Profilin Promotes Barbed-end Actin Filament Assembly without Lowering the Critical Concentration*

The mechanism of profilin-promoted actin polymerization has been systematically reinvestigated. Rates of barbed-end elongation onto Spectrin·4.1·Actin seeds were measured by right angle light scattering to avoid confounding effects of pyrenyl-actin, and KINSIM was used to analyze elongation progress curves. Without thymosin-β4, both actin and Profilin·Actin (P·A) are competent in barbed-end polymerization, and kinetic simulations yielded the same bimolecular rate constant (∼10 × 106 m −1 s−1) for actin monomer or Profilin·Actin. When measured in the absence of profilin, actin assembly curves over a 0.7–4 μmthymosin-β4 concentration range fit a simple monomer sequestering model (1 μm K D for Thymosin-β4·Actin). The corresponding constant for thymosin-β4·pyrenyl-Actin, however, was significantly higher (∼9–10 μm), suggesting that the fluorophore markedly weakens binding to thymosin-β4. With solutions of actin (2 μm) and thymosin-β4 (2 or 4 μm), the barbed-end assembly rate rose with increasing profilin concentration (0.7–2 μm). Actin assembly in presence of thymosin-β4 and profilin fit a simple thermodynamic energy cycle, thereby disproving an earlier claim (D. Pantaloni and M.-F. Carlier (1993)Cell 75, 1007–1014) that profilin promotes nonequilibrium filament assembly by accelerating hydrolysis of filament-bound ATP. Our findings indicate that profilin serves as a polymerization catalyst that captures actin monomers from Thymosin-β4·Actin and ushers actin as a Profilin·Actin complex onto growing barbed filament ends.

Determining how cells regulate actin assembly is vital for understanding motility. Resting cells contain high concentrations of unpolymerized actin, typically 200 -400 M, or about 600 -1200 times greater than the critical concentration for assembly of pure actin (1,2). Actin-sequestering proteins prevent spontaneous assembly of monomeric actin, and nonmuscle cells contain two such sequestering proteins: the 15-kDa protein profilin and the 5-kDa thymosin-␤4 (3,4). These proteins were mainly thought to bind actin monomers with a one-to-one stoichiometry and to sequester actin monomers, and both have a higher affinity toward Actin⅐ATP as compared with Actin⅐ADP. The current view is that profilin and thymosin-␤4 are likely to play complementary roles in the cell. Thymosin-␤4 has a single mode of action (5)(6)(7)(8), namely binding actin monomers to create a sequestered pool of monomers to supply Actin⅐ATP during periods of active filament growth. The mode of action of profilin, however, is more complex and in many respects has remained controversial and elusive. Although the Profilin⅐Actin complex was initially also thought to be strictly a monomersequestering protein (9 -11), Pollard and Cooper (12) later confirmed the inference that it adds to the barbed ends of actin filaments (13). Initial rate and steady-state assembly measurements showed that profilin can efficiently bind actin monomers (K d Ͻ 5 M) and can weakly interact with filament barbed ends (K capping ϳ 100 M). Profilin catalyzes exchange of free ATP with actin-bound ADP to form actin-bound ATP and free ADP (14 -16). Profilin also exhibits affinity toward oligoproline modules (17), another unique property that facilitates transfer of actin monomers to the polymerization zone (16) lying immediately behind motile Listeria (18), Shigella (19), and vaccinia (20). By concentrating Profilin⅐Actin complex in regions of active filament assembly, explosive rates of filament growth (Ն500 monomers s Ϫ1 ) are readily catalyzed. By contrast, thymosin-␤4 does not bind to oligoproline and fails to concentrate in the polymerization zone.
Highly motile cells contain high concentrations of both profilin and thymosin-␤4, and these proteins probably act in concert to promote rapid filament assembly. Pantaloni and Carlier (21) investigated the role of profilin in the absence and presence of thymosin-␤4. They suggested that profilin reduces the critical monomer concentration needed for filament assembly, and they suggested that this is accomplished by accelerating the irreversible hydrolysis of polymer-bound Profilin⅐Actin⅐ATP complex. However, their experimental design was compromised in several significant ways: 1) they used pyrenyl-actin polymerization and had to correct the observed data for the low affinity of profilin for pyrenyl-actin (9,22,23); 2) all of their determinations of the steady-state extent of polymerization rested on the results of single-point assays; and, most seriously, 3) their chosen conditions incorrectly assumed that it is possible to bias assembly in favor of barbed-end filament growth. In the latter case, steady-state assembly assays on filaments with both ends free must always reflect monomer gain and loss from both ends, irrespective of the experimentally chosen monomer concentration. Moreover, such conditions permit ATP-dependent treadmilling, a nonequilibrium condition that may have unnecessarily complicated their thermodynamic analysis.
The oft-quoted proposal that profilin reduces the critical actin monomer concentration by promoting irreversible ATP hydrolysis bears directly on virtually all efforts to understand the underlying mechanisms of actin-based motility. In the light of the aforementioned limitations on the earlier study, we decided to conduct an independent examination of how profilin promotes actin assembly in the absence and presence of thymosin-␤4. To monitor the progress of actin polymerization, we relied on right angle light scattering, thereby avoiding all complications encountered with the use of pyrenyl-actin. To examine barbed-end actin assembly exclusively, we used the erythrocyte Spectrin⅐4.1⅐Actin complex to "seed" polymerization. Finally, to achieve self-consistent kinetic and equilibrium parameters, we utilized the KINSIM/FITSIM software to analyze large data sets defining the polymerization progress curves. Our experimental findings indicate that (a) the Profilin⅐Actin complex participates in an assembly scheme defined by a closed thermodynamic cycle, whether or not thymosin-␤4 is present, and (b) there is no perceptible reduction in the actin critical concentration, even when profilin and thymosin-␤4 are both simultaneously present.

