THE JOURNAL OF BIOLOGICAL CHEMISTRY 0 1984 by The American Society of Biological Chemists, Inc

Vol. 259, No. 16, Issue of August 25, pp. 10430-10438,1984 Printed in U.S.A.

The Kinetics of Microtubule Assembly EVIDENCE FOR A TWO-STAGE NUCLEATION MECHANISM*

(Received for publication, February 1, 1984)

William A. Voter and Harold P. Erickson From the Department of Anatomy, Duke University Medical Center, Durham, North Carolina 27710

A model describing the nucleation and assembly of purified tubulin has been developed. The novel feature of this model is a two stage nucleation process to allow the explicit inclusion of the two-dimensional nature of the early stages of microtubule assembly. In actin as- sembly the small starting nucleus has only one site for subunit addition as the two-stranded helix is formed. In contrast, microtubule assembly begins with the for- mation of a small two-dimensional section of microtu- bule wall. The model we propose is a modification of the work of Wegner and Engel (Wegner, A., and Engel, J. (1975) Biophys. Chem. 3,215-225) wherein we add a second stage of nucleation to directly account for lateral growth, i.e. the addition of a small number of subunits to the side of an existing sheet structure. Subsequent elongation of the sheets is treated in the usual way. The experimental system used to test this model was the Mg2+/glycerol induced assembly of pu- rified tubulin. The computer simulation of the polym- erization time courses gave a fairly good fit to experi- mental kinetics for our model, where the primary nu- cleus comprises two protofilaments, of four and three subunits, and lateral growth requires a three-subunit nucleus to initiate a new protofilament.

Microtubule assembly can be obtained in uitro from a solution of purified tubulin subunits in the appropriate buffer. The assembly reaction is generally described as comprising two phases, nucleation and elongation. During the nucleation phase, new microtubule ends are generated spontaneously from subunits. Once these nuclei are large enough to be stable, polymerization of subunits onto the ends produces elongation. The elongation phase continues until the subunit pool is reduced to the concentration in equilibrium with microtu- bules. Nucleation plays a significant role only in the earlier part of the assembly, when the concentration of free subunits is highest.

The simplest pathway of assembly is a sequence of bimo- lecular reactions in which polymers are built by sequential addition of monomers. The initial bimolecular reactions are thermodynamically unfavorable in the sense that each larger polymer is less stable than its precursor. As the polymer grows, however, the addition of monomers becomes less un- favorable. Once a certain size, n monomers, is reached, the addition of the next monomer results in a polymer that is more stable than its precursor. From this point, the addition of successive monomers is thermodynamically favorable and

* This work was supported by National Institutes of Health Grant GM-28553. The costs of publication of this article were defrayed in part by the payment of page charges. This article must therefore be hereby marked “advertisement” in accordance with 18 U.S.C. Section 1734 solely to indicate this fact.

assembly will continue spontaneously. The n-mer is com- monly referred to as the “critical nucleus,” or, in the termi- nology we will use, simply the nucleus. In this definition, the nucleus is the first polymer whose growth is thermodynami- cally favorable, in the sense that the (n + 1)-mer is more stable than the n-mer; the n-mer itself is, however, the least stable species in the pathway. The thermodynamic aspects of nucleation are discussed in more detail in Ref. 1.

The standard treatment of nucleation uses a mixture of thermodynamics and kinetics. The nucleus is assumed to be in equilibrium with subunits, and it exists at a very low concentration because it is thermodynamically unstable. This low concentration in turn limits the overall kinetic? of assem- bly, because all larger polymers have to come from nuclei. Information on the size of the nucleus can thus be obtained from the kinetics of assembly.

The simplest indication of the size of the nucleus is obtained from a log-log plot of the tenth-time of polymerization uers’sus concentration (2, 3). More detailed analysis involves fitting the curves for the entire time course of polymerization to ones generated by computer for a particular assembly model and nucleus size. The definitive treatment of this approach is that of Wegner and Engel (4), who introduced simplifying assump- tions that made the computer simulations practical for routine use. The original theory has been expanded and used to study nucleation of actin polymers under a variety of conditions (3, 5 , 6). It has also been applied to sickle cell hemoglobin polymerization (2, 7). The present study is the first applica- tion to microtubule assembly.

The polymerization of actin is conceptually simple because the nucleus is small and the helical polymer has only a single site for elongation. The polymerization of microtubules is much more complex. Although the intact microtubule is a helical polymer, its assembly involves primarily the formation of a two-dimensional polymer, the microtubule wall. We ob- served in some of our earliest experiments (8) that microtu- bule assembly began with the formation of a small sheet, a segment of the wall comprising a few protofilaments. This two-dimensional polymer grew wider and longer until it at- tained its full complement of thirteen protofilaments. At this point, the sheet could close and form the intact helical micro- tubule. Similar observations were made independently by Kirschner et al. (9) and subsequently confirmed by a number of laboratories for a variety of assembly conditions. In some studies the small two-dimensional polymers have not been seen as intermediates, but in these cases lateral growth and closure probably occurred very rapidly.

These observations and other arguments have led us to conclude that microtubule assembly, especially the nucleation phase, is essentially a two-dimensional polymerization, the formation of the microtubule wall (1,9). Closure of this sheet to form the intact helix occurs when possible, but if closure

10430

Kinetics of Microtubule Assembly 10431

is prevented, the assembly will still proceed and will generate long, two-dimensional polymers. In our model developed be- low, we will treat the assembly of the two-dimensional poly- mer, and closure will be implicitly assumed by limiting the lateral growth to thirteen protofilaments.

For this study we have chosen a M$+/glycerol buffer, and we use a temperature jump from 0 to 37 “C to initiate micro- tubule assembly. In this buffer, normal microtubules and, under some conditions, large sheets can be obtained from highly purified tubulin. We felt it was important in the initial studies to work with the simplest system, and the purified tubulin avoids the complication of the MAPs.’ In addition, we used conditions that avoided the formation of tubulin rings, since solutions containing rings were found to have slower assembly kinetics. We therefore believe we are dealing with the simplest possible system, nucleation and assembly of microtubules directly from tubulin dimers.

