Quasielastic Light Scattering Study of Thermal Excitations of F-Actin Solutions and of Growth Kinetics of Actin Filaments TH. PIEKENBROCK and E. SACKMANN

Physik Department (Biophysics Laboratory),Technische Universitat Munchen, D-8046 Garching, Germany

SYNOPSIS

In the first part of this work we report quasielastic light scattering (QELS) studies of the internal dynamics of transient actin networks over a time range of 10-6-10-2 s, scattering angles between { = 20" and 150°, and a concentration range of 0.015 (0.3) to 0.7 mg/mL (15 p M ) . We confirm our previous result that (1)the dynamic structure factor g ( q , t ) is determined by the thermally excited undulations of the actin filaments and ( 2 ) that the initial decay of g ( q , t ) scales as g ( q , t ) K exp( - q " t ) while the long time decay scales as g ( q , t ) cc exp[- (Aq"t)"'] with 01 = 2.75. The deviation of 01 from the theoretical value of 01 = 3 predicted for Rouse-Zimm chains is similar to that found for high molecular weight macromolecular solutions by QELS. A refined analysis of the dynamic structure factor showed that it can be interpreted in terms of three relaxation processes (besides the contribution of the residual monomer diffusion) : ( 1) the dominant Rouse-Zimm dynamics, which comprises between 65 ( a t high concentrations) and 85% of the signal; ( 2 ) a fast relaxation process with a decay constant of r = 9 X lo3 s-', which contributes at all concentrations with the same amplitude; and ( 3 ) a nonexponential ultraslow contribution ] 1'4. The third contribution appears only at high concenof the form g,, K exp [ ( - rust) trations and increases strongly with decreasing scattering angles. It is thus attributed to fluctuations of the mesh size of the transient actin network. In the second part we show that high sensitivity QELS may be applied to follow the actin polymerization process at low temperatures ( 10'C). The apparent diffusion coefficient and the static scattering intensity of the actin filaments were determined as functions of polymerization time tpol.We show that the process consists of the rapid growth of a few filaments that become very long ( x 1 0 pm; even at actin concentrations of 0.04 pg/mL) near the critical growth concentration of 0.012 pg/mL, as is expected for a growth process determined by nucleation. Finally, we studied actin networks polymerized in the presence of complexes of gelsolin with actin. By application of the CONTIN program we could determine the length distribution of the filaments. The very broad length distribution is nearly exponential, quite analogous to the distribution predicted for polymers grown by the polycondensation process; that is the association of monomers and oligomers. 0 1992 John Wiley & Sons, Inc.

INTRODUCTION Solutions of actin polymers exhibit viscoelastic behavior even a t very low actin concentrations, which is a consequence of the transient entanglement of the very long filaments.'-4 At low actin concentrations ( t l mg/mL) the actin solutions behave like Biopolymers, Vol. 32, 1471-1489 (1992) Q 1992 John Wiley & Sons, Inc.

CCC 0006-3525/92/111471-19

* To whom correspondence should be addressed.

dilute and semidilute Newtonian polymer solution^,^ whereas a t higher concentrations ( > 5 mg/mL) one observes pronounced nonlinear effects such as shear thinning' and thixotropy.* According to Kerst et a L 5 this is the consequence of the heterogeneous microstructure of the polymer solutions consisting of partially ordered liquid crystalline domains. As demonstrated by dynamic light scattering experiments6x7 and measurements of polystyrene latex sphere diffusion, the actin solutions exhibit a homogeneous structure characterized by a well-defined mesh size 1471

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PIEKENBROCK AND SACKMANN

in the low concentration regime.g In this case the internal dynamics of the separate actin filaments may be evaluated quantitatively by dynamic light s ~ a t t e r i n g ~and , ~ .low ~ shear rate measurements of the viscoelastic constants: Such measurements allow the detection of variations of the actin-chain flexibility by the method of polymerization (e.g., absence or presence of A T P ) ' or of subtle changes caused by actin binding proteins (e.g., tropomyosin) or cross- linker^.^ T h e present work is a continuation of a previous quasielastic light scattering (QELS) study of the internal dynamics of actin filaments where we provided strong evidence that F-actin solutions exhibit close analogies to solutions of Rouse-Zimm chains and posess features typical of the Rouse-Zimm model.4 In particular, we found that the actin filaments are much more flexible than anticipated hitherto.'" T h e close analogy between transient networks of F-actin and high molecular weight macromolecular solutions was further verified in recent studies of the frequency dependence of viscoelastic moduli of transiently and chemically cross-linked actin networks by Muller et al.4 At low shear rates and low actin concentrations (volume fractions of C#I = we observed two relaxation regimes: ( 1) a terminal process corresponding to the transition from a rubber-like plateau to viscous flow (characterized by a terminal relaxation time T d ) and ( 2 ) a relaxation process a t a frequency w > 7L1 determined by the internal dynamics of the filaments. A number of power laws were established. At w > 7 i 1 the storage and the loss moduli scale with the square root of the frequency a s is typical for Rouse chains. T h e terminal relaxation time 7 d scaled with the fifth power of the length L , that is, the exponent is slightly higher than found for macromolecular solutions ( 7d K L34 ) . T h e concentration dependence of the plateau modulus Gg is characterized by a power law Gfi K CA7'O2 as is typical for solutions of flexible polymers. From the power laws we obtained a value of the persistence length, L, N 0.10.3 pm, which was found to be in good agreement with values based on electron microscopy measurements of end-to-end distances of actin filaments cleared by severin. In the present paper we first present a more detailed QELS study of the internal dynamics of the actin network. We extended the measurements t o much lower actin concentrations: that is, just below the critical concentration of association. The light intensity correlation function was measured over a much larger time scale ( 10p6-101 s ) ,which enabled

us to explore systematic deviations from the RouseZimm dynamics in the long- and short-time regimes. In the second part we first report measurements of the growth kinetics of actin filaments by QELS. Simultaneous measurements of the apparent diffusion coefficient of the filaments and the scattering intensity show that the process consists of the rapid growth of a few chains (after the well-known nucleation period) rather than of a simultaneous growth of many chains, in agreement with the Oosawa model. In the third part we study the dynamics of Factin solutions grown in the presence of gelsolinactin dimer complexes (GAC). We show that the width of the length distribution can be obtained by analysis of the dynamic structure factor by the CONTIN algorithm.