EXPERIMENTAL PROCEDURES
Protein Preparations-Actin was purified from rabbit muscle acetone powder by the method of Spudich and Watt (24) and gel-filtered through a 16/60 Superdex-200 column (Amersham Pharmacia Biotech). Purified actin was maintained in G-buffer (5 mM Tris-HCl, 0.1 mM calcium chloride, 0.2 mM dithiothreitol, 0.2 mM ATP, 0.02% sodium azide, pH 7.9) and used within a week. Pyrene-labeled actin was prepared as described previously (25) and was then gel-filtered as indicated above.
Recombinant human profilin was expressed in Escherichia coli (16) and purified by poly-L-proline affinity chromatography (26) and Superdex 200 gel filtration to remove any profilin dimers or oligomers. Profilin monomer fraction was kept in 10 mM Tris-HCl, pH 7.5, containing 1 mM dithiothreitol, 20 mM sodium chloride, and 0.02% sodium azide and used within a week.
Appropriate restriction sites were introduced into the coding sequence of rat T␤4 1 by polymerase chain reaction, and the modified DNA was ligated into the NdeI-BamHI sites of PET-12a vector (Novagen, Madison, WI). The sequence was verified by double-strand DNA sequencing. The construct of T␤4 was transformed into the E. coli BL21 (DE3) cells. The cells were grown at 37°C in 500 ml of LB medium (containing 50 g/ml carbenicillin) until A 600 reached a value of 0.6, whereupon expression was induced by adding isopropylthiogalactoside (0.5 mM). The cells were harvested 3 h later by centrifugation, and the pellet was disrupted in 25 ml of ice-cold 0.5 M perchloric acid with sonication for 30 s. After centrifugation at 30,000 ϫ g for 30 min at 4°C, the supernatant fluid was adjusted to pH 7.0 with potassium hydroxide. The sample was then kept on ice for 30 min, and the potassium perchlorate precipitate was removed by brief centrifugation. The resulting supernatant was enriched in T␤4, based on analysis by SDS-PAGE and gel shift assay for G-actin (7). The further purification of T␤4 was carried out as described previously (27,28). The T␤4-enriched sample was adjusted to pH 4.0 with formic acid and dialyzed overnight against 20 mM formic acid (pH 4.0). The sample was loaded onto a 1-ml HiTrap SP column (Amersham Pharmacia Biotech) and preequilibrated with 20 mM formic acid (pH 4.0). Column-bound proteins were eluted with a linear gradient (0 -1 M sodium chloride). Thymosin-␤4-containing fractions were pooled and dialyzed against 10 mM Tris-HCl (pH 7.5) containing 100 mM sodium chloride, 0.02% sodium azide. Further purification was achieved using a 10/30 Superdex 75 gel filtration column (Amersham Pharmacia Biotech). The resulting recombinant T␤4 was virtually pure, based on amino acid composition analysis.
The Spectrin⅐Band 4.1⅐Actin complex was prepared from erythrocyte ghosts as described by Young et al. (29), stored in G-buffer at 20°C, and used within a week.
Protein Concentration Determinations-Actin and profilin concentrations were measured by UV absorbance at 290 and 280 nm, using extinction coefficients of 0.026 and 0.015 M Ϫ1 cm Ϫ1 , respectively (30).
Seeded Elongation Kinetics at the Barbed End-Spectrin⅐Band-4.1⅐Actin complex only contains filaments with free barbed ends, and no pointed end assembly can occur (31). Therefore, we used this erythrocyte preparation as polymerization seeds to study actin barbed-end elongation by measuring light scattering and/or fluorescence in a Spex Fluorolog-3 fluorometer (Spex Instruments, Inc, Edison, NJ). All experiments were carried out at 22°C using Mg 2ϩ -actin. First, Ca 2ϩ -actin was converted to Mg 2ϩ -actin by incubating the Ca 2ϩ -actin sample in a buffer containing 10 mM Tris-HCl (pH 7.5), 0.2 mM dithiothreitol, 0.2 mM ATP, 0.05 mM magnesium chloride, 0.2 mM EGTA, 0.01% sodium azide for 10 min. The incubation of actin was continued for an additional 5 min in the presence or absence of profilin and/or T␤4. Seeded actin filament elongation was then initiated by adding Spectrin⅐4.1⅐Actin seeds, potassium chloride and magnesium chloride (final concentrations, of 0.1 and 2 mM respectively). For unlabeled actin, the time course of polymerization was determined by right angle light scattering at 350 nm. For pyrenyl-actin, the time course was monitored as an increase in fluorescence emission intensity at 387 nm (excitation at 366 nm with shutter on anti-bleaching mode). Polymerization of pyrenyl-actin was also measured by fluorescence and light scattering methods contemporaneously for the same sample by timed-switching between the two sets of excitation/emission wavelength settings.
Our finding that the intensity of light scattering at a 90°angle is linearly proportional to polymer weight concentration is fully supported by earlier investigators (32)(33)(34). In particular, Cooper and Pollard (32) state that 90°light scattering is proportional to polymer weight concentration and that light scattering is insignificant below the critical concentration. The work of Wegner and Engel (33) indicated that light scattering is insensitive to length distribution, when the polymer length reaches a critical length *. The parameter * equals /(4 sin), where is the wavelength of incident light; thus, when is 90°, * is approximately 90 nm. In practice, * is only a rough estimate, and those experienced with molecular light scattering find that linearity is often achieved when polymers reach a length of 0.5 *. In this regard, the average degree of polymerization of the actin filament seeds in Spectrin⅐4.1⅐Actin is 25-35, corresponding to a length of approximately 50 nm at the very beginning of the assembly process. Moreover, the concentration of actin seeds in our experiments was approximately 0.6 nM, whereas the initial actin monomer concentration was typically in the 2 M range. At steady state, the filaments should reach an average length of nearly 1.2 m (i.e. 450 monomers/filament ϫ 2.8 nm/monomer). This polymer length is around 14 times greater than *, indicating that light scattering intensity should be strictly proportional to polymer mass after the first 3% of polymerization has occurred, a condition that is most likely achieved within 15 s after mixing.
