Biochimica et Biophysica Acta, 400 (1975) 263-274 © Elsevier Scientific Publishing Company, Amsterdam- Printed in The Netherlands

BBA 37109

KINETIC ANALYSIS OF THE POLYMERIZATION PROCESS OF ACTIN

FUMIO ARISAKA*, HARUHIKO NODA and KOSCAK MARUYAMA Department of Biophysics and Biochemistry, Faculty of Science, University of Tokyo, Hongo, Tokyo 113 and Department of Biophysics, Faculty of Science, University of Kyoto, Sakyo-ku, Kyoto 606, (Japan) (Received January 13th, 1975)

SUMMARY

The polymerization process of actin was examined by measuring the amount of flow birefringence and by analyzing release of labeled inorganic phosphate from the bound [7-32p]ATP upon polymerization of G-actin to F-actin.

Comparison of the above experimental results with the electron microscopic data of Kawamura and Maruyama (J. Biochem., 67, 437-457, 1970) suggested that growth and redistribution steps occurred simultaneously during polymerization.

Attempt was made to simulate the polymerization process of actin by calcu- lating the kinetic equations numerically. The results of simulation suggested that it was necessary to take into consideration the association and dissociation between F-actin particles as well as the association and dissociation between F-actin and G-actin.

INTRODUCTION

Changes in the distribution of particle length of F-actin during polymerization have been observed with electron microscopy by Kawamura and Maruyama [1 ]. They also measured flow birefringence under the same conditions. The experimental results were explained by a model of polymerization process which included three steps; nucleation, growth and redistribution [2, 3]. In their succeeding report [4], they derived kinetic equations for polymerization of actin from Mg-polymer. In those kinetic equations, the reactions between polymers were neglected and the rate con- stants k+ and k_, i.e. the rate constants of the forward reaction of association and the reverse reaction, respectively, were assumed to be independent of the length of polymer.

In the present study, the change in flow birefringence was reexamined and the release of labeled inorganic phosphate during polymerization from [~,-32p]ATP bound to G-actin was analyzed. The time course of both experiments coincided well, and showed that only a third of G-actin had polymerized during the initial 1 h of incu-

* Present address: Department of Biochemistry and Biophysics, Oregon State University, Corvallis, Oregon 97331, U.S.A.

264

bation, when the actin concentration was about 0.1 mg/ml, which was the concen- tration at which the electron microscopic data were obtained. According to the elec- tron microscopic examination, however, the redistribution step of polymerization had already started five or ten minutes after the addition of salt. This information indicates that the growth and redistribution steps occur simultaneously.

We formulated kinetic equations for the polymerization of actin according to the usual scheme which includes nucleation, growth and their reverse reactions. From these equations we computed theoretical curves for the time course of the poly- merization using various values of the rate constants. In these calculations we first assumed that the forward reaction of polymerization and its reverse reaction were independent of the length of F-actin, and neglected the association and dissociation between F-aetin particles, it became clear that as long as we used the kinetic equations mentioned above, we could not choose suitable values for parameters so that the growth step and the redistribution step occur simultaneously.

Other terms were then added to the equations to represent the association and dissociation of F-actin particles, and the dependence of rate constants on length was taken into consideration. This modification improved the results of the calculations by allowing the growth step and redistribution step to occur at about the same time. Although these considerations are not enough to cover all aspects of the experimental results, we should like to present the simulation procedure.

EXPERIMENTAL PROCEDURE

Actin was prepared from acetone powder of rabbit skeletal muscle, according to Supudich and Watt [5]. [y-a2P]ATP was purchased from Radiochemical Centre, Amersham.

Flow birefringence and extinction angle were measured at 25 °C in an appa- ratus of Edsall type (Rao Instrument Co.) at a velocity gradient of 100 s -~. Poly- merization was initiated by mixing equal volumes of salt-free actin solution and 0.2 M KCI, 20 mM tris-maleate (pH 6.8) to make final KC1 concentration of 0.1 M.