MATERIALS A N D METHODS Proteins

Actin was prepared from rabbit muscle according to the method of Spudich and Watt." Gelsolin-actin (1 : 1) complex was provided by Dr. A. Wegner (Universitat Bochum). Special care was taken to avoid contamination by actin binding proteins. Material was only collected from the end of the gel chromatography band, while that of the leading end was discarded. Evidence for the absence of substantial amounts of actin binding proteins is also provided by the fact that the nucleation time found in this work agrees well with that reported for pure actin.'" Buffer

T h e F buffer contained 10 m M imidazol, 1 m M ATP, 1 m M EGTA, and 2 m M M g Z f .T h e G buffer consisted of 2 m M Tris containing 1 m M ATP, 0.5 m M DDT, and 0.2 m M Ca2+.For the experiments with gelsolin, 2 m M Ca2+was used instead of EGTA in the F buffer. light Scattering Apparatus

The basic setup of the light scattering apparatus has been described e l ~ e w h e r eT . ~h e following modifications were made: T h e 514.5-nm spectral line of a n "Innova 70-4" (Coherent) laser was used as a light source, with a n output power of 0.05-0.5 W. Instead of a 64-channel linear time scale digital au-

QELS OF THERMAL EXCITATIONS

tocorralator in a multiplexing operation mode, we used a new digital autocorrelator "ALV3000" ( ALV Langen/Frankfurt ) . This allowed us to record photon autocorrelation functions in real time on a blockwise logarithmic time scale of 192 channels covering a time span from 0.8 ps up t o about 50 s within a single measurement. The output curve has no breaks or kinks; therefore there is no adjustment necessary between individual blocks. This yields a remarkable improvement of data statistics, and measuring times can be reduced by a factor of about 8. Moreover, the time range is so wide that it could be kept constant in all measurements, and no dependence upon the choice of the sample time can arise. Detailed descriptions of this correlator are given by Schatzel.'*

Sample Preparation and Measurement Cylindrical screw cap test tubes of 1.5 cm diameter were cleaned exhaustively with deionized filtered water. In a dust-free environment water, F-buffer concentrate, gelsolin, and actin were filtered into the test tubes through precleaned Milex-GV (Millipore) filters. All samples were stored overnight a t 4°C in order t o reach a stable state. Light scattering experiments were performed a t 10°C in runs of 10 min duration. The scattering intensity remained constant within this measuring time. In order t o perform measurements during the polymerization process, G buffer was used instead of F buffer. Test tubes were not covered with screw caps. Instead, a 1-mL plastic syringe with a Millex GV filter unit, filled with F-buffer concentrate, was mounted on top and sealed with parafilm. The first measurement was performed under G-buffer conditions, then two drops of F buffer were added from the syringe. After shaking carefully for a few seconds the sample was put back into the light scattering apparatus. Correlation functions were subsequently recorded until the polymerization process apparently arrived a t a stable state. All measurements were performed a t excess ATP.

EXPERIMENTAL RESULTS QELS Measurements of F-Actin Solutions Figures 1-3 summarize our QELS studies of F-actin solutions of various concentrations and scattering angles. Figure 1shows the time correlation function g ( q , t ) of F-actin solutions as a function of the mono-

1473

meric actin concentration CA obtained for a scattering angle of [ = 90". The time of polymerization was 24 h in all cases; the values of the concentrations are given in the insert. The g( q , t ) vs t curves may be divided into three classes. At C A = 0.01 mg/mL, g ( 4 , t ) decays very rapidly to zero with a correlation time of r , N 3.3 X s (decay constant I', x 30 000 sP1). T h e long time tail a t t > 0.3 X lop3 s in Figure 1cannot be considered as reliable or has to be attributed to residual dust particles. It thus follows that the actin has not polymerized as expected from the fact that the critical concentration of actin polymerization is CAN 0.012 mg/mL under the present condition^.^.' At CA = 0.015 mg/mL the decay curve is drastically slowed down, but the decay rate becomes only slightly larger if the concentration is increased further by a factor of six ( t o C A = 0.1 m g / m L ) . At, increasing CA from 0.2 t o 0.7 mg/mL the slowing down of g ( 4 , t ) is first more rapid, but saturates a t CA 2 0.4 mg/mL. In order to exhibit the long time behavior of the correlation function more clearly and to demonstrate the q dependence of the correlation function, we plotted a family of curves g ( q , t ) on a logarithmic time scale in Figure 2. For the theoretical interpretation of the light scattering data it is essential to clarify whether the correlation function obeys some general power law with respect to the scattering vector. For that purpose we plotted the curves of Figure 2 on a reduced time scale T cc q " t . In Figure 3 the time scale of g( t ) for the scattering angle [ = 90" is real, while for [ = 20", 40°, 60°, and 150" the time has been reduced by factors 41.77, 8.075, 2.592, and 0.525, respectively. These reduction factors were obtained by a trial and error procedure yielding best agreement between all g ( q , t ) curves for a common exponent a = 2.7 k 0.1. Deviations are only observed for small scattering angles ( [ I 40" ) and in the long time behavior. The latter deviation increases strongly with increasing actin concentration, suggesting that these long time tails are determined by fluctuations of the mesh size. T h e long time tail is expected to be determined by another relaxation process than the universal curves. I t is thus helpful to separate these processes. At { = 90" the universal part of g ( q , t ) is practically decayed to zero a t t 2 20 ms. The long time ("ultraslow") tail [gus( q , t ) ]decays very slowly and appears to be nonexponential. It is therefore plotted a s a function of powers of time, namely t " , where a is varied between 0.1 and 1. Fitting a straight line to the g,, ( 4 , t ) vs t" plots by linear regression yields

