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`THE JOURNAL OF BIOLOGIC~L CHEMISTRY
`Vol. 258. No. 5, Iasue of March 10, pp. 3207-3214,1983
`Printed in U.S.A.
`
`The Kinetics of Actin Nucleation and Polymerization*
`
`Larry S. Tobacman and Edward D. Korn
`From the Laboratory of Cell Biology, National Heart, Lung, and BZood Institute, Bethesda, Maryland 20205
`
`(Received for publication, August 25, 1982)
`
`The polymerization kinetics of rabbit skeletal muscle
`actin was studied by following the increase in fluores-
`cence of tracer amounts of actin conjugated to N-pyr-
`enyl iodoacetamide. The observed polymerization ki-
`netics could be precisely fit by numerical integration of
`equations describing a nucleation-elongation process.
`Under all conditions tested, the rate of nucleation was
`proportional to the fourth power of the actin concen-
`tration; therefore, the actin nucleus is a tetramer. In
`buffers containing either MgCl, or CaC12, but not both,
`the observed kinetics accurately fits the unique polym-
`erization time course derived from the actin concentra-
`tion, the critical concentration, and the product of the
`nucleation and filament elongation rate constants.
`When MgClz was added to G-actin in a buffer also
`containing CaCl,, polymerization did not follow simple
`nucleation-elongation kinetics because divalent cation
`exchange preceded nucleation. Relative filament elon-
`gation rate constants were 12,24,31,79,100, and 67 for
`actin in 1 l l l ~ CaCl,, 0.1 m CaClz + 0.1 M KC1, 1 m
`MgC1~,2 m MgCl,, 1 n m MgCl, + 0.1 M KC1, and 50 CM
`MgClz + 0.1 M KCl, respectively. The relative rate con-
`stants for filament-monomer dissociation differed less
`than 2-fold, but the relative nucleation rate constants
`varied dramatically: 1.7, 46, 550, 10,000, 10,000, and
`2,900, respectively, under these conditions of polymer-
`ization. These data strongly support the validity of the
`nucleation-elongation theory of actin polymerization
`and establish the nucleus size as four. The
`rate of
`nucleus formation is the major variable determining
`the rate of actin polymerization. The high degree of
`sensitivity of the nucleation rate to the concentration
`of actin and to ionic conditions may indicate the way in
`which intracellular polymerization kinetics are regu-
`lated.
`
`tion of actin monomers. Small aggregates of actin monomers
`are unstable and tend to dissociate. The number of intermo-
`lecular bonds in an aggregate increases with its size until the
`aggregate is favored to grow. The nucleus for filament growth
`is defined as the smallest aggregate which is more likely to
`grow than to dissociate. All aggregates of the nucleus size (n)
`or larger grow by sequential addition of monomers until the
`concentration of unpolymerized monomers falls to its steady
`state value.
`A formally complete description of polymerization depends
`upon each rate constant for monomer addition to and
`loss
`from each size aggregate. The complete analysis requires an
`infinite set of inter-related differential equations which are
`mathematically unwieldy and impossible to relate to experi-
`mental data. Wegner and Engel (3) have pointed out a much
`simpler and more useful special case in which: 1) the aggregate
`of size n - 1 is in rapid pre-equilibrium with monomer, i.e. it
`forms rapidly and is much more likely to dissociate than to
`grow; 2) the forward rate constant for monomer addition is
`the same for all aggregates of size n - 1 or larger; and 3) the
`rate constant for one monomer to dissociate is the same for
`all aggregates of size n or larger. For this special case, the rate
`of polymerization (dAF/dt) can be expressed as:
`dAF - - dAl= k’CAI - K-C = k’C(AI - A;)
`dt
`
`dt
`because Ay = k-/k’, where A I is the concentration of G-actin,
`k’ is the sum of the rate constants for monomer addition at
`the two filament ends, k - is the sum of the rate constants for
`two filament ends, C is the
`monomer dissociation at the
`concentration of polymers (aggregates
`larger than the nu-
`cleus), and AT is the concentration of G-actin at steady state
`(critical concentration). The nucleation rate is the rate of
`change of the number of filaments per unit volume:
`
`( 1)
`
`”
`
`~~
`
`~
`
`-
`
`In non-muscle cells, the interconversion between mono-
`meric actin and polymerized actin is a reversible, highly
`regulated process which
`is critically important for a wide
`variety of cellular functions (1). A large number of intracellular
`proteins which can modulate actin polymerization has been
`isolated. To understand actin polymerization in the cell, how-
`ever, one must first elucidate the polymerization of purified
`actin.