Whereas the above considerations indicate that accurate elongation rates are being measured over virtually the entire time course of elongation, we offer the following empirical evidence that is specifically relevant to our experiments. First, each successive pair of data points on any of our polymerization time courses corresponds to the instantaneous elongation rate with respect to a given incremental change in actin concentration. All of our polymerization data almost exactly match a first-order kinetic process, from start to end. Such a finding further validates the conclusion that incremental increases in light scattering intensity are proportional to polymer mass. Second, plots of light scattering intensity versus actin monomer concentration are linear, and when extrapolated to zero light scattering intensity, they yield critical concentration values matching those obtained with pyrenylactin in the absence of profilin and thymosin-␤4. This observation also indicates that the total light scattering amplitude is linearly proportional to polymer mass. Third, as will be shown under "Discussion," all of the polymerization rate data in this report fit a single linear plot. This finding by itself demonstrates that there is a linear correspondence between light scattering increment and polymer mass over a broad range of actin, profilin, and thymosin-␤4 concentrations.
Kinetic Simulation and Data Analysis-Actin polymerization kinetics in the presence or absence of profilin and T␤4 were analyzed with the KINSIM software (35) and fitted to experimental data with the Marquardt-Levant algorithm of FITSIM (36). Each individual run (i.e. each uninterrupted series of data points obtained from the fluorescence spectrometer in a single elongation experiment) could be fitted by a first-order reaction progress curve, and based on the deviation of the actual data points from individual exponential curves, we find that the range of experimental error in all light scattering data presented in this study was Ϯ1.8%. Because error bars on each point would only tend to obscure the actual progress curve, we set the size of the symbols in each graph to correspond to Ϯ2% of the overall amplitude. The reader will also note that some of the computer-generated progress curves presented within any one figure may actually deviate more than Ϯ2% from the data for an individual run. The computer-generated curves presented in each plot are those returned by the KINSIM software based on a single set of rate constants applied globally to all of the data sets. This statistical treatment constitutes a far more rigorous test of the rate constants than simply analyzing individual runs with an adjustable set of rate constants.

RESULTS
The proposal that profilin promotes filament assembly by stimulating irreversible ATP hydrolysis (21) was based on the inability of experimentally derived equilibrium constants to yield a closed thermodynamic cycle. Such closed cycles (or "energy squares") constitute valuable quantitative tests of models for binding interactions. If one can rigorously demonstrate the energy changes depend on the path taken around such a cycle, then there is prima facie evidence for the occurrence of one or more irreversible steps. Thus, the problem of actin polymerization in the simultaneous presence of thymosin-␤4 and profilin reduces to a single question: are the constants that define actin elongation in the presence of profilin and thymosin-␤4 consistent with a closed thermodynamic cycle? To answer this question, we have conducted a stepwise investigation: (a) beginning with the least complicated process, namely the barbed-end elongation of actin only (see Fig. 1A); (b) proceeding next to examine actin polymerization in the presence of profilin (Fig. 1B); (c) then, examining the effects of thymosin-␤4 in the absence of profilin (Fig. 1C); and (d) finally, evaluating elongation in the simultaneous presence of profilin and thymosin-␤4 (Fig. 1D).

Direct Comparison of Actin Filament Elongation Kinetics Using Pyrene Fluorescence and Right Angle Light Scattering-
Previous investigators demonstrated that profilin binds more weakly to pyrenyl-actin than to unlabeled actin (9,10,21,23). This behavior requires a series of calculations to correct all primary data to account for the differential affinity of profilin for unlabeled and labeled actin monomers. To assess the effects of profilin on actin filament assembly in the absence of this confounding effect, we employed right angle light scattering, which is a direct measure of polymer weight concentration. Light scattering is a useful means for studying actin polymerization (9, 10, 32-34), especially when care is exercised to minimize protein aggregation or dust. The availability of highly stable, digitally controlled fluorescence spectrometers has greatly improved the sensitivity of 90°light scattering meas-urements. However, because pyrenyl-actin has become the standard polymerization assay method, we directly compared fluorescence and light scattering data to identify significant differences in ability of these two assays as quantitative measures of actin filament assembly. To initiate assembly only onto free barbed ends, we added Spectrin⅐4.1⅐Actin nuclei (final protein concentration, 29.5 g/ml; estimated filament number concentration, 0.66 nM) to G-actin (2 M) at a mole fraction of 0.6 pyrene-conjugated monomer. Simultaneous monitoring of fluorescence and light scattering intensity was achieved over 10-s intervals by switching from the fluorescence mode (365 nm excitation, 387 nm emission) to light-scattering mode (350 nm for both excitation and emission). The photon count rate (cps) was higher in the light scattering mode (see values presented in Fig. 2); however, when plotted as ⌬F/⌬F maximum or ⌬Light-Scattering/⌬Light-Scattering maximum , both curves describe first-order processes with virtually identical rates and extents of elongation. Furthermore, the t1 ⁄2 values were 98 and 103 s from fluorescence and light scattering, respectively. Slight differences between fluorescence and light scattering were only evident at the earliest time points and were without effect on the entire progress curve analysis. The symbols are as follows: A, monomeric (or G-) actin; F n and F nϩ1 , filamentous actin containing n and (n ϩ 1) monomers; T␤4, uncomplexed thymosin-␤4; T␤4⅐A, thymosin-␤4⅐actin monomer complex; P, uncomplexed profilin; P⅐A, Profilin⅐Actin monomer complex; and P⅐F nϩ1 , filament capped with profilin. The cycle shows two pathways connecting free actin A with the elongated filament F nϩ1 , and in the absence of an irreversible process, the additivity principle requires that a cycle at equilibrium must obey certain thermodynamic constraints. A-D are the segments and relevant rate or equilibrium constants determined in the experiments described below.