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PIEKENBROCK AND SACKMANN

1.o

I

-

Actin concentration

~

-c-

-

0.01 mglml

0.015 mg/ml 0.02 mg/ml 0.03 mg/ml 0.04mg/ml 0.06mg/ml 0.10 mg/ml 0.20 mg/ml

0.30 mg/ml 0.40 mg/ml

0.50 mg/ml 0.70 mg/ml

0.2 -

\

'

0.0

I

I

1x i 0-3 time 1 sec

2x1 0-3

I

3 x i 0-3

Figure 1. Square root of time correlation function g ( g , t ) of scattered light intensity (for scattering angle { = 90' or q = 22.6 pm-') for F-actin solutions of various monomer actin concentrations C A . The concentrations (in milligrams of G-actin per milliliter) are given in the inset. For long time behavior see Figure 2. The baseline has been subtracted.

a.

best fit for a range of a values: -?, I CY IAs a second step the range of a values is further confined by determination of the relative contribution of the ultraslow component g,, ( q , 0 ) to the correlation function g ( q , 0 ) . This is achieved by fitting the total range of the correlation function with the procedure described in the discussion section [ cf. Eq. ( 3 ) and Table I]. This procedure yields the best fit for a

a

= and gus(q , 0) = 0.323 determined from data recorded for C A = 0.4 m g / m L a t { = 90". Three examples are presented in Figure 4 where Ing,,(q, t ) vs t1'4plots are presented for scattering angles { = 60°, go", and 150". Clearly the average decay constant rus{ defined by g,, ( q , t ) = exp [ A ( rust) } increases strongly with the scattering angle (or 4 ) . The slopes of the straight lines in Figure 4 scale as

QELS OF THERMAL EXCITATIONS

1475

1.00

0.80

-

0.60

-

at) 0.40

0.20

-

Scattering angle

-

-

30" 40" 50"

60"

75" 90" I 110" 130" --150" -,

time I sec Figure 2. The scattering vector dependence of dynamic structure factor g ( q , t ) for Factin solution of CA = 0.1 mg/mL. The g ( q , t ) is plotted on a logarithmic time scale to exhibit long time behavior. The scattering angles are given in the insert. The temperature was 10°C.

I';L4 cc q 2 ,t h a t is, cc 4 8 .

rUsis strongly q dependent: I',,

QELS Study of F-Actin Solution During Polymerization Process

In the following we report the application of dynamic light scattering t o study the growth kinetics of actin filaments. This was possible owing to the high sensitivity of our instrument and by performing the experiments a t 10°C. In Figure 5a and b we present a family of g ( q , t ) vs t curves that were taken SUCcessively after starting the polymerization. The curve a t 0 min corresponds to pure G-actin, which was taken prior t o the addition of the F buffer. The record of the curve denoted as 5 min was started 4 min after addition of two drops (about 100 pL) of F-buffer concentrate and the time averaging lasted for 2 min. T h e second g( q , t ) curve (denoted as 10 min) was started 9 min after addition and the time averaging process lasted again 2 min. The averaging run time was increased t o 3 min after 20 min of

polymerization and t o 5 min after 60 min €or the subsequent records. For such short run times the increase in scattering intensity during the measurement would not affect the data. Measurements were continued until the correlation functions did not change any further. In our previous study we found t h a t prolonged irradiation of one single measuring spot resulted in local birefringence and a slow decrease in scattering intensity. By separate temperature measurements we could exclude a remarkable local heating and we therefore assume that the above changes are due t o photochemical effects. Such effects were avoided by rotating the cuvette every 10 min by a small angle. The most drastic changes are observed during the first 40 min. In particular, one can clearly see that the initial decay ( t h a t is during the first few s) corresponds to t h a t of the monomers (cf Figure 1) . This suggests that only a few chains grow rapidly, and that g ( q , t ) consists of a superposition of this growing chains and the residual monomers. This is a consequence of the well-known fact that the for-

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PIEKENBROCK AND SACKMANN

Scattering angle

-

-0-

'

1

0.0 10-5

10-4

60"

90" 130"

I

time / sec

10-3

Figure 3. Evaluation of the wave vector dependence of the correlation function. The correlation function g( q , t ) is plotted on a reduced logarithmic time scale: In 42.7t}for scattering angles { = 20°, 40°,60°,go", and 150'. The q values are normalized with respect to the value at = go", that is, a reduced time scale T = [ q ( { ) / q ( 9 0 ' ) ] " t is introduced. ( a ) Actin concentration CA = 0.04 mg/mL. ( b ) Actin concentration CA = 0.10 mg/mL. Note that the splaying of the curves at long times becomes more prominent at increasing concentrations.


1 ms).

In order to explore the effect of the chain length on the dynamic structure factor, we studied F-actingelsolin complexes. As is well known, in the presence of C a 2 + ,gelsolin can act as a capping protein and as such leads t o the formation of a population of chains with a reduced filament length L . According to Janmey e t al.," the number weighted average length is proportional to the ratio of the number of actin molecules to that of gelsolin molecules. This is attributed t o the binding of gelsolin to the barbed end of the actin filaments. Light scattering studies of actin chain length without gelsolin have been recently published by Seils e t a1.l' I n the present work shortened actin filaments were prepared by polymerization of F-actin in the presence of 1 : 1 complexes of gelsolin and actin provided by Prof. A. Wegner. These complexes (abbreviated as GAC in the following) act a s nuclei that accelerate the initial stages of actin polymerization.13 Figure 7 exhibits a family of time correlation functions g ( q , t ) of gelsolin-F-actin solutions (for CA= 0.16 mg/mL) of various actin to gelsolin ratios. The form of the curves g (4, t ) is clearly distinguished from that of partially polymerized pure F-actin (shown in Figure 5 ) . The initial slope is less steep, but in a longer time regime ( 2 X lop3 s < t < 3 X lop3s ) it decays faster. This shows that no monomers coexist with the F-actin. The curves for rAc 2 2000 agree approximately with those of pure Factin a t the same concentration CA.At low actin to gelsolin ratios ( r A G I30) the g ( q , t ) curves decay t o zero a t about 2 X s, which shows that the chains are rather short.