`The theory describing actin polymerization kinetics was
`developed by Oosawa and co-workers (2) and, more recently,
`was augmented by Wegner and collaborators (3-5). According
`to the theory, polymerization of actin is a cooperative process
`resembling the condensation of a gas. An aggregate containing
`any number of actin molecules is formed by sequential addi-
`
`* The costs of publication of this article were defrayed in part by
`the payment of page charges. This article must therefore be hereby
`marked “aduertisement” in accordance with 18 U.S.C. Section 1734
`solely to indicate this fact.
`
`(2)
`
`~ = K , - ~ ~ + ( A ~ Y ~ ( A ,
`- A ? )
`dt
`where K , - is the association constant relating the concentra-
`tion of monomer to the concentration of the aggregate one
`smaller than the nucleus.
`Equations 1 and 2 are inter-related differential equations
`which together predict the time course of actin nucleation and
`polymerization. Numerical
`integration of these equations
`yields a polymerization
`curve dependent upon the nucleus
`size, the actin concentration, the critical concentration, and a
`parameter designated k+k,,,, which is the product of the
`filament elongation rate constant, k’, and an apparent nu-
`cleation rate constant, k’K,, - l = k,,,. (3, 5).
`Recently, Kouyama and Mihashi
`(6) have developed an
`extremely sensitive fluorescence assay of actin polymerization.
`We have used this assay to obtain precise kinetic data on the
`conversion of G-actin to F-actin at several concentrations of
`actin under different conditions. Although a single polymeri-
`zation curve could be fit by more than one set of values, the
`
`3207
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`3208
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`The Kinetics of Actin Nucleation and Polymerization
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`6000
`Time (s)
`FIG. 1. Polymerization of M&+-G-actin in 1 m~ MgCl, and
`0.1 M KCl. G-Actin (6% pyrenyl actin) was dialyzed into a buffer
`containing 50 p~ MgC12 and no added CaCL. Polymerization of 2.1,
`4.2, 6.8, 9.0, and 10.9 PM actin was induced by addition of 40.5 pl of a
`75028.5:30 mixture of 4 M KCL, 1 M MgC12, and 10 m~ CaCh. Final
`sample volumes were 1.5 ml. Especially during the early stages of
`polymerization, only some of the experimental points (X) are plotted.
`Theoretical curves (- - -) were drawn for n.= 4, k+k,,, = 3.96 X l O I 5
`s? M-4, A? = 0.21 p ~ ,
`and the appropriate constant relating the
`
`arbitrary units of change in fluorescence intensity to the known
`concentration of polymerized actin at steady state. The inset shows
`the same data for 6.8, 9.0, and 10.9 p~ actin curves, displayed with a
`different time scale to show the nucleation phase to better advantage.
`
`
`
`5600
`
`0
`
`1200
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`2400
`Time (s)
`FIG. 2. Polymerization of M&+-G-actin in 1 m~ MgCl,. Po-
`lymerization of 11.3, 14.4, 18.3, and 22.2 p~ actin (6% pyrenyl actin)
`was initiated by addition of 30 pl of 47.5 mM MgC12, which increased
`the sample volumes to 1.5 ml. Especially during the initial stages of
`polymerization, only some of the experimental data are shown (X).
`Theoretical curves (- - -) were derived for n = 4, AT = 0.58 p ~ ,
` and
`
`h'k,,, = 7.1 X
`S-' M-4.
`
`Polymerization kinetics was highly dependent upon the actin
`concentration; polymerization was nearly complete in 10 min
`for 10.9 PM actin, but required about 3 h for 2.1 PM actin.
`The experimental data are fit remarkably well by theoreti-
`cal curves derived from the nucleation-elongation model (Fig.
`1). With a nucleus size of four and a single computer-chosen
`parameter, k'k,,,,
`the theoretical and experimental curves
`coincide whether polymerization required minutes or hours.
