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Elongation of Actin Filaments is a Diffusion-limited Reaction at by oek76922


              OF                                                                            Vol. 261, No. 27, Issue of September 25, pp. 12754-12758,1986
0 1986 by The American Society of Biological Chemists, Inc.                                                                               Prrnted in U.S.A.

Elongation of Actin Filaments is a Diffusion-limited Reaction at the
Barbed End and Is Accelerated Inert Macromolecules*
                                                                                                    (Received for publication, May 20, 1986)

                 Detlev DrenckhahnS and Thomas D. Pollard
                 From the Department of Cell Biology and Anatomy, JohnsHopkins University School of Medicine, Baltimore, Maryland 21205

   We used a fluorescence method to measure the rate where k is a unitless steric factor, is a unitless electrostatic
constants for the elongation of pyrene-labeled actin factor, b is the interaction radius in centimeters, DMand DF
filaments in a number of different solvents. The abso- are the diffusion coefficients of the two reactants (monomers
lute values of the rate constants were established by and filament ends,                 respectively), and No is Avogadro’s number
electron microscopy. Using glycerol, sucrose, or eth- (see Berg and von Hippel (1985) fordetails). For the addition
ylene glycol to vary the solution viscosity, the associ- of an actin monomer to the endof an actin filament,D M = 5
ation rate constant (k,) was lo7M” s” viscosity” (in                X                                         and
                                                                             cm2 s-’ (Lanni et al., 1981; Tait Frieden,     1982),
centipoise). Consequently, plots of l l k , versus viscosity DF < lo-’ cm’ s (Tait and Frieden, 1982), and is assumed
                                                                                     ”                                    b
are linear and extrapolate to near the origin as ex-
pected for a diffusion-limited reaction where the rate to be about 2 x 10” cm based on the shape of the actin
constant approaches infinity at zero viscosity. By elec- molecule (Smith et al., 1984). If k and felec havevalues of