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PIEKENBROCK AND SACKMANN

1 .o

1 .o

(b)

0.8

0.8

0.6

0.6

Polymerisation time

g(t) g(t) 0.4

0.4

0.2

0.2

- gz

-f

11OmR

14omm

-

-f

0.0

-f

Mmln

-

Polymerisation time

420mk

immn

1XI 0-4

0

2x1 0-4

0.0

time I sec

2x10-3 1x i 0-3 time I sec

3x1 0-3

Figure 5 . Study of actin polymerization kinetics ( a t 10°C) by QELS. The evolution of the correlation function with time of polymerization is presented for the short time regime in ( a ) while the long time regime is given in ( b ) . The curves g ( q , t ) were recorded successively, after addition of the polymerization buffer to the G-actin solution at time t = 0. The measuring intervals (of 2 min) a t the initial state of polymerization (0-16 min) were limited by the time required to obtain an acceptable signal to noise ratio. The times of polymerization are given in the inserts of both figures.

Since it is expected that the dynamic structure factor of short actin chains is determined by the self-diffusion of the whole chain, the power law I’ cc q“ of the initial decay rate is expected to go over from the q2.7to the q 2 law characteristic for diffusion determined processes. In order to clarify this point we measured the initial decay rate :’I as a function of the scattering angle for a number of actin to gelsolin ratios. In Figure 8a we exhibit a number of double-logarithmic plots of I’: vs q . Clearly, straight lines are obtained in all cases. The slopes decrease with decreasing actin to gelsolin ratios r A G . In Figure 8b the exponent a is plotted as a function of the reciprocal actin to gelsolin ratio. A q 2 dependence is only observed for the shortest chains rAG = 7.5 (or L = 25 nm). The increase to a = 2.2 for rAc 2 9

is attributed to the polydispersity of the system. This causes an angle-dependent weighting of signal contributions from long chains due to the static structure factor P ( q , L ) (which is defined in the Appendix). The a increases further as rAGbecomes larger, and at rAG = 2000 it arrives at the limiting value for “infinitely long immobilized chains” as determined for actin alone: a = 2.7. These experiments provide further evidence that dynamic light scattering probes the internal dynamics of the F-actin filaments.

DISCUSSION The present QELS experiments, which were performed over a much larger time scale ( 10-6-10’ s

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1.o

1479

I

Polymerisation time

---e

0.8

-c

-

-c

0.6

45 rnin

35 min 27 rnin 20 rnin

10 min

5min

g(t) 0.4

0.2

0.0

1x i 0-3

2x10-3

3x1 0-3

time / sec Figure 6. Growth kinetics of F-actin in presence of 1 : 1complexes of gelsolin with actin dimers (GAC), which act as nuclei for filament growth. The G-actin concentration was C, = 0.16 mg/mL, the scattering angle { = go", and temperature 10°C. The GAC concentration was 1 mole% with respect to G-actin.

instead of 10-5-10-2 s ) and range of concentrations than in our previous study,g confirm our previous conclusion that the q-dependence of the dynamic structure factor is determined by a power law of the form g ( q , t ) cc F ( q " t ) , a = 2.7 k 0.05

(1)

In the previous work we showed that the short time behavior can be represented by a single exponential law g ( q , t ) cc exp{ - A q 2 . 7 t } while the long time behavior is well represented by an exponential function with fractal exponent g ( q , t ) cc exp{ - B ( I'qt)2'3} with rqcc q2.7.In particular, the finding of the latter law led to the conclusion

that the internal dynamics of the actin filaments exhibits Rouse-Zimm behavior. This model indeed predicts a power law r4 cc q 3 ,but QELS studies of high molecular weight polyethylene ( M = 2.4 X lo6) yield also an exponent smaller than three (about 2.75 2.85) .20,21 The theoretical exponent of three has only been approximately confirmed by spin-echo studies of polymethylacrylate22 in which q was, however, only varied by a factor of three, and qdependent ultraslow decay components such as described in the section on QELS measurements of Factin solutions and Table I may have influenced the data. The deviation from the Rouse-Zimm power law could, however, also be due to the fact that, besides

-

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PIEKENBROCK AND SACKMANN

1.o

Molar ratio GAC : actin

0.8

0.6

g(t) 0.4

0.2

0.0

Figure 7. Time correlation function of F-actin solutions polymerized in the presence of GACs presented for various GAC to actin molar ratios, r A G . The curve for r A G = 2000 is identical with pure F-actin. The scattering angle was { = go", the G-actin concentration C A = 0.16 mg/mL, and the temperature 10°C.

the Rouse modes, other relaxation processes contribute to the dynamic structure factor. In order to explore this possibility we tried to represent g ( q , t ) for times smaller t h a n 3 X s as a superposition of the Rouse-Zimm modes and additional relaxation processes that were expected to contribute mainly to the short and long time regime of g ( q , t ) . The procedure is restricted t o the scattering angle [ = 90". T h e dynamic structure factor for the RouseZimm model g R Z ( q , t ) is given in the Appendix. In order t o facilitate the numerical fitting procedure, Eq. (A4) was formally expanded into a sum of exponentials with one decay constant r as adjustable parameter:

gRZ(

+

q , t ) = 0.29981e-0.3995rt 0.40450e-0.7245rt

+ 0.25172e-1.4162rt+ 0.04337e-4.4072rt ( 2 ) I'was then determined by fitting Eq. ( 2 ) t o the experimental g ( q , t ) curves. The second column of Table I gives the result for different actin concentrations. We find a systematic decrease of I? with increasing concentrations. Whereas for 0.02 mg/mL 5 CA I0.1 mg/mL the decay constant remains nearly stationary ( r = 900 k 50 s-l) large deviations are observed for CA I0.02 mg/mL and CA > 0.2 mg/mL. Therefore additional fast ( al, I',) and slow ( u2, I'2 ) relaxation terms are added t o g R Z as

QELS OF THERMAL EXCITATIONS

9

a

7

In r 6

5

... .. ..