`Similar results were obtained when actin was polymerized
`by the addition of 1 mM MgClz without KC1 (Fig. 2). Although,
`at all actin concentrations, polymerization was slower when
`the KC1 was omitted, the polymerization curves had the same
`sigmoidal shape as when KC1 was present. With a nucleus size
`
`set of experimental curves obtained at different actin concen-
`trations (but under the same polymerization conditions) could
`be fit by only one nucleus size and one product of rate
`constants. The experimental data were then used to determine
`the effects of MgC12, KC1, and CaClz on the rates of nucleation,
`filament growth, and filament dissociation.
`
`MATERIALS AND METHODS
`Actin was prepared from rabbit fast skeletal muscle by the proce-
`dure of Eisenberg and Kielley (7), followed by gel
`fitration over
`Sephadex G-200. Monomeric actin was stored on ice in a buffer
`containing 5 mM Tris.HC1, 0.2 mM dithiothreitol, 0.2 mM ATP, 0.1
`mM CaC12, and 0.01% sodium azide, pH 8.0. The actin concentration
`was measured by absorbance at 290 nm, using an extinction coefficient
`of 0.62 ml. mg". All reagents were the highest commercially available
`grade.
`Modification of actin with N-pyrenyl iodoacetamide, purchased
`from Molecular Probes, was performed by the procedure of Kouyama
`and Mihashi (6), with minor modifications.' The protein concentra-
`tion of solutions containing modified actin were determined by the
`procedure of Bradford (8), using unmodified actin as a standard.
`G-actin free of added Ca2+ was prepared by dialyzing a mixture of
`pyrenyl G-actin' (6%) and unmodified G-actin (94%) for 24 h against
`buffer containing 5 mM Tris. HC1,0.2 mM dithiothreitol, 0.2 mM ATP,
`50 p~ MgC12, and 0.01% sodium azide, pH 8.0. Experiments were
`completed within 48 h after concluding the dialysis.
`Similarly, actin in a buffer with low added CaCl' was prepared by
`overnight dialysis of 1-2 mg/ml of G-actin, 6% modified with N-
`pyrenyl iodoacetamide, against a buffer containing 5 mM TriseHCl,
`0.2 mM dithiothreitol, 0.2 mM ATP, 40 PM CaCl', and 0.01% sodium
`azide, pH 8.0. Minutes before beginning a polymerization experiment,
`the CaClz concentration was further lowered to 10 p~ by mixing the
`actin with a buffer identical with the dialysate except for the absence
`of CaCL
`An SLM 4000 spectrofluorometer was used to monitor fluorescence.
`The excitation and emission wavelengths were 368 nm and 388 nm,
`respectively. Polymerization was initiated by addition of mixtures of
`small aliquots of concentrated KCl, MgCL, and/or CaCl' to 1.5-ml
`samples prewarmed to 25 "C. To avoid bleaching the fluorophore,
`pyrenyl actin was stored in the dark. During polymerization, the
`sample was exposed to the lamp intermittently. A circulating water
`bath maintained sample temperature at 25 "C. The digitized fluores-
`cence data were analyzed with the MLAB software package.
`Critical concentrations were measured by first polymerizing actin
`(5% pyrenyl actin) at a single concentration between 6 and 24 p ~ .
`The fully polymerized actin was then diluted in the same buffer to
`multiple lower actin concentrations. After a 16-h incubation in the
`dark at 25 "C, the steady state fluorescence was measured on 0.4-ml
`samples in cuvettes (3 X 10 mm).
`Intermolecularly cross-linked F-actin was obtained as a byproduct
`of the preparation of covalently cross-linked actin dimer. The proce-
`dure has been described elsewhere (9). Briefly, F-actin was cross-
`linked with N,N'g-phenylenebismaleimide, concentrated by ultra-
`centrifugation, and dialyzed against a buffer in which F-actin normally
`depolymerizes. The fraction of cross-linked F-actin that did not
`depolymerize was pelleted and the pellet was homogenized and re-
`suspended at 8.4 mg/ml.