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tron microscopy, we found that this inhibitory effect unity, then the rate constant for elongation will equal that
ofglycerol is almost entirelyatthefastgrowing,                      for collision and have a value of 7.5 X lo8 M” s”. However,
barbed end. For the pointed end, plots of 1/k+ versus               the correct orientation of the reacting molecules is essential
viscosity extrapolate to a maximum rate of about lo6 so that k is more likely to have a value of low4 10” (Berg          to
M” s” at zero viscosity, so that elongation at the                  and von Hippel, 1985),so k+ will be in the range of 7.5 x lo4
pointed is not limited by diffusion. In contrast to these to 7.5 X lo7 M” s-’. These predictions put the              observed values
small molecules, polyethylene glycol,            dextran,      and for k+B and k+’ well within the range for diffusion-limited
ovalbumin all cause a concentration (and therefore reactions.
viscosity)-dependent increase in k,. At any given vis-                  Since D is an inverse function of viscosity, one can test for
cosity, their effects are similar to each other. For ex- a diffusion-limited process by evaluating the dependence of
ample, at 3 centipoise, k , = 2.2 % lo7 “’ s”. We the reaction rate onviscosity (see Berg and von Hippel (1985)
presume that this is due to an excluded volume effect for a review of diffusion-limited processes). To do so for actin
that causes an increase in the thermodynamic activity polymerization, we measured the elongation rate and rate
of the actin. If the proteins in the cytoplasmic matrix constants in several different solutes with a range of viscosi-
have a similar effect, the association reactions of actin and also  ties           over a rapge of temperatures. We also evaluated
in cells may be much faster than expected from exper-
                                                                    felec by determining the rate constants over a wide range of
iments done in dilute buffers.
                                                                    salt concentrations.
                                                                        We found that the rate constants for both subunit associ-
                                                                    ation and dissociation at the barbed end of actin filaments
   The large association rate constants for elongation at both scale with inverse viscosity in glycerol, sucrose, and ethylene
ends of actin filaments raised the question of whether these glycol, as expected for a diffusion-limited            process. The de-
reactions are diffusion-limited processes (Pollard and Moo- pendence of the rate on the temperature explained in part
seker, 1981). The filaments polymerize bidirectionally with         by the viscosity, but thereis also a chemical activation process
the rate at the fast end about 10 times than at the           slow with an activation energy of about 3.5 kcal/mol. Above 50
end. The fast endis the barbed (B), end and the slow end is mM KC1, salt inhibits elongation in a concentration-depend-
the pointed (P) end (Woodrum et al., 1975), relative to the ent fashion so electrostatic interactions are           also an important
arrowhead-shaped complexes of myosin with actin filaments factorinthe process. Elongation at the pointed end                      also
(Huxley, 1963). By electron microscopy, the association rate depends on theviscosity but is not diffusion-limited. Several
constant for ATP-actin at the barbed end is k+B = 10’ M”            macromolecules (ovalbumin, polyethylene glycol, anddex-
s-I, whereas k,’ = lo6 M” s” in KC1 and MgC12 at pH 7 tran) all increase the elongation rate in spite of the fact that
(Pollardand Mooseker, 1981; Bonder et al., 1983). These they also increase the viscosity. This is presumably caused by
values can be compared with theoretical valuescalculated            an excluded volume effect that increases the thermodynamic
from theDebeye-Smoluchowski expression:                             activity of the actin (reviewed by Minton(1983)). Conse-
                 k , = 4~ k feiec b(D, + DF) N o                (1) quently, actin polymerizationinside cells may occurmuch
                                                                    morerapidly than expectedfromprevious              work in dilute
_ _ ~
   * This work was supported by a grant from the Muscular Dystrophy
Association of America and National Institutes Health Grant GM-                          MATERIALS AND METHODS
28338 (to T.D. P.) and Dr 91-4-3 (to D. D.). The costs of publication
of this article were defrayed in part by the payment of page charges.     The solutes used in these experiments came from the following
This article must therefore be hereby marked “uduertisement” in suppliers and were used without further purification: dextran T-40
                                                                      (&fr = 40,000) (Pharmacia Fine Chemicals, Uppsala, Sweden); ethyl-
accordance with 1 U.S.C. Section 1734 solely to indicate this fact.
   $ Permanent address: Institut fur Anatomy und Zellbiology, Uni- ene glycol, glycerol, polyethylene glycol 6000, and sucrose (J. T. Baker
versity of Marburg, 3550 Marburg, Federal Republic of Germany.        Chemical Co., Phillipsburg, NJ); ovalbumin (Sigma). Actin was Pu-

                                        Elongation of Actin
                                                   Filaments                  Is Diffusion-limited                                     12755
rified from rabbit skeletal muscle using gel filtration on Sephadex G-                                   TABLE    I
150 in 2 mM imidazole (pH 7.0), 0.5 mM dithiothreitol, 0.2 mM ATP, Effect of solutes on the critical concentration for the polymerization of
0.1 mM CaCI2 as the final step (MacLean-Fletcher and Pollard,         1980).                               actin
Acanthamoeba actinwas purified from sucrose extracts(Pollard,                                                muscle actin except for twosamples
                                                                               All samples are rabbit skeletal
1984). Part of theactin waslabeled withpyrene iodoacetamide                  of Acanthamoeba actin, which are indicated by theasterisk.cP,
(Pollard, 1984) and useda t 1-5% of the total actin in the     fluorescence centipoise.
assay for polymerized actin (Kouyama and Mihashi, 1981). Unless
noted otherwise, the polymerization reactions were carried out in                                                      Critical concentration
standard buffer (50 mM KCI, 1 mM MgCI2, 1 mM EGTA,' 0.5 mM
dithiothreitol, 0.1 mM CaCI,, 0.1-0.2 mM ATP, 10 mM imidazole (pH
7). The steady state extent      of polymerization andinitialrate        of
                                                                                         Solute           Viscosity _____-__
                                                                                                                     Steady stateE'o:ztion
elongation from unlabeled actin filamentnuclei were measured fluo-                                           CP                  PM
rometrically, usually a t 20 "C (Pollard, 1983). Most of the solutes            Standard buffer             1.o          0.16,
                                                                                                                     0.15,          0.08, 0.10,
affected the fluorescence intensity of pyrene-labeled actin, so all data                                                0.12           0.08
are corrected. All plots of elongation rate versus concentration of
actin monomer were linear, except for those in high concentrations              Ethylene glycol (5%)
of ovalbumin. In thatcase, the elongation rate constantswere calcu-               15                        1.5      0.14           0.08
latedfromthecriticalconcentrationsandrates               a t 1.2 PM actin.        25                        2.0      0.16           0.08
Absolute rates of elongation were measured by electron microscopy                 35                        2.5                     0.08
using bundles of actin filaments from Limulus sperm as nuclei (see                40                        3.0      0.16           0.08
Pollard and Cooper (1984)) except that the reaction was carried out
directly on the electron microscopy grid and was stopped by inversion           Glycerol (%)
of the grid onto a large volumeof 5 mM spermine (Sigma) in standard               10                        1.31     0.16
buffer to aggregate the filamentsgrown from the ends the Limulus                  19*                       1.65     0.10           0.10
bundles. The details of this method will be presented in another                  27.5*                     2.2      0.10           0.10
paper., The viscosities of glycerol- and sucrose-containing solutions             30                        2.5      0.16