FIG. 2. Contemporaneous determination of barbed-end actin elongation using light scattering and fluorescence. The time course of actin elongation at the barbed ends was monitored using 2 M actin (60% labeled pyrenyl-actin and 40% unlabeled actin). Seeded assembly was initiated by addition of Spectrin⅐4.1⅐Actin complex, immediately after which the solution was adjusted to 2 mM MgCl 2 and 100 mM KCl. Fluorescence and light scattering measurements were automatically carried out by programming the instrument to switch automatically between two different settings of excitation and emission wavelengths at 10-s intervals. For fluorescence intensity measurements, the excitation wavelength was 366 nm, and the emission wavelength was 387 nm. For light scattering, we used the same excitation and emission wavelength (i.e. 350 nm). A, fluorescence intensities; B, light scattering intensities. The lines are progress curves generated by simulating the first-order filament elongation process. We obtained half-lives of 98 Ϯ 6 and 103 Ϯ 6 s from the fluorescence and light scattering data, respectively. Note that the size of the symbols was chosen to reflect the range of experimental error. Fig. 1A, we conducted experiments on filament assembly as a function of actin monomer concentration at a fixed concentration of Spectrin⅐4.1⅐Actin seeds. As expected, elongation kinetics were first order, and the rate and extent of polymerization depended on the initial G-actin concentration. The results presented in Fig. 3 were reproduced in three separate experiments. The rate of seeded assembly of actin at the barbed end obeys first-order kinetics as follows

Elongation at Actin Filament Barbed Ends in the Absence and Presence of Profilin-To characterize the elongation reaction shown in
where C critical is the critical actin concentration (equal to K 1 or k Ϫ1 /k ϩ1 for the scheme shown in Fig. 1A). The polymerization rate equals the product of k ϩ1 and the concentration of polymerization-competent ends. The observed rate constant (k obs ϭ k ϩ1 [F n ]) was determined by fitting a first-order curve to the time course data using Origin software (Microcal Software, Inc., Northampton, MA). We set k ϩ1 at 10 M Ϫ1 s Ϫ1 , based self-consistent values obtained in earlier studies (12,22). We estimated that the Spectrin⅐4.1⅐Actin concentration (i.e. [free barbed end] or [F n ]) to be about 0.66 nM. Moreover, combined use of the FITSIM/KINSIM software permitted us to estimate k Ϫ1 at 0.3 s Ϫ1 . The values of k ϩ1 and k Ϫ1 allowed us to obtain a value of 0.03 M for the critical concentration (C critical or K 1 ) for the actin monomer interaction with the barbed ends. Finally, because the observed kinetics remain strictly first-order over the entire assembly time course, the polymer number concentration must not change appreciably. This finding suggests that all of the filaments remain assembly-competent and that little or no spontaneous nucleation occurred during the elongation measurements.
To evaluate how profilin influences the rate and extent of polymerization, we obtained complete time courses for polymerization of 2 M actin in the presence of 0, 0.7, 1.2, 2.0, 3.5, and 6.0 M profilin (Fig. 4). With the aid of the KINSIM/FITSIM software, we simultaneously analyzed the data for all progress curves to extract self-consistent, global values for rate and equilibrium constants. As shown in Fig. 1B, both A and P⅐A complex add to barbed ends by different routes, each resulting in net elongation. We began by using k ϩ1 ϭ 10 ϫ 10 6 M Ϫ1 s Ϫ1 and k Ϫ1 ϭ 0.3 s Ϫ1 for actin monomer addition to and release from the barbed end, respectively. We allowed the FITSIM program to determine the best-fit values by minimizing the sum of the squares of the deviations. This approach put k ϩ2 at 10 ϫ 10 6 M Ϫ1 s Ϫ1 , k Ϫ2 at 50 s Ϫ1 , K 3 at 1.5 M, k ϩ4 at 5 ϫ 10 6 M Ϫ1 s Ϫ1 , and k Ϫ4 at 1.25 ϫ 10 3 s Ϫ1 . Furthermore, when 0.7-2 M profilin (final concentration) was added to the actin solution, no significant change in the initial rate of assembly occurred, suggesting that profilin does not lower the critical concentration of barbed-end actin filament assembly (Fig. 4, inset).
Effect of Thymosin-␤4 on Seeded Assembly of Actin-We next chose to analyze actin polymerization in the presence thymosin-␤4 (see Fig. 1C) for two reasons. First, the quantitative analysis is best accomplished when one analyzes all polymerization steps under identical experimental conditions, rather than relying on a range of literature values obtained by using other assays. Second, whereas Weber et al. (8) reported that that thymosin-␤4 acts exclusively as a monomer-sequestering protein, Carlier et al. (37) suggested that thymosin-␤4 may not strictly act as a sequestering protein. To obtain the rate constants for T␤4⅐Actin complex formation, we conducted polymerization experiments in the absence and presence of thymosin-␤4. Because the model in Fig. 1C only accounts for monomer sequestration, failure to fit the T␤4 data to the model would indicate that T␤4 does not act strictly as a sequestering protein. Shown in Fig. 5 are the progress curves for actin assembly at five different T␤4 concentrations. Both the rate and extent of actin polymerization are reduced in the presence of T␤4. The solid lines were again generated with the aid of the KINSIM/ FITSIM kinetic modeling software. We obtained the following parameters: k ϩ ϭ 8 ϫ 10 6 M Ϫ1 s Ϫ1 , k Ϫ ϭ 0.6 s Ϫ1 , and K ␤ ϭ 0.92 M. In situations in which there are several parameters, one can also employ the KINSIM/FITSIM software to test the uniqueness of a chosen set by identifying other sets of parameters that also fit adequately. We obtained excellent fits using the following values: k ϩ ϭ 10 7 M Ϫ1 s Ϫ1 , k Ϫ ϭ 0.3 s Ϫ1 , K ␤ ϭ 0.64 M, and filament number concentration ϭ 0.66 nM. We likewise analyzed the case for k ϩ ϭ 2 ϫ 10 6 M Ϫ1 s Ϫ1 , k Ϫ ϭ 0.06 s Ϫ1 , and K ␤ ϭ 0.64 M, with [F n ] ϭ 3.3 nM, again obtaining an excellent fit. This exercise indicated that the process is more sensitive to the value of the equilibrium constant than to the precise values of the on-and off-rate constants or the filament number concentration.
