers have performed extensive numericalsimulations that require the properties of tubulin in theterminal region of the microtubule to be defined expli-citly. Their treatment involved assumptions about therelationship between microtubule elongation and GTPhydrolysis leading to the presence of a terminal 'cap' oftubulin-GTP, which would be particularly extensiveunder assembly conditions. The experimental determi-nation of the size of such a structure is controversial(Carlier et al. 1987a; O'Brien et al. 1987; Voter et al.1987; Schilstra et al. 1987; Caplow et al. 1984, 1985).

The approach described here avoids these problems.We reproduce the steady-state dynamic behaviour ofmicrotubules by numerical simulations using only exper-imentally observed rates, without explicit considerationof the mechanism of GTP hydrolysis or of the transitionsbetween growing and shrinking microtubules. We adoptthe simplest formulation of the dynamic behaviour ofmicrotubules; namely, that in a steady-state population(constant polymer mass) individual microtubules areeither growing or shrinking and can switch between thesestates. On the basis of the original demonstration of theextensive incorporation of [3H]GTP-tubulin intosteady-state tubulin microtubules (Martin et al. 19876;Bayley et al. 1988), we now develop the treatment toexamine the quantitative implications of the dynamicinstability mechanisms for microtubules under a varietyof conditions.

The basic formulation requires only four kinetic para-meters, of which three are independent. Even with thissimplified approach, many steady-state microtubuleproperties are amenable to investigation, including: thetime-dependent behaviour of individual microtubules,their length redistribution, changes in microtubule num-ber concentration, GTP hydrolysis rate, tubulinexchange kinetics, and isotopic nucleotide pulse-chasekinetics.

This approach emphasizes the importance of thechanges at steady state in microtubule number concen-tration, since microtubule 'ends' are the site forelongation and disassembly. We therefore also examinethe effects on microtubule populations of introducingdiscontinuous changes in the number of nuclei in asimulation of the control of renucleation of growingmicrotubules from an organizing centre, which could

occur during the cell cycle (Kirschner & Mitchison,1986a).

Treatment of microtubule dynamics under non-steady-state conditions requires the explicit inclusion of theconcentration dependence of the microtubule switchingfrequencies. From this we examine the behaviour ofdynamic microtubules: (1) in rapid disassembly at highdilution; and (2) in oscillatory behaviour at high concen-tration. This treatment illustrates the role of dynamicinstability in microtubule oscillations, and accounts forthe observed oscillatory properties.

We discuss the factors that could control the rate ofcooperative transitions between microtubules in differentstates, and consider the cellular implications of theseresults.

Materials and methods

MethodsPhosphocellulose-purified tubulin was obtained from bovinebrain as described (Clark et al. 1981). Exchange of tubulin intosteady-state microtubules was measured as described (Martin etal. 19876). In brief, tubulin (30 fiM) was assembled by incu-bation at 37°C (0-lmM-GTP) for 60 min in PEM-G buffer(0-lM-Pipes buffer, pH6-5, containing OlmM-EGTA and0-SmM-Mg with 1 M-glycerol) in the presence of a GTP-regenerating system. The solution was then sheared by passagethrough a 23 gauge syringe needle and divided into two parts. Apulse of [3H]GTP was added to one part 3 min after shearingand an identical pulse was added to the other part 93 min aftershearing. Samples were taken at fixed time intervals forestimation of [3H]GDP by HPLC and for electron-microscopic(EM) measurement of the length distribution (Martin et al.1987a).

Numerical simulationThe kinetic formulation suggested by the experimental obser-vations of Horio & Hotani (1986) is described by Scheme I:

Scheme I. Dynamic behaviour of microtubules at steady-state,based on the observations of Horio & Hotani (1986). Anindividual microtubule end is in either the growth state, G (inwhich it adds T u - G T P (T) with bimolecular rate constant, k+)or in the shrinking state, S (in which it loses Tu-GDP (D) withrate constant /t_). Interconversions between states occur withrate constants &GS ar>d &SG. which at steady-state are constants.Free D is converted to T with first-order rate constant, k*

In this scheme, G and S represent growing and shrinkingmicrotubules. The rate constants are k+, k- for addition(tubulin-GTP) and loss (tubulin-GDP), respectively,and &GS> ^SG f°r transitions between states. We thusassume that growth and shrinkage are uniform processesand for simplicity (cf. Chen & Hill, (1987)) we treat

242 P. M. Bayley et al.

activity at only one microtubule end (see Horio & Hotani,1986). (In fact the treatment is readily extended to twodifferentially active ends.)