`
`RESULTS
`Nucleus Size-G-actin, 6% labeled with N-pyrenyl iodo-
`acetamide, was dialyzed into a buffer containing 50 PM MgClz
`and no CaC12. The fluorescence intensity increased about 15-
`fold after polymerization was induced by the addition of KCl,
`MgClZ, and CaClz to final concentrations of 0.1 M, 1 mM, and
`10 ,UM, respectively (Fig. 1). The presence of 10 PM CaClz did
`not significantly change the kinetics, but was added for com-
`parison to another experiment described below. A sigmoidal
`polymerization curve was observed at each of five actin con-
`centrations, regardless of the overall rate of the process.
`' S. L. Brenner and E. D. Korn, manuscript submitted for publi-
`cation.
`The abbreviation used is: pyrenyl actin, N-pyrenylcarboxyami-
`domethyl actin.
`
`Page 2
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`

`

`The Kinetics of Actin Nucleation and Polymerization
`
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`L L
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`900
`
`1800 0
`Time ( s )
`FIG. 3. Polymerization of M$+-G-actin in 0.1 M KC1 and determination of a unique nucleus size. Actin,
`10.7 ( X ) and 20 (0) p ~ ,
` was polymerized by addition of 37.5 p1 of 4 M KCl, raising the final sample volumes to 1.5
`ml. Each half of the figure shows the experimental data at both actin concentrations and three pairs of theoretical
`curves. A: . . . . , n = 3 and the k'k,,,
`required to fit the data for 10.7 p~ actin; ---,
`n = 5 and the k'k.,,
`required to fit the data for 10 p~ actin; - - -, n = 4 and a k'k,,, that fits the data for 10.7 p~ and 20 p~ actin. E :
`n = 3 and the k'k,,, required to fit the data for 20 p~ actin; --,
`n = 5 and the k'k,,,
`required to fit the
`-,
`data for 20 p~ actin; - - -, n = 4 and a k'k,,,
`that fits the data for 10.7 p~ and 20 p~ actin.
`
`9 0 0
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`of four, a single value for k'k,,, allows generation of theoret-
`ical curves, accurately corresponding to all four experimental
`curves.
`For each ionic condition studied, only a nucleus size of four
`can be used to fit a family of theoretical curves to the family
`of experimental curves obtained from a series of actin concen-
`trations. This is true for polymerization in 1 mM MgClz plus
`0.1 M KC1 (Fig. l), in 1 mM MgClz alone (Fig. 2), as well as for
`0.1 M KC1 alone which is the example
`polymerization in
`illustrated in Fig. 3. The data for 10.7 IJM actin in 0.1 M KC1
`can be accurately described by any of three almost superim-
`posed theoretical curves which assume nucleus sizes of three,
`four, or five monomers, respectively, and use different com-
`puter-generated values for k'k,,,.
`(Fig. 3A). When the as-
`sumed nucleus (n) is a tetramer, the same value for k+k,,,
`produces a theoretical curve that also fits the data for 20 IJM
`actin. When n = 3 or n = 5, however, the values for k'k,,,
`that fit the data for 10.7 IJM actin predict polymerization
`curves which do not fit the data at 20 IJM actin. When n = 3,
`the theoretical curve describes a slower polymerization than
`was observed experimentally for 20 IJM actin. When n = 5, the
`computer-solved curve is faster than the experimental data
`for 20 IJM actin.
`Similarly, the data for 20 IJM actin alone can be fit by
`theoretical curves for assumed nucleus sizes of 3,4, or 5 using
`for each curve (Fig. 3B). However,
`different values of k'k,,,
`when n = 3, the computer-solved curve using the same value
`for k'k,,, predicts faster polymerization than was observed
`for 10.7 IJM actin. Assuming n = 5, the k'k,,, which fits the
`experimental curve at higher actin concentration generates
`too slow a polymerization curve to fit the data at lower actin
`concentration. Only when n = 4 can theoretical curves be fit
`to the polymerization data at both actin concentrations using
`the same value of k*k,,,.
`Relation between Nucleation and Polymerization-The
`relationship between fractional polymerization and fractional
`nucleation depends only on the nucleus size and, of course, on
`the validity of the model. The relationship is independent of
`ionic conditions, the nucleation rate, the Fiament elongation
`(if it is much greater than
`rate and the actin concentration
`the critical concentration). Therefore, knowing that the actin
`nucleus consists of 4 monomers, the temporal relationship of
`
`800
`
`1000
`
`0
`
`200
`
`600
`400
`Time (s)
`FIG. 4. Comparison of polymerization and nucleation kinet-
`ics. The theoretical curve which fits
`the fluorescence data for the
`sample of 10.9 p~ actin in Fig. 1 was normalized to extend between 0
`The numerical solution of Equations
`and 100% polymerization (-).