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were taken from standard    tables. The viscosities of the other solutions        39                        3.4      0.10           0.10
were measured in a Cannon-Manning semimicroviscometer (Cannon
Instruments Co., State College, PA) at 20 "C. All linear plots were fit         Sucrose (%)
by least squares linear regression.                                               13.5                      1.5                     0.08
                                                                                  21                        2.0      0.18           0.08
                                                                                  25.5                      2.5      0.19           0.10
                                RESULTS                                           29                        3.0      0.19
    Sucrose, glycerol, and ethylene glycol all inhibit the elon-
                                                                         Dextran (%)
gation of rabbit muscle actin filaments but have little (su-                 5                     2.8                   0.10
crose) or no (glycerol and ethylene glycol) measurable effect                7                     4.25     0.12
on the critical concentration for      polymerization evaluated by          10                     7.5                   0.10
either steady state or initial rate methods (Table I). Similar
results were obtained with glycerol solutions using Acantha-             Ovalbumin (%)
moeba actin. The rate constants association and dissocia-
                                       for                                   5                     1.1                   0.10
                                                                             7                     1.25     0.10
tion were determined from the dependence of the initial rate               10                      1.5                   0.10
of elongation from actin filamentnuclei on the actin monomer               15                      2.3                   0.10
concentration. In all three solutes, k , scales a s a n inverse            20                      3.3                   0.10
function of viscosity (Fig. 1, open symbols). Plots of k+-' uersus
viscosity are linear, and all extrapolate to near the (Fig.
                                                          origin         Polyethylene glycol (%)
2B). The origin corresponds to an infinite rate constant a t                 1                     1.25                  0.10
                                                                             2                     1.60                  0.10
zero viscosity. Since the critical concentration is constant,     k-         3                     1.80                  0.10
also scales with viscosityin exactlythesame way as k,,                       4                     2.7      0.08
otherwise the system        would violate the principleof reversible         6                     3.6      0.08
thermodynamics. The absolute         value of k , in standard buffer         7                     4.25                  0.10
a t 20 "C was io7 M" s-' inparallelelectron               microscopy
experiments, and all of these plots were normalized to this
value.                                                               but do notextrapolatethroughthe         origin(Fig. 5 ) . Above
   Similar experiments by electron microscopy using glycerol 20 "C, k , is larger than expected from the temperature de-
to vary the viscosity showed that elongation at the barbed           pendence of the viscosity alone; whereas below 20 "C, the
end is an inverse function of viscosity that plots of ( k ~ , ~ ) " opposite is true. If the values for k, as a function of temper-
uersus viscosity extrapolate to near the origin (Fig. 2A (0) ature are corrected by a factor of viscosity", the average
and Fig. 3). Elongation at the pointed end is only weakly activation energy is 3.5 kcal/mol.
dependent on the viscosity. Plots of (k,')" uersus viscosity             Contrary to theeffect of small solutes, all three macromo-
do not extrapolate to the origin (Fig. 2A ( 0 ) )Instead, the
                                                        .            lecular solutes that we tested increased the rateof elongation,
value of k,' at zero viscosity is about IO6 M" s-*.                  in spite of the fact that they increase the viscosity (Fig. 1,
   In standard buffer, the elongation rate but not the critical      filled symbols). When plotted asa function of their viscosity,
concentration depends on the temperature(Fig. 4, A and B ) . all three macromolecules had similar effects on elongation.
This is true for both muscle and Acanthamoeba actin. From Note that thedifference between the small solutes andmac-
thedependence of k , onthetemperature,theactivation                  romolecules can be considerable. For example, at a viscosity
energy forelongation is8.5 kcal/mol (Fig. 4B, inset, and Table of 3 centipoise, elongation is almost6 times faster in dextran,
11). Using the temperature to vary the viscosity in standard ovalbumin, or polyethylene glycol than in ethylene glycol,
buffer ,lo% glycerol, plots of k+-' uersus viscosity are linear glycerol, or sucrose. Dextran and ovalbumin did not change
                                                                     the critical concentration for polymerization, whereas poly-
   ' Theabbreviation                                                                                                       to
                           used is: EGTA, [ethylenebis(oxyethylene- ethylene glycol decreased the critical concentration a small
nitrilo)]tetraacetic acid.                                           extent as described previously (Stromqvist et al., 1984). Our
     T. D. Pollard, manuscript submitted for publication.            observations with polyethylene glycol differ from previously
12756                                                        Filaments
                                                  Elongation of Actin        Is Diffusion-limited