Release of inorganic phosphate (Pi) was measured as follows. G-actin in 0.2 mM ATP and 0.3 mM sodium bicarbonate was mixed with 1/15 volume of Dowex 1 suspension at 0 °C for 10 min with gentle stirring to remove excess free ATP [6]. The mixture was then filtered to remove the resin, and [7-azP]ATP was added to the filtrate to make total radioactivity of 0.2 #Ci. Incubation was continued for 20 min at 0 °C [7]. The [7-3EP]ATP which had not been taken up by G-actin was removed with Dowex 1 by the same procedure as above. About 7 0 - 8 0 ~ of the [7-32P]ATP was incorporated into G-actin, estimated by the comparison of the radio- activity before and after the resin treatment.

The G-actin solution was then polymerized in the usual way. Aliquots were taken from the solution during polymerization, and trichloroacetic acid was added to a concentration of 5 ~ . The solution was then centrifuged, filtered and P~ was ex- tracted from the filtrate by benzene-isobutanol according to the method of Martin and Doty [8].

A dioxane-based scintillation fluid was used with 2,5-diphenyloxazole(PPO) and 1,4-Bis-2-(4-methyl-5-phenyl-oxazolyl)-benzene (dimethyl POPOP) as scintil- lators.

265

Protein concentration was determined by the biuret method. Computation of the assumed kinetic formulae (see Numerical Calculation and

Discussion) was carried out on a HITAC 8800/8700 computer at the Computer Center, University of Tokyo, using the Runge-Kutta-Gill method [9].

RESULTS

Flow birefringence The time course of the increase in the amount of flow birefringence during

polymerization at various concentrations of actin is shown in Fig. 1. As the concen- tration of actin is lowered, the rate of increase of flow birefringence diminishes. At around O. 1 mg/ml of actin, at which point the observation by Kawamura and Maru- yama [1] was made, the manner of increase of birefringence agrees well with their results.

Measurement of P, release The time course of P~ release during polymerization is shown in Fig. 2. In a

manner similar to the increase in flow birefringence, the rate of P~ release decreases as the concentration of actin becomes lower. During the initial 45 min, about 1/3 of actin was polymerized at a concentration of 0.14 mg/ml.

The above experimental results of flow birefringence and Pl release show that,

5O

o

w 3 0

~ 2 0

!,0 X m,i

0

// -o-

-X -

.~x- //

! -a-

I I

X, x

1'0 z'0 6 5'0 40 TIME (rain)

Fig. 1. Changes in the amount of flow birefringence and extinction angle during polymerization of actin at a velocity gradient of 100 s -1. Actin concentration: O - - O , 0.35 rng/ml; × - - x , 0.21 mg/ml; Z~--A, 0.14 mg/ml.

266

0.5

o 0 3 o

~BO.2

E -

-O-

H

-x-

-A-

//

s'o o o IO 20 30 4 0

T I M E (rain)

Fig. 2. Increase of Pt release during polymerization of actin. (cpm)0 is the background, and (cpm)~ is measured at 180 rain or later. Actin concentration: ©--(), 0.50mg/ml; × - - × , 0.37 mg/ml; A - - ~ , 0.14 mg/ml.

under these experimental conditions, i.e. at an actin concentration of 0.1 mg/ml, monomer G-actin is consumed very slowly. The results justify the assumption in our calculations (see below).

Changes in the length distribution In electron microscopic measurements, the absolute concentration of F-actin

cannot be determined. The length distribution observed in electron micrographs is therefore shown as percentage of the number of F-actin of each length observed in the field. For this reason, the histograms of different stages of reaction cannot be compared. The histogram in percentage, however, may be transformed into that of relative molar concentration with the help of Sadron's formula [10], which gives the birefringence of a polydisperse system. By comparing the apparent flow birefringence based on the electron micrograph and Sadron's formula, with the experimental data of flow birefringence, the scale of each histogram may be calibrated. It was thus possible to compare the relative quantity of F-actin particles of different lengths at each step of polymerization. The resulting histograms are shown in Fig. 3. The results of the model calculation (see Numerical Calculation and Discussion) are to be com- pared with these histograms.