ASSEMBLY MODEL AND EQUATiONS FOR KINETICS

There is some confusion in the literature over the exact species designated by the word “nucleus.” The definition that we use here, the least stable intermediate, has been used by Eaton and Hofrichter (2), Ferrone et al. (7), Erickson and Pantaloni (I), Wegner and Savko ( 5 ) , Cooper et al. (6 ) , and Frieden and Goddette (10). Some authors define the nucleus as the next polymer in the pathway, ix. as the first polymer that is itself more stable than its precursor. This usage was employed by Oosawa and co-workers (11, 23), in the earlier work of Wegner and Engel (4), and most recently by Tobac- man and Korn (3). We prefer the former definition because it is being used in most of the recent biochemical studies and because the theory and the term critical nucleus is well established in studies of nucleation in physical systems (12, 13).

A complete description of the kinetics of polymerization requires rate constants for monomer addition and loss from each size polymer, and an infinite set of related differential equations. The classic treatment of Oosawa and co-workers (11,23) introduced approximations that simplified the math- ematics, but it is only applicable for subunit concentrations much greater than the critical concentration. A modified form of this approach, using the tenth-time instead of the half- time suggested by Oosawa, has a greater range of validity and has been used in recent studies (2,3). The definitive treatment of the kinetics of nucleation is that of Wegner and Engel (4). Their analysis is more detailed and difficult than the Oosawa (11, 23) approach, but the theory is correct and the approxi- mations are valid for a broad range of self-nucleating systems.

The simplest model for the Wegner and Engel (4) approach is for the case in which 1) the nucleus is assumed to be in rapid equilibrium with subunits throughout the reaction, and 2) the rate constants for formation and for decay of polymers by gain or loss of a monomer is the same for all polymers larger than the nucleus (n-mer). With certain simplifying assumptions they reduced the mathematics to a pair of cou- pled differential equations. These equations can be solved by numerical integration, requiring computation times of only a few minutes on a microcomputer. In recent applications the theory has been expanded to include generation of new ends (growing points) by fragmentation of polymers ( 5 ) ; a first order step of monomer activation preceding assembly (3, 6); and heterogeneous nucleation, in which assembly is acceler-

’ The abbreviations used are: MAPs, microtubule-associated pro- teins; MES, 2-(N-morpholino)ethanesulfonic acid; EGTA, ethylene glycol bid 6-aminoethyl ether)-N,N,N’,N’-tetraacetic acid.

ated by facilitated nucleation of new polymers along the length of polymers formed by homogeneous nucleation (7).

Microtubule assembly, involving two-dimensional growth, is more complicated than actin. We have modified the theory for a model of microtubule assembly, the major new feature being an explicit step for nucleated lateral growth. Lateral growth is treated as a second step of nucleation, similar to the heterogeneous nucleation model of Ferrone et al. (7).

Our model for the nucleation and growth of microtubules is shown schematically in Fig. 1. It is essentially the same as we proposed in our previous work on the thermodynamics of nucleation (1). The primary nucleus is assumed to be a poly- mer consisting of two protofilaments. We will assume that the two protofilaments are equal in length (to within one odd subunit), and the length will be specified after the total size of the nucleus is determined by curve fitting. Lateral growth requires the nucleation of another protofilament along the side of the two-filament sheet. This second nucleation event becomes increasingly favorable as the sheet grows longer, because the number of sites for nucleation of the new proto- filament is proportional to the length of the sheet. The growth to each successively wider sheet is assumed to involve the same mechanism of lateral nucleation. The number of sub- units required to nucleate a new protofilament is to be deter- mined from the curve fitting.

For the computation we assume a population of sheets from 2 to 13 protofilaments wide. The simulated assembly consists of the following three steps: (1) nucleation of two-filament sheets, which is assumed to involve an n-mer nucleus in rapid equilibrium with free subunits; (2) lateral growth, which is assumed to involve a second nucleus, a short protofilament of m subunits attached to the side of an existing sheet, also in

FIG. 1. Model for nucleation, lateral growth, and elonga- tion. 6-S tubulin dimers are indicated as dumbbell-shaped subunits with lateral and longitudinal bonds protruding. The most recently added subunits are shown in black. a, the primary nucleation mech- anism is assumed to be formation of a two-filament sheet. The nucleus is in rapid (compared to the subsequent elongation steps) equilibrium with the pool of free subunits. The heptamer shown here is the best fit from our computer modeling. b, lateral growth is assumed to involve the formation of a short protofilament, in equilibrium with subunits, on the side of a sheet. Once the protofilament reaches a critical size, indicated as three subunits by our computer fitting, it is stable and grows rapidly to the length of the sheet. c, elongation is assumed to involve a second order addition of subunits onto the end of protofilaments. For simplicity in the present theory, the proto- filaments are all treated as equivalent, ignoring the variable cooper- ativity.

10432 Kinetics of Microtubule Assembly

rapid equilibrium with monomers; and (3) elongation, which is assumed to involve a second order association and a first order dissociation of subunits at the end of protofilaments. These steps will be detailed after definition of the parameters involved, which are: Co, the concentration of free monomers; Ci, the concentration of sheets of width i (i = 2 to 13); Ci*, the concentration of subunits in sheets of width i, libt, the total length of sheets of width i = C,*/i); l:"? the average length of sheets of width i (l? = li"/C, = C$/iCJ; K,, the equilibrium constant for formation of the two-filament nu- cleus, an n-mer; K,,,, the equilibrium constant for nucleation of the new protofilament, containing m subunits, on the side of a sheet; k+, the second order rate constant for elongation, association of a subunit at the end of a protofilament; and k- , the first order rate constant for dissociation of a subunit from the end of a protofilament.