We were also interested in investigating the dependence of the initial polymerization rate on the free monomer concentration [Actin] free at various concentrations of thymosin-␤4. One must use the quadratic formula to calculate [Actin] free from the values for the dissociation constant K ␤ , [Actin] total , and [T␤4] total . The initial actin assembly rate was obtained by drawing the best tangent line to the initial, linear phase of each polymerization progress curve. As shown in Fig. 6A, the initial rates in the absence (solid circles) of thymosin-␤4 clearly exhibit a linear dependence on uncomplexed monomeric actin [Actin] free . From the T␤4 polymerization data in Fig. 5, we estimated the concentrations of uncomplexed monomer in each sample using K ␤ values of 0.9 -1 M. All data points (open squares) fell on the same regression line generated for the data from Fig. 5, a finding that strongly indicates that thymosin-␤4 acts strictly as a monomer-sequestering protein under the conditions chosen in this investigation.
Thymosin-␤4 Interacts More Weakly with Pyrenyl-Actin as Compared with Unmodified Actin-As noted earlier, the lower affinity of profilin for pyrenyl-actin compared with unmodified actin (9, 22, 23) necessitates numerical corrections of observed data to analyze polymerization quantitatively. We were inter-  Fig. 5). For each sample, we calculated the corresponding free actin concentration by assuming a particular value for the dissociation constant for Actin⅐T␤4 complex (K ␤ ϭ 1.0 M). B, plot of the initial rates of pyrenyl-actin assembly versus the free actin monomer concentration at various total concentrations of thymosin-␤4. The time course of actin elongation was monitored using 2 M actin (60% labeled pyrenyl-actin and 40% unlabeled actin). Assembly was initiated by adding Spectrin⅐4.1⅐Actin complex, after which the solution was adjusted to 2 mM MgCl 2 and 100 mM KCl. The closed circles represent initial rates in the absence of T␤4, as determined by light scattering (350 nm excitation and emission wavelengths); these data constitute a standard curve relating initial assembly rate to the free actin monomer concentration. Note that all fluorescence rate data can be brought into correspondence with this standard curve assuming a K ␤ value of 1 M for T␤4⅐Actin and a K ␤ Ј value of 9 -10 M for T␤4⅐pyrenyl-Actin. The other symbols (upward triangles, downward triangles, and squares) are the data points obtained in three separate experiments in the presence of thymosin-␤4. Inset, initial rate versus free actin monomer concentration using identical values of 1 M for K ␤ and K ␤ Ј. Note that this assumption of equivalent affinity results in the systematic, upward deviation from the standard curve, indicating that K ␤ Ј is significantly higher than K ␤ . Note that the size of the symbols was chosen to reflect the range of experimental error. ested in learning whether pyrenyl-actin and unmodified actin bind equally well to thymosin-␤4, because any error in the magnitude of K ␤ is likely to compromise the quantitative analysis of polymerization in thymosin-␤4 solutions in earlier studies relying on pyrenyl-actin assembly assays. For example, Pantaloni and Carlier (21) assumed that pyrenyl-actin and unlabeled actin have the identical affinities for thymosin-␤4, and they used a 2.0 Ϯ 0.3 M value for K ␤ (7, 21). From the light scattering measurements, however, we consistently obtained a 0.9 -1.0 M dissociation constant for T␤4⅐Actin complex. We therefore analyzed the effectiveness of T␤4 as a sequestering agent by comparing samples containing various total concentrations of T␤4 and pyrene-actin to those containing various total concentrations of T␤4 and unlabeled actin. We took advantage of the fact that the polymerization behavior of pyrenelabeled actin and unlabeled actin are indistinguishable by light scattering measurements (see Fig. 2). Therefore, by using Spectrin⅐4.1⅐Actin complex to maintain the same filament number concentrations in each measurement, we could estimate the concentration of uncomplexed actin present as the x,y-coordinates from plot of the observed rate of elongation versus [Actin] (see Fig. 6B). We assumed a simple monomer sequestration model and successively chose different values of K ␤ for and until we achieved a good agreement between the observed polymerization data and the above-mentioned curve obtained in the absence of labeled actin (Fig. 6A). When K ␤ and K ␤ Ј were both set at 1 M, the resulting values did not fall on the standard curve, showing that at each calculated [Actin] free , the measured rate was too high (see Fig. 6, inset). Best agreement was achieved when K ␤ ϭ 1 M and K ␤ Ј was 9 -10 M. Our use of light scattering as a calibrated polymerization standard represents the first direct demonstration that thymosin-␤4 binds more weakly to pyrenyl-actin.