3 rain 5 rain I 0 min 8 0 mln

,,,,ItllL,,, . . . . J , , ............. ............. ldd,l,t, . . . .

I 2 0 I 2 3 4 0 I ~ 3 4 O I 2 3 4

PARTICLE LENGTH (ym) 0 ! 2 3 4 5

Fig. 3. Normalized curves for the polymerization process of actin. The shape was from electron microscopic data [1 ]. Values for concentration were deduced from data of flow birefringence by using Sadron's formula.

267

NUMERICAL CALCULATION AND DISCUSSION

Assuming that the polymerization reaction proceeds through steps of nucle- ation, growth and corresponding reverse reactions, we may write

dC1 f+l e Z k+C1C~ -]- Z k _ C ~ (1)

d t i=io 1=i0+1 _ _ - - iok + C~ ° q- iok_ C~ o - -

dC2__ __-- dClo-1 = 0 dt . . . . . dt

d C l o * ~ * : k + C l ° - - k - C ~ o - - k + C 1 C l o + k - C l o - 1

d t

de, dt

d C ~ _ k + C ~ C f _ ~ - - k _ C ~ d t

- - k + C 1 C l - 1 - - k _ C ~ + k _ C l + l - - k + C 1 C l (i----i0 + 1, it -1- 2 . . . . . . . f - - 1)

! (2) dm = k + F m [

d t J

where C~ is the molar concentration of F-actin of i-mer, it is the number of monomers in a nucleus, f i s the upper limit of i, and k~., k+, k_*, k_ are the rate constants. The last equation and the upper limit of the summation were arbitrarily chosen to make calculations possible. In this case, it was assumed that it was 3 [1, 11] a n d f w a s 200. The value of f was much smaller than the actual number, considering that F-actin of 4 or 5 #m long, whose degree of polymerization was about 1600 or 2000 might be observed in electron micrographs. We may, however, make the results of calculations correspond to the experimental data, if we assume that each monomer in the above equations represents a length o f F-actin, e.g. a 50-mer in the above equations is sup- posed to represent a polymer of 400 to 500 monomers.

In the above equations, we neglected the polymer-polymer association and scission of a polymer, i.e. we considered only association and dissociation of a polymer and a monomer. Furthermore, we assumed that all the k+'s and k_'s were respec- tively the same. The change in time of the distribution of particle length of F-actin during polymerization was obtained by solving the above equations numerically with appropriate initial conditions. We first set the initial conditions as CI = 2 #M, C~ = 0 (i ---- 2, 3, . . . . . . 200) at t ---- 0 and estimated roughly the rate constants from experimental data, such as the critical monomer concentration, the number average particle length of F-actin and the rate of polymerization. Afterwards, we sought better values for rate constants by a trial and error method, comparing the results of calcu- lation with the experimental data.

We also calculated from Fig. 2 the value of it, i.e. the number of monomers composing a nucleus. We may describe the initial reaction of polymerization approxi- mately as follows:

dF * t ) d t = k + m o [

268

where F is the molar concentration of polymers and m is that for monomers. The above equations may be solved with the initial conditions [11, 12] of m m0 and F = 0 a t t = 0 .

V (1/ ) ~'" +" '0 ] . iO F = i~ok~- tanh ~ iok+k+mot

1 . * io m m0 cosh ~ t o k + k + m o t

We may approximate ;7; within experimental error (t is not very small):

V . In m 2 2k+k+ • - - - - ~ - - In 2 -- . m'o°/2 l

mo io to

Then

( dm In - - -d~- i t=o = (1 + @ - ) In mo + constant (3)

From the above Eqn 3, we obtain the value of i0 from Fig. 3 of approximately 2.8. This value is to be compared with a value of 2.55 obtained from the dependence on protein concentration of the number average particle length of F-actin at equilib- rium [1]. In that calculation, however, it was assumed that the number of F-actin particles which were formed by nucleation did not change by reactions between polymers. A value of 3.5 for i0 has also been reported by measuring the concentration dependence of initial Pi release [11 ]. In this case, however, Pi release corresponded to - - d m / d t , but not to dF/dt , and the value of 3.5 should correspond to 1 + (i0/2), i.e. i 0 = 5.0. This value seems to be too large. However, the value of (dm/dt) t .~o was obtained from measurements at much higher protein concentrations and over a longer period of time by Kasai et al. [11] than by our technique. Eqn 2 may be a poor approximation at higher concentrations of F-actin.