The progress of assembly is calculated for 12 separate species, representing sheets of width i, with i from 2 to 13. By limiting lateral growth to i = 13, we implicitly assume that the sheet closes and further growth is by elongation alone. The differential equations will be written simply as incre- ments (e.g. dCJ calculated in the numerical integration for a small time interval, dt. The equations are considered in three sections, nucleation of two-filament sheets, nucleation of a new protofilament, and elongation.

Step 1. Nucleation of Two-filament Sheets-We assume that the n-mer nucleus is in rapid (relative to elongation, C&+) equilibrium with monomers. C2 is the concentration of stable (larger than the critical nucleus) two-filament sheets. The increase in Cz is equal to the number of critical nuclei that gain a monomer in the time dt. CZ is decreased when an (n + 1)-mer loses a subunit and decays into the pool of labile nuclei. The decrement of two-filament sheets lost by growth into three-filament sheets is handled in a later step.

dCZ = {(Cok+ - k-)K,,(Co)"]dt (1)

The product K,( Co)" represents the equilibrium concentration of n-mers, the nuclei. An important assumption in this equa- tion, introduced and validated by Wegner and Engel (4), is that the reaction is always near steady state and the concen- tration of (n + 1)-mers is approximately equal to the concen- tration of n-mers ( i e . the concentration of both n-mers and ( n + 1)-mers is K,(Co)").

Step 2. Lateral Growth by Nucleation of New Protofila- rnents-Formation of a three-filament sheet (3-sheet) re- quires the addition of a new protofilament to the side of a 2- sheet, a step that is thermodynamically unfavorable and therefore requires a second step of nucleation. The nucleus for this lateral growth is assumed to be a short protofilament of m subunits in rapid equilbrium with monomers. Since this protofilament nucleus will be attached to the side of a 2-sheet, the rate of this step should also be proportional to the total length of 2-sheets. Generalizing for the formation of a sheet of width i (i-sheet), for i = 3 to 13,

dC; = {I?w,(Ca)mldt. ( 2 4

At this point it is useful to assume that all of the newly formed i-sheets have a length equal to the averuge length of the ( i - 1)-sheets from which they were formed. Furthermore, we assume that the newly formed protofilament is rapidly (in- stantaneously) filled in to form a smooth edged sheet, i subunits wide by 1:Il subunits long. The increase in C? (the subunit concentration in i-sheets) from this step of lateral growth is given by

dC,* = dciIy1i = dcic:-li/ci-l(i - 1). (2b)

Finally, the decrement in number and subunit concentration

of ( i - 1)-sheets is calculated, using the newly calculated values for C, and Ci*.

dC,, = -dC, (2c)

d C L = dCi*[(i - l)/i] ( 2 4

The calculation is done for i = 3 to 13, completing the calculation of lateral growth for one cycle time, dt. It may be noted that C2* is not incremented in the first nqcleation step. The contribution from nucleation of 2-sheets is negligible when compared to that from elongation so Cz* is only incre- mented in the elongation step (see Ref. 4 for justification). The second nucleation step, lateral growth, involves a signif- icant fraction of the total subunits polymerized. T o account for this fraction and for transfers from one width category to the next, C,* is incremented and CEl is decremented for i = 3 to 13. These sheets are also incremented to account for elongation.

Step 3. Elongation of All Sheets-In principle, elongation is much more complicated than for the case of actin, where there is only a single growth point/polymer end. Addition of a subunit to the end of a sheet can involve a single longitudinal bond or both a longitudinal and a lateral bond. Thus, there is a variable cooperativity to elongation depending on the width of the sheet and the location of the subunit (1). In the present analysis we will ignore the cooperativity of elongation and assume that addition and loss of subunits at the ends of sheets occurs independently on each protofilament. As justification, we may note that (a ) in the second half of the assembly, when nucleation is negligible, the elongation reaction does appear to be a simple second order reaction, and ( b ) the model is already sufficiently complex that addition of more parameters would not enhance its credibility. We therefore make the simplifying assumption that the elongation reaction is a sec- ond order addition of subunits to the end of each protofila- ment. The elongation of a sheet will be simply proportional to the width of the sheet, i. The increment in subunit concen- tration in each size sheet is

d C t = {iCi(k'Co - k-)]dt. (3)

This increment in polymer concentration is calculated for i = 2 through 13 following each cycle of lateral growth.

In the numerical integration, the kinetic parameters are not entered separately and ex2licitly as shown here. They are grouped into one experimentally fixed and two variable pa- rameters. The fixed parameter is k-/k+, which is equal to the experimentally determined critical concentration (2.85 pM for the buffer conditions used in these studies). The variable parameters are P, = (k-)'K,, and P,,, = k-K,,,. The parameters P, and P,,, are determined from the curve fitting by adjusting their values to give the best fit. The kinetic and nucleation equilibrium constants cannot be determined separately, how- ever, without additional and independent information that would specify k-.

EXPERIMENTAL PROCEDURES

Purification of Tubulin-Microtubule protein was purified by a procedure similar to that of Shelanski et a&. (14). The buffer used during the three cycles of polymerization was 100 mM MES, pH 6.5, 1 mM EGTA, 0.5 mM MgSO,, and 0.5 mM GTP, with 0.5 g of glycerol added per ml of solution prior to each assembly step. Pellets from the third cycle were resuspended in buffer as above but containing only 25 m M MES. Following the cold spin, the protein was passed over a phosphocellulose column (15) which had been equilibrated with the 25 mM MES buffer as above but with no GTP. The phosphocellulose was pretreated with 100 mM MgSO, as described by Williams and Detrich (16). To concentrate the purified tubulin from the phospho- cellulose column, solid L-glutamic acid (monosodium salt, Sigma) was

Kinetics of Microtubule Assembly 10433

added with gentle stirring in the cold to 1.0 M, as well as 1 mM EGTA and 0.5 mM GTP. The solution was then put to 37 "C for 20 min and centrifuged to collect the microtubules. The pellets were resuspended a t about 5 mg/ml in buffer A with 0.6 to 0.75 mM GTP. After 25 min on ice the tubulin was centrifuged at 4 'C and the supernatant was made 3.4 M in glycerol, aliquoted, and stored a t -80 "C until needed.