                                                                             A-                         7       B-
                          -   1
                                                  0 Ovalbumin
                                                  A PEG 6 K
                                                  W Dextrsn40K
                                                  0 Glycerol
                                                  A Sucrose
                                                  0 Elhyleneglycol

                      0           2           4            6

   FIG.1. Dependence ofthe elongation rate constant (12,)
          on the viscosity (centipoise,cP) using lowmolecular
(~LM-'s - I )
weight solutes (opensymbols) or macromolecular solutes
(closed symbols).  These rate constants were measured by the fluo-
rescence method. 0, rate constant in standard

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                                                KCl/MgCl, buffer
without added solutes. PEG 6K, polyethylene glycol 6000.

                1.5               I                 I            I
                      A                                                     FIG. 3. Electron micrographs of actinfilaments grown from
                                                                         the barbed (B) and pointed ( P ) ends of Limulus acrosomal
                                                                         processes. A, standard KC1/MgC12 buffer, 3pM actin for 20 s; B , 30%
                                                                         glycerol in standard buffer, 3 p~ actin for 20 s. Magnification 30,000;
                                                                         inset, X 80,000. Bar = 0.5 pm; inset, bar = 0.2 pm.

                                                                                      1       I     l       a


                                                                                     0.5 1.5 1.0                       1.5 1.0 0.5
                                                                                                                Actin, uM

                                                                            FIG. 4. Effect of temperature on polymerization. A, 7 "C; 0,
                                                                         18.4 "C; 0, "C; 0, 36.8 "C. A, steady state extent of polymeriza-
                                                                         tion as a function of actin concentration at four different tempera-
                                                                         tures; B, rate of elongation as a function of actin concentration at
                                                                         the fourdifferent temperatures indicated in A . These rates were
                                                                         corrected for the effect of temperatureon fluorescence intensity
                                                                         shown in A . Inset, an Arrhenius plot of In k , uerscu inverse absolute
                                                                         temperature. The slope gives an activation energy of 9.0 kcal mol"
                                                                         in this experiment.