4

0

1.5

2.5

2.0

1:75

2.00 2.22 2.28 2.w 2.44 2.47 2.50 2.69

l:o

1:22

1:so 1:7s 13222 1:900 1:2wo

3.0

3.5

In q

constant is I' = 850 spl. The fast process with a decay constant of rl = lo4 spl contributes about equally to all concentrations, whereas the slow process with r2= 30 s-' appears only at high concentrations ( CA > 0.06 mg/mL). The decay constant rl of the fast process increases strongly at very low concentrations CA and the mean square deviation X 2 becomes very large. This is due to the fact that contributions of residual actin monomers have not been considered yet. Therefore an additional term a, exp { - rmt1 is added to Eq. ( 3 ) .The decay constant can be calculated according to I', = D, q 2 , while D, is obtained from D, = k B T / 6 q r , ( rm is the average radius of a monomer, which is r, N 5 nm). With 17 = 0.013 Poise one obtains for { = 90" a decay constant of r3= 3 X lo4 s-'. The slow process can be attributed to the ultraslow decay component discussed above (section on proteins and Figure 4). Since only data points for t 5 3 ms are fitted, the ultraslow component can be assumed to be almost constant, i.e., it is well represented by a single exponential term: g,, ( q , ~exp ) [ - ( rust) 1/4]. We finally represent g ( q , t ) as g ( q , t ) = gFtz(q, t ) + ale

- I', t

+ ameprmt + ause-ru,t 2.8

1

2.6-

1 :1

Molar ratio

GAC :actin

Figure 8. ( a ) Variation of power law of initial decay rate I'" cc q* of F-actin/GAC solutions with actin to GAC ratio. Note that the slope of the straight line observed in the InI'" vs Inq plot decreases with increasing rAGfrom CY = 2.0 to N = 2.7. ( b ) Values of a plotted as a function of ~AG.

The amplitudes al ,a2and decay constants rl,r2obtained by fitting Eq. ( 3 ) to the dynamic structure factors for j- = 90' are given in columns 4-9 of Table I. Table I shows that the Rouse-Zimm process contributes between 66 and 95% to g ( q , t ) and the decay

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(4)

The final amplitudes and constants are summerized in Table 11. While the monomer diffusion (a,, r,) contributes only at low concentrations and the ultraslow component (aus,Fus) only at high concentrations, the fast component ( al , ) is essential for all cases. Moreover, column 10 of Table I1 shows that the ratio of the amplitudes of the Rouse-Zimm modes and the fast component are nearly constant. Owing to the different dependence of the fast and the ultraslow process on the actin concentration, we think that they are quite significant and contain interesting information on the actin dynamics. Our analysis showed that by taking into account the contribution of the residual monomer diffusion, the dynamic structure factor at times shorter than 3X s is dominated by the Rouse-Zimm modes, but that a fast relaxation process with a relaxation time of about lop4s contributes at all concentrations while an ultraslow process [ exp { - ( rust) } ] contributes with growing strength at increasing concentrations. This strongly suggests that the fast process corresponds t o an internal relaxation process of the chains while the slow mode is due to collective motions of the chains. The ultra slow mode is certainly required to fit our data but we do not have an explanation for the q 8 dependence of ruS.

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PIEKENBROCK AND SACKMANN

Table I Simulations of Dynamic Structure Factor for 9 = 90” Rouse-Zimm Model of Chain Dynamics According to Dubois-Violette and de G e n n e ~and , ~ ~First Correction at Long and Short Time Limit”

0.015 0.02 0.03 0.04 0.06 0.06 0.10 0.20 0.30 0.40 0.50 0.70

1121 967 955 945 900

.0320 .0095 .0105 .0087 .0135

849 743 696 569 537 478

.0193 .0323 .0316 .0393 0.445 .0469

0.843 0.933 0.938 0.943 0.918 0.882 0.839 0.780 0.797 0.661 0.660 0.666

843 836 846 840 760 976 917 929 869 859 870 884

0.157 0.067 0.062 0.057 0.082 0.059 0.068 0.057 0.065 0.073 0.068 0.069

23680 12530 16150 14560 8812 15350 11320 10910 11510 8931 9557 10155

0.059 0.093 0.153 0.138 0.266 0.272 0.265

10 75 16 0 40 11 5

0.00468 0.00055 0.00152 0.00101 0.00081 0.00040 0.00072 0.00038 0.00032 0.00022 0.00019 0.00015

a First column: G-actin concentration CA,of solutions, Second and third columns: decay constant r and mean square error obtained by fitting of Rouse-Zimm model to g ( q , t ) curves. Fourth to tenth columns: amplitudes and decay constants obtained by fitting of a superposition of Rouse-Zimm model (aRZ,rRZ) and two exponential functions consisting of a fast component (al, r,) and a slow component ( a 2 ,r2)to the experimental g ( q , t ) curve.

It was indeed shown by B r ~ c h a r dthat ~ ~ in good solvents the dynamic structure factor can be described in terms of the internal dynamics of a nonentangled ideal chain for q f > 1 mesh size). Following Brochard, 24 the ultraslow mode can be attributed to the cooperative diffusion of ‘‘blobs’’ of size { exhibiting a diffusion coefficient D = kbT/ 67r17{. In our case these blobs correspond to the slightly coiled stretches of filament extending between the transient cross-links. The contribution of these collective excitations to the time correlation function increases with decreasing scattering angles in accordance with our experiments and becomes quite pronounced a t angles < 40”.

