Microtubule dynamic instability: numerical simulation of … · 2005-08-26 · Microtubule dynamic instability: numerical simulation of microtubule transition properties using a Lateral

Microtubule dynamic instability: numerical simulation of microtubule

transition properties using a Lateral Cap model

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

Division of Physical Biochemistry, National Institute for Medical Research, The Ridgeway, Mill Hill, London NW7 IAA, England

* Author for correspondence

Summary

We present a numerical formulation for the dy-namic instability of microtubules involving thestabilisation of growing microtubules by a singlelayer of tubulin-GTP, with GTP hydrolysis effec-tively coupled to tubulin-GTP addition. This Lat-eral Cap model provides a readily visualised, work-ing mechanism for the co-existence and inter-conversion of growing and shrinking microtubules.

This class of model is specified in terms of ahydrolysis rule, whereby the addition of tubulin-GTP causes hydrolysis of GTP on a previouslyterminal tubulin-GTP molecule as it becomes incor-porated into the microtubule lattice. A specificformulation is illustrated, though this is not unique.A limited set of parameters defines the kinetics andaffinity for tubulin-GTP at the binding sites at agiven end of the microtubule. The rate constants area function of the nucleotide composition of thebinding site, principally comprising the two tubulinmolecules, which interact laterally and longitudi-nally with the incoming tubulin-GTP molecule.

The Lateral Cap formulation demonstrates that a

single terminal layer of tubulin-GTP is sufficient toreproduce the apparently complex behaviour of adynamic population of microtubules. It differs sig-nificantly from the fluctuating tubulin-GTP capmodel of Chen and Hill (1985). It gives a moleculardescription to the switching of individual micro-tubules between growing and shrinking states interms of the composition of the multi-start terminallayer of the microtubule, and provides a generalmechanism for the differential kinetic behaviour atopposite ends of dynamic microtubules. It repro-duces the essential features of microtubule lengthexcursions, and predicts detailed characteristics ofmicrotubule dynamics, including the basis of theapparently cooperative nature of the transition be-haviour as a function of the concentration of tubu-lin-GTP. It is readily amenable to further exper-imental test and refinement.

Key words: tubulin, kinetics, Monte-Carlo methods.

Introduction

Microtubules are formed by the assembly or polymerisa-tion of tubulin-GTP (T-GTP*). During assembly, tubu-lin-GTP is hydrolysed, apparently irreversibly, to tubu-lin-GDP. The polymer is a dynamic assembly, sustainedat a steady state of polymer mass by excess GTP; when

the GTP is used up, the microtubule (T-GDP polymer)disassembles. Recent observations have shown that thesteady state of tubulin microtubules is characterised bythe unusual property of dynamic instability (Mitchisonand Kirschner, 1984a,b). The population of micro-tubules is composed of growing and shrinking micro-tubules; a given microtubule interconverts relatively

* Abbreviations and definitions: T-GTP, tubulin-GTP, is tubulin with GTP at the exchangeable nucleotide binding site; [T-GTP], concentrationof tubulin-GTP; Cc, critical or steady state concentration of tubulin-GTP; m, microtubule number concentration.G,S, growing or shrinking state of an individual microtubule; To, mean lifetime of the growing state; Tg, mean lifetime of the shrinking state; RQ,growing rate of an individual microtubule; R$, shrinking rate of an individual microtubule.{R}, bulk growth rate for microtubules; { }, indicates an average over the population. {/?}>0 assembly; {R}<0 disassembly. ({/?} = 1 /immin"1

corresponds to 27.1 eventss"1 per microtubule end for a microtubule of 13 protofilaments with tubulin repeat 8nm.)So, standard set of rate constants for simulation; Si to Ss, modified sets of rate constants.b,c,d,e, microtubule lattice positions, adjacent to binding site X; bd, bde, designation of microtubule binding site in terms of composition ofpositions b,d,e (e.g. TTD,TDV,DD* etc; T=(tubulin-GTP), D = (tubulin-GDP), V=(vacant); * designates T,D or V).n, number of association (or dissociation) sites in a given microtubule end configuration; (ti) average value of n (=3.8235).Pniax. maximum number of T-GTP in the lateral cap.k(+T); association rate constant for addition of tubulin-GTP; k(—T),- dissociation rate constant for loss of tubulin-GTP; k( — D),- dissociation rateconstant for loss of tubulin-GDP, where subscript (i) designates a given binding site (bde).

Journal of Cell Science 95, 33-48 (1990)Printed in Great Britain © The Company of Biologists Limited 1990 33

infrequently between these states. These properties havebeen well characterised in vitro (Cassimeris et al. 1987;Farrell et al. 1987; Horio and Hotani, 1986; Mitchisonand Kirschner, 1984a,6; Walker et al. 1988), recentlyreviewed (Engelborghs, 1989), and also observed in vivo(Cassimeris et al. 1988; Sammak and Borisy, 1988;Schulze and Kirschner, 1988).

Dynamic instability may have a functional role incontrolling the disposition both in time and space of themicrotubule arrays at different times of the cell cycle(Kirschner and Mitchison, 1986a). The fast disassemblyprovides a means for the rapid re-organisation of themicrotubules in response to changing cellular require-ments as well as having a potential role in the establish-ment of the correct spatial distribution of specific micro-tubules. It is therefore of great interest to examinepossible mechanisms for these dynamic properties. Ouraim is to clarify the fundamental principles giving rise tothis behaviour, and to evaluate the capacity for control ofthese properties via the basic physical mechanisms ofmacromolecular self assembly.

The original concept of dynamic instability was de-rived from observations of length changes of populationsof microtubules in vitro (Mitchison and Kirschner,1984a,6) and was largely substantiated by the directobservations of individual microtubules using dark-fieldmicroscopy (Horio and Hotani, 1986). The implicationsof these properties have recently been discussed in termsof a simple formulation of dynamic instability (Bayleyet al. 19896). More extensive kinetic data on the growthof individual microtubules on axonemal seeds, in vitro(Cassimeris et al. 1987; Walker et al. 1988), haveconfirmed the basic principles of the steady growth andshrinking of microtubules together with an infrequentlyobserved 'pause' state (Sammak and Borisy, 1988; Walkeret al. 1988) and has also shown that the interconversionbetween growing and shrinking is dependent upon theT-GTP concentration in a complex way (Walker et al.1988).

A physical model for the dynamic behaviour of micro-tubules should ideally account quantitatively and consist-ently for the following properties: (1) the co-existenceand relative proportions of growing and shrinking micro-tubules, designated G and S, respectively; (2) the rates ofinterconversion (or 'switching') between states G and Sof an individual microtubule; (3) the dependence of theswitching frequencies on the concentration of T-GTP(Walker et al. 1988); (4) the possible origins of a 'pause'state (Sammak and Borisy, 1988; Walker <?f al. 1988); (5)the expected effect of solution variables such as Mg2+

(Gal et al. 1988; O'Brien et al. 1989), temperature(Caplow et al. 1988a; Walker and Salmon, 1988) etc. ondynamic behaviour; (6) differential activity at the twoends of the polar microtubule lattice (Cassimeris et al.1987; Sammak and Borisy, 1988; Walker et al. 1988).

The fundamental principle of microtubule assemblyunder a wide range of experimental conditions in vitro isthat T-GTP assembles whereas T-GDP does not. Theaddition of T-GTP to the growing end of a microtubuleresults in stoichiometric hydrolysis of one mole of GTPper mole of T-GTP added (Carlier, 1982). The key

question is then the relationship between microtubuleelongation (i.e. the addition of T-GTP) and GTPhydrolysis. Data were originally presented suggestingthat there was a significant uncoupling between the twoprocesses (Carlier and Pantaloni, 1981), so, during therapid elongation phase of assembly, when the associationrate exceeded the assumed hydrolysis rate, the growingmicrotubule would contain a substantial proportion ofnon-hydrolysed GTP (Carlier etal. 1984). This principlewas adopted in the modelling of Hill and co-workers(Chen and Hill, 1985, 1987; Hill, 1984), and has becomewidely known as the GTP-Cap model. Random fluctu-ations in the size of this cap from several hundred tubulinmolecules down to zero were implicated as responsiblefor the conversion from a stable, growing microtubuleinto a rapidly shrinking one (Chen and Hill, 1985).