                                      Viscosity,                                                            is
                                                                         ratio, the critical concentration,relatively constant a t 0.16-
  FIG.2. Plots of 1/k+   (pM-' s-')-' versus viscosity. A, measure-      0.34 pM.
ments at the two ends of the filament by electron microscopy using
glycerol to vary the viscosity; B, measurements by the fluorescence                                     DISCUSSION
method. 0, standard KC1/MgC12 buffer without added solutes.
                                                                           In the small solutes that we tested, the elongation rate at
published data describing only a small effect on the rate of             the barbed end(R') of an actin filament is
elongation (Stromqvistet al., 1984).                                                        RB = ( / z + ~ ( A J kB)viscosity"
                                                                                                               -                             (2)
  T o evaluate the electrostatic factor( felec), the rate constants
were measured over a wide range of KC1 concentrations (Fig.              with the viscosity in centipose. A, is the concentration of
6). The optimal concentration KC1 for elongation in 1 mM                 monomeric actin. This relationship, the extrapolation h   "
                                                                                                                                of ,
MgC12 is 50 mM, and k, declines a t higher concentrations of             versus viscosity to near the origin, and the large absolute
KCl. Both h, and k- vary with salt concentration, but their              value of h, all support the conclusion that the association
                                       Elongation of Actin Filaments Is Diffusion-limited                                               12757
      Activation energy for the elongation of actin filaments
 The conditions were standard polymerization buffer and other
methods were as described for Fig. 4.
          Actin              Temperature      Activation energy
                                  "C             kcal mol"
Rabbit skeletal muscle         7.0-36.5                      6.6
                               7.6-37.0                      8.0
                               7.0-36.8                      9.0
                               7.0-22.5                      9.9

Acanthamoeba                   7.5-25.0                  10.1
                               5.2-28.5                   7.1
                                                Mean     =x
                                                  S.D. = 1.5

              .    1
                   I                             ,/'         1                 0
                                                                                        Critical concentration, U M
                                                                                           100         200            300
                                                                                                      UCI, m M

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         -a 0.2                                                       FIG. 6. Dependence of     elongation rate constants and critical
                                                                   concentration on the concentration of         KCl. 0, k,; 0, k-; 0,
                                                                   critical concentration. All samples contained 1 mM MgC12, 10 m   M
        x                                                          imidazole (pH 7), 0 1 mM ATP, 0.25 m dithiothreitol, 0.1 m CaCl,.
                                                                                       .                 M                   M
                                                                   Measurement was by the fluorescence method. Values were normal-
                                                                   ized t o k+ = 10 PM-' s" in 50 mM KC1 based on independent electron
                                                                   microscopic measurements.