(r:

r

A superposition of a Rouse-Zimm and a fast process was postulated for helical wormlike chains by Yamakawa and c o - ~ o r k e r s .In ~ ~this model the polymer chains are represented by elastic wires determined by bending and torsional elasticity. The latter are introduced to account for the fast local motions. This model could for instance explain the high-frequency plateau of the dynamic intrinsic viscosity25in terms of a coupling of the Rouse-Zimm and the local modes. This model cannot be applied to the case of actin filaments. On the other hand, it is now well known from electron microscopy studies that the contact between the two strands of the helically shaped

Table I1 Simulation of Dynamic Structure Factor by Four Relaxation Processes: (1) The Rouse-Zimm Model of Internal Chain Dynamics (aRZ, r R Z ) , (2) a Fast Component (ul, rl),(3) the Contribution of Diffusion of Residual Monomers (am,I’,,,), and (4) a Slow Process (u,,., rus). Note that the Amplitude Ratio ul/uRZis Practically Constant Over the Whole Range of Concentrations X2

C.4 0.015 0.02 0.03 0.04 0.06 0.10 0.20 0.30 0.40 0.50 0.70

0.812 0.905 0.909 0.915 0.901 0.864 0.790 0.771 0.691 0.662 0.664

0.089 0.083 0.083 0.082 0.084 0.079 0.082 0.075 0.069 0.072 0.076

0.099 0.012 0.008 0.003

30000 -

0.015 0.057 0.178 0.154 0.240 0.266 0.260

10 -

-

0.110 0.092 0.091 0.090 0.093 0.091 0.104 0.097 0.100 0.109 0.114

.00212 .00168 .00115 .00105 .00067 .00077 .00046 .00038 .00027 .00020 .00016

QELS OF THERMAL EXCITATIONS

chains are locally broken yielding unraveled stretches (of a few hundred Angstrom length) both within and at the end of filaments.28These local "defects" are driven by thermal fluctuations and are thus expected to be associated with (overdamped) torsional modes. Of course nothing is known yet about the correlation time of these modes but it should be of the same order of magnitude as those of the short wavelength bending modes. We thus attribute the high-frequency modes of the dynamics structure factor to these local torsional modes.

Growth Kinetics of F-Actin In Figure 5 we showed that the growth of actin filaments can be followed by QELS at low temperature. This makes it possible to explore this process not only on the basis of intensity measurements (cf. Wegner and SavkoZ6)but also by measurements of the apparent diffusion coefficient. In the following we attempt to answer the question of whether the polymerization process consists ( 1) of the simultaneous growth of many chains (called model I ) or ( 2 ) of the growth of a few chains coexisting with the (decreasing amount of) monomers (model 11). Let us first consider model I: The correlation function for short filaments of equal length is g( q , t ) = exp( - rt),I' = q2D and is thus determined by the filament d h s i o n . Considering these as rigid rods of length L the diffusion coefficient D can be calculated for any given filament length ( d is the thickness of the rod)27:

kBT D ( L ) = -[ l n ( L / d ) 374

+ 0.3831

1483

In this equation P ( L , () is the angle-dependent static structure factor that is due to interference of light scattered from different points within the macromolecule. In case of rigid rods, it can be calculated theoretically and the result is given by Brochard.27 In the case of rodlike molecules and a length per monomer repeat unit of b = 2.5 nm, one obtains an asymptotic value of Z( co ) = 54.7 Z, . This allows us to calculate dimensionless values Z"(L) = I ( L ) / Z(co ) for any given chain length. We can now determine expectation values D O (1' ) as a function of the time of polymerization tpOlfrom measured values Z" ( tpol)by combining Eqs. ( 6 ) and (7). Let us consider now model 11: Let xp,l be the molar fraction of monomers attached to F-actin and (1 - xpol)that of the residual monomers. The scattering intensity as a function of xpOlis then

The number of 53.7 corresponds to the asymptotic value of I ( co ) /I, - 1 as calculated in Eq (A9 ) . By comparing this equation with the measured scattering intensity, one can estimate xpol. The apparent diffusion coefficient as calculated from the initial decay of the correlation function is

(5)

This calculation ignores any kind of internal motions and thus holds only for very short filaments. In the case of longer chains it yields a lower limit for the apparent diffusion coefficient D = r0/q2, where I'" is the initial slope of the correlation function. For very long chains (L + co) the autocorrelation function is entirely determined by internal motions, and I'" can be obtained experimentally from a measurement of the fully polymerized sample. We found a value of D (a,) = D, / 21 ( monomer diffusion coefficient: D, = k ~ T / 6 a ~ r ,r,, = 2.5 nm) . This value is necessarily a minimum value for all shorter filaments. We may calculate a dimensionless relative diffusion coefficient D O ( L )= D ( L )/ D (co) for any given chain length L . We also calculate the static scattering intensity for molecules of a given chain length, according to Eq. ( A 7 )

Normalized values are Z" (xpol) = Z ( x p o l ) / I ( ~and )

D " ( 3cpol ) = D ( xp,l ) / D ( co ) . This allows a calculation of the expectation values xpol( tpol) and D o ( tp0l) or D o ( I " ) from measurements of I" ( t p o l ) , based upon the experimental values of Z(co ) and D (co ) . In Figure 9 we plotted the measured normalized intensity values I" over the polymerization time tpol, together with the expectation values of the normalized reciprocal diffusion coefficient 1/D " according to model I and model 11. The theoretical curves were calculated for Z ( c o ) / I , = 54.7 and D,/ D(co) = 21. Experimental values for l / D o as determined from dynamic light scattering are plotted as triangles ( A ) . The data of Figure 9 show that model I1 fits the experimental L ( tpol)vs tpolcurve better than model I. The growth of only a few chains becoming very

0.0

0.2

0.4

0.f

0.'

1.

"V

1

1

I

a

1I D

10

(calculated,model II) (calculated, model I)

100

Polymerisation time / min

-0-

4- 1I D

lo (experimentalvalue)

- - - -

g

*1I D (experimentalvalue from QELS).

It"

Figure 9. Evaluation of growth of actin filaments ( a ) for C, = 0.6 mg/mL and ( b ) CA = 0.15 mg/mL. ( - A -) Increase of normalized inverse diffusion coefficient, of growing filaments. ( . 0 . . ) Increase of scattering intensity Z, a t j- = 90" with time of polymerization tpol.Both curves are normalized to one at tpol+ a.In addition, the inverse diffusion coefficient as predicted from the intensity data by two different models are presented. ( - 0 -) Model I: simultaneous growth of many filaments; ( - 0 -) model 11: growth of very few chains coexisting with monomers. Note that best agreement is obtained for model 11.