The steady-state mass balance at a given end may beexpressed in terms of fG and / s , the fractions of growing(G) and shrinking (S) microtubules, and the observedrate of growth RG and of shrinkage Re, (in /zm per unittime) as:

dCp/dt=fG.RG-fs.Rs = O, (1)

where Cp is the concentration of polymer (=L.N.m,where L is the mean microtubule length in /im, N(= 1625) is the number of subunits per /zm of micro-tubule, Amos & Klug (1974), and m is the concentrationof microtubule ends). RG could be written as (k+G.Cc-k-G)/N but, following the observation (Mitchison &Kirschner, 1985a) that k-G is small, we obtain RG

(=k+G.Cc/N). Likewise, R$ could be equated with thedissociation rate constant, k-s/N. Substituting inequation (1), and multiplying by m.N, we obtain anexpression for Cc, the molar concentration of free tubulinat steady state, in terms of k+ and k- (omitting their G,Ssubscripts):

6Cp/dt = fG.k+.Cc.m -fs.k-.m = 0 (2)

Thus, treating the microtubule population at constanttotal polymer mass, as a two-state model of interconvert-ing sub-populations as in Scheme I, gives a rate equation(eqn (2)) formally similar to that of the classical Oosawamechanism, (see Johnson & Borisy, 1977).

RG and /?s are the measured rates of growth andshrinkage in ;Um per unit time. The other measuredparameters are TG and Ts> the lifetimes of the growingand shrinking states, which are the reciprocals of thetransition frequencies kG$ and kSG. The values obtainedby Horio & Hotani (1986) were RG = 0-63/zm min"1,/?s = 7-9/zmmin"1, 71c = 2-9min and 7"g = 0-3 min.Although these values are not completely balanced thesystem is close to steady state and we enforce thiscondition by using:

/ G / / S = RS/RG = TG/TS = *SG Acs .

Because of these equalities, only three independentparameters are needed to .define the steady-state kineticbehaviour.

In the simulation we use 5000 microtubules, with aninitial Gaussian length distribution centred at the initialmean microtubule length (L(0)) (width, dL = L(0)/l0).This population is divided randomly into G and S statesin the ratio / G / / S .

a s determined by the chosen values ofRG and R$. We also choose a value for Tg, from which wecalculate TG, &SG a nd ^GS a 9 above. At each time step inthe simulation (dt), each G-microtubule is tested to see ifit undergoes a transition to the S-state, based on thecomparison of a random number with PG$, the prob-ability that the G—»S transition will occur. When dt <

PGS = {1-0 - exp(-df/TG)} = dt/TG.

Each S-microtubule is tested to see if it undergoes atransition to the G-state, based on the probability Pgc

(= di/T$). Transitions between states are assumed to beinstantaneous. Values of dt used were generally in therange 0-02-0-005 min, the simulated behaviour did notdepend on the value of dt in this range. During thesimulation, the total population is not allowed to fallbelow 2500; this is accomplished numerically by doub-ling the effective 'simulation volume' at this point, andduplicating the existing microtubule population.

The transition from S to G is the process by which amicrotubule is 'rescued' before shrinking to zero length.If PSG equals zero, a microtubule entering the S-statewould disappear (i.e. 'catastrophe': Kirschner & Mitchi-son, 19866). Experimentally, PSG is not equal to zero(Horio & Hotani, 1986) and rescue occurs on a randombasis. In the simplest mechanism, a microtubule thatreaches zero length simply disappears, and the endconcentration is reduced by one unit. This irreversibleloss of microtubules is presumably the situation for self-nucleated assembly of tubulin, where the 'nucleus' is ashort microtubule fragment indistinguishable in struc-ture from the elongated microtubule.

The simulation permits continuous monitoring of eachmicrotubule and provides data on the properties of thepopulation in terms of length distribution, average lengthand m. Thus we can simulate the behaviour of the wholepopulation and assess how experimentally observableproperties would vary as we change the three indepen-dent kinetic parameters and the initial length distri-bution. The process of subunit exchange at steady state issimulated by monitoring length changes after label 'ad-dition', and this approach is readily extended to thesimulation of pulse-chase experiments.

This formulation may be extended to represent micro-tubule growth from recognizable nucleation centres suchas axonemes and centrosomes. In this case, a shrinkingmicrotubule that reaches zero length (So) becomes aninactive nucleation centre. This is allowed to 'renucleate'(to form an active nucleus Go) with a probabilitydetermined by the first-order rate constant, k,,. (seeScheme II).

•tSo

Scheme II. Dynamic behaviour of microtubules at steady state,including microtubule renucleation. Scheme I is extended toallow a shrinking microtubule end that reaches zero length (So)to transform into an active nucleus (Go) by a first-order process,rate constant, kn\ the species Go then grows by the normalelongation reaction involving T and k + .