Experimental support for a substantial cap resultingfrom uncoupled hydrolysis depends principally uponindirect evidence of phosphate determination duringmicrotubule assembly (Carlier and Pantaloni, 1981; Car-lier et al. 1987a). More-direct attempts to identify andmeasure such a cap have been unsuccessful (O'Brien etal. 1987; Schilstra et al. 1987; Stewart et al. 1989).Under a range of experimental solution conditions, theextent of GTP hydrolysis, assessed from the nucleotidecomposition of assembling microtubules, has been foundto proceed in parallel with the assembly of tubulin dimer.These results are inconsistent with the concept of un-coupled or vectorial hydrolysis. They strongly suggestthat GTP hydrolysis is closely coupled to the addition ofT-GTP to the microtubule end, both in the elongationphase and at steady state. Hence the validity of theuncoupling of GTP hydrolysis is seriously questioned asa general mechanism in microtubule assembly anddynamics.

These considerations have led us to re-examine theprinciples of the modelling of microtubule assemblyaccording to the Hill formulation (Chen and Hill, 1985;Hill, 1984; Hill and Chen, 1984) and to examine theconsequences of a fully coupled mechanism linkingT-GTP addition and GTP hydrolysis. We take accountof the more recent quantitative evidence obtained fromthe direct observation of individual microtubules; ofmajor importance is the observation of the dissociation ofT-GTP during elongation, which led to the idea of'stochastic dissociation, coupled hydrolysis' (Walker et al.1988). The switching from S to G is known as 'rescue'and that from G to S as 'catastrophe' (Kirschner andMitchison, 19866; Williams et al. 1986). We will not usethe latter term, since it derives from the concept that,once into the S state, disassembly is irreversible. This isnot generally found either in vitro or in vivo (Cassimeriset al. 1988; Sammak and Borisy, 1988; Schulze andKirschner, 1988).

We therefore seek to establish the minimum terminalstructure necessary to develop the observed dynamiceffects, together with the simplest practical kineticscheme. The basic principle is that the addition of asingle molecule of T-GTP causes the hydrolysis of asingle previously terminal T-GTP (hence retaining thestoichiometry of hydrolysis), which thereby becomes

34 P. M. Bayley et al.

buried in the lattice. We (Bayley and Martin, 1989;Bayley et al. 1988; Bayley et al. 1989c), and others(Walker et al. 1988; O'Brien et al. 1987) have concludedthat a cap composed of a single terminal layer of T-GTPis probably sufficient to stabilise the steady-state micro-tubule, but yet to permit the transitions between growingand shrinking behaviour closely resembling the exper-imentally observed dynamic properties.

We now evaluate the detailed properties of one 'LateralCap' model. An important feature of this formulation (inwhich the microtubule contains several sites forelongation within the lateral cap) is that the mean lifetimefor either growing or shrinking microtubules is predictedto be a strong function of [T-GTP]. We comparesimulated results with the available experimental data forgrowth of individual microtubules and oscillations ofmicrotubule populations. The model accommodates theknown microtubule polarity (Amos, 1979), and thusaccounts, in principle, for the differential properties ofthe two microtubule ends. This study suggests criticalrequirements in observations on individual microtubulesand also suggests several areas of interest for furtherinvestigation. In particular, the effects of anti-mitoticdrugs, such as podophyllotoxin, which appear to act sub-stoichiometrically in suppressing microtubule dynamics(Schilstra et al. 1989), provide further data for testingand refining the formulation.

The model

We consider the dynamic processes in terms of theaddition of tubulin a'/J-heterodimer to the idealised 'five-start' helical microtubule lattice (Amos, 1979), as shownin Fig. 1. The minimum requirement for the binding siteX is the presence of molecules b, c and d (cf. Chen andHill, 1985). With this definition the configuration of themicrotubule end shown in Fig. 1 has« = 3, i.e. three sitesfor binding and three for dissociation. The averagenumber of sites for binding (during growth) or fordissociation (during shrinkage), (n) equals 3.8235 (Chenand Hill, 1985). The binding step involves interactionwith at least one laterally and one longitudinally relatedtubulin molecule.

We define rate constants (and hence affinities) princi-pally in terms of the nature of the residues at positions b(lateral) and d (longitudinal relative to X), depending onwhether the tubulin at these positions carries GTP orGDP at the exchangeable nucleotide binding site. Thuswe define the bimolecular rate constant for addition ofT-GTP as k(+T)bd,

t n e rate constant for dissociation ofT-GTP as k( — T)bd and the rate constant for dissociationof T-GDP as k(-D)bd, where bd = DD, DT, TD or TT,(where T=tubulin-GTP and D=tubulin-GDP in sites band d as above). These parameters are given specificvalues, which reflect the greater affinity and associationrate for T-GTP when the binding site is richer inT-GTP, and conversely, a lower affinity, and greaterdissociation rate, when the site is richer in T-GDP. Therate constants typically are in the order:

k(+T)rr>k(+T)ro,H+T)DT>k(+T)DD

f /

10 111213

a

Fig. 1. Schematic drawing of the arrangement of moleculesat the end of a 13-protofilament microtubule, theprotofilaments are numbered 1 through 13. Growth is on thefive-start helical lattice, with the growing helices designated Athrough E (thus F is contiguous with A, G with B etc.).Each box represents the alignment of the tubulin aj3-heterodimer in the lattice, with the /3-subunit oriented(arbitrarily) to the top. At the /3-end an incoming T-GTPmolecule binds at X and the five nearest neighbours for thisbinding site are designated a, b, c, d and e. At the ff-end theincoming T-GTP binds at X' and the nearest neighbours aredesignated a', b ' , c', d', and e'. The small arrows indicatesites for association and dissociation.

Table 1. Rate constants for different binding siteconfigurations: standard set, So

Site 10-6x/;(+T)bd *(-D) b d * ( - T ) M

bd (M 's ') (s ') (s ')

DDDT,T T

TD011

.33

.00

.25

111.66 66.668.331.25

Notes: T=T-GTP; D=T-GDP; and V=vacant. The siteconfigurations, DTD and TTD, do not exist. &(+T)bd is thebimolecular association rate constant for T-GTP at a site withconfiguration bd. £( -T) b d is the dissociation rate constant for T-GTP at a site with configuration bd. A(-D)bd is the dissociation rateconstant for T-GDP at a site with configuration bd. Occupancy ofsite e (by T or D) gives a twofold increase in k(+T) values, and atwofold decrease in k(—T),k(—D) values, i.e.A(+T)bde=2xA(+T)bd, etc.

*The event cannot occur.

A ( - T ) T T < * ( - T ) T D I * ( - T ) D T < * ( - T ) D D < * ( - D ) D D

For simplicity, the situations where bd=TD andbd=DT are taken as equivalent, so that the set iscomposed basically of seven rate constants for a givenend: see Table 1 for the standard set (So). In addition,sites a and e can further modulate the properties. Forexample, occupancy of site e is modelled as a second-order effect, so that when the site is occupied (by either

Dynamic instability of microtubules 35

T-GTP or T-GDP) the value of k( + T) is doubled andthe values of k( —T) and k( — D) are halved relative to thevalues that apply when it is vacant (V). The rateconstants are then defined in terms of a site withcomposition bde. We have also investigated the kineticbehaviour for eight modified rate constant sets (Si to Ss).

The 'hydrolysis rule' is specified so that the addition ofa T-GTP molecule at position X (Fig. 1) causes thehydrolysis of what was previously a terminal residue insite c, and creates a new configuration of sites for furtherassociation and dissociation. This mechanism involvestight coupling in time between the addition of T-GTPand the hydrolysis of the previously terminal T-GTP toT-GDP and maintains unit stoichiometry. Spatially, itrestricts the T-GTP molecules to the terminal positionsof the lattice, irrespective of whether this occurs duringnet polymerisation (i.e. 'assembly') or at steady-state.This principle limits the number of T-GTP molecules inthe terminal layer to a maximum of pm3*=l8 duringoptimal growth. This number is derived from 13 laterallyand (maximally) 5 longitudinally related terminal mol-ecules, irrespective of the end configuration (seeFig. 2,1).