Effect of the Simultaneous Presence of Thymosin-␤4 and Profilin on Barbed-end Elongation-To complete our kinetic analysis, we determined the time courses for polymerization as a function of profilin concentration (0 -2 M) in the presence of two different thymosin-␤4 concentrations (2 and 4 M). In these experiments, we maintained the total actin monomer concentration at 2 M (see Fig. 7). All data points shown in this figure are the uncorrected photon counts obtained from right angle light scattering measurements. Furthermore, in our kinetic analysis of these data, we sought to determine whether there was any change in the mechanism of profilin action in the presence of thymosin-␤4; accordingly, we deliberately used the values for k ϩ1 , k Ϫ1 , k ϩ2 , k Ϫ2 , K 3 , k ϩ3 , k Ϫ3 , and K ␤ obtained above. Were any change in mechanism to have occurred, then the earlier values would fail to provide any acceptable fit between the KINSIM/FITSIM theory line and the observed data. As shown in Fig. 7, we achieved excellent agreement between calculated and observed time courses without altering any kinetic and equilibrium constants. As the concentration of profilin was increased, the initial rate of actin assembly also increased (see Fig. 7, insets). Thus, when T␤4⅐Actin is used to keep [Actin] free constant, the addition of profilin increases the total concentration of monomeric actin ([P⅐A] ϩ [A] free ) capable of barbedend addition. As will be discussed below, promotion of actin assembly by profilin in the presence of thymosin-␤4 does not require the previously proposed model in which profilin lowers the critical concentration for actin assembly (21).
Thermodynamic Analysis of Actin Polymerization-Upon obtaining all necessary rate and equilibrium constants under identical experimental conditions, we were in a position to determine whether there was any need to invoke a nonequilibrium mechanism for profilin-promoted assembly. 2 As pointed out by Pantaloni and Carlier (21), the best approach is to attempt to construct a closed thermodynamic cycle of the sort shown in Fig. 1D. The experimenter then assesses the additivity of energy changes along any two paths connecting the same species (i.e. ⌬G 1 ϩ ⌬G 4 should equal ⌬G 2 ϩ ⌬G 3 , such that Ϫ{RTln K 1 ϩ RTln K 4 } ϭ Ϫ{RTln K 2 ϩ RTln K 3 }). One therefore determines whether K 1 ϫ K 4 is less than, equal, or greater than K 2 ϫ K 3 . In the absence of thymosin-␤4, we found that all constants for actin polymerization in the presence of profilin were found to be completely compatible with a closed thermodynamic cycle. For example, the polymerization data are defined by the following equilibrium constants: K 1 ϭ 0.03 M, K 2 ϭ 5 M, K 3 ϭ 1.5 M, and K 4 ϭ 250 M. The path from A ϩ F n 3 F nϩ1 and then proceeding along the path F nϩ1 ϩ P 3 P⅐F nϩ1 corresponds to the product K 1 ϫ K 4 (or 0.03 M ϫ 250 M, or about 7.5 M 2 ). Likewise, the path from A ϩ P 3 P⅐A and then proceeding from F n ϩ P⅐A 32 3 A⅐P⅐F nϩ1 corresponds to the product K 3 ϫ K 4 (or 1.5 M ϫ 5 M, or about 7.5 M 2 ). Therefore, because K 1 ϫ K 4 ϭ K 3 ϫ K 4 to within experimental error, the cycle is at or very close to equilibrium, and there is no evidence for invoking the occurrence of an irreversible step. We also employed the KINSIM/FITSIM software to evaluate the case in which k ϩ1 ϭ 2 ϫ 10 6 M Ϫ1 s Ϫ1 , k Ϫ1 ϭ 0.06 s Ϫ1 (such that K 1 ϭ 0.03 M) and k ϩ2 ϭ 2 ϫ 10 6 M Ϫ1 s Ϫ1 , k Ϫ2 ϭ 10 s Ϫ1 (such that K 2 ϭ 5 M), k ϩ4 ϭ 1 ϫ 10 6 M Ϫ1 s Ϫ1 , k Ϫ4 ϭ 250 s Ϫ1 (such that K 4 ϭ 250 M), while keeping K 3 at 1.5 M. Even in this case, K 1 ϫ K 4 ϭ K 3 ϫ K 4 . For this last set of calculations, the filament number concentration was set at 3.3 nM. Those experienced in kinetic investigations will recognize that the first-order on-rate constant is the product of the bimolecular rate constant and the filament number concentration (i.e. k ϩ1 [F n ]). Therefore, one can express the apparent first-order rate constant of 6.6 ϫ 10 -3 s Ϫ1 either as the product 2 ϫ 10 6 M Ϫ1 s Ϫ1 ϫ 3.3 nM or as 10 ϫ 10 6 M Ϫ1 s Ϫ1 ϫ 0.66 nM. Finally, we were also able to account 2 These equilibrium constants are calculated in terms of experimentally determined off-and on-rate constants. Although such derived constants are useful for testing the thermodynamic cycle, we should emphasize that the various paths in the cycle do not satisfy a rapid equilibrium condition. This fact is most directly indicated by the ability of profilin to stimulate actin polymerization.  Fig. 1D. Insets, initial assembly rate versus profilin concentration at 2 and 4 M thymosin-␤4. Note that the size of the symbols was chosen to reflect the range of experimental error.
quantitatively for profilin-promoted actin polymerization in the presence of thymosin-␤4 by including the dissociation constant for T␤4⅐Actin. This is true because all other dissociation constants remained unchanged, even when thymosin-␤4 was present, and therefore Under all conditions, we obtained a closed thermodynamic cycle. Thus, because there is no evidence of the occurrence of an irreversible step as proposed by Pantaloni and Carlier (21), we can reject their suggestion that profilin somehow promotes nonequilibrium filament assembly by accelerating hydrolysis of filament-bound P⅐A⅐ATP complex. Additional support for our conclusion is provided in Fig. 8, in which all of the polymerization data are presented as single plot of observed polymerization rate on the ordinate versus the sum {[A] free ϩ [P⅐A]} on the abscissa. Because all the data are well described by a single curve, there is no significant departure from a single critical concentration value, irrespective of the concentration of profilin or thymosin-␤4. The linearity of this plot again confirms that the monomeric actin species A and P⅐A are kinetically indistinguishable with respect to their ability to add to filament ends.