`the polymerization curve also yields
`1 and 2 which gives
`a curve
`which is proportional to the concentration of filaments, i.e. cumulative
`nucleation. This curve was also normalized to 100% (- - -).
`
`nucleation kinetics to polymerization kinetics can be calcu-
`lated. The cumulative extent of nucleation during polymeri-
`zation is plotted in Fig. 4 . Nucleation precedes polymerization,
`as it must, but is not confined to the earliest portion of the
`polymerization curve. When 7% of the actin is polymerized,
`the lag phase is over and the polymerization rate has reached
`75% the maximal rate. But at this point nucleation is still only
`50% complete, i.e. nucleation is not confined to the lag phase.
`Not until 45% of the actin is polymerized does nucleation
`reach 95% completion.
`Ca2+-G-Actin versus Mgzf-G-Actin-A subtle change in
`a significant change in
`experimental conditions can cause
`polymerization kinetics and deviation from the nucleation-
`elongation model. Fig. 5 shows two polymerization
`curves
`obtained in buffer containing 1 II~M MgC12, 0.1 M KC1, and 10
`IJM CaC12. One of these curves is of data taken from Fig. 1 for
`Mg*+-G-actin. The other curve is for an identical experiment
`except that the G-actin had
`been equilibrated with buffer
`
`Page 3
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`

`

`The Kinetics of Actin Nucleation and Polymerization
`
`ory was achieved, however, by assuming that only Mg2+-G-
`actin nucleates at a measurable rate and that G-actin ex-
`changes bound Ca2+ for Mg2+ with first order kinetics. With
`these assumptions, the nucleation rate would be
`
`3210
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`120
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`0
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`0
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`30
`
`60
`Time (s)
`FIG. 5. Comparison of the polymerization kinetics of Ca2+-
`G-actin and M$+-G-actin. G-actin, 10 PM, 6% pyrenyl actin, in 10
`PM CaCh (X) or 50 PM MgC12 (0), was polymerized by adding 1 mM
`MgCh + 0.1 M KC1 (X) or 0.95 m~ MgClz + 0.1 M KC1 + 10 PM CaClz
`(0). Fluorescence data were converted to F-actin concentration to
`facilitate comparison between the two samples. Theoretical curves
`(---) were derived using Equations 1 and 3 and kinetic constants
`from Fig. 6 (for Ca*+-G-actin) or using Equations 1 and 2 and kinetic
`constants from Fig. 1 (for Mg2+-G-actin). The data for Mg"+-F-actin
`are taken from the experiment in Fig. 1.
`
`containing 10 p~ CaClz instead of 50 p~ MgClZ. When MgClz
`was added to G-actin in the
`buffer containing CaClZ, the
`length of the initial, slow phase of polymerization was pro-
`longed. For the conditions used in Fig. 5, the polymerization
`of 10.9 p~ Ca2+-G-actin lagged about 20 s behind polymeriza-
`tion of 10.9 p~ Mg2+-G-actin. The polymerization curves have
`the same shape except for the initial portion.
`The prolonged lag phase for polymerization of Ca2+-G-actin
`was more obvious at higher actin concentrations
`(9.3, 10.9,
`
`and 12.6 p ~ ; Figs. 5 and 6). At lower actin concentrations (4.9
`
`and 7.0 p ~ ; Fig. 6), where polymerization was slower, the
`shapes of the polymerization curves for CaZC-G-actin were
`indistinguishable from thoese obtained for Mg2+-G-actin. The
`theoretical curves which fit all of the data for Ca2+-G-actin ip
`Figs. 5 and 6 were derived from a polymerization model
`described below (Equation 3).