The concept of the Lateral Cap means that thecomputation is simplified to consideration of associationand dissociation events at a limited and well-definednumber of sites on the lattice, whose properties arerepresented by a small set of rate constants. Even so, thebehaviour in time of an individual microtubule end has alarge number of possibilities, owing to the nature of thelattice, and Monte-Carlo methods are used to deal withthis multiplicity (cf. Chen and Hill, 1985).

The simulation procedure

The behaviour of a single microtubule is simulated over along period of time, thereby allowing it to exploregrowth, shrinking and the transitions between the twobasic states. The intrinsic step is one of addition ordissociation of a tubulin subunit, allowing for the multi-plicity of binding sites («), and their continuouslychanging configuration (geometric) and composition (Tversus D). The full sequence of association (and dis-sociation) events is followed, and the length of anindividual microtubule monitored as a function of time.Details of the end configuration are noted continuouslyfor further analysis. The net growth of a single micro-tubule over a long time parallels the net behaviour of amicrotubule population, and is used to calculate the bulkgrowth rate {R} (positive for net assembly, negative fordisassembly) at a given (fixed) value of [T-GTP].(Brackets { } indicate an average over the population.)This condition is equivalent to the experimentally ob-served growth of microtubules from a very low concen-tration of seeds or nuclei, i.e. w ^ [ T - G T P ] , and shouldbe distinguished from a conventional microtubule as-sembly curve, where the free [T-GTP] decreases pro-gressively until Cc is reached.

The Monte-Carlo method is used to determine, on a

probabilistic basis, which of a number of differentpossible alternative steps is the first to occur. For each ofa total oij sites (for association and dissociation) the timetj is calculated at which the event would occur fromequation (1):

(1)tj.kj — —1 n(1 — pj),

where the value of kj is simply the rate constant k(—T) ork(-D) for the dissociation sites, or k(+T).[T-GTP] forthe (bimolecular) association step, with values taken froma set such as Table 1 according to the nature of positionsb and d (and e) in each case; pj is a random number in therange 0 to 1. The physical event occurs at the site whichhas the smallest value of tj {tmm). The elapsed time isextended by tmm, and the new configuration (plus orminus one molecule depending on the step taken) is thenused, with its appropriately modified kinetic properties,as the origin of the next step. This shows how the balanceof reactions changes, on a stochastic basis, as a function ofchanges in [T-GTP], increases in which naturally favourassociation reactions.

Thus for a given set of rate constants, the onlyadditional parameter is the concentration of free T-GTP.Simulations are performed for a range of values of [T-GTP] in order to examine the dependence of the netgrowth rate (molecules of tubulin per microtubule endper unit time) versus [T-GTP]. This defines the netassembly region ({R}>0 at [T-GTP] >CC), the netdisassembly region ({R}<0 at [T-GTP]<CC, and thecritical concentration, [T-GTP] =CC, where the netgrowth {R} is zero.

To investigate the transition behaviour, the timecourse of a single microtubule at a given value of [T-GTP] was analysed over an extended time-period to givevalues of growing and shrinking state lifetimes, calculatedby evaluating the time and extent of each growing andshrinking excursion. An excursion is defined as involvinga net length change of greater than 0.15|tim correspond-ing to the best limits achievable by direct microscopicobservation (Walker et at. 1988).

It became apparent that the model showed an interest-ing dependence of the individual growth and shrinkagerates for a microtubule as a function of the externalvariable, [T-GTP]. The number of events per secondduring growth, i.e. the individual growth rate, RQ, andthe nature of the site at which these events occurred weretherefore monitored continuously, by examining thedetails of the excursions occurring purely in growthphase; similarly, information on the individual shrinkagerate, Ra,, was monitored for excursions during theshrinking phase. These data were obtained for differentvalues of [T-GTP], to examine how the variations inindividual growth and shrinkage rates could be related tochanges in the composition of the microtubule end as afunction of [T-GTP].

Finally, the behaviour of the model was investigatedfor selected variants of the basic kinetic parameter set(Table 2), to determine the quantitative relationshipbetween specific rate constants and the observable exper-imental parameters such as bulk and individual growthrates, and lifetimes of growing and shrinking states.


Table 2. Modified rate constants: sets S, to S$

Set

s,s2S3

S4

s5SoS7

s8

• Indicates anyDDT, where V=

Rate constantmodified

* ( - T ) T I -&(+T)DD-* ( - T ) D T -* ( - T ) T D -* ( + T ) D D .* ( -T) T T .*(-D)D D.* ( -T)DD«k(+T)TT.* ( + T ) T D «

* ( + T ) D T -

Factor

X3.0XO.SxO.5xO.SxO.5X3.0X2.0xO.5X2.0XO.SxO.5

occupancy, i.e. DD* represents DDV, DDD andvacant.

A qualitative illustration of microtubule switching

It is informative to consider a hypothetical sequence ofevents involved in the transitions of a microtubule end.Fig. 2 shows a continuous series of dissociation events,chosen for illustrative purposes. Normally, any suchsequence would comprise a number of association anddissociation events, interspersed at random. The initialconfiguration (Fig. 2,1) has three sites (w = 3), all of themwith bde=TTT. Loss of T from E7 via k(-T)TTT

(Fig. 2,2) gives ;/=4 (2XTTT, 2XTTV). Subsequentevents are detailed in Table 3: this shows how thenumber of sites (ti) varies continuously, as does thecomposition of the sites in terms of their T or D content.

The mixed configurations with terminal T and D occurthroughout the transition. That of Fig. 2,10 occurs insimulations, as an apparent intermediate state betweenshrinking and growing: when the configuration isreduced to n = \, a state of apparent 'inactivity' or 'pause'is often seen. Rescue could occur by addition of T at F4via 6( + T ) D D D (Fig. 2,11) giving «=2, (TDD, DTV),and subsequent association reactions at E4 and F5.Alternatively, the dissociation could continue (fromFig. 2.10) by dissociation of T from B3 and C3, anddissociation of D from D3, E3, and F3 (Fig. 2,10 to2,13). Loss of the final layer of T from B2, Bl, and G13to G7 via appropriate k( — T) processes would give theconfiguration (Fig. 2,14), totally lacking T-GTP fromwhich the dissociation of T-GDP would proceed rapidlyvia £ ( - D ) D D V , with the number of sites (e.g. «=4,Fig. 2,15), fluctuating about the mean (n)=3.8235.

Partial steps in rescue from Fig. 2,15 would occur asshown with the addition of T via & ( + T ) D D V at E2 andH l l , and via k( + T)DDD at G3 and 112 (Fig. 2,16), thusslowing down the dissociation reaction. Rescue of the endwould only be complete when association reactions hadpredominated as a result of the random processes, and afull cap was restored. The rescue reaction could proceed,e.g. as from Fig. 2,11 (plus E4 and F5) (Fig. 2,17) givingn = 3; (DTV, TTV, TDT) , with T adding at D4, via£ ( + T ) D T V >

a t C4 and B4 via A ( + T ) T T V g ' v m g(Fig. 2,18), M = 2 (TTV, TDT) . Addition of T at E5 via£(+T) T T V and at F6 via &( + T ) T D T would give(Fig. 2,19) « = 3; (2XTTV, TTT) . Addition of T at E6and F7 would then give a complete terminal layer ofT-GTP, with « = 3 (Fig. 2,20).