                                                                      may be the chemical rate constant for a rate-limiting step.
              0                           I          I                The existence of these additional steps may account for the
                 0            1             2        3                                                             end
                                                                      slower rate compared with the barbed where the collision
                                Viscosity, C P                                       be
                                                                      rate seems to the limiting step.
    FIG. 5. plots of Ilk+ uersus viscosity with viscosity varied         The activation energy of 8.5 kcal/mol is higher than ex-
bytemperature in standardbuffer (0)or 10% glycerol in pected for a diffusion-limited reaction (-5 kcal/mol for reac-
standard buffer (0).     Measurement was by the fluorescence method.  tions in waterfrom 0 to 37 "C). When corrected for the effect
The doshed line is the average of the three plots in Fig. 2.
                                                                      of temperature on viscosity, actin filament elongation has a
                                                                      small activationenergy of 3.5 kcal/mol that canbe attributed
reaction at thebarbedendis            highly favorable and under      to chemical activation of the reactants. Presumably, temper-
optimal conditions limited only by diffusion. Assuming that ature affects the conformation and/or              dynamics of either the
felec is 1, the steric factor k is rather large, about 0.02. Inter- monomers or filament ends or both. Although temperature
preted physically, 2% of the collisions of monomers with the affects the rates the elongation reactions, it does not change
end of a filament lead to binding, high efficiency        considering thecriticalconcentration      in ourstandard polymerization
that the incoming subunit rarely be oriented correctly to buffer as it does under other conditions (see Korn (1982)).
bind to the end the filament.                                         Hence, in MgC12, the apparent equilibrium constant is tem-
    At first glance, it seems remarkable that        dissociation of perature-independent, A H = 0, and the reaction is driven by
subunits from the barbed endof the filament is also diffu- A S as shown previously by others (see Korn (1982)).
sion-limited process. However, there are at leasttwo reason-             Electrostatic forces are a factor in the bindingof subunits
able explanations. First, the dissociation reaction must in-          to the endof filaments since k+ declines toward zero at high
volve diffusion of the exiting subunit away from the vicinity concentrations of salt. Consequently, the felec term in Equation
of the filament end. To the extent that this movement is                                                             (k
                                                                      1 is less than 1, and the unitless steric factor ) in Equation
limited by the viscosity of the medium, the molecule may fail 1 is actually larger than 0.02.
to move away fast enough to escape rather than rebinding.                In terms of the biological functions of actin, the effect of
Second, the dissociation reaction may require that the ter-           macromolecules on elongation ratesis probably our most
minal subunit be in a particular conformation to break its            important new observation. At a total protein concentration
bonds with itsneighbors. Since some intramolecular motions like that in the cytoplasm,             elongation may be 10 times faster
are driven by collisions of solvent with the surface, the rate than expected for a solution of that viscosity. As in the case
of conformational fluctuations and therefore the         frequency of of some other biochemical reactions that are accelerated by
some states will depend in part on viscosity of the medium. inert macromolecules, this is probably an excluded volume
    At the pointed end of actin filaments, theelongation reac- effect (reviewed by Minton (1983)). The inert         macromolecules
tionisprobablynotdiffusion-limitedsince               k,' is not an   increase the thermodynamic activity the actin and thereby
inverse function of viscosity and its absolute value is much increase its effective concentration. This effect can be sub-
smaller than at the barbed end. The mechanism canonly be stantial, on the order of 10-fold. Consequently, inside cells,
guessed at, but it likely to be a multistep process with some both association and dissociation may occur at vastly higher
steps independent of the viscosity. The extrapolated value of rates thanexpected from previous measurements of rate con-
the rate constant at zero viscosity is about lo6 M-' s-'. This stants in dilute       buffers. For example, Tilney andInoue (1982)
12758                                Actin of
                                   Filaments                                Is Diffusion-limited
calculated that diffusion alone could just barely account for           Huxley, H. E. (1963) J . Mol. Biol. 7 , 281-308
therapidelongation      of actinfilamentsinthe     acrosomal            Korn, E. D. (1982) Physiol. Reu. 6 2 , 672-737
                                                                        Kouyama, T., and Mihashi, K. (1981) Eur. J . Biochem. 1 1 4 , 33-38
process of Thyorze sperm, but this required some generous               Lanni, F., Taylor, D. L., and Ware, B. R. (1981) Biophys. J. 35,351-
assumptions about the actin monomer concentration at the                  364
base of the process. If the thermodynamic activity of actin is          MacLean-Fletcher, S., and Pollard, T. D. (1980) Biochem. Biophys.
10 times higher than its concentration, then conservative
                                             even                         Res. Commun. 96,18-27
estimates regarding the monomer concentration are consist-              Minton, A. P. (1983) Mol. Cell. Biochem. 5 5 , 119-140
                                                                        Pollard, T. D. (1983) Anal. Biochem. 1 3 4 , 406-412
ent with the growth rate that theyobserved.                             Pollard, T. D. (1984) J. Cell Biol. 9 9 , 769-777
                                                                        Pollard, T. D., and Cooper, J. A. (1984) Biochemistry 2 3 , 6631-6641
  Acknowledgments-We thank Dr. Gary Ackers and Dr. Alan Min-            Pollard, T. D., and Mooseker, M. S. (1981) J . Cell Biol. 88,654-659
ton for their comments on our results.                                  Smith, P. R., Fowler, W. E.,.Pollard, T. D., and Aebi, U.~(1984).  J
                                                                          Mol. Biol. 1 6 7 , 641-660
                          REFERENCES                                    Stromqvist, M., Backman, L., and Shanbhag, V. P. (1984) J. Muscle
                                                                          Res. Cell Motil. 5 , 443-455
Berg, 0. G . , and von Hippel, P. H. (1985) Annu. Reu.Biophys.          Tait, J. F., and Frieden, C. (1982) Biochemistry 21,3666-3674
  Biophys. Chem. 1 4 , 131-160                                          Tilney, L. G., and Inoue, S. (1982) J. Cell Biol. 9 3 , 820-827
Bonder, E. M., Fishkind, D. J., and Mooseker, M. S. (1983) Cell 3 4 ,   Woodrum, D. T., Rich, S. A,, and Pollard, T. D. (1975) J. Cell Biol.
  491-501                                                                 67,231-237

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