Preparation of Tubulin /or Assembly Studies-On the day of use the tubulin stored at -80 "C was taken through another cycle of assembly to eliminate any protein not competent to assemble. This involved addition of 5 mM MgSO, followed by a 20-min incubation at 37 "C, centrifugation to pellet the microtubules, and resuspension in assembly buffer (50 mM MES, pH 6.6, at room temperature, 3.4 M glycerol, 5 mM MgS04, 1 mM EGTA, and 1 mM GTP, buffer B) at 37 "C in a homogenizer prewarmed to 37 "C. After the microtubules were fully resuspended, the solution was put on ice for 20-30 min and then centrifuged at 4 "C. The supernatant was used as the starting material for assembly runs and was found to be pure tubulin by electrophoresis (17), (30 pg loaded on a 1.5-mm thick slab gel). In some experiments the microtubule pellet was cooled on ice for 30 min and ice-cold buffer was then added to resuspend the pellet. This procedure resulted in a high concentration of rings and slower assem- bly kinetics.

Assembly Procedures-Tubulin polymerization was monitored tur- bidimetrically at 350 nm with a Shimadzu UV-240 recording spectro- photometer equipped with a thermally jacketed cuvette holder. The cuvette had a 10-mm pathlength and was 2-mm wide internally. Prior to starting the assembly run, the diluted tubulin solution was degassed on ice for 10 min with a water aspirator and a 0.6-ml sample was transferred quickly to a glass culture tube (15 X 85 mm) sitting in a 40 "C water bath. After 10 s of gentle swirling, the solution was transferred immediately to the cuvette using a prewarmed plastic transfer pipette. It was verified with a small thermocouple probe that this procedure warmed the tubulin to 37 "C. The total warming time prior to the start of recording was approximately 15 s.

To determine the effect of the aging of the freshly prepared tubulin on the assembly kinetics, the foHowing experiment was performed. Starting at t = 0, two assembly runs were performed (9.4 and 14 p~ tubulin). At t = 3 h, the runs were repeated using the same concen- trations of tubulin. Comparison of the turbidity plots showed only a slight slowing of the early phases of the polymerization and in the 14 p~ case the slope of the middle portion of the reaction decreased slightly. Overall, the aging effect was much less than the difference between the neighboring curves.

Critical Concentration-After a full cycle of assembly to remove inactive protein, tubulin samples at 14, 22, and 32 pM in buffer B were incubated at 37 'C for 30 min. Then the microtubules were pelleted and the supernatants were analyzed by Lowry (18) assay. The tubulin concentration in the supernatants (3.0 to 3.25 pM) extrapolated to a critical concentration of 2.85 pM assuming a molec- ular weight of 100,000 for the dimer. The line through these points had only a slight positive slope indicating the presence of very little inactive tubulin.

Analysis of the Turbidity Plots-A Numonics 1224 digitizing plan- imeter was used to enter the turbidimetric time scans into a computer. The A350 values were converted to molarity of polymerized tubulin subunits by an empirically derived formula using a plot (Fig. 2) of plateau value of the turbidity uersw the Lowry-determined (18) total protein in each sample. The small departure from linearity was accounted for by incorporating a third order term in the curve (see legend to Fig. 2). From this curve the total tubulin concentration can be estimated from the plateau Am value. For fitting the theoretical curves we generally used the A350 value as the most accurate measure of the total active tubulin.

Electron Microscopy-Conventional negative staining (using car- bon-coated grids and 2% uranyl acetate) was used to examine the tubulin structures formed during the polymerization. Samples were taken carefully out of the cuvette in the spectrophotometer with a thin plastic pipette. To examine the very early stages of the reaction, say at 10 S, 60 p1 of tubulin solution was placed into a glass test tube (10 x 75 mm) held in ice. At t = 0 the tube was placed directly into a 37 "C water bath and agitated gently. At t = 10 s a specimen was quickly made SO that by about t = 25 s the uranyl acetate was applied.

Mkcellaneous Procedures-Tubulin concentrations were deter- mined by the method of Lowry et al. (18), using bovine serum albumin as a standard. Corrections were made for the differences between the Lowry extinction coefficients of bovine serum albumin and tubulin and for the effect of glycerol on the assay. The value A&'$Y = 1.19,

20.0 1 I I 0

FF 0.0 0.0 0.2 0.4 0.6 0.8

PLATEAU VALUE OF

FIG. 2. Relationship of the turbidity plateau at the end of assembly to total starting protein concentration. The total amount of tubulin in each sample (from Lowry assays done in triplicate) is plotted against the maximum A m value obtained for that sample upon polymerization. The curue drawn through the points was empirically derived with the constraint that it pass through the point 2.85 p M tubulin (the critical concentration) at A m = 0. The equation of the line is C," = 1.90 A3 + 19.22 A + 2.85 where Corn is the total concentration of tubulin subunits and A is Am. This equa- tion was routinely used to calculate C," from the plateau A3w, and to calculate the t o t a l polymer, C*, for each time point in Figs. 5-8.

10.01 1 I 1

1 I

4-1.0

0 . o L o ' I I 1 0

0 5 IO 15 20

TIME ( M m ) FIG. 3. The kinetics of microtubule assembly followed by

turbidity. The total polymer, C*, determined from turbidity using the relationship of Fig. 2, is shown (0) and the data are also plotted as In ( C L - C*) (0). The nucleation phase continues until about 6 min, after which polymerization is by elongation alone. The straight l i n e of the log plot for the elongation phase indicates that elongation is a second order reaction.

determined by refractive index increment studies, was used to quan- titate the tubulin initially.