Table 3. Typical sequence of events involved in the transition of a microtubule end, as illustrated in Fig. 2,starting with n=3 sites (all TTT: Fig. 2,1)

Loss ofTTTDTTTDTD

Addition ofT

TT

TTT

TT

TT

Position

E7E6F7F6CSC4D5D4E5E4FSF4

F4

F5E4

D4C4B4

ESF6

E6F7

Rateconstant

A ( - T ) T T V

* ( - T ) T D T£ ( - T ) D T T*(-D)D D T* ( - T ) T T V* ( - T ) T D T&(-T)DTv* ( - D ) D D T*(~T)DDV

*(-D)D D D

*(+T)DDD

*(+T)TDD£ ( + T ) D T T

*(+T)DTV*( + T)TTV*(+T)TTV

*(+T)TTV*(+T)TDT

*(+T)TTV*(+T)TTT

See

Fig- 2,2Fig. 2,3Fig. 2,4Not shownFig. 2,5Fig. 2,6Fig. 2,7Fig. 2,8Fig. 2,9

Fig. 2,10

Fig. 2,11

Fig. 2,17

Fig. 2,18

Fig. 2,19

Fig. 2,20

n

433232323

1

2

3

2

3

3

Result

New sites

(2XTTT, 2XTTV)(2XTTT, TDT)(TTT, TDV, DTT)(TTT, DDT)(2XTTV, DDT)(TDT, DDT)(TDV, DTV, DDT)(2XDDT)(2XDDV, DDT)

(DDD)

(DTV, TDD)

(TTV, DTV, TDT)

(TTV, TDT)

(TTT, 2XTTV)

(2XTTT, TTV)


13

17

14

18

is

15

19

8

12

-i 2

20

i

Fig. 2. Illustrative sequence of individual events at a microtubule end. The idealised five-start helix (see Fig. 1) is shown withT-GTP represented by # and T-GDP by *. 1-15. A sequence of dissociation events; 16, initial steps in the rescue process; 17,an alternative path from 11 leading to a full Lateral Cap at 20. See text for comments and Table 3 for full details.


This progression illustrates some interesting qualitat-ive aspects of how apparent cooperativity in the tran-sitions could result from the single-layer configurations.There is clearly a kinetic barrier to 'breaking into' theintact surface layer: the initial step, dissociation of T viak{—T)TT* (where * indicates T, D or V, vacant), is theleast favourable dissociation step for T-GTP, and thisdissociation generally increases the potential number ofsites. Only when TT* sites have become less dense, andthe predominant form is TD* or DT*, with consequentlessening of the elongation rate via 6( + T)Trj* or£(+T)rjT«, is there a marked shift towards theT-depleted end configurations. Configurations withmixed T and D residues in terminal positions arepopulated in the course of the transitions in both direc-tions G to S and S to G. The (mixed) configuration withn = \ with a DDD site (Fig. 2,10) represents a partiallystabilised state where both growth (T onto DDD) anddissociation (T off TTV) are relatively unfavourable.

Conversely, the initial step in 'rescue' is essentiallyunfavourable (Fig. 2,16); subsequent steps in com-pleting the cap (Figs 2,17 to 2,20) are progressively morefavourable. Another potentially cooperative feature isthat the G to S transition requires a succession of stepsinvolving dissociation of T from TD*, involving thelateral stripping away of a given terminal layer, and alsothe sequential dissociation of a longitudinal series of Tfrom DT*. Obviously, either of these processes could beemphasised at the expense of the other by appropriatechoice of k( — T)rD* and k(—T)DT.. To minimise thenumber of parameters, we have equated these two rates.An alternative way of differentiating them would be toinclude differential effects of sites a and e (as definedabove), which could emphasise either lateral or longitudi-nal interactions selectively. The effect of occupancy ofsite e, used here (Table 1) to increase the affinity forT-GTP when site e is occupied, gives a slight weightingin favour of dissociations that increase n and associationsthat conserve n.

The only perfectly defined states are G, with a fulllateral T-GTP cap, and S, with zero T-GTP cap. Themixed states (cf. Fig. 2) do not by themselves signify theoccurrence of a transition. Depending upon the value of[T-GTP], the microtubule end can in general entermixed configurations from G (or S) without necessarilycompleting the transition G to S (or S to G, respect-ively). Addition of T and dissociation of both T and Dwill be occurring during what would appear (e.g. in videomicroscopy) as continuous microtubule growth or shrink-age; mixed states will then cause significant perturbationof individual growth and shrinking rates. This is con-sidered quantitatively below.

The progression of events typified in Fig. 2 can readilybe realised as a computer graphic presentation, showingthe end composition and the length behaviour as afunction of time. This is easily implemented on a PC(XT, AT, 286, 386 versions) and has been filmed fordemonstration purposes. We will now show that evenwith the imposed simplifications to ensure the minimumnumber of parameters, the model is able to simulate

Fig. 3. Simulated length changes in a microtubule as afunction of [T-GTP] for rate constant set Si (see Tables 1and 2). T-GTP concentrations are: A, 7.8 ^M; B, 6.85 /AM; C,9.2(IM; D, Ofiu; and E, 25 //M. CC for this rate constant set is7.8 m-

dynamic aspects of the experimental microtubule proper-ties with remarkably good quantitative agreement.

Results: quantitative predictions of transitionproperties of dynamic microtubules using theLateral Cap model

Microtubule assembly rate as a function of [T-GTP]As described above, the assembly is assessed from thebehaviour of a single microtubule, simulated for a givenset of rate constants, at a range of values of [T-GTP].Fig. 3 shows the simulated time course of the length of anindividual microtubule at different [T-GTP] values. Thebehaviour ranges from exclusive growth at [T-GTP] >CC

through extensive transition behaviour at [T-GTP] closeto Cc to exclusive shrinkage at [T-GTP] =0. This figureillustrates the relatively narrow concentration range avail-able for experimental determination of transition behav-iour, within practical time-limits. It also shows thattransitions from G to S state occur even under net growthconditions at [T-GTP] >CC, and that transitions from Sto G state occur even under net shrinking conditions at[T-GTP]<CC. The simulation also illustrates that therates of both growing and shrinking phases are not strictlyconstant, but show fine variation within a given excur-sion. This is discussed further below.

Simulation was continued until the net growth rate,i.e. total length change/total time, was constant to 5 %.In the vicinity of the Cc, where the behaviour is domi-nated by transitions between G and S states, the simu-lation has to be continued for longer times. This rate {R}(positive=assembly; negative=disassembly) corre-sponds to the experimental J(c) plot (Hill, 1984). Fig. 4shows that the characteristic experimental behaviour ofdynamic microtubules is well reproduced. At high [T-GTP] a constant slope is seen; the Cc is defined as the


concentration giving zero net growth; at [T-GTP]<CC,net disassembly occurs at elevated rates. The linearity at[T-GTP] >CC suggests a back extrapolation may bemade to [T-GTP] =0 to give an apparent dissociation rateconstant, &'_, characteristic of the dissociation ofT-GTP from a steadily elongating microtubule accordingto a relationship resembling the behaviour of an equilib-rium polymer, namely equation (2):

{R} =k\. [T-GTP] -k'. (2)

However, unlike a true equilibrium polymer (Oosawaand Asakura, 1975) the true Cc value is not given by theratio k'_/k' + .

Fig. 4A shows the net growth rate of a population ofmicrotubules at constant microtubule number concen-tration, m, for the standard rate constant set (So) and forthe modified sets Ss and S7. Doubling the values of£ ( - D ) D D . (set S5) doubles the value of the net rate at [T-GTP]=0 compared with set So, as expected. However,the effect on Cc is surprisingly small, increasing only from5.9 to 6AfiM (Table 4). Also there is no effect on the netrate at high values of [T-GTP], since dissociation ofT-GDP via A( —D)DD« is a rare event at high concen-trations of T-GTP. Doubling the value of * (+T) T T * (setS7) has a markedly different effect. The net rate at [T-

10-

-12

T-GTP concentration ((J.M)

Fig. 4. Net growth rate simulated for a population ofmicrotubules as a function of T-GTP concentration. A. Rateconstant sets So (standard), S5 and S7. B. Rate constant setsSo (standard), Si, S3 and S4. See Tables 1 and 2 fordefinition of rate constants.

