Early nucleation events in the polymerization of actin, probed by time-resolved small-angle x-ray scattering

Nucleators generating new F-actin filaments play important roles in cell activities. Detailed information concerning the events involved in nucleation of actin alone in vitro is fundamental to understanding these processes, but such information has been hard to come by. We addressed the early process of salt-induced polymerization of actin using the time-resolved synchrotron small-angle X-ray scattering (SAXS). Actin molecules in low salt solution maintain a monomeric state by an electrostatic repulsive force between molecules. On mixing with salts, the repulsive force was rapidly screened, causing an immediate formation of many of non-polymerizable dimers. SAXS kinetic analysis revealed that tetramerization gives the highest energetic barrier to further polymerization, and the major nucleation is the formation of helical tetramers. Filaments start to grow rapidly with the formation of pentamers. These findings suggest an acceleration mechanism of actin assembly by a variety of nucleators in cells.


SAXS intensity profiles with liquid-