By allowing S-microtubules that have reached zero lengthto 'renucleate', we can examine the effects of discontinu-

A simple formulation of microtubule dynamics 243

or

Effect of kinetic parameters on the extent ofredistributionIncreasing the absolute values of either (R(TQ,TS) increases the extent of length redistribution(Table 1) by increasing the size of individual length

35-0

0-10

140-0

Time (min)

Time (min)

Fig. 3. Simulation of microtubule dynamics: the effect of theinitial length distribution on microtubule properties.A. Time-dependent changes in average length. B. Time-dependent changes in microtubule number concentration.C. Steady-state GTP hydrolysis: the production of phosphateion. Computer simulations were performed withRc = 1-26/immin"1, Rs = 7-9/im min"1 and Ts = 0-3min.Initial lengths used in the simulation were 2 (trace a), 4(trace b), 8 (trace c), 16 (trace d) and 32 Jim (trace e).

Table 1. The effect of changes in standard kineticparameters on simulated changes in average length attime, L(t), for a microtubule population with an initial

mean length, L(0) = 8 /JJII, over a period of 30 min,using the model of Scheme I

Re(/an min ') (/un min ) (min)

L(30)(length in /mi)

A 1-261-261-261-26

B 0-631-262-52

7-907-907-907-90

3-957-90

15-80

0-10-30-61-0

0-30-30-3

10-715-219-523-2

9-915-227-3

A. Variation in lifetime of the shrinking state (Ts), with the ratioT3/TG kept constant.

B. Variation in rates of growth (Rc) and shrinkage (Rs)-

excursions, the average value of which is AL = Rc-Ts-Fig. 4A shows that a microtubule with Tg = 0 3 under-goes approximately 45 transitions in 40 min; withTs = 0-05, this number is some 10-fold greater, and theexcursion amplitude is lower. For low values of Tg,individual microtubules could appear to be effectivelyinactive (cf. Cassimeris et al. 1987). Thus increasedtransition frequency suppresses the extent of lengthredistribution, and hence reduces the extent of thetubulin exchange into steady-state microtubules. How-ever, such exchange measurements reflect a bulk popu-lation property. The simulation emphasizes that thebehaviour of individual microtubules is stochastic andtherefore highly heterogeneous, particularly over shortperiods. Fig. 4B shows the behaviour of four micro-tubules over a period of 5 min; although these micro-tubules show little change in total length, the individualextent of label incorporation (for label added at / = 0)varies from 4 to 79 %.

Implications of dynamic instability for pulse-chaseexperimentsBoth the rate and extent of exchange of T u - G T P(tubulin with GTP at the exchangeable nucleotide bind-ing site) into steady-state microtubules are stronglyinfluenced by the initial length distribution, (Fig. 1).The time-dependence of m (and thus average length) hasimportant implications for pulse-chase experiments, i.e.that the average length at the point of the chase can besubstantially different from that at the time of the pulse.

Simulation allows us to assess the importance of thiseffect. Fig. 5A shows the kinetics of the chase for amicrotubule population pulsed for various times. Thetime course of both pulse and chase processes arecomplex non-exponential functions (cf. Fig. 1). In par-ticular, the amount of slowly exchanging label increasesas the pulse length increases. Fig. 5B shows the effect ofvarying the initial length distribution for a simulatedpulse-chase experiment. Long microtubules incorporateproportionately less label in the pulse, and show substan-tial retention of the label at long times after the chase. Weemphasize that these properties are derived for processes

A simple formulation of micmtubule dynamics 245

52-000-80

34-00

9-00

Time (min)

00

0-00

Time (min)

Fig. 4. Simulation of microtubule dynamics: time-dependentbehaviour of individual microtubules. Values of Re and R^were 1-26 and 7-9/imin~ , respectively. A. Simulations for7s = 0-3 (trace a) and 005 (trace b) min. B. Heterogeneity ofbehaviour of individual microtubules: although the finallengths are closely similar, the extent of regrowth (and hencetubulin exchange) varies widely.

occurring at one end of the microtubule only; including asecond active end would result in the superposition of anadditional distinct but similar time-course. The import-ant fact is that the exchange process is progressivelyslower with extended time, for the dynamic instabilitymechanism operating at a given microtubule end. Theincompleteness of the exchange is due to the stochasticnature of both pulse and chase; 100% chase will beshown only by those individual microtubules thatundergo excursions to a length shorter than their lengthat the point of label addition. Thus, complete displace-ment of label will not, in general, be observed, particu-larly under conditions where the average length is in-creasing due to the dynamic instability mechanism.