`When polymerization curves obey nucleation-elongation ki-
`netics, the size of the nucleus can be determined approxi-
`mately by measuring the time required to reach a certain
`extent of polymerization, e.g. 5% polymerization (t5%), as a
`function of actin concentration (2, 10). The slope of In
`( 4 % )
`uersus In [actin] is one-half the nucleus size. This analysis
`assumes irreversible polymerization, which is a reasonable
`approximation when the actin concentration is much greater
`than A?. The results of treating the data in Figs. 1,5, and 6 in
`this way are plotted in Fig. 7. For the data from Fig. 1, the
`slope is 2; therefore, the nucleus is a tetramer. The data from
`Figs. 5 and 6, however, give a slope of 1.46; therefore, the
`apparent nucleus for polymerization of Ca2+-G-actin in KC1
`+ MgClz is a trimer by this analysis.
`This would appear to be different from the nucleus size of
`four obtained for Mg2+-G-actin polymerized in MgClz + KC1
`(Fig. l), MgClz (Fig. 2), or KC1 (Fig. 3) and, in addition, Ca2+-
`G-actin in 0.1 mM CaClz + 0.1 M KC1 (data not shown). But,
`in fact, analysis by a ln/ln plot of the data for CaZC-G-actin in
`Figs. 5 and 6 results in an incorrect nucleus size because the
`kinetics of polymerization is inconsistent with the nucleation-
`elongation model. No values for k+k,,, and n yield a numeri-
`cally integrated theoretical curve that corresponds to the
`experimental data. Successful matching of experiment to the-
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`"
`
`1 (AI - A 3
`
`(3)
`
`dC - k A;- '(1 - e(-%p,t) n
`dt
`where k,,, is the rate constant for conversion of slowly nu-
`cleating CaZf-G-actin to the much more rapidly nucleating
`Mg2+-G-actin. With a nucleus size of four and appropriate
`values for k,,, and k'k,,,, Equations 1 and 3 yield the theo-
`retical curves displayed in Figs. 5 and 6. The theoretical curves
`conform to the experimental data very well. Therefore, a
`minimal scheme for the polymerization of Ca2+-G-actin by
`addition of MgClz is: Ca"-monomer + monomer + Mg2+-
`monomer + nucleus + polymer.
`Effect of Ionic Conditions on Rates of Filament Nuclea-
`tion, Elongation, and Dissociation-Comparisons of polym-
`erization kinetics in different buffers can be analyzed by
`
`1
`0
`900
`1200
`300
`600
`Time (s)
`FIG. 6. Polymerization of Ca2+-G-actin in 0.1 M KC1 + 1 m~
`MgC12. Actin, 4.9, 7.0, 9.3, and 12.6 PM in 10 ,UM CaClz and containing
`6%' pyrenyl actin, was polymerized by addition of 39 pl of a 75030
`mixture of 4 M KC1, 1 M MgClZ. Final sample volumes were 1.5 ml.
`Theoretical curves (- - -) were derived from Equations 1 and 3, with
`n = 4, k+k,, = 4.3 X l O I 5 s? M-4, and k,, = 8.0 X IO-*S-~.
`
`2.4
`
`2.0
`
`1.2
`
`1.6
`
`2.0
`In Actin (,uM)
`FIG. 7. Estimation of nucleus size by the time to 5% polym-
`erization as a function of actin concentration. The time to reach
`5% of the final change in fluorescence (t5%,) was determined for the
`curves in Figs. 1, 5, and 6. The slope of the plot of In ( t w ) uersus In
`[actin] was determined by least squares analysis to be 2.01 for Mg2+-
`G-actin ( X ) and 1.46 for Ca2+-G-actin (0).
`
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`experiments in the absence of added nuclei showed that
`negligible polymerization could have occurred by elongation
`of spontaneously formed nuclei. Presumably, the lag reflects
`the time required either for binding of the added cations to
`actin or for a conformational change of actin following binding.
`The maximum rate of polymerization (ClA,~/dt),,. is related
`to the rate constant for filament elongation, k':
`
`where A , (tmax) is the concentration of G-actin at the time of
`the maximal rate of polymerization. The validity of the appli-
`cation of Equation 4 to the data in Fig. 8 was demonstrated
`by a control experiment in which the actin concentration was
`varied. In samples with equal amounts of cross-linked F-actin,
`but differing up to 4-fold in their concentrations of G-actin,
`the maximum rate of polymerization was directly proportional
`to the final concentration of F-actin (data not shown).