DISCUSSION
The findings presented in this report permit us to reach the following conclusions regarding the mode of action of profilin: (a) profilin promotes assembly without lowering the critical concentration for actin assembly; (b) the thermodynamic cycle defining A and P⅐A addition (Fig. 1) is a closed equilibrium process (i.e. there is no evidence of irreversibility, thereby excluding the proposal (21) that profilin accelerates irreversible hydrolysis of actin-bound ATP); (c) thymosin-␤4⅐A enhances the ability of profilin to promote actin assembly by buffering the concentration of free actin monomer; and (d) because the monomeric species A and P⅐A add to barbed ends at virtually the same rate, P⅐A complex is apt to be the principal polymerizing form of actin monomer in living cells. The validity of the first three conclusions is supported by our kinetic and thermo-dynamic analysis of barbed-end actin polymerization in the absence and presence of profilin and/or thymosin-␤4, and the fourth conclusion is justified below.
If profilin possessed the ability to lower the critical concentration, then its presence would have changed the magnitude of one or more of the rate and/or equilibrium constants in the scheme shown in Fig. 1. However, because all of our polymerization data can be accounted for in terms of a single, selfconsistent set of constants, profilin must not alter critical concentration. Why then have our experiments provided such clear answers in the face of conflicting results in earlier investigations? We believe that the chief reason is that we adopted a straightforward experimental design, one that avoided potentially ambiguous corrections for the weak binding of profilin to and thymosin-␤4 to pyrene-actin. We also chose to focus on barbed-end elongation, and we used computer software to analyze large data sets obtained in our polymerization measurements. Another, perhaps even more fundamental, problem limited the approach of Pantaloni and Carlier (21), namely their assumption that it is possible to bias actin monomer addition to favor elongation at the barbed ends by limiting the actin monomer concentration range. Although such an assumption might be sound for initial rate measurements, the same cannot be said for experiments conducted at steady state. The measured extent of polymer assembly at steady state must always reflect the rates of monomer gain/loss at both ends, irrespective of the actin concentration range. This condition is an absolute requirement for attainment of the steady state. Furthermore, when polymerization of filaments with both ends uncapped reaches the steady state, actin participates in the nonequilibrium ATP-dependent process known as treadmilling; the Gibbs free energy of ATP hydrolysis drives unidirectional flux of monomer gain/loss whenever both ends of the filaments are uncapped (38). In this respect, the occurrence of treadmilling in the Pantaloni and Carlier (21) experiments precludes any possibility of obtaining equilibrium parameters, again explaining why they could not obtain equilibrium constant values conforming to a closed thermodynamic cycle. Perelroizen et al. (39) subsequently claimed that bovine spleen profilin or exchangeincompetent pollen profilin weakly promoted the treadmilling at magnesium-actin concentrations below the critical concentration. This observation was taken as evidence that profilin lowers the critical concentration of the barbed end. It is again worth emphasizing that treadmilling is a nonequilibrium flux, reflecting differences in the binding energies of actin monomers to barbed and pointed ends. Because our experiments used spectrin-actin seeds, no assembly occurred at the pointed end, and our conclusions about barbed-end elongation are not encumbered by the same limitation.
Use of pyrenyl-actin fluorescence to assay filament assembly/disassembly deserves further comment. Although proteins containing a covalently attached fluorescent reporter group have proven to be extremely powerful tools in biochemistry, the implicit assumption is that the tracer (here, pyrene-actin) replicates the properties of the tracee (in this case, unlabeled actin). As shown in Fig. 2, contemporaneous determinations of actin polymerization using the fluorescence mode and right angle light scattering mode are virtually indistinguishable. One can therefore use the equality of Ϫd[pyrene-A]/d t and Ϫd[A]/d t when defining the rate of filament elongation. Nevertheless, our experiments demonstrate that severe problems arise whenever one (or more) monomer-binding proteins display disparate affinities for labeled and unlabeled actin. Weak binding of pyrene-actin to profilin is well documented, and the magnitude of fluorescence changes at steady-state actin assembly must be corrected to obtain reliable estimates of the extent For example, the higher equilibrium dissociation constant for P⅐pyrene-A complex, as compared with that for P⅐A, probably reflects a correspondingly higher rate constant for P release from the P⅐pyrene-A complex. If the same holds true for the rate for profilin departure after P⅐A adds to a growing filament (i.e. if the rate of P⅐pyrene-A⅐F n 3 P ϩ pyrene-F nϩ1 is faster than the rate of P⅐A⅐F n 3 P ϩ F nϩ1 ), then a disproportionately greater amount of pyrenyl-actin would be incorporated into elongating filaments. Such behavior would manifest itself as increased fluorescence, resulting in an artifactual increase in the amount polymer observed at each actin concentration. Extrapolation to the abscissa would then give the false appearance of a lower critical concentration (i.e. the point of intersection would appear to shift to the left). The above investigators also reported that the apparent critical concentration decreased as the total profilin concentration was increased. Such a result would also be consistent with our explanation, because at higher profilin concentrations, a greater amount of pyrene-actin would be ushered onto each filament. We now document that labeling of actin with pyrenyl-iodoacetamide also reduces the affinity for thymosin-␤4 by a factor of around 10. This makes the situation impossibly complex. Another technique, such as light scattering, is required when measuring polymerization in the presence of both profilin and thymosin-␤4. The reader will note that all of our polymerization time courses are presented as the raw light scattering data sets, not as secondary, or corrected, data sets. Finally, we believe it is worth emphasizing that we actually began our investigations using pyrene-labeled actin. The results of our initial trial experiments indicated that pyreneactin binding to thymosin-␤4 was much weaker than that of unmodified actin (see Fig. 6), and this fact convinced us that experiments of a quantitative nature should not involve unnecessarily complex corrections. We also wish to stipulate that we have no argument with the data presented in the earlier study of Pantaloni and Carlier (21), and there is absolutely no need to replicate their results. We do, however, recommend that experimenters avoid fluorescence assays with pyrene-actin when investigating the simultaneous action of profilin and thymosin-␤4.