I f we set k+ --~ 6 . 0 " 1 0 9 M-2"min - I , k+ ~ 1.8"107 M - l ' m i n -1 and k_* k_ = 0 in Eqn 1, where all the reverse reactions are neglected and only the nucleation and the growth reaction are taken into consideration, the polymerization ceases when all the monomers are incorporated into the polymers as shown in Fig. 4. The resulting distribution of particle length has a maximum.

When we maintain k~ and k+ the same as above, and set k_ -- 6.0 min -~ and k_* = 0.0, the peak in the distribution curve begins to collapse at about 10 min and the distribution curve becomes almost exponential at 100 min, as shown in Fig. 5. I f we calculate the time course of the number average degree of polymerization, < i > , , , it decreases as the length distribution approaches an exponential curve as shown in Fig. 6a. This does not agree with the electron microscopic data. With k ~, k+ and k_ the same as above and k_* ~ 0.2 rain -~, the qualitative feature of the redistribution remains unchanged, but < i ) , does not decrease during redistribution as shown in Fig. 6b.

269

IO

8 2 _o~ X

G

# i . 5 rain

: \ /

0 5'0 \" i- MER

10

IO0 O- 160

IO mln

/ .

5"o i - MER

Fig. 4. Calculated polymerization process of actin, k~. = 6.0" 109 M- Z 'min -~, k+ = 1.8.107 M -1- min -1 and k* = k_ = 0. Actin concentration was 2/~M.

In order to compare qualitatively the calculation with the experimental data, two parameters R and T were introduced. R is a quantity related to the sharpness of the maximum that appear in the length distribution, and T represents the rate of redistribution.

R=bT _ t a t o

where a is the concentration of trimers (nuclei) and b is the concentration at the peak in the distribution, to is the time when the highest peak occurs and t is the time when the concentration at the peak and that of trimers become equal, as shown in Fig. 7.

I f we set k+ = 1.8. l0 T M - l - m i n -1, k* = 0.2 min -1 and vary k~_ and k_, the parameters R and T defined above varies as shown in Table I. Values of k~_ and k_ are determined so as to make values of R and T to fit experimental data. The value of k* is also determined to fit the time comse of < i > , , as mentioned previously.

i 2.5rain IO IO

o / ,%, 0 b . 0 50 I00 ( o i

Z i011 40mtn ! I " , ,o 5 8,

IOmin I° l 20min

col '.

IOOmin

\ , "'%.~_

5 0 I00

IC 1 200mln \

¢ ~ ~o IOO Fig. 5. Calculated polymerization process of actin, k* = 6.0.109 M-Z.min -1, k+ = 1.8. l0 T M -1. min -1, k_ = 6 .0min -1 and k*_ = 0.

270

6O

4 0 X/

20

60

4 0

2 0

a ratio

I&O 2bo b

~ o o

o ebo TIME (min)

Fig. 6. T ime course o f average degree o f polymerizat ion. © I © , n u m b e r average; O - - O , weight average; × - - × , ratio of weight average to n u m b e r average.

t-t, t-t

R- b T =_t_- CI , t o

Fig. 7. Defini t ion o f paramete rs R and T.

T A B L E I

A D J U S T M E N T O F P A R A M E T E R S , R A N D T

Values o f R and T were compu t ed by sett ing k + = 1.8.10 7 M-1 . min-1 and k* ~ 0.2 min-~, and by varying the values for k* and k_ .

k~ R T

k_ 6.0 8.0 6.0 8.0

6 ' 108 O. 17 0.29 19.9 9.6 3.109 0.21 0.34 9.6 3.9 6.109 0.23 0.39 6.8 2.9 Exper imenta l

values 0.1 ~ 0.3 5 ~ 6

271

b

o

o so ,~o o so ,~o o 50 Ioo Z O

~ , o Z IM O Z 0 5

i 0 flO I00 150

I 1 9 0 ra in

0 5 0 I00 150

Fig. 8. Calculated polymerization process of actin, k~. = 1.5.101° M-2.min -t, k+ = 4.5.10 7 M -1. min -1, k_ = 15.0 min -1 and k*- = 0.4 min -1.