RESULTS

On the day of use, tubulin from the freezer was taken through a cycle of assembly to eliminate all inactive protein. The particular choice of buffer and temperature conditions used to resuspend the pellets was eventually selected to avoid rings in the tubulin solution. In early experiments, we found that tubulin preparations containing a large number of rings at 0 "C had considerably slower nucleation kinetics than preparations lacking rings. Preparations containing rings were obtained when microtubules were disassembled, by cool- ing on ice, at a protein concentration above 50 pM. We found that we could reproducibly obtain preparations with very few rings if the pellets were first resuspended at 37 "C to a tubulin concentration of 30 PM or less, and then cooled to 0 "C to depolymerize the microtubules. There is apparently a critical concentration for ring formation upon the depolymerization

10434 Kinetics of Microtubule Assembly

assembly do fit a straight line on the log-log plot, the slope being 3.77. This implies that the nucleus is a 6-mer.

To obtain a more detailed analysis of nucleus size we used the procedure of Wegner and Engel (4) to fit the entire set of kinetic data with computer-generated curves. We first tried the simplest model of homogeneous nucleation followed by elongation. This simplest model is the one used for actin, and it ignores the complexities of two-dimensional polymeriza- tion. Different values of n, the size of the nucleus, were tested to determine the best fit to the microtubule kinetic data. The best fit, shown in Fig. 5, was obtained for n = 6. The concen- tration dependence of the kinetics is matched fairly well, as the theoretical curves fall near the experimental points for the whole range of concentrations. This fit and the value n = 6 are consistent with the analysis by the log-log plot of tloa versus concentration. The fit of each individual curve as a function of time is not very good, however. The considerable sigmoidal character of the experimental data is not reflected in the calculated curves. We also tried a modified theory with an added term for fragmentation, (5 , 19) but this did not give a significant improvement in the fit. We conclude that the simple model of homogeneous nucleation cannot fit the ki- netics of microtubule assembly.

We obtained a much better fitting using a more complex model that included heterogeneous nucleation. In this model, which was developed by Ferrone et al. (7) for hemoglobin S assembly, a term was added to the homogenous nucleation scheme to generate additional nuclei along the sides of exist- ing polymers. This heterogeneous nucleation is proportional to both the total length of the existing polymers and to the mth power of the monomer concentration. This model gave a much improved fit to the microtubule assembly kinetics. The best results were obtained using a hexameric primary nucleus and a hexameric species for the heterogeneous nucleation

LN [TOTAL TUBULIN CONC] (&MI

FIG. 4. Log-log plot of tenth-time uersus total starting protein concentration. The In of the time required for the turbidity to reach 10% of its maximum value ( t m ) is plotted against the In protein concentration (derived from the plateau value of the turbidity) for the two sets of eight assembly runs. The solid line was generated by linear regression and has a slope of -3.77 indicating a nucleus size of about 6 (see Footnote 2).

of microtubules. Once the rings were formed, however, they appeared to be stable for several hours at 0 "C, even after dilution to low protein concentration. Studies are currently underway to investigate this more fully. All of the kinetics measurements reported here used the ring-free tubulin prep- aration.

A typical plot of assembly as a function of time is shown in Fig. 3. The turbidity readings were converted to polymer concentration using the empirical relationship derived in Fig. 2, and the data are plotted both directly and on a logarithmic scale. Two phases of the reaction may be distinguished most clearly on the logarithmic plot. In the first phase, nucleation, the (negative) slope of the curve increases. In the second phase, elongation, the curve is a straight line. In the nuclea- tion phase, the number of microtubule ends is increasing, resulting in accelerated reaction kinetics. Nucleation becomes insignificant after about one-fourth to one-half of the tubulin is polymerized. The remainder of the assembly, the elongation phase, appears to be a simple second order addition of sub- units to microtubule ends. The linear logarithmic curve im- plies that the number of microtubule ends is constant in this phase.

A set of eight tubulin assembly runs was generated using protein concentrations from 6.7 to 19.0 FM. The whole set of eight runs encompassed a very large range of total assembly times (from 8 to 300 min), although the concentration of tubulin changed by less than a factor of three. A second, independent set of assembly runs was also obtained and analyzed. The assembly curves for these two data sets are shown in Figs. 6 and 9, respectively.

A simple indication of the size of the nucleus was obtained from a log-log plot of the tenth-time (tlo%, the time required for the turbidity to reach 10% of its maximum value) versus protein concentration. As discussed in the theory section, for a simple nucleation system the points on the log-log plot are expected to fall on a straight line of slope ( n + l ) /2 (however, see Footnote 2), where rz is the number of subunits in the nucleus (2, 3). Fig. 4 shows that the data from microtubule

We actually found that a slope of (n + 1.5)/2 was more accurate for our particular values of the critical concentration ( k - / k + ) and tubulin concentrations. If the critical concentration is reduced by a factor of 10, the slope becomes the expected (n + 1)/2.

I I I I L I

L I ^ ^ 1

10 15 2U I

4 0

0 0 1 I I I L I 0 50 100 150 200 250

T I M E (Mlnl FIG. 5. Experimental kinetics of tubulin polymerization

and an attempt to fit the data by a simple (one-step) homoge- nous nucleation model. The circles represent the experimental values for C* determined from AaW values. The starting tubulin concentrations Corn were 6.7, 8.3, 9.4, 10.1, 11.7, 13.8, 17.0, and 19.0 FM. Note the change in time scale between panels. The solid curues are the computed assembly plots based on the homogenous nucleation model with a hexamer nucleus as described in the text.

Kinetics of Microtubule Assembly 10435

component as well (data not shown). Notwithstanding the reasonably good fit to the experimen-

tal data, we felt that the heterogeneous nucleation model was not a realistic description of microtubule assembly. The path- way of assembly deduced from electron microscopy is, how- ever, closely related to this model. Nucleation of a new pro- tofilament to produce lateral growth can be treated as a heterogeneous nucleation. To obtain a more realistic model we modified the approach to relate the two nucleation steps explicitly to our proposed pathway for microtubule assembly. In this model, described in detail under “Assembly Model” above, the primary nucleation event is the formation of a two- filament sheet of n subunits, and the secondary nucleation event is the nucleation of a new protofilament of m subunits to achieve lateral growth.