These effects are also important in double-label exper-iments (cf. Jordan & Farrell, 1983). If assemblingmicrotubules, labelled uniformly (with [14C]nucleotide),are allowed to achieve steady state in which dynamicinstability is operating, and then pulsed with a secondlabel ([3H]nucleotide), a cold chase shows different rates

o-oojTime (min) 1 2 0

0-90

u-

0-00

Time (min)

Fig. 5. Simulation of pulse-chase experiments: the effect ofdifferent factors on the extent of tubulin incorporation.Kinetic parameters were RQ =1-26 Jtfn min~',Rs = 7-90 /Jm min"1 and Ts = 0-30 min. A. Effect on extent ofmicrotubule labelling of variation in pulse duration (5, 10,20, 40, 60 min) followed by 'cold' chase. The initial meanlength is centred at 8fim. The average lengths at the time ofthe chase are 9 1 , 10-4, 12-9, 17-4 and 20-8/tfn. B. Effect onextent of microtubule labelling (for a pulse of 20 minduration) of variation in initial length distribution, centred at2 (trace a), 4 (trace b), 8 (trace c), 16 (trace d) and 32/im(trace e), respectively.

and amplitudes of displacement of the two isotopes(Bayley et al. 1989) with extended kinetics. Such slowprocesses have been interpreted in terms of microtubuletreadmilling for microtubule-associated protein (MAP)-rich microtubules in which dynamic instability is sup-pressed (see Wilson & Farrell, 1986) but also for MAP-depleted microtubules that show length redistribution(Farrell et al. 1987). However, the stochastic nature ofboth 3H-pulse incorporation and cold chase resultingfrom dynamic instability alone, as illustrated in Fig. 5,produces this slow kinetic behaviour, even in a systemwith no treadmilling. The slow exchange is very similarin form to the time course associated with unidirectionalflux or treadmilling. Thus, if treadmilling and dynamicinstability are operating simultaneously (Farrell et al.1987), we conclude that it will be extremely difficult frompulse-chase data alone to evaluate the relative contri-butions of the two mechanisms.

The effects of microtubule renucleationWe now consider the effect of introducing the re-nucleation step in Scheme II. Instantaneous renucleation(high kn) produces a system in which m remains effec-

246 P. M. Bayley et al.

tively constant and ensures that short microtubules arealways present in substantial quantity, giving an expo-nential length distribution. For intermediate values of kn,a peaked length distribution develops and stabilizes. Thisis more in keeping with the requirements of a biologicalsystem. The rate and extent of the redistribution are afunction of k,,, plus the main kinetic parameters, RQ,RS,TG and Ts.

This scheme illustrates how changes in the extent ofnucleation can affect the microtubule length distributionin a dynamic population at constant polymer mass. Thevalue of m is varied discontinuously, by changing thevalue of kn at different times (Fig. 6). Starting with apopulation of average length 4/im under conditions ofinstantaneous renucleation (kn high), an exponentiallength distribution is produced. After lOmin, renuclea-tion is effectively suppressed by reducing kn to0-05 min"1, when m rapidly falls to approx. 20% of itsinitial value. The existing microtubules grow out in aconcerted fashion (Fig. 6B,C), to give a much longer(eventually exponential) distribution, (Fig. 6E). (Ifk,, = 0, a typical non-exponential distribution is gener-ated, cf. Fig. 2.) At 30min, the high renucleation rate isre-introduced. This rapid change in m causes many shortmicrotubules to appear (Fig. 6F,G) which then grow atthe expense of longer microtubules, until the originaldistribution is re-established, (Fig. 6J). This behaviour(short microtubules growing and long microtubulesshrinking) is the reverse of the classical dynamic insta-bility phenomenon, but is expected when the lengthdistribution is perturbed by a rapid increase in thenumber of nuclei (i.e. m). Although a variety of enzymicprocesses are probably operating in vivo, the simulationillustrates the capability for a far-reaching response of adynamic microtubule population at constant polymermass to a transient change in the number of nucleationcentres (cf. Gliksman et al. 1987).

Implications of dynamic instability for microtubulesunder non-steady-state conditions: microtubuleoscillationsTwo lines of argument suggest that the transition fre-quencies, kG§ and k$G, must be functions of [Tu-GTP] ,as follows.

(1) Microtubule disassembly induced by extensiveisothermal dilution: the observed dissociation amplitudeand kinetics show that the whole microtubule populationmust be converted rapidly to the S-state. Thus kcs mustincrease as [Tu-GTP] decreases. The value of k-obtained from such dilution experiments (Farrell et al.1983; Karr et al. 1980; Martin & Bayley, unpublished) isin good agreement with that used here. These argumentssuggest that the ratio / G / / S is indeed controlled by[Tu-GTP] . A similar argument (Schilstra et al. 1989) isused to explain the rapid disassembly of tubulin micro-tubules induced by addition of excess podophyllotoxin;removal of T u - G T P as an inactive Tu-GTP-drugternary complex causes a rapid switch from G- to S-states, and disassembly proceeds at the same value of k-(160 - 200 s~') as observed at steady state (Horio &Hotani, 1986).