Accordingly, if we take the overall rate of the polymerization reaction into consideration, the four parameters of the rate constants are determined almost uniquely. The resultant rate constants are k+ = 4.5.107 M - l . m i n -1, k_ = 15.0 min -1, k+ ---- 1.5.10 l° M-2 .m i n -1, k* ---- 0.5 min -~ and the calculations with these constants are shown in Fig. 8. In Fig. 9a, the number and weight average degree of polymerization and their ratio are plotted as a function of time, and in Fig. 9b, the time course of monomer concentration is shown. In Fig. 9c, the time course of flow birefringence and extinction angle expected from the calculation are shown.

It is clear from Figs. 8 and 9, that in this calculation the growth step in poly- merization is completed very early in the process and the redistribution follows the growth step. However, as is stated already, experimental results suggest that the growth and redistribution proceed simultaneously (see Figs. 1, 2 and 3). Further- more, as shown in Fig. 9b, the computed concentration of monomers at equilibrium, i.e. the critical concentration, is much higher than that obtained experimentally. I f we give a smaller value of k_, in order to lower the critical concentration, the time course of polymerization will not agree well with experiments. We also solved the Eqn 1 with other values of initial concentration of G-actin, but the qualitative features of the result remained unchanged.

It was next assumed that the longer the F-actin particles were, the smaller the values of k+ 's were. The results showed the effect we expected, but were insufficient. Therefore, at tempt was made to take the reaction between F-actin particles into consideration.

I f polymer-polymer reactions are included, the kinetic equations are as follows :

- - io~+C~ ° + io~:-C,o - s k I + c l c , + z k~_c , l=lo 1 = 1 0 + 1

(

__ dC3 __ -- dC~°-~ = 0 [ dC2 dt dt . . . . . dt l

(4)

272

(a )

60 ~ / /

• ~ 40 ~ o- . - - .o.- / /

0 30 6 0 90

"~ I

i --x o ab

~, 4 5 (.9

- 0 -

- O -

° X -

19o

g ao 0 1 . -

~=" i5 p ~

x x - / / - X -

6'o 9'0

(G) g

2~_-"

o m

X la

o , 'o i, o ' ' 180 440 TIME ( nlin )

Fig. 9. Time course of several quantities calculated from Fig. 8. (a) The number average, ( (3 - -0 ) and weight average, (I~--Q); degree of polymerization and their ratio, ( × - - × ) . (b) The monomer concentration. (c) Flow birefringenee (0---0) and extinction angle (©--©) calculated by using Sa- dron's formula.

_ _ * ' * L I o + l ( - v i o _ _ io dCio k+C~ ° -- k -C lo q- r~l- ~io+1 -- kl+CICio ~' kl+CioCl dt ]=i0

- k '° c 2 + S k~_ +mc,0+. (5) i o+ io m = i o

+ k2'°C2,o IO

dCl

dt

2J<~i [ - - 1 i i " i - - j

-- kl+ ClCi-1 k l - C i -- k l+ClCi ~- k~+-lCi+l -~- Z' k3+ CjCi_ , J=lO

2k< i i i i 2 l,.i + ml,~ 2i

-- k i- C2i - - kk_ Ci S - - k l + C t C i ki+Ci + S ,~,,_ ~i+,,, ÷ k = i o 1 = 3 m = i o

( i z i o q - l , io ~ 2 . . . . . . )

(6)

where k~+ and k~._ represent the rate constants for the association of i -mer and j -mer to produce (i + j ) -mer and the dissociation of i-mer to p roduce j -mer and (i - - j ) -mer , respectively. It is obvious that

kJi + ~ i i i kj+, k j_ ~ k , _ j ) _

273

Eqns 4--6 must satisfy the following relationship, because of the conservation of total mass of actin (Eqn 1 also satisfy Eqn 7).