Id

0.2

0.4

0.6

0.8

1.o

1000

%U

QELS OF THERMAL EXCITATIONS

long is also verified by negative staining electron microscopy of very dilute F-actin solutions. An example is shown in Figure 10 for the case of CA = 43 pg/mL. T h e micrograph shows clearly that nearly all of the actin has polymerized into one chain of a contour length of the order of 70 pm. It is also interesting to note that the few shorter chains present (of L = 5-10 pm) form random coil-like structures rather than rigid rods. This is in agreement with our previous conclusions that actin filaments are quite flexible with persistence lengths of the order of L, x 0.1-0.5 pm rather than L, rci 10 pm (calculated by Yanagida e t a1.l”). O n the length Distribution of F-Actin Crown in the Presence of GACs

T h e growth of only a few chains in the case of pure actin polymerization is a consequence of the very small probability of nucleus formation.’5 This is also clearly verified by the finding that in the presence of GAGS acting as nuclei many chains start t o grow immediately after adding the F buffer. We found that the duration of the polymerization process is reduced by a factor of about 10, and the scattering intensity increases linearly with time. Now let us consider the autocorrelation functions and chain length distributions of completely polymerized GAC-actin probes (equilibrium state). At actin to GAC ratios rAc 5 20, the initial decay of g ( q , t ) is determined by the diffusion of actin oligomers ( rs a q 2 ) ,suggesting that the solution is less polydisperse t h a n uncompletely polymerized pure

1485

actin (compare Figures 5 and 7). In order t o explore the polydispersivity of these systems, we attempted to analyze the length distribution by CONTIN analysis of the dynamic structure factor. The dynamic structure factor of a distribution of chains of length L = nb is

where r, = q 2 D (n ) is the decay constant of the n mer and x , the fraction of chain composed of n monomers. This equation is equivalent to Eq (A3) for the special case of a polymer consisting of n = 1 t o n = co monomer units. By assuming a continuous length distribution, one can write a

g ( { x ( n ) } , q t, ) =

dnx(n)nP(n,q)e-r‘n,q’t

where x ( n )d n is the weight fraction of chains between length n and n dn. Inverse Laplace transformation yields

+

Figure 10. Negative staining electromicrograph of actin (CA = 43 pg/mL) after 14 h of polymerization at 4°C. Note that only a few long chains are present. The chains shown are the only ones found in a total area of 70 X 70 pm. Also note the presence of many short highly flexible chains indicated by arrows.

1486

PIEKENBROCK AND SACKMANN

I 1.0 - \

I

1 :9

0.4

0.6

1 :7.5 0.3*-

‘-

0.0;

J

I

10’

03

1 L1

number of monomers per chain

0.8 C

0 .c

0

(b)

0.4

E

.-w

a,

0.0

5

.I3

(c)

!.

0.8

0.4 E .-aCI) 3 0.0

a>= 21

cn, = 9

1 05

J

QELS OF THERMAL EXCITATIONS

The distribution of the decay constants G ( I') can thus be determined from the measured dynamic structure factors by inverse Laplace transformation. Finally, the length distribution function x ( n ) and the average length can be obtained from G ( r ). For t h a t purpose t h e well-known C O N T I N procedure was applied. T h e CONTIN distribution function was calculated on a logarithmic time scale that was divided into 25 discrete r values ranging from I'l = 1 s-' t o r25= 3 X lo5 s-I. This function was represented by a sum of exponentials: 29930

In Figure 10a we present amplitude (a,) vs decay constant (I',) plots for a few low values of actin to GAC ratios. Finally, the distribution of chain lengths has been determined from these an-rndistributions by application of Eqs. ( 5 ) and Eq. ( A3 ) . To simplify this procedure, the chain lengths were divided into groups of n 2 500, 200,97,46, 21.8, and 3. I t should be noted, however, that values of n = 500 ( L = 1.25 pm) do not correspond t o real chain lengths since for such chains the approximation of a n exponential dynamic structure factor with r,, = D,q2 does not hold. Figure 10b shows histograms in which the mass fraction of chains of length n are plotted versus the monomer number n. The absence of certain chain lengths (e.g., n = 8 for r A C = 7.5) is a n artifact of the CONTIN program. Reliable chain length distributions are certainly obtained for r A C I 15 where chains of n > 500 are practically absent. It is interesting to note that the chain length distribution of Figure l l b exhibits a close analogy to the length distribution of polymers formed by polycondensation. Following Flory, 31 the number weighted probability of formation of a chain composed of n monomers is

1487

corresponds to a n exponential distribution of the number density of chain lengths. T h e mass distribution of chain lengths as obtained from Eq. (13) is given in Figure l l c for average chain lengths ( n ) = 21 and ( n ) = 9. As in the case of actin, one observes a rather broad length distribution characterized by the presence of rather long chains (with n = 200 in the case of ( n ) = 21). The close analogy of the actin polymerization t o the polycondensation is surprising. However, a n exponential number distribution as described in Eq. (13) is also found in polymer solutions of a nonterminated polymerization-depolymerization reaction a t thermodynamical equilibrium. Since this condition holds for GACbound actin, the final state should be a n exponential distribution such a s we found by CONTIN analysis. This does not contradict the assumption that simultaneous filament growth starts a t all GAC nuclei in the beginning of the reaction, leading to a transiently narrow Gaussian distribution of filament lengths during the polymerization process. We did not observe a redistribution of filament length during the measuring time. Further studies are required to clarify this point. A comment concerning the CONTIN procedure should be added. Since the inverse Laplace transformation is a n ill-defined problem, CONTIN only gives reasonable results making the physically necessary conditions that all a, values are positive. Depending on the assumed smoothness of the resulting distribution function a ( n ), there is a wide range of possible solutions. A general tendency of CONTIN is t o set as many a ( n ) values as possible to zero, yielding a multimode rather than a continuous distribution. The data can thus only give information on the broadness of the distribution, but not on its shape.