`Table I presents the quantitative comparison of relative
`polymerization kinetics in different buffers containing either
`Ca2+ or MgZ+ with and without KCl. For each condition,
`and relative values of k'
`absolute values for AT and k'k,,,
`were experimentally determined. From these three values,
`relative values of k - and k,,, were calculated. The units for
`k+ and k,,, are arbitrary, unrelated, and chosen for best
`comparison among buffers.
`The time required for spontaneous polymerization in the
`absence of added nuclei is dependent upon k'k,,,. The higher
`this value, the faster is polymerization. For concentrations of
`actin much larger than the critical concentration, the time to
`any fractional polymerization is approximately inversely pro-
`portional to (k+knuc)L'2 (2, 10). By comparing these values
`(Table I, column 3), it can be estimated that it took 30 times
`longer for actin to polymerize in 1 mM CaC12 than in 1 mM
`MgC12. Similarly, in samples containing 0.1 M KCl, polymeri-
`zation was about 14 times faster with added 50 PM MgClz than
`with the addition of 100 PM CaC12. Either in the presence or
`absence of KC1, polymerization was accelerated by raising the
`MgClz concentration. Also apparent from Table I is that
`
`TABLE I
`Effect of ionic conditions on the rate constants for the
`polymerization of actin
`G-actin, containing 6% pyrenyl actin, was polymerized in six differ-
`ent ionic conditions in the presence or absence of cross-linked actin
`nuclei. Values for k'k,,,
`(column 2) were determined from experi-
`ments such as those described in Figs. 1-3 and the square roots were
`calculated (column 3). The values for A;" (column 4) determined by
`light scattering and fluorescence were indistinguishable but the fluo-
`rescence data were more precise and these were used. Relative values
`for k'
`(column 5) and k - (column 6) were determined from the
`experiment described in Fig. 8 and were normalized to k' = 100 in 1
`mM MgClz + 0.1 M KCl. To determine the relative nucleation rate
`constants, k,,, = k'K,
`I , the values for k'k,,, were divided by the
`corresponding relative values for k' and the results normalized so
`that the highest value was 10,OOO.
`
`~
`
`Condition
`
`k+k.,
`
`(k'k,,)'''
`
`A?
`
`X IO'"
`4.6 X IO"
`
`-2
`
`-4
`
`p M
`S""'
`2.9 X lo5 2.0
`2.1 X lo6 0.76
`2.9 X lo7 0.50
`8.2 X lOI4
`7.1 X loJ3 8.4 X lo6 0.58
`3.3 X lot5 5.7 X io7 0.25
`4.2 X IOi5 6.5 X 10' 0.21 100
`
`Relative values
`
`k'
`
`k -
`
`knur
`
`12 24
`24 18
`
`1.7
`46
`
`67 34
`
`2,900
`
`31 18 550
`
`79 20
`
`10,000
`21 1O,O00
`
`considering how the rate constants for filament growth, k',
`filament dissociation, k - , and nucleation, k,,,, depend upon
`the polymerization conditions. Unfortunately, bulk polymeri-
`zation experiments as in Figs. 1-3 do not determine
`the
`absolute rate constants, because the number of filaments
`cannot be accurately measured. Relative (not absolute) fda-
`ment elongation rate constants, however, can be determined
`under a variety of buffer conditions by comparing the polym-
`erization of samples to which an equal number of nuclei have
`been added, Nuclei which are stable in a nonpolymerizing
`buffer can be added prior to initiation of polymerization in
`large enough quantities so that spontaneous nucleation can
`be neglected. Once relative values for k' are known, a more
`complete comparison among polymerization conditions only
`requires measurement of A;" and k'k,,,. As A? = k - / k + ,
`simple algebra then yields relative values of k', k - , and k,,,,
`the elongation, dissociation, and nucleation rate constants.