We should also note that use of human erythrocyte Spectrin⅐4.1⅐Actin complex (or "seeds") offered three advantages in our polymerization studies. First, Spectrin⅐4.1⅐Actin complex only promotes barbed-end actin filament elongation, and data analysis becomes far more tractable when dealing with growth at one filament end. Second, seeded assembly allowed us to work at a constant filament number concentration, and the time-evolution of actin elongation was strictly first-order under these conditions. That no nucleation occurred during assembly is indicated by the fact that all assembly reactions fitted by a single exponential process. Third, most investigators would agree that barbed-end polymerization is the chief pathway for actin filament assembly within cells, and experiments that avoid the simultaneous occurrence of pointed end assembly are apt to be more useful in extrapolating in vitro behavior to in vivo situations.
The KINSIM/FITSIM method (35) for kinetic simulation and data analysis proved to be invaluable in our quantitative assessment of the effects of profilin and/or thymosin-␤4 on actin polymerization. This software becomes particularly useful when analyzing the hundreds of individual data points that define our polymerization progress curves. We simultaneously fitted all reaction progress curves with the same set of rate constants and equilibrium constants, and the consistently excellent fits of theory and experiment fortify our data analysis. Although the scheme shown in Fig. 1D is the simplest mechanism that reconciles all observed rate behavior, other, more complicated schemes might also work as well. By choosing the simplest scheme that satisfies the complete set of experimental findings, we defer to Occam's Razor. Pring et al. (22) previously used KINSIM to investigate the role of profilin in actin assembly, but they did not consider any additional effects of thymosin-␤4. In their investigation, pyrenyl-actin was used to measure the initial rate of filament assembly. Shown in Table I are the specific components, sources, methods of analysis, and individual reactions relating to the studies of Pantaloni and Carlier (21), Pring et al. (22), and this study. Most noteworthy, our data for barbedend polymerization in the absence of thymosin-␤4 agree with the thermodynamic analysis of profilin-promoted actin assembly by Pring et al. (22). We obtained a value of unity for the ratio K 1 K 4 /K 2 K 3 , as compared with their value of 1-2.3. These data, which clearly support an equilibrium model, are at odds with those presented by Pantaloni and Carlier (21), who obtained a value of 14 for this ratio. Furthermore, an earlier study by Pollard and Cooper (12) put this value at 20, although the results of a later investigation from the same laboratory (40) were found by Pring et al. (22) to be much closer to unity. In this respect, there appears to be no basis for invoking the occurrence of an irreversible step in profilinpromoted actin filament elongation.
Because a substantial fraction of the total intracellular actin monomer pool is sequestered as T␤4⅐Actin complex (4,5), the free actin monomer concentration is well buffered in most cells. Under such conditions, the cytoplasm will maintain the concentration of P⅐A complex, the monomer species that is apt to be the major polymerizing form of actin in nonmuscle cells.  (1), such that the barbed-end critical concentration is not a factor. Therefore, whenever actin assembly is activated by filament uncapping, the instantaneous polymerization rate will be the sum of the velocity of profilin-mediated actin addition (i.e. k on [P⅐A][Barbed ends]) and the rate of uncomplexed G-actin addition (i.e. k on [A] free [Barbed ends]). Because the P⅐A concentration is about 11-12 times higher than free actin, one may infer that more than 90% of all monomers adding to filaments in vivo do so as P⅐A. By mobilizing G-actin from the T␤4⅐Actin pool and by ushering subunits onto barbed ends, profilin acts like a polymerization catalyst. The off-rate constant for profilin release from the barbed end (i.e. P⅐A⅐F n 3 P ϩ F nϩ1 ) must be sufficiently high to allow profilin to depart quickly from the filament. This would permit a newly freed end to commence another round of elongation. Our value for this rate constant is ϳ1250 s Ϫ1 , and based on a rate of 1 m/s, the corresponding rate constant for actin polymerization in cells is in the 800 -1200 s Ϫ1 range. Our experiments also indicate that the presence of profilin does not change the G-actin critical concentration for barbed-end elongation. In this respect, profilin acts as a catalyst by accelerating the polymerization rate without altering the equilibrium position of the reaction. Moreover, although one might argue that the ability of profilin to sequester actin monomers does not appear to be consistent with the action of a catalyst, Enzyme⅐Substrate complexes are known to accumulate when the enzyme is present at concentrations comparable to that of the substrate (41). We also prefer to view profilin as a molecular usher, because the cardinal feature of an usher is the ability to conduct something from one place to another. The specific binding of profilin to oligoproline modules located within the primary structure of such actin regulatory proteins as VASP and Mena is an important clue about its role as an usher. We previously demonstrated that the GPPPPP sequences within the primary structure of VASP bind profilin with a 0.09 mM dissociation constant (16). We also estimated that Listeria ActA surface protein can concentrate sufficient amounts of VASP, such that the local concentration of GPPPPP sites can rise as high as 0.8 -1.0 mM during Listeria motility. These sites serve to attract P⅐A complex into the polymerization zone, where barbed-end elongation ensues. To achieve the actin-based motility normally required by nonmuscle cells, zyxin or vinculin, two focal adhesion proteins also known to contain ActA-like sequences, probably attract VASP to regions of active filament assembly.
In conclusion, profilin promotes assembly without lowering the actin critical concentration, and there is no requirement for an irreversible step during profilin-mediated elongation in the presence or absence of thymosin-␤4. Profilin promotes actin assembly by capturing actin monomers from thymosin-␤4 and by ushering these monomers to the uncapped barbed ends of actin filaments.