(2) Tubulin assembly at high protein concentration: athigh [Tu-GTP] oscillatory behaviour is observed, andhas been attributed to slow processes in the regenerationof T u - G T P from Tu-GDP, either by conformationalchange (Carlier et al. 19876), nucleotide levels (Pirollet etal. 1987) or oligomer formation (Mandelkow et al. 1988;Lange et al. 1988). Slow nucleotide exchange is probablypromoted by oligomer formation; the suppression ofnucleotide exchange in oligomeric species is well-estab-lished (Caplow & Zeeberg, 1980). Oligomer formation isselectively enhanced by Tu-GDP (Howard & Timas-heff, 1986; Melki et al. 1988), and by the high concen-tration of Mg24" generally used, (Carlier et al. 19876;Mandelkow et al. 1988).

The primary assembly 'overshoot' requires a mechan-ism whereby [Tu-GTP] is depleted below its eventualsteady-state value. The dynamic instability mechanismshows how this overshoot arises. In the growth phase,effectively all microtubules will be in state G, i.e. fG = 1.If &GS is low at high [Tu-GTP] , the conversion ofgrowing to shrinking microtubules is delayed, and theassembly will exceed the plateau value implied by thecritical concentration. With suitable power dependencein the transition frequencies, as [Tu-GTP] decreases,&GS increases. Growing microtubules eventually switchto the S-state, and the system is disassembled rapidly andproduces Tu-GDP in excess of the Cc. The slowness ofthe regeneration of T u - G T P from Tu-GDP (by k»,Scheme I) delays the point at which the switch back to Gstate occurs (Bayley et al. 1989). The result is periodicassembly, the oscillations becoming progressivelydamped as the initially concerted behaviour becomesuncorrelated.

In simulation, the concentration dependence of kGsand &SG c a n be represented by various (arbitrary) powerfunctions such as:

and

°. [ T U - G T P ] "

= *SG°- [Tu-GTP] +-v,

where superscript (°) indicates steady-state values (seeBayley et al. 1989). Alternative functions were used byChen & Hill (1987). The precise concentration depen-dence implied by either set is as yet without experimentalfoundation; recent data suggest that overall effects maybe less pronounced (Walker et al. 1988).

We have used the formulation of Scheme I to investi-gate the importance of dynamic instability in four areas ofnon-steady-state behaviour: (1) high dilution, when thesystem is disassembled without lag as expected (notshown); (2) effects of Mg2"1" and Ca , which enhance &_and hence R$ (Gal et al. 1988); (3) possible effects on thekinetics of regeneration of Tu -GTP; and (4) the effect ofTu-GDP being present when assembly is initiated.

Oscillatory behaviour has been simulated for SchemeI, i.e. the simplest formulation without renucleation.The wide variation in [Tu-GTP] during assemblyaffects RG as k+. [Tu-GTP] with k+ = 5x 1 0 6 M - 1 S" ,and controls kGS and kSG (see Chen & Hill, 1987). We

A simple formulation of microtubule dynamics 247

0-00 2-000-00 2-00 4-00

Fig. 7. Simulation of microtubule oscillations according toScheme I: effect of variation in the dissociation rate constant,k-. Simulations for k. = O-fWs"1 and k- = 100 (A), 400 (B)and 800 s~ (C). The plots show concentrations of polymer(O). free T u - G T P ( • ) , free Tu-GDP ( + ) and percentageof microtubules in the growing G-state (X) as a function oftime in min.

polymer, T u - G T P and Tu-GDP, as a function of time.Also shown is the percentage of G-state microtubules,which is seen to vary dramatically. Since this is a dynamicsystem without renucleation, m decreases progressively,particularly in the first disassembly phase. Fig. 7 showsthe way in which oscillatory behaviour is affected bychanging k- from 100 to 800 s~ . The variation in / G ismost marked for low k- values; the absolute value of[Tu-GTP] during oscillation remains relatively low, andthe effect on oscillation period and amplitude is relativelysmall. By contrast, Fig. 8 shows that, for a given value ofk-, the effect of variation in £. is more dramatic.Increasing k* from 0-02 s"1 to 0-2 s"1 markedly increases

Fig. 8. Simulation of microtubule oscillations according toScheme I: effect of variation in the regeneration rateconstant, k. = 0-02 (A), 0-10 (B) and 02 s"1 (C). Symbols asfor Fig. 7.

the period, and decreases the amplitude of oscillation.The relative change from G- to S-state becomes lessmarked, and as expected, the amplitude of oscillation ofTu-GDP decreases strongly. In all these cases T u - G T Pand Tu-GDP oscillate with similar period, but out ofphase by an amount that decreases as £. increases. Underall conditions where Tu—GTP is regenerated solelythrough k. (except for the highest values of k.), thepredominant disassembled form is Tu-GDP.