-- 2~ i dCl d Z iC~ = = 0 (7) dt dt

Since the particle length of F-actin shows an exponential distribution at equilibrium [1 ], there exists the following relationship between rate constants:

k~+ C~+j ~I+ ~ c, cj

- - -- K (a constant independent of i and j ) (8)

We calculated the Eqns 4-6 with the same initial conditions used in the calcu- lation of Eqn 1, setting the upper limit of polymerization at 100-mer with a tentative dependence of rate constants on the degree of polymerization as follows:

- l+J- r - e- ati+J), k~+ and Kl- a c proportional to where a is a constant. (9)

The relationship in Eqn 9 satisfies Eqn 8. In performing the numerical calculations,

O ~

x ~

Z o

0 Z

8 0

o

: : t - l . 5

0 &

• • t - 2 . 0

t= 7 ' . 5

t -1.0

- . . - . .

. . "... . °% _ :o %, °.

i%1, i i .... i i "'r,l, i

l o g o o zo" 20 o lo zo 3o o I'o 20 3 o 4 o i

,d

rn o~.0

!o

]] • o~

2 4 t 6 8

Fig. 10. (a) Calculated polymerization process of actin by Eqns (2-4). (b) Time course of monomer concentration and flow birefringonce calculated from Fig. 9.

274

those terms in the summations in Eqns 5 and 6 which contribute predominantly were selected.

In Fig. 10a, a result of the calculation is shown. In this calculation, we set 0.08 for a, 1.5.107 M-1 .min -~ as the proportionality constant for the forward reac- tion of polymerization, 1.0 min -1 for the reverse reaction, k ; : 6. l09 M-2 min-1 and k_* ~ 0.05 min- ~. The time course of the change of monomer concentration and the expected values of flow birefringence calculated from Fig. 10a using Sadron's formula are shown in Fig. 10b.

The results in Fig. 10 show that we could improve in the following two respects: 1. The growth and redistribution occurred almost simultaneously. 2. The monomer concentration at equilibrium could be lowered. The second point was achieved because the additional terms of polymer-

polymer reaction facilitated the collapse of the peak, and the value of k_ could be reduced.

However, the results are not in good agreement, since the way of collapse of the peak was not exactly that of the experimental results. In Figs 3 and 8, the position of the peak of concentration did not change during redistribution, but in Fig. 10 the position of the peak moved to the left during redistribution. This point may not be improved by adding more terms of polymer-polymer reaction. Therefore, it appears that actual polymerization process ofactin is not as simple as has been suggested [3].

ACKNOWLEDGMENT

This work was partly supported by the Science Research Grant from the Ministry of Education.

REFERENCES

1 Kawamura, M. and Maruyama, K. (1970) J. Biochem. 67, 437-458 2 Waugh, D. F. (1957) J. Cell. Comp. Physiol. 49, 145-164 30osawa, F. (1970) J. Theoret. Biol. 27, 69-86 4 Arisaka, F., Kawamura, M. and Maruyama, K. (1973) J. Biochem. 73, 1211-1215 5 Sl0udich, J. A. and Watt, S. (1971) J. Biol. Chem. 246, 4866--4871 6 Asakura, S. (1961) Arch. Biochem. Biophys. 92, 140-149 7 Martonosi, A., Gouvea, M. A. and Gergely, J. (1960) J. Biol. Chem. 235, 1700-1703 8 Martin, J. B. and Doty, D. M. (1949) J. Anal. Chem. 21,965-968 9 Iri, M. and Matsutani, Y. (1967) Johoshori 8, 103-107

10 Sadron, C. (1938) J. Phys. Radium (7)9, 381-384 i 1 Kasai, M., Asakura, S. and Oosawa, F. (1962) Biochim. Biophys. Acta 57, 22-31 12 Katsura, I. and Noda, H. (1971) J. Biochem. 69, 219-229