For the numerical integration, four parameters were speci- fied for each computer run as follows: n, the number of subunits contained in the starting two-protofilament wide nucleus; m, the number of subunits needed to nucleate a new protofilament on existing sheets; and P, and P,, parameters proportional to K,, and K,. This best fit (Fig. 6) was obtained with n = 7 for the primary nucleus and m = 3 for the secondary nucleation of new protofilaments. The overall fit is reasonably good, both in matching the concentration dependence of assembly times and in matching the time course of each assembly curve. The fit is poorest for the very early assembly times, where the computer curves rise faster than experimen- tal data. Assembly number 2 (the second curve from the bottom) had an especially long apparent lag time not matched by the computer curve.

To obtain the best fit shown in Fig. 6, we made many computer runs with different values of n, m, and the nuclea- tion parameters. It is instructive to show the best fits obtained with different values of n and m. In Fig. 7 the value of m is kept equal to 3, but n is reduced to 5 and P, is also reduced

16 0

12 0

8 0 n E 3 - 4 0 e W

>- E

0 L

J 0 0

8 0

4 0

0.0 I

I I I I 1

I I I I 5 1 0 15 20

TIME (Mln) FIG. 6. The best fit to the experimental kinetics data ob-

tained from the model of two-dimensional nucleation and polymerization (Fig. 1). The circles are the experimental C’ values as in Fig. 5. The solid curves are the computed polymerization plots with the parameters n = 7, m = 3, P, = 4.0 X IOz2, and P,,, = 5.0 X 1O’O.

12 0

8 0 A

t a - 4 0

W

>- E

0 a _L 0 0

4 0

0 0 ~~

I I I I 0 50 100 150 200 250

TIME ( M l n )

FIG. 7. Computed polymerization as in Fig. 6 but with n = 5 and P. = 4.0 X 10”.

12 0

8 0 A

E - 4 0 a

E W E >-

II 0 -J 0.0

8.0 I- -I

4 0

0 0 I I I I I I 0 50 1 0 0 150 200 250

TIME ( M l n ) FIG. 8. Computed polymerization as in Fig. 6 with n = 5, m

= 5, P. = 4.0 X lo”, and P, = 5.0 x 1020.

to re-establish the best fit. Note that the overall spread or range of slopes of the curves has decreased, making the lowest and the upper three curves deviate substantially from the experimental points. Changing only the parameter m from 3 to 5, (with an appropriate change in E‘,) as shown in Fig. 8, causes the curves to reach plateau levels more slowly and causes a steeper initial rise for the higher protein concentra- tions. Fig. 8 is also interesting because it demonstrates that even though the sum of n and m is 10 as in Fig. 6, the curves generated are quite different. The overall spread of the slopes is inadequate to parallel the experimental data.

A completely separate set of tubulin assembly curves was

10436 Kinetics of Microtubule Assembly

1 6 . 0 cLq

8 . 0 r\

z 1 - 4.0 Clf W z > 0 -I 0.0 a

8 . 0

4 0

0 . 0

-

-

- I . I I I 5 10 15 20

0 50 100 150 200 250 T IME (Min)

FIG. 9. Experimental and computed polymerization plots form a second set of assembly runs as described in the text. The tubulin concentrations were 6.5, 7.6,9.3, 9.6, 11.1, 13.4, 16.0, and 17.9 WM. The solid curues are the polymerization plots generated by computer using the two-dimensional nucleation and polymerization model. The parameters were identical to those in Fig. 6 except that P, = 7.5 x IOZ2.

analyzed to verify the reproducibility of the experimental kinetics measurements. Fig. 9 is a plot of the experimental data and the computer-generated assembly curves for the best fit. Once again the best fitting was obtained with n = 7 and m = 3. The nucleation parameter was P , = 5 x IO1’, the same as for Fig. 6. The parameter P , had to be increased from 4 to 7.5 x lo2* to obtain the best fit, but a compromise value for this parameter would fit both sets of data fairly well.

The subunits incorporated into polymers are added by one of three mechanisms, initial formation of the nuclei, widening of sheets, and elongation. Ignoring the contribution of the first mechanism (which cannot be directly calculated and is assumed to be negligibly small), the percentage of C* due to elongation, and that due to the nucleation and “rapid” com- pletion of new protofilaments (see “Assembly Model”), was calculated for the theoretical curves in Fig. 6. A t the highest protein concentrations, lateral growth and filling in newly nucleated protofilaments accounted for a substantial fraction, 30-40%, of the total polymer formed. The elongation mecha- nism always accounted for more than half of the polymeriza- tion, and at the middle and lower protein concentrations 80- 90% of the subunits were incorporated into polymers in elongation.

We also determined the relative number and the average length of sheets of each size ( i = 2 to 13, the 13-filament sheets assumed to be intact microtubules) generated by the computer model. In summary, these model calculations showed that a significant fraction of polymers should be sheets of 2-12 protofilaments, and that at higher protein concentra- tions the sheets are shorter and narrower than at low protein concentration. This may be understood as reflecting the in- creased nucleation of two-filament sheets, which depends on a higher power of subunit concentration than either lateral growth or elongation.

Electron microscopy of negatively stained specimens con- firmed that the assembly began with the formation of sheets.

At low protein concentration (7.8 pM, similar to the second lowest curve of Fig. 6), a few rings and rare sheets were observed at 2 min. By 6 min the sheets had lengthened to a few micrometers and were more numerous (Fig. loa); the appearance was similar at 12 min. At 26 min (approximately tlow) the number of sheets was greatly increased and there were also many intact microtubules. A t 135 min (the turbidity plateau) the polymers were almost entirely intact microtu- bules, with 10-20% sheets, Fig. lob. Thus, the proposed pathway from small sheets to larger sheets to intact micro- tubules was confirmed for these assembly conditions at low protein concentration.