These calculations show that the kinetic parameters ofexchange and dissociation interact in a complex way, andthat oscillations occur for a large range of k* and k-values. This is due to the imposed condition that thetransition frequencies are a function of the absolute valueof [Tu-GTP] . Because of the cooperativity of thetransitions with protein concentration, the oscillatory

A simple formulation of microtubule dynamics 249

0-00 2-00 4-00

Fig. 9. Simulation of microtubule oscillations according toScheme I: effect of variation of the regeneration rateconstant, k,, in the presence of an initial 25 % fraction ofTu-GDP. Simulations for k- = 400s"1 and k. = 001 (A),0-02 (B) and 0-04s"1 (C). Symbols as for Fig. 7.

behaviour diminishes at low concentration unless theexchange process k' becomes very slow (Carlier et al.19876).

In experiments at high protein concentration, it isoften difficult to obtain complete conversion of Tu-GDPto T u - G T P before assembly, because slowly exchangingoligomers are formed, particularly with Tu-GDP at high[Mg2*] (Howard & Timasheff, 1986). We thereforestudied the predicted effect of including significantamounts of Tu-GDP in the initial reaction mixture.Fig. 9 shows that the initial oscillation amplitude ispredicted to be sensitive to the presence of Tu-GDP (itdecreases further with increasing [Tu-GDP] or decreas-ing A.: not shown) and is only free of this effect when k*becomes large. Again, the levels of [Tu-GTP] attained

during oscillation are relatively small. This emphasizesthat, owing to the concentration-dependence of thetransition frequencies, only a small variation in [Tu-GTP] is needed to activate the switching.

This treatment of the complex oscillating system isundoubtedly an over-simplification. At high proteinconcentration, both Tu-GDP and T u - G T P associateinto oligomeric complexes (stimulated by Mg2"1"; Howard& Timasheff, 1986) and thereby increase the amount ofnon-assembled tubulin, as well as increasing the kineticcomplexity. Our simplified approach illustrates some ofthe important kinetic factors, showing the basic require-ment for a kinetically limiting regeneration of theelongating species (Tu-GTP), and a non-linear (feed-back) process, e.g. the dependence of &GS a n d ^SG o n t n e

same species, Tu-GTP. The frequency and extent of theoscillations is determined by several interacting variables.Solution conditions affect k- (and k»)\ excess GTP, ifpresent, may promote re-formation of T u - G T P ; andMAPs, which change the dimer/oligomer balance, maybe expected to influence the mechanism of assembly/disassembly in a complex way.

Discussion

General formulation of steady-state dynamicsWe have shown here that a simple kinetic formulationderived from the experimental observations of Horio &Hotani (1986) is capable of rationalizing many of thesurprisingly complex properties observed with steady-state and non-steady-state microtubules. For illustrativepurposes we have limited our basic presentation to asingle microtubule end at overall steady state of polymermass. It is clearly very difficult from direct visualobservation to determine the experimental parameterssufficiently accurately to determine whether both ends ofthe polar microtubule are in fact at steady state, orwhether there is net growth and net shrinkage correlatedwith the microtubule polarity (cf. Walker et al. 1988). Afurther question is the adoption of just two states, G andS. The possibility exists that microtubules may exhibitapparent phases of inactivity (Cassimeris et al. 1987;Schulze & Kirschner, 1987,1988; Sammak & Borisy,1988), which could be due to enhanced transition fre-quencies, or could indicate a truly inactive intermediatestate of the end. Recent evidence (Schilstra et al. 1989)suggests similar behaviour in the presence of podophyllo-toxin. Although these simplifications may present somerestrictions, they nonetheless permit examination ofmany of the implications of the general principles ofdynamic instability in an accessible form.

For a given microtubule population at steady state,only three experimentally observable kinetic parametersare required for the simulation of the properties of a givenend. No assumptions are made concerning the precisemolecular mechanisms for transitions between the twostates, and the controversial questions about the compo-sition of ends are avoided. These simulations provide aparticularly clear indication of the effects of changes inparameters TQ, T$, RQ and Ro, in determining bulkproperties of the microtubule population, as well as

250 P. M. Bayley et al.

demonstrating the behaviour of individual microtubules.Considering the simplicity of the approach, the calcu-lations are in remarkably good agreement with exper-imental data for length changes and isotopic exchange(Martin et al. 19876).