`To nucleate actin polymerization, we used F-actin which
`had been intermolecularly cross-linked by phenylenebismal-
`eimide and was resistant to depolymerization. It has been
`shown that elongation occurs at both ends of these filaments
`(11). Equal amounts of cross-linked F-actin were added to
`each of six samples of G-actin. Polymerization was initiated
`by addition of KCl, MgC12, and/or CaC12. To facilitate com-
`parisons among samples, the fluorescence data were converted
`to concentrations of F-actin. For each sample, the maximum
`fluorescence change was the result of the polymerization of
`an amount of F-actin equal to the total actin concentration
`minus the critical concentration, which was determined inde-
`pendently. Fig. 8 shows the kinetics of nucleated polymeriza-
`tion that were obtained in a range of conditions including one
`where spontaneous polymerization is comparatively fast (0.1
`M KC1 and 1 mM MgC12) and one where spontaneous polym-
`erization is very slow (1 mM CaC12). The maximum polymer-
`ization rates occurred 10-20 s after addition of salt. This slight
`delay was not due to spontaneous nucleation because control
`
`
`
`4 -
`
`3
`
`P
`3
`c .-
`+
`9
`L L 2
`
`1
`
`0
`0
`
`20
`
`80
`
`100
`
`"
`
`60.
`40
`Time (s)
`FIG. 8. Polymerization rate of G-actin mixed with cross-
`linked nuclei; dependence upon MgCla, KC1, CaC12. Samples (1.5
`ml) containing 5% pyrenyl actin and 95% unmodified actin plus 0.017
`mg/ml of cross-linked F-actin were polymerized at 25 "C by addition
`of mixtures O f 4 M KCl, 10 nm CaCh, 0.1 M MgC12, and/or 1 M MgCl2.
`1 mM CaC12 8.4
`0.1 mM CaC12 +
`Four samples of actin were first dialyzed against buffer containing 50
`p~ MgClz and polymerized by the indicated additions. -,
`6.8 p~ G-
`actin plus 0.95 m~ MgCb; - - -, 6.8 p~ G-actin plus 1.95 rnhf MgClz;
`0.1 M KC1
`50 p~ MgClp +
`, 6.8 p~ G-actin plus 0.95 nm MgClz and 0.1 M KCI; - - -,
`0.1 M KC1
`6.8 p~ G-actin plus 0.1 M KC1. Two samples of actin were fmt dialyzed
`1 mM MgC12
`against buffer containing 100 p~ CaCh and then polymerized with
`the following additions. . . , 20.4 p~ G-actin plus 0.9 mM CaCl2; 0,
`2 mM MgC12
`1 MgC12 +
`6.8 p~ G-actin plus 0.1 M KCl. Each fluorescence curve was separately
`0.1 M KC1
`normdied to convert the data to F-actin concentration.
`
`Page 5
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`http://www.jbc.org/
`
` at SUNY at Stony Brook on May 2, 2017
`
`32'12
`
`The Kinetics of Actin Nucleation and Polymerization
`
`polymerization of actin in 1 mM MgClz was accelerated about
`ing and fluorescence measurements.' Finally, if there were any
`8-fold by addition of 0.1 M KC1 or 7-fold by raising the MgClz
`minor differences between pyrenyl actin and unmodified actin,
`concentration to 2 mM. There was a 50,000-fold difference in
`the consequences would be minimized by using only tracer
`k+k,,, or approximately a 200-fold change in the time required
`amounts of the pyrenyl actin in that the tracer, which is below
`for polymerization, between the fastest (0.1 M KC1 + 1 mM
`its critical concentration, serves as a measure of the polym-
`MgC12) and the slowest (1 mM CaC12) conditions used.
`erization of the bulk unmodified actin.
`Column 4 of Table I lists the critical concentration in each
`Oosawa and Asakura (2) discussed the use of the polymer-
`salt condition. Alteration of the MgCL, CaC12, or KC1 concen-
`ization kinetics of actin to estimate the size of the nucleus,
`trations affected the critical concentration much less than it
`which they judged to be a trimer or a tetramer. Wegner and
`changed k'k,,, or (k+k,,,)1/2. In other words, the ionic condi-
`Engel (3) improved the theoretical basis for the analysis by
`tions chosen to induce polymerization have a much greater
`showing how reversible, instead of irreversible, polymerization
`influence on the kinetics of polymerization than on the con-
`can be considered, using numerical integration techniques.
`Wegner's kinetic data (5, 15), however, do not fit the simple
`centration of G-actin remaining when polymerization is com-
`plete.
`nucleation-elongation model we have successfully applied to
`The fifth column of Table