APPENDIX A Theoretical Basis of QELS of Dilute and Semidilute F-Actin Solutions

where ( n ) is the average monomer number. This

T h e time correlation function of the scattered light amplitude g ( 4 , t ) (which is also denoted as dynamic

Figure 11. Evaluation of length distribution of actin filaments grown in the presence of GACs. ( a ) Amplitude ( ai) vs decay constant ( ri ) distribution obtained by expansion of dynamic structure factor into exponentials according to Eq. ( 1 2 ) by application of CONTIN procedure. Distributions are given for four actin to GAC ratios rAG = 22, 15, 9, 7.5. ( b ) Histogram of chain lengths obtained from uj-ri distribution in Fig. ( a ) . The chain lengths are divided into groups n = 500, 200,97,46, 21,8, and n = 3. The ordinate gives the mass fraction of the chains of length n , the abscissa gives the chain length. ( c ) Histogram of mass distribution of macromolecules polymerized by polycondensation calculated according to Flory."

1488

PIEKENBROCK AND SACKMANN

scattering factor) of a chain of N monomers is given by Berne and P e ~ o r a ~ ~ :

and

lV n,m

where the scattering vector q is related t o the scattering angle { as

Simple power laws are expected for two limiting situ a t i o n ~For . ~ ~r,t @ 1 (initial decay) one obtains a simple exponential law (Doi and Edwards,33 Eq. 4.113):

1 ~ =1 4H sin(;) { 3,( t )-

x, ( 0 )} is a measure of the deformation

of the chain in time interval t between the positions gnand g,,,. T h e angle brackets denote the ensemble average. For the present work we have t o consider two situations: 1. T h e case of dilute solutions of polymers with radii of gyration R, (or chain lengths L ) smaller than the reciprocal scattering vector q-'. For polydisperse solutions [ P L cf. Eq. ( IS)]

with

For &'I % 1 (long time behavior) one obtains a n exponential function with a fractal power of t . gRz(q,t ) = exp{-1.35 (rRZt)2/3}

(A6)

Static Scattering Factor of Filaments

2 L2nLPL

(A3)

L

where nL is the number of chains with variable length L and DL is the diffusion coefficient of a chain of length L . PI,(q ) is the static structure factor t h a t depends on the shape of the chain. 2 . The case of dilute (or semidilute) solutions of flexible chains with radii of gyration large compared to 9-l and with strong hydrodynamic coupling between monomers. The center of mass diffusion is so slow that only internal motion is observed by dynamic light scattering. The dynamic structure factor has been calculated by Dubois-Violette and de Gennes (cf. Doi and Edwards34) for the case of Rouse chains (absence of hydrodynamic interaction between monomers) and for Rouse-Zimm chains (case of hydrodynamic interaction). In the latter case,

The static scattering intensity of macromolecules with radii of gyration (or chain lengths L ) comparable to 4-l is expressed in terms of a form factor P ( q ) as

where Z,,,is the scattering intensity of L / b freely diffusing monomers and b is the chain length per monomer unit, with b 4 q-l. The form factor has been calculated for polymers of various shapes (cf. BurchardZ7).For thin rod-like polymers it is P(q, L)

=

--dt 2 sin t qL t

lim L . P ( q , L )

with

h ( ~ ( r , , t ) - ~ /(A4) ~)}

(d;'"f2:,

)

(A8 )

P ( q , L ) approaches zero for large L , but L a P(q , L ) is a monotonically increasing function of L , which approaches a n asymptotic value: L- a

X

-

=

3.O9-1Oy/q

(A9)

For { = 90" this asymptotic value is 136.8 nm, which is 54.7 X b in the case of actin, where b = 2.5 nm. First of all we are most grateful to Prof. A. Wegner for the gift of gelsolin and for many helpful discussions concerning actin growth kinetics. Helpful discussions with Dr. M. Schleicher, Prof. P. Janmey, and Dr. M. Barmann

QELS OF THERMAL EXCITATIONS

are also gratefully acknowledged. The work was supported by the Deutsche Forschungsgemeinschaft (SFB 266) and by the Fonds der chemischen Industrie.

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18. Janmey, P. A., Peetermans, J., Zaner, K. S., Stossel, T. P. & Tanaka, T. (1986) J. Biol. Chem. 261,83578362. 19. Seils, J., Jockusch, B. M., & Dorfmuller, Th. (1990) Biopolymers 30, 677-689. 20. Adam, M. & Delsanti, M. ( 1977) Macromolecules 10, 1229-1237. 21. Nemoto, N., Makita, Y., Tsunashima, Y. & Kurata, M. (1984) Macromolecules 17,425-430. 22. Richter, D., Hayter, J. B., Mezei, F. & Ewen, B. (1978) Phys. Rev. Lett. 41, 1484-1487. 23. Yamakawa, H. & Yoshizaki, T. (1981 ) J . Chem. Phys. 75, 1016-1030. 24. Brochard, F. (1983) J. Phys. 44, 39-44. 25. Yoshizaki, T. & Yamakawa, H. (1987) J . Chem. Phys. 88, 1313-1325. 26. Wegner, A, & Savko, P. (1982) Biochemistry 21, 1909-1913. 27. Burchard, W. (1983) in Advances in Polymer Science 48, Springer-Verlag, Berlin, Heidelberg, New York. 28. Bremer, A., Millonig, R. C., Sutterlin, R., Engel, A., Pollard, T. D. & Aebi, U. (1991) J . Cell Biol. 115, 689-702. 29. Provencher, S. W. (1979) Makromol. Chem. 180,201209. 30. Provencher, S. W., Hendrix, J., de Maeyer, L. & Paulussen, N. (1978) J. Chem. Phys. 69, 4273-4276. 31. Flory, P. J. (1953) Principles of Polymer Chemistry, Cornell University Press, Ithaca, NY. 32. Berne, B. J. & Pecora, R. ( 1976) Dynamic Light Scattering, Wiley, New York. 33. Doi, M. & Edwards, S. F. (1986) The Theory of Polymer Dynamics, Oxford University Press, Oxford. 34. Dubois-Violette, E. & de Gennes, P. G. (1967) Physics 3,181-198. 35. Reihanian, H. & Jamieson, A. M. (1979) Macromolecules 12, 684-689.

Received June 17, 1991 Accepted April 17, 1992