In this work we have generally retained the self-consistent set of experimental values of T s , TG, RG andRs observed by Horio & Hotani (1986). The value ofRs = 7-9/immin"1 corresponds to k- values close to200 s~l. Significantly larger values of k- have beenobserved on addition of Ca (Karr et al. 1980), by directobservation in vitro (Cassimeris et al. 1987; Walker et al.1988) and with measurements in vivo (Schulze &Kirschner, 1987; Sammak & Borisy, 1988). Some of thisvariation may be due to solution conditions, and particu-larly to ionic composition. We observed a fivefold en-hancement by [Mg2+] in the range 0-5-2-5 mM on therate of microtubule disassembly at low temperature(Martin et al. 1987a). Higher concentrations causegreater enhancement in isothermal disassembly (Gal etal. 1988). On the dynamic instability model theseenhanced values presumably imply a strongly reducedfraction of shrinking microtubules (/s) if k+ and/or Cc

are not to change disproportionately.As indicated above, the question of net growth and net

shrinking at opposite ends of the microtubule becomes ofparticular interest with dynamic microtubules. Thisproperty, equivalent to a difference in apparent affinityfor T u - G T P at the two ends of the microtubule gives riseto microtubule treadmilling (cf. Wegner, 1976; Margolis& Wilson, 1978), but this mechanism alone would notaccount for length redistribution at steady state. We showhere that simulation of a microtubule population in termsof dynamic instability without treadmilling reproducesthe observed length changes. It also reproduces observedtubulin exchange behaviour, again without treadmilling.

This analysis is based on the observations that: (1)pulse incorporation is stochastic and shows multiphasickinetics (Fig. 1); and (2) the (cold) chase is intrinsicallyinefficient, since (Fig. 4) the level of incorporation inindividual microtubules is inevitably heterogeneous, andthe mechanism of the chase is also stochastic. Thus, adeeply labelled microtubule will necessarily be moreresistant to the chase. It also follows that in a double-labelexperiment, when microtubules initially labelled with[^CjGTP are pulsed with [3H]GTP and then coldchased, the displacement of the two isotopes will followdifferent kinetic courses (Bayley et al. 1989) with dis-placement of [14C]GTP invariably exceeding that of[3H]GTP. The simulations show that this necessarilyoccurs at a single end of a dynamic microtubule; it is notnecessary to postulate treadmilling, differential affinity orend-dependent polar growth and shrinking. This dispar-ity in kinetics is in fact observed experimentally; how-ever, where both dynamic instability and treadmillingeffects could occur simultaneously (cf. the phase 'dy-namics' mechanism; Farrell et al. 1987), it will beextremely difficult to distinguish between the relativecontribution of each mechanism from exchange dataalone. We therefore see the relative importance of thesetwo effects as being an unresolved question, but conclude

that incorporation experiments designed to measuretreadmilling rates must be interpreted with great caution,given the stochastic nature of the pulse and chaseprocesses. The magnitude of treadmilling rates inferredfrom electron microscopy of both microtubule proteinand tubulin microtubules, e.g. 0-7-3 / imh" ' (Bergen &Borisy, 1980; Rothwell et al. 1985; Hotani & Horio,1988) would correspond to a differential addition of aboutone subunit per end per second. Such treadmilling couldbe a significant mechanism of exchange only whendynamics are strongly suppressed.

A further complication in studying microtubule dy-namics is that rapid changes in microtubule numberconcentration, m, may be produced by the process ofannealing (Rothwell et al. 1986). This process is gener-ally found to be of importance under conditions where inis high, and following the generation of very shortmicrotubules formed by extensive shearing. In order tominimize such effects, we use relatively low tubulinconcentrations, and allow a period of stabilization follow-ing shearing (cf. Mitchison & Kirschner, 1984a), so thatin the exchange experiments reported here, the isotope isadded only when the original polymer mass has been re-established. This addition point is the time-origin forlength measurements, and this initial population ofmicrotubules has a mean length of approximately 5 /im.We have shown that the incorporation kinetics are wellfitted to the mechanism simulated for microtubulesundergoing dynamic length changes over the ranges ofmean length of 5-19 im (0-90 min) and 19-30 fim(90-150 min). We therefore do not believe that annealingmakes a major contribution to length redistributionunder these (steady-state) conditions.