GTP]=0 is the same as for the standard set, the value ofCc is slightly reduced (from 5.9 to 4.6 jitM) and the net rateat high [T-GTP] is greatly increased. This is becauseaddition of T to TT* sites is the predominant processunder these conditions (see below). All three sets give thesame (small negative) intercept (&'_) (not shown) fromextrapolation to [T-GTP] =0 of the data at high values of[T-GTP]; this will be discussed further below.

Fig. 4B shows changes in net rate for the standard setSo and three other modified sets (Si, S3 and S4). Severalfeatures may be noted here. (1) All four plots tend to thesame limiting disassembly rate at [T-GTP] =0, charac-teristic of the values of k( — D)DD*> which are the same foreach set. (2) Each curve has the same limiting slope athigh [T-GTP], since the rate of elongation at high [T-GTP] is controlled by the value of &(+T)TT». (3) Theintermediate behaviour shows that the Cc value is sensi-tive to the values of rates and affinities for T-GTP atdifferent sites, and varies qualitatively as expected. Adecreased dissociation rate for T-GTP decreases Cc, anda decreased association rate increases Cc. However, in allcases, the magnitude of the effect is less than thenumerical change in the rate constant, indicating that theobserved behaviour is the result of multiple opposingeffects. It is thus unreliable to deduce information aboutthe magnitude of rate constants for T-GTP associationand dissociation simply from experimental Cc values. (4)Extrapolation of the linear high concentration region ofthe plot to [T-GTP] =0 yields one value for curves S3 andSo, and a different (more negative) value for curves Sjand S4. Because this intercept is simply a summation ofdissociation events occurring during sustained growth,we can see (Table 2) that increasing k( — T ) T T * (sets Siand S4) increases the number of such dissociation events,as expected. In contrast, decreasing the values ofk(—T)DT* and k( — T ) T D * does not reduce the number ofsuch events. This will be discussed in more detail below.

Lifetimes of the growing and shrinking states as afunction of [T-GTP]Lifetimes for the individual growing and shrinking states

Table 4. Values of critical concentration, C o and statelifetimes, T$ and TQ, for standard, So, and modified

rate constant sets, S/-Ss

Set

SoSiS2

S3S4

s5s6s7S»

Cc

(/<M)

5.97.87.34.79.96.4S.O4.67.6

Lifetimes

Ts

(s)

10.76.7

14.711.37.68.09.8

23.013.2

at Cc

rG(s)

43.015.046.072.019.062.056.050.042.0

Lifetimes[T-GTP] = 5

Ts(»)

10.7>15.0>30.0

6.7>100.0

11.06.29.0

>25.0

at.9/IM

7'G

(•»)

43.0<4.0

<10.0>450.0

<?5.036.0

>180.0190.0<3.0

See Tables 1 and 2 for the definitions of the rate constants. Valuesof Ts and 7'G are given at the Cc appropriate for that set, and at [T-GTP]=5.9;<M, the Cc for the standard set.


are evaluated from an extensive simulation of the lengthbehaviour of an individual microtubule. In this case,since the transition frequencies become small at extremesof [T-GTP], the simulation has to be prolonged so thatreasonable estimates can be made of average propertiesover a period of time. Even so, the calculated lifetimesstill show some variation, owing to the finite length of thesimulation, as seen in Fig. 5. Such variation could bereduced by longer calculation; but, more importantly,this feature illustrates the difficulty of obtaining accurateexperimental values of average lifetimes and hence fre-quencies of interconversions in experimental measure-ments, which for practical reasons must be of finite andrelatively short duration.

The calculated state lifetimes are shown as a functionof [T-GTP] in Fig. 5A and B for the different sets of rateconstants. In the case of the standard set So, it is seen thatthe lifetime of the growing state increases rapidly abovethe Cc (5.9 (IM), and correspondingly the shrinking statelifetime increases strongly below this Cc.

Fig. 5A and B shows that the model is robust, in that

0 -6 8

T-GTP concentration12

Fig. 5. Variation of microtubule state lifetimes as a functionof T-GTP concentration for: A, the shrinking state (T$); andB, the growing state (Tc), simulated for rate constant sets So(standard) and Sj through Ss. See Tables 1 and 2 fordefinition of rate constants.

the phenomenon of the concentration dependence doesnot depend on any individual rate parameter of themodel, but individual rates can affect the absolute valuesof lifetimes in a dramatic way via changes in the value ofthe Cc. Although these changes are small on an absolutebasis, the effect on lifetimes are significantly amplified bythe non-linearity of the response of the system to T-GTPas seen in Fig. 5. It may be noted here that shorter statelifetimes produce shorter average length excursions,which are the product of mean lifetime and growth rate.Thus increased transition frequencies result in less di-rectly observable dynamic activity of individual micro-tubules, and slower average incorporation of free tubulindimer into the bulk population (Schilstra et al. 1989).

Table 4 shows the values of the state lifetimes fordifferent parameter sets at the appropriate Cc values(corresponding to the data of Fig. 4) and, for compari-son, at [T-GTP] =5.9/JM, which is the Cc for thestandard set So. The former values show that the valuesof the lifetimes at the individual Cc differ significantly,indicating an influence of individual rate constants on thetransition properties. The latter values are of more directpredictive value, since they indicate the expected effecton lifetimes of a particular change in a specific rateconstant at constant??/ and constant (absolute) [T-GTP].These conditions correspond to the direct experimentalobservations by video microscopy of microtubule growthoff seeds under different solution conditions.

The complex effects of these permutations on thelifetimes of the two states at [ T - G T P ] = 5 . 9 ^ M , the Cc forthe standard set S0) are seen in Table 4. Reducingk(—T)DT. and k(—T)TD. (set S3) slightly reduces theshrinking-state lifetime and greatly increases the grow-ing-state lifetime. Conversely, (set Ss) reducing6 ( + T ) T D . and £( + T) D T * lengthens Tg and shortens TQ.These results suggest that reactions of T at mixed sitesare of primary importance in the transition from G to S.Increasing &(-T) T T * (set Si) increases T s somewhat andreduces TQ significantly. A combined increase ink( — T ) T T *

anc> reduction in £ ( + T ) Q D . (set S4) makes Tgvery long and TQ very short. Thus either increaseddissociation of T from full-T sites or decreased additionof T to full-D sites has a pronounced effect on thetransition behaviour. The lack of reciprocal effect can beseen in sets S2 and S(,, owing to the non-linearity of theresponses shown in Fig. 5. Reduction in &(+T)rjD» (setS2) leads to a slight increase in Cc, a large increase in thevalue of To, and a large decrease in TQ. By contrast,reduction in k( — T) D D * (set S(,) leads to a slight decreasein Cc, a slight decrease in the value of Tg and a largeincrease in TQ.

Molecular events during periods of growthSince the simulation can provide a detailed description ofthe microtubule end configuration at each step of aMonte-Carlo simulation, as illustrated in Fig. 2 andTable 3, detailed analysis can be made of the way inwhich the end composition changes as a function of [T-GTP]. The gross effect is clearly that the end is fully T at[T-GTP] >CC giving the optimal elongation rate andfully D at [T-GTP] <?CC giving the optimal shrinking


rate. Examination of the end composition as a function of[T-GTP] over this range provides a molecular descrip-tion of the basis of the phenomenon of microtubuledynamic instability. We have therefore examined thenumber of events occurring at sites of different nucleo-tide composition during periods of sustained growth as afunction of [T-GTP], using a single set of rate constants

Fig. 6A shows the total number of association events toTT* sites as a function of [T-GTP]. At the highest

\~ \ If allconcentration (25,itM) the rate is approx. 225 sevents were at T T T sites this rate would be 240 s(7?G,TTT=[T-GTP]xfc(+T)TTTx<M>), or 120s"1 atTTV sites. In fact, the simulation confirms that 84% ofthe association events to T T * are to T T T , as expectedstatistically; the optimal rate is theni?G,opt=0.92x7?G T T T . Inspection of Fig. 6A clearlyshows that significant deviation from linearity occurs atlower [T-GTP]. This is because the number of TT* sitesavailable for association reactions begins to decrease atlower [T-GTP], owing to the increased occurrence ofmicrotubule ends of mixed configuration, which containT-GDP, although the microtubule continues to grow.

The occurrence of 'mixed' microtubule ends at lowvalues of [T-GTP] is shown more clearly by the data of

200 -

100 -

0 -I r

12 16T-GTP concentration

Fig. 6. Analysis of microtubule end composition as afunction of [T-GTP] during the growth phase: rates ofevents occurring at specific site configurations during periodsof sustained microtubule growth. A. Binding of T to TT*(D • ) ; and linear least squares fit to data from 17.S to251.M ( ). B. Dissociation of: T from TT* ( • • ) ;and D from DD* (+ +).

4 6T-GTP concentration

Fig. 7. Analysis of microtubule end composition as afunction of [T-GTP] during the shrinking phase: rates ofevents occurring at specific site configurations during periodsof sustained microtubule shrinking. A. Dissociation of Dfrom DD*. B. Dissociation of T from DD* ( • • ) ;binding of T to DD* (H h); and the limiting slope at[T-GTP] =0( ).

Fig. 6B, which show the total number of dissociationevents for T from TT* and D from DD* occurringduring periods of growth. Dissociation of T from TT*remains virtually constant at about nine events/s"1,although it falls off at low concentrations. More dramati-cally, dissociation of D from DD* starts to become a verysignificant event at concentrations below about 15 /1M.Since dissociation of D can only occur (see Fig. 2) whenthe terminal T layer is significantly disrupted, theenhanced rate for this event is a direct indication of thefact that the microtubule spends increasing amounts oftime in a mixed configuration as the concentration islowered.

Molecular events during periods of shrinkingA similar analysis of the composition of the microtubuleend as a function of [T-GTP] during shrinking showshow the transition into an optimally shrinking micro-tubule occurs. Fig. 7A shows the total number of dis-sociation events from DD* sites as a function of [T-GTP]. At [T-GTP] =0 there are approx. 280 dissociationevents per second. If all dissociation events were fromDDD sites this rate would be 223 s~' (/?S,DDD =

10


k(—D)DDDx(«)), or 446s ' if all events were fromDDV sites. The simulation confirms that 27 % of thedissociation events are from DDV and the rest are fromDDD; the optimal rate is given by RStOpt=l.27x^S.DDD- Fig- 7A clearly shows that the rate falls mono-tonically as [T-GTP] increases. This is because thenumber of DD* sites available for dissociation decreasesas [T-GTP] increases. Thus, although the microtubulecontinues to shrink, there is a substantial kinetic contri-bution from the increased occurrence of microtubuleends of mixed configuration, which contain significantproportions of T-GTP in terminal positions.

This is illustrated more clearly by the data of Fig. 7B,which show the total number of association and dis-sociation events for T at DD* sites occurring duringperiods of sustained microtubule shrinking. If the shrink-ing microtubule contained only DD* sites for the ad-dition of T, the number of association events for Tshould be a simple linear function of [T-GTP]. In fact,the plot shows significant deviation from linearity for [T-GTP] greater than approx. 4^tM. The curve for dis-sociation of T from DD* is practically coincident withthat for the association of T except at the highestconcentrations of T-GTP investigated. This is expectedbecause the affinity of T for a DD* site is low, since thevalue of k( —T)DD* 'S high (see Tables 1 and 2).

Rates of elongation during growth phases andshortening during shrinking phases: comparison withbulk propertiesIt is clear that the model predicts that the rate of growthof an individual microtubule is not a truly linear functionof [T-GTP], and that it approaches its optimal value onlyasymptotically at high [T-GTP], as seen in Fig. 6A. Thedeviations from this value will be most marked close tothe critical concentration. Unfortunately, this is therange in which the state lifetimes have generally beendetermined, being the most accessible experimentally. Itis thus important to evaluate how this non-linearityaffects the interpretation of the observed growth rates ofindividual microtubules in terms of the individual kineticconstants of the system. Similar considerations apply tothe observed shrinkage rate of individual microtubules.The model predicts this rate to vary significantly, accord-ing to the exact end composition, which is in turndetermined by [T-GTP] as shown in Fig. 7A.

Fig. 8 shows the calculated dependence on [T-GTP]for the mean rate of elongation (during sustained growth)and the mean rate of shortening (during sustainedshrinking), derived from the data of Figs 6 and 7, andcompared with the net microtubule growth rate, allcalculated for the kinetic constants of modified set Si,with CC=7.8JMM. At high [T-GTP] (>17.5JUM) themicrotubule growth rate is optimal, being a linear func-tion of [T-GTP] and following equation (2), so thatlinear extrapolation of this region to [T-GTP] =0 yields atrue value of £'_, the dissociation rate constant for T-GTP from a growing microtubule. Using a linear least-squares fit to the above data gives A'_=0.33 fimmin"'1, or9s~ ' (cf. Fig. 6B). For reasons discussed above, theobserved rate begins to deviate significantly from this

c1I

-12 -

-16 -

- 4 -

10 20T-GTP concentration (JJM)

Fig. 8. Comparison of simulated net growth rate for amicrotubule population and the rates of individualmicrotubules in growth and shrinking phases. Rate constantset S]. Net growth rate from Fig. 4B ( • • ) ; individualgrowth rate ( O O)i an£J individual shrinking rate(-1 H); linear least squares fit to curve D —• from17.5 t

linear extrapolation at [T-GTP] < 15/iM, i.e. substan-tially above the Cc. Extrapolation of the elongation ratedata in the region of the Cc gives an intercept at [T-GTP]=0 of 2,ummin~1, or 54 s"1, which is thus >five-fold overestimated. The value of k' + is 25% overesti-mated. Thus extrapolation of individual growth ratesdetermined in the vicinity of the Cc may give unreliablevalues of both k' + and, more seriously, k'_.

Finally, it is seen that the net rate of shortening (duringperiods of shrinking) decreases monotonically as theconcentration of T-GTP is increased in the range 0<[T-GTP]<CC for reasons discussed in the previous section.Extrapolation of these data from the vicinity of Cc will notgive a reliable estimate of k(-D)DD* at [T-GTP] =0. Itmay be noted that the net growth of the microtubulepopulation is defined in terms of individual rates andmean lifetimes (cf. Horio and Hotani, 1986) as[TQ .RQ — T^.RS] at a given [T-GTP] and is zero at [T-GTP]=CC. This simulation shows that state lifetimes,and to a lesser but still important extent, ratesRQ and/?s,are complex functions of [T-GTP].

These observations impose critical limits on the num-bers that can be extracted experimentally from obser-vation of individual microtubules; the observable (mean)growth and shrinking rates in the region of the Cc appearlikely to overestimate the optimal values of the growthrate constant and likewise to overestimate the optimalvalue of the shrinking rate constant, both by significantamounts. We attribute this effect to the presence of mixedmicrotubule ends, which are neither fully T-GTP (ingrowth) nor fully T-GDP (in shrinking). This conditionexists in a broad range of [T-GTP] values around the Cc.These factors are important in trying to estimate steady-


state properties of individual ends, and particularly inseeing whether the magnitude of any imbalance in Cc atthe two ends, which would lead to the bulk property ofdirectional flux or 'treadmilling' (cf. Walker et al. 1988),as established for actin (Wegner, 1976; Pollard, 1986;Pollard and Cooper, 1986).

The modelling presented here indicates that there is alimitation in principle to assigning the values of optimalelongation and dissociation rates from data obtained nearto the critical concentration, or at steady state, sinceunder these conditions the microtubule ends are pre-dicted to spend a significant proportion of time in sub-optimal states with mixed end configurations.

Discussion

The model described here demonstrates conclusively thatan extensive T-GTP cap is not necessary to confer theproperties of microtubule dynamic instability. The con-cept of a Lateral Cap of T-GTP has several attractivefeatures. First, it corresponds to the experimental data inwhich the amount of T-GTP present in the microtubuleat steady state and during microtubule elongation isbelow the limits of experimental detection (O'Brien et al.1987; Schilstra et al. 1987; Stewart et al. 1989). Second,it confines the spatial range over which the microtubulelattice affects the properties of the end to a single tubulinmolecule in depth (Bayley and Martin, 1989; O'Brien etal. 1989). While this postulate is unproven, it allows therealistic possibility that it is in the act of completion ofspecific contacts constituting the set of interactionscharacteristic of the microtubule lattice that hydrolysis ofT-GTP is triggered. Third, as shown here, the principlesof the model as specified are sufficient to reproduce themain experimental features of dynamic microtubules,and predict a strong T-GTP concentration dependenceof transition lifetimes characteristic of a cooperativeprocess. Fourth, as will be argued, a terminal-layermodel shows how different dynamic behaviour at theopposite ends of the polar microtubule may be ration-alised, on the basis of the intrinsic a-/3 polarity of thetubulin heterodimeric molecule.

This is a coupled hydrolysis model, which emphasisesthe primary role of interactions in the terminal layer ofthe microtubule. The condition for GTP hydrolysis isthat this particular tubulin molecule is fully incorporatedinto the microtubule lattice. There are several ways inwhich this could occur. The formulation used here is notunique; we have developed the specific Hydrolysis Ruleof Fig. 1 to illustrate the principle of restricting T-GTPto the terminal layer. Other rules have been examined,e.g. those in which binding at X causes hydrolysis ofT-GTP at position b (lateral hydrolysis) or at position d(longitudinal hydrolysis). (These lead to the conditionthat pmax<l8.) Although these models could be con-strained to produce similar dynamic behaviour, they werequite sensitive to changes in individual rate constants.The present model is extremely robust to such changes,as seen in Fig. 5. This is an attractive feature, consideringthat dynamic instability has been observed under a wide

range of experimental conditions, which may be expectedto cause variation in both association and dissociation rateconstants.

The Lateral Cap model differs from the fluctuatingT-GTP cap model of Hill and co-workers (Chen andHill, 1985; Hill, 1984; Hill and Chen, 1984) in both theprinciples of the hydrolysis rule and the nature of thebinding interactions, and in the specific detail of thevalues of parameters involved. In that model, GTPhydrolysis was represented by a set of 24 hydrolysis rateconstants, varying over four orders of magnitude.Although this generated dynamic instability, it alsoproduced large T-GTP caps during assembly. Thecoupled hydrolysis mechanism is consistent with theabsence of detectable caps under a range of assemblyconditions, and obviates questions of the mechanism ofvectorial hydrolysis by which a T-GTP/T-GDP inter-face passes along the microtubule.

This coupling has important effects on the magnitudeof the association and dissociation rate constants, whichalso differ considerably from those of Chen and Hill(1985). No attempt has been made here to refine values ofthese parameters (Table 1) in order to improve fits toexperimental data. The parameter for the elongation rateof a growing microtubule by T-GTP, (6 + T ) T T T , is2.5xlO6M~ s~ , corresponding to an optimal micro-tubule elongation rate RG,opt at [T-GTP] = 10[iM of 80subunitss" (per microtubule end), i.e. approximately3jWmmin~1. The other association constants are derivedconsistently from this with a range of less than one orderof magnitude depending on the site (bde). Dissociationrate constants for T-GTP are scaled from the lowestvalue of k(—T)XTT of 0.6 s~ . The intrinsic affinity ofthe site (bde=TTT) is thus 4X 1 0 6 M ~ ' . The dissociationrate constant for T-GDP, k( — D)DDD is set at 58 s"1,corresponding to an optimal dissociation ratc/?g)Opt at [T-GTP]=0 of 280 subunitss"1 (per microtubule end), or ashrinking rate of 10/immin"1. Thus the basic set ofparameters used corresponds to physically realistic valuesfor observed microtubule properties, as well as beingintrinsically reasonable in magnitude.

The effects of changing individual rate constants havebeen described in detail in the previous sections. Increas-ing any of the dissociation rate constants or decreasingany of the association rate constants increases Cc. Therobustness of the model is illustrated in the results withthe modified sets Ss and S7, compared with set So(Table 4). The effect of doubling k(-D)D0», (Ss), is tochange Cc, Tg and TQ from 5.9/iM, 10.7s and 43 s to6.4 jiiM, 8 s and 62 s. By changing the fraction of shrinkingmicrotubules from 19.9 to 11.5 %, Cc is maintained veryclose to the original value. Similarly, the fraction ofshrinking microtubules is not halved, because the in-crease in Cc increases the growth rate (RQ) by some8.5 %. The effect of increasing & ( + T ) T T » by a factor of 2(S7) is to change Cc, T$ and 7<3 to 4.6 fJM, 23 s and 50s.In this case the fraction of shrinking microtubulesincreases from 19.9 to 31.5 %, the growth rate RQ falls bysome 22% and the shortening rate /?g increases some-what. This behaviour demonstrates the remarkable ca-pacity of a microtubule (within this model) to respond to


changes in solution conditions affecting the values ofparticular rate constants. The three-way response interms of Cc, TQ and T$ makes the system very robust.Biologically, such properties would ensure the continu-ation of dynamic behaviour under a range of environmen-tal conditions.

It should be pointed out that, with the exception ofoptimal growth and optimal shrinking, the observableparameters such as state lifetimes are not simply deter-mined by any single parameter. This is shown clearly inFig. 5. Thus the 'rescue frequency' ( l /Ts ; transition S toG) or 'catastrophe frequency' (1/TG; transition G to S),as sometimes designated, cannot easily be related to theserate constants. However, they are accessible by numericalsimulation of a specific model, as illustrated here. In thiscontext it is very interesting that doubling &( —D)DD# (setS5, Table 4) has little effect on the state lifetimes, andhence the transition frequencies, at a given [T-GTP].Recent data (O'Brien et al. 1989) show relatively littleeffect of increased [Mg2+] on state lifetimes, althoughthis ion can increase the dissociation rate constant signifi-cantly (Gal et al. 1988). By contrast, one of the mostsensitive parameters is 6 ( + T )DD», since this contributesto steps in the rescue reaction (Fig. 2,16), but is not thesole determinant of it. In fact the progressive increase inaffinity of the site bde for T-GTP as it becomes richer intubulin molecules carrying GTP (shown by the rateconstants in Table 1) is the central feature of the tran-sition behaviour. It is the operation of stochastic pro-cesses involving this multiplicity of sites, and theirdistribution both in relative lattice position and inguanine nucleotide composition that produce the appar-ent co-operativity of the transitions.

In comparing the concentration dependence of tran-sition behaviour with experimental results, data areavailable from two sources. Under appropriate conditionsmicrotubule populations are observed to undergo sus-tained oscillations of assembly and disassembly (Carlieret al. 19876; Chen and Hill, 1987; Lange et al. 1988;Mandelkow et al. 1988; Melki et al. 1988; Pirollet et al.1987). In this case, the concentration of free T-GTPvaries significantly owing to nucleotide utilisation, slownucleotide replenishment and tubulin self-associationreactions. The coupling of these effects via a strongconcentration dependence of switching rates (&GS a n dk$o) leads to oscillations. Second, observations of thetransition behaviour of individual microtubules (Walkeret al. 1988) showed a progressive decrease in the fre-quency of catastrophe (kGs) and an increase in thefrequency of rescue (&SG) with increasing [T-GTP].These observations are necessarily limited in number andare in the concentration range accessible to experiment,and while these data give strong qualitative support to theabove concepts, additional quantification would benecessary to provide a critical experimental test of thenon-linear dependence predicted by the simulations.Because of this, the data sets have deliberately not beenrefined to 'fit' the data, but have been chosen to illustratethe general behaviour of the model and the principlesunderlying this formulation. Thus the modelling isrelatively insensitive to the microtubule shrinking rate;

k(—D)DD» can be increased significantly (Fig. 4A; setsSo and S5) with relatively small changes in Cc (Table 4)and state lifetimes (Fig. 5A,B). The fact that the concen-tration dependence of the state lifetimes is maintainedthrough all the parameter sets examined adds substanceto the proposition that this effect is indeed an importantprinciple underlying oscillatory behaviour of microtubulepopulations, and switching of individual microtubules.

The question of the role of microtubule polarity in thedynamic properties remains an intriguing one. It isgenerally accepted that the protofilaments in assembledmicrotubules of normal morphology are parallel in orien-tation (as opposed to the anti-parallel orientation of Zn-induced sheets; see Amos, 1979). Thus the intrinsicasymmetry of the tubulin heterodimer is reflected in thetwo microtubule ends, which, for simplicity we willdesignate as effectively '<*-end' and '/3-end'. It is also wellestablished that the two ends of the microtubule showdifferential dynamic properties (Horio and Hotani, 1986;Walker et al. 1988, 1989). This asymmetry is sometimesexpressed as an 'active' and an 'inactive' end, referring tothe size of microtubule length excursions. In fact thismay be at variance with the true underlying kineticactivity, since high transition frequencies will lead toshort excursions. On models with extensive T-GTP caps,where the assumption is that the activity of the endreflects the property of the whole cap, it has been difficultto rationalise how a T-GTP within such a cap shouldhave different kinetic properties at either end.

A particularly attractive feature of single-layer capmodels, independent of the exact formulation, but cer-tainly consistent with the Lateral Cap model shown inFigs 1 and 2, is that the difference between the micro-tubule ends is reduced to the difference of the <x-end andthe /3-end of the tubulin heterodimer in the terminal layerat opposite ends. Inspection of Fig. 1 shows that sites Xand X' are not identical, given the polar orientation of thetubulin heterodimer at X and X'. Likewise, the interac-tions of X with positions a,b,c,d,e are intrinsicallydifferent from those of X' with a' ,b' ,c ' ,d' ,e ' . It istherefore reasonable to postulate that the two ends willhave at least some kinetic parameters different from eachother, and that these will, in the sense of the modeldeveloped here, give rise to different dynamic properties.In the case of the extreme asymmetry observed afterultraviolet microbeam severing of individual micro-tubules (Walker et al. 1989), it appears that the severedminus (inactive) end rescues relatively rapidly, comparedto the severed plus end, then continues in slow growth.This is apparently consistent with the model used here,on the condition that there is a narrower distribution of&( + T)bde values at the minus end, rescue being stronglydetermined by £( + T)rjrj», and growth rate determinedby the other £( + T)b d e parameters.

It is not possible at present to correlate the (a—/3)structural asymmetry of the end with the asymmetry ofthe kinetic properties. Likewise, it is not the intention ofthis work to resolve any kinetic steps occurring betweenthe initial addition reaction of T-GTP and the conversionof a previously terminal T-GTP to an internal T-GDPresidue. Such steps could include the bond breakage of


the gamma-phosphate bridge oxygen bond of GTP (Oseiet al. 1988), and the release of phosphate. There is noreason from the present simulation to postulate thateither of these is rate-limiting, since kinetically theaddition of T-GTP(free) is well represented by a singlestep process formalised as:

T-GTP(free) + T-GTP,. T-GDP,-, . . .- • T-GTP, + , . T-GDP/. T-GDP,-, . . .

Recent evidence has shown that microtubules can beformed as polymers of T-GDP .A', where X is BeF3 orAIF3, which in hydrated form resemble phosphate ion(CarliereJ al. 1988, 1989). Related to this is the postulatethat the stabilising terminal residue of normal micro-tubules might be T-GDP. P, (Carlier et al. 1988). How-ever, the experimental basis of the interaction of micro-tubules with phosphate is controversial (Caplow et al.1989). It is noteworthy that the hydrolysis of GTPcoupled to microtubule assembly appears irreversible,i.e. phosphate is not incorporated back into GTP. Weconclude that the simulation confirms that a simpleformulation is at present sufficient, although the detailedmolecular mechanism of GTP hydrolysis linked to as-sembly is still not fully resolved.

Perhaps the most useful aspect of the simulation is theinsight it gives into the interpretation of direct obser-vations of individual microtubule dynamics. This isshown by the indirect relationship between intrinsickinetic parameters and the experimentally observablefeatures of Cc, and state lifetimes (or frequencies), whichappear to constitute a robust and highly interactive set ofproperties. An experimentally verifiable prediction is thatthe rates of elongation and shortening of individualmicrotubules will be complex functions of [T-GTP] andsignificantly different from either the linear dependenceof association rate or the concentration-independentdissociation rate, which are expected as asymptotic be-haviour at extremes of [T-GTP]. While the experimentalaccuracy required to define these properties may not yetbe available, the more general point that caution isrequired in extrapolation of individual rate data obtainedin the intermediate region of the Cc appears well-founded.

The simulations also suggest important criteria for thedesign of experiments on individual microtubules.Measurements clearly need to be made on as wide a rangeof [T-GTP] as possible. At the extreme values, transitionfrequencies will be strongly perturbed. Observationsneed to be taken to at least one order of magnitude longerthan the mean state lifetime in order to give a reasonableaverage for these parameters, and stability of the systemmay prove a limiting factor.

Finally, the effect of different solution conditions andadded proteins may be considered. In all cases wherethere is an effect on Cc, state lifetimes will inevitably bechanged, sometimes dramatically. Microtubule-associ-ated proteins (MAPs) reduce the critical concentrationfor assembly, and have been reported in general tostabilise microtubules against dynamic behaviour (Far-rell et al. 1987; Horio and Hotani, 1986). The dynamicsof microtubules reconstituted from tubulin and purified

MAP components is very varied (E.D. Salmon, unpub-lished). Although our modelling is explicitly concernedwith the purified tubulin system, it is clear in principlethat the reduction of Cc by any MAPs will lead to asignificant increase in TQ. It thus appears necessary tomake such measurements over a range of [T-GTP] (tocover the new Cc), but at suitable MAP: tubulin ratios toensure constancy of the [T-GTP]. Given the structuralcomplexity and biochemical heterogeneity of MAPs, thiswill be a forbidding task.

More tractable is the use of the simulation to study theeffects on microtubule dynamics of variation in physicalconditions, such as temperature (Caplow et al. 1988a,b;Walker and Salmon, 1988), glycerol, and ionic compo-sition (Schilstra, M. J., Martin, S. R. and Bayley, P. M.,unpublished data). Mitotic drugs affecting dynamicsinclude colchicine (Salmon et al. 1984; Wilson andFarrell, 1986), Vinca alkaloids, taxol and podophyllo-toxin (Bayley et al. 1989a; Schilstra et al. 1988). Thislatter drug acts substoichiometrically to suppress bulkdynamics (Schilstra et al. 1989) and simulation of thepossible mechanism of action using the principles of thelateral cap is the subject of current investigation (Martin,S. R., Schilstra, M. J. and Bayley, P. M.).

Thus, in conclusion, the Lateral Cap formulation givesa physically realistic representation to the mechanism ofmicrotubule dynamics, based almost exclusively on theproperties of the terminal layer of tubulin molecules inthe microtubule. It presents a readily visualised qualitat-ive picture of the process of dynamic instability, and itprovides a quantitative formulation to the phenomena oftransitions between growing and shrinking microtubules.It shows how dynamic instability can persist over a rangeof physical conditions, a property with potential biologi-cal significance. It provides a quantitative basis for theprinciple that growth and shrinking of dynamic micro-tubules generally occur at sub-optimal rates, due to theoccurrence of end configurations with mixed and varyingcomposition of T-GTP and T-GDP. It constitutes acritical basis for the evaluation of experimentally deter-mined individual length measurements, and it providessome specific predictions testable in future investigations.The successful application of this formulation suggeststhat the principles underlying the Lateral Cap modelconstitute a plausible molecular basis for the phenom-enon of microtubule dynamic instability.

We thank Dr Yves Engelborghs for useful discussion, and DrTed Salmon for access to unpublished data. This work wassupported in part under E.E.C. Twinning Grant no. 852/002SS/UK/05PUJU1 and Nato Collaborative Research Grantno. 86/0499.

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