At higher protein concentration (19 p ~ , similar to the top curue of Fig. 6) the reaction proceeded much more rapidly. By 45 s, long sheets were already present in large numbers. At the turbidity plateau, the polymers were almost all long sheets, with very few intact microtubules. Many of the sheets were wider than 13 protofilaments and contained inversions that would prevent closure (20). The sheets observed by micros- copy are considerably wider than those predicted by the theory, but the interpretation of width is complicated by the inversions. The electron microscopy results agree with the model calculation in showing mostly intact microtubules at low protein concentration and sheets at high concentration.

The smallest assembly intermediates were observed in a specimen prepared approximately 10 s after warming, using the special procedure described under “Experimental Proce- dures.’’ The protein concentration was relatively high (16.5 p ~ ) . Fig. 1Oc shows the small polymers observed in this specimen. The substructure is not always clearly resolved, probably because it is obscured by the large amount of unpo- lymerized protein, but in favorable views the polymers are seen to be small sheets of two to five protofilaments and variable length. This image does not show the proposed two- filament critical nucleus (and indeed we do not expect to visualize this rarest intermediate) but it is important for confirming the general assembly pathway in which polymer- ization begins with very small sheets and proceeds by lateral growth and elongation.

DISCUSSION

The reasonably good fit of the computer curves to the experimental kinetics does not prove that our model is correct, but certain aspects seem well established. Perhaps the best established conclusion is that the size of the nucleus is about seven subunits. This value is essentially independent of the model used for interpretation. It is shown by the log-log plot of tenth-time uersus concentration (Fig. 4); by the fit to the simple homogeneous nucleation model, which, although never very good, was best for a hexamer nucleus (Fig. 5); and by the much better fit to the two-step nucleation model (Figs. 6 and 9) using a heptamer nucleus. It is clear that our fitting of experimental data to model curves is subjective, and our judgement that the fit is “reasonably good cannot establish the model as correct. Notwithstanding the subjectivity, the fitting does show that the two-step nucleation model is better than simple homogeneous nucleation and establishes the numbers 7 and 3 as the best estimates for the sizes of the primary and secondary nuclei.

The large size of the nucleus is in contrast to the dimer or trimer determined for actin assembly by essentially the same curve fitting procedure (3,5,6). Another significant difference between actin and tubulin is that the actin kinetics could be fit very well by a model of simple homogeneous nucleation, while the tubulin assembly required a heterogeneous nuclea- tion or two-step model. Two aspects of our interpretation are

”- . ”

FIG. 10. Electron micrographs of negatively stained specimens made at various stages of assembly. Details of the specimen preparation are given under “Experimental Procedures.” a, sheet structures observed at t = 6 min. Partially closed microtubules may be forming at this stage. The tubulin concentration was 7.8 p ~ . Magnification X 70,000. b, microtubules observed at 135 min in the same sample as in a. By this time the turbidity has reached its maximum value. Magnification X 70,000. c, small assembly intermediates. The specimen was prepared as described under “Experimental Procedures” starting at about 10 s after warming the tubulin to 37 “C. The tubulin concentration was 16.5 p ~ . Magnification X 120,000.

very dependent on the model, the structure of the primary nucleus and the size and structure of the second nucleus. These features are less well established than the size of the primary nucleus, but fit nicely into a unified interpretation, discussed in the final paragraph.

Carlier and Pantaloni (21) studied tubulin polymerization under conditions very similar to ours. They analyzed their results by a modified form of the equations of Oosawa and co-workers (11,23) and concluded that the size of the nucleus was probably 9-11 subunits (based on our definition of nu- cleus; 10-12 subunits based on the Oosawa definition). This is somewhat larger than the nucleus size 6-7 that we deter- mine here. To compare our results with theirs we made a log- log plot of kapp (the apparent rate constant for elongation) versus concentration for our data (Fig. 9), according to their method. Our data fit a straight line up to 14 p~ with a very slight decrease in slope from 14 to 18 pM. This range of linearity is more extensive than for the data of Carlier and Pantaloni, which fell off sharply above 12 p ~ . The slope of our line up to 14 p~ was 3.86, essentially the same as the slope of 3.75 of our log-log plot of tenth-times. Thus, the larger slope reported by Carlier and Pantaloni apparently reflects a difference in their data or in the way data was processed. There are several differences in experimental pro- cedure between the two laboratories. For example, they do not report any preprocessing of tubulin before the kinetics

measurement, whereas we did a cycle of assembly and disas- sembly to eliminate inactive tubulin and rings. Also, their temperature jump procedure and processing of the turbidity data are different from ours. Since our data give the same value for nucleus size both from log-log plots and by our more definitive curve fitting procedure, we believe our number of 6-7 subunits is the best estimate for the size of the primary nucleus under these experimental conditions. The previous study of Carlier and Pantaloni (21) is in qualitative agreement in demonstrating a cooperativity much larger than for actin.

The polymerization of pure tubulin in 8% dimethyl sulf- oxide was studied by Robinson and Engelborghs (22). Based on a plot of kapp (the apparent rate constant for elongation) versus the tubulin concentration, they concluded that the stoichiometry coefficient of nucleation, n, was equal to 2. Interpretation of their system was complicated, however, by the requirement that the dimethyl sulfoxide be added at high concentration and mixed with the tubulin a t 37 “C. Nucleation was probably occurring at the interface between the tubulin solution and the dimethyl sulfoxide, and therefore would not be expected to fit a model for homogeneous nucleation.