The behaviour of steady-state microtubules is gov-erned by the rate of growth (addition of Tu-GTP) andshrinkage (loss of Tu-GDP) and the proportions of G-and S-state microtubules. Therefore, the questions ofcentral importance are: what factors determine whether aparticular microtubule grows or shrinks, and what con-trols the rate of the transitions between these states, andhence determines the distribution of microtubules be-tween the two states. The steady-state formulation ofScheme I successfully treats the behaviour of the systemat a constant concentration of free T u - G T P (Cc), inwhich interconversions occur between growing andshrinking microtubules. With the additional assumptionthat the transitions between G and S are highly co-operative processes, it also illustrates the dynamic behav-iour under extremes of non-steady-state conditions,where, principally under the influence of [Tu-GTP] , thefrequency and extent of synchrony of the switchingprocesses determines the properties of the system. Thistreatment shows the role of dynamic instability in theinitiation of oscillations: likewise, the slowness of T u -GTP regeneration, for which several potential mechan-isms exist, has a marked effect on oscillation frequency.Also the initial amplitude of oscillation is seen to bestrongly sensitive to the presence of Tu-GDP.

Detailed modelling of microtubule dynamicsA detailed model for microtubule dynamics must provide

A simple formulation of microtubule dynatnics 251

a mechanism for the phenomena of apparent cooperat-ivity in k$o and kcs ar>d the possibility of stationarystates, and must account for the distinctive properties ofthe two ends of the dynamic microtubule. A majorquestion is the specification of the nature of the micro-tubule end in terms of the appropriate tubulin-nucleotidespecies. The principle of a substantial 'cap' of T u - G T Phas been proposed by Carlier & Pantaloni (1981), andextensively modelled (Hill, 1984; Chen & Hill, 1987a,fc).Evidence for the cap depends principally upon indirectevidence of phosphate determination following hydroly-sis; attempts more directly to identify and measure such acap have been unsuccessful (O'Brien et al. 1987; Schil-stra ef a/. 1987; Stewart et al. 1989).

We have derived an alternative treatment of dynamicmicrotubules, which we term the 'lateral-cap' model,(Bayley et al. 1988; Bayley & Martin, 1989). Thequantitative aspects of this model will be presented indetail elsewhere. It is sufficient to say that a short (singlelayer) cap has the advantage of providing a relativelysimple formulation of the transition behaviour amenableto computer simulation, which demonstrates a pro-nounced dependence on the tubulin dimer concentrationas postulated above (Walker et al. 1988). Thus, such amodel is able to simulate all the steady-state propertiesdiscussed in this paper and illustrates possible mechan-isms for microtubule transitions under steady-state andnon-steady-state conditions.

Cellular implicationsThe recent direct observations of dynamic instability incellular systems (Sammak & Borisy, 1988; Schulze &Kirschner, 1988; Cassimeris et al. 1988) indicates thegenerality and potential function of microtubule dy-namics in vivo. The formulation as given here permitssome comment on the behaviour of microtubules in thecytoplasmic environment. Microtubule dynamics areclearly strongly influenced by variations in concentrationsof T u - G T P and GTP. Decreases in either will lead tomicrotubule disassembly, whereas at steady state itappears quite possible for two microtubule ends, adjacentin space and sampling the same local concentrations, toshow opposing shrinking or growing behaviour on astochastic basis. Major increase in ionic concentration,affecting the shrinking rate constant k-, appears likely tocause macroscopic disassembly; effects of specific agentssuch as mitotic poisons may, by contrast, be able toinhibit dynamics, while causing relatively little disassem-bly.

Microtubule dynamics as observed in vitro appear tobe largely explicable in terms of the principles of dynamicinstability, as formulated here. More extensive variationsin microtubule behaviour in vivo may result from macro-scopic changes in tubulin pools, GTP/GDP ratios,capping proteins and number of active microtubulenucleating sites. It is an intriguing possibility that thedistribution of microtubule arrays could be changedrapidly through the exercise of a relatively small numberof endogenous cytoplasmic factors such as kinases andphosphatases. These could increase or decrease thenumber of active microtubule nucleating centres, and,

coupled to the intrinsic dynamic instability of the micro-tubule population, could determine the subsequent tem-poral and hence spatial distribution of microtubules in acontrolled manner.

Thus our results indicate that the basic physicalproperties of the microtubule provide a number ofpotential control mechanisms that can regulate the basicmachinery of microtubule dynamics. Superimposed onthese will be the effects of biochemical modifications, forexample, post-translational acetylation and tyrosinyla-tion, and phosphorylation-dephosphorylation, whichcan exert additional regulation of overall dynamic proper-ties and, which may, potentially, provide fine tuning ofthe basic switching mechanisms.

This work was supported in part by E.E.C. Twinning grant(852 00255 UK 05 PU JU1) and N.A.T.O. CollaborativeResearch grant (RG 86/0499). We thank Miss Felicity Butlerfor length measurements, and Dr Yves Engelborghs, Dr LeslieWilson, and Dr Vera Gal for useful comments. We thank DrsEngelborghs, Erickson and E. Mandelkow for access to manu-scripts prior to publication.

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(Received 17 January 1989 - Accepted 22 February 1989)

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