The simplest assumption for the structure of the nucleus is that it is simply a small piece of the microtubule wall and that intersubunit bonds are identical to those in the micro- tubule. On the basis of this assumption, we have previously concluded that any two-filament sheet would elongate spon-

10438 Kinetics of Microtubule Assembly

taneously above a certain “critical supersaturation” (1). In this case one would expect the critical nucleus to be a trimer. Our conclusion here that the nucleus is a heptamer suggests that the assembly is more complicated than this simplest model. Recent studies of actin assembly have demonstrated a similar disagreement with the simplest model. One would expect from the helical structure of actin that its nucleus would be a dimer. Fitting the kinetics to computer curves, however, demonstrated a dimer, trimer, or tetramer nucleus, depending on solution conditions (3,5,6). The larger nucleus found for both actin and microtubules suggests that the simplest model of rigid bonds, identical in the smallest inter- mediates and the larger polymers, is not completely valid. Smaller polymers appear to be less stable than predicted by summation of bonds, but the factors involved are not under- stood.

Since the critical nucleus is the least stable intermediate, it exists at a lower concentration than any other polymer. At- tempts to capture, isolate or even to visualize the nucleus in electron micrographs are therefore futile. Nevertheless, elec- tron microscopy has been extremely valuable in demonstrat- ing that the pathway of assembly is essentially as described in the model, in particular, that assembly begins with the formation and growth of small two-dimensional polymers.

We should point out that to obtain micrographs of the smallest intermediates it is important to work with the fastest assembly times and therefore the highest tubulin concentra- tions. The reason is that nucleation increases as the sixth or seventh power of concentration, while elongation increases only as the first power. Thus, at high protein concentration there will be a large number of nuclei formed, and if a specimen is prepared in the first few seconds the intermedi- ates will be captured before they have had a chance to grow very large. At low protein concentration the formation of nuclei is much rarer, especially relative to elongation. No polymers are found in the first few seconds, and after several minutes of assembly one sees only long polymers. This ration- ale of using the highest protein concentration to visualize the smallest intermediates should be of use in the study of other nucleated polymerization reactions.

The rapid equilibrium assumed for the nucleus does not mean that the heptamer is formed by a true seventh order reaction. The nucleus must be built up through a sequence of bimolecular reactions, either stepwise addition of single sub- units, or association of preformed intermediates. It is possible, for example, that the nucleus is formed by association of a three- and a four-subunit filament. The rapid equilibrium hypothesis simply means that the pathway before the nucleus has no effect on the kinetics.

Furthermore, the kinetics data and computer fit do not give

information on the structure of the nucleus, only on its size. Our interpretation that it comprises protofilaments of three and four subunits is a reasonable speculation. An attractive aspect of the complete theory is that the second nucleation step also involves of a protofilament of three subunits. This suggests that both steps of nucleation may be closely related. Any three subunits attached to the side of another filament or sheet can serve as a nucleus for a new protofilament. The primary nucleation is special in that the first protofilament has to be assembled also, and the whole complex is in rapid and unstable equilibrium with subunits.

Acknowledgment-We thank Lisa Doberstein for her expert tech- nical assistance.

REFERENCES 1. Erickson, H., and Pantaloni, D. (1981) Biophys. J. 34,293-309 2. Eaton, W. A., and Hofrichter, J. (1978) in Proceedings of the

Symposium on Biochemical and Clinical Aspects of Hemoglobin Abnormalities (Caughey, W. S., ed) pp. 443-457, Academic Press, New York

3. Tobacman, L. S., and Korn, E. D. (1983) J. Biol. Chem. 2 5 8 ,

4. Wegner, A., and Engel, J. (1975) Biophys. Chem. 3,215-225 5. Wegner, A., and Savko, P. (1982) Biochemistry 21,1909-1913 6. Cooper, J. A., Buhle, E. L., Jr., Walker, S. B., Tsong, T. Y., and

7. Ferrone, F. A., Hofrichter, J., Sunshine, H. R., and Eaton, W. A.

8. Erickson, H. P. (1974) J. Cell. Biol. 60, 153-167 9. Kirschner, M. W., Williams, R. C., Weingarten, M., and Gerhart,

J. C. (1974) Proc. NatLAcad. Sci. U. S.A. 71, 1159-1163 10. Frieden, C., and Goddette, D. W. (1983) Biochemistry 22,5836-

5843 11. Oosawa, F., and Kasai, M. (1962) J. Mol. Biol. 4,lO-21 12. Abraham, F. F. (1974) Homogeneous Nucleation Theory, Aca-

13. Zettlemoyer, A. C. (ed) (1969) Nucleation, Marcel Dekker, Inc.,

14. Shelanski, M. L., Gaskin, F., and Cantor, C. R. (1973) Proc. Natl.

15. Weingarten, M. D., Lockwood, A. H., Hwo, S.-Y., and Kirschner,

16. Williams, R. C., Jr., and Detrich, H. W., 111 (1979) Biochemistry

17. Laemmli, U. K. (1970) Nature (Lo&.) 227,680-685 18. Lowry, 0. H., Rosebrough, N. J., Farr, A. L., and Randall, R. J.

19. Wegner, A. (1982) Nature (Lord.) 296,266-267 20. Erickson, H. P. (1978) in Electron Microscopy 1978, Vol I11

(Sturgess, J. M., ed) pp. 483-494, Microscopical Society of Canada, Toronto

21. Carlier, M. F., and Panaloni, D. (1978) Biochernitry 1 7 , 1908- 1915

22. Robinson, J., and Engelborghs, Y. (1982) J. Biol. Chem. 257 ,

23. Oosawa, F., and Asakura, S. (1975) Thermodynamics of the PO- lymerization of Protein, pp. 41-55, Academic Press, New York

3207-3214

Pollard, T. D. (1983) Biochemistry 22,2193-2202

(1980) Biophys. J. 32 ,361380

demic Press, New York

New York

Acad. Sci. U. S. A. 70,765-768

M. W. (1975) Proc. Natl. Acad. Sci. U. S. A. 72,1858-1862

IS, 2499-2503

(1951) J. Biol. Chem. 193,265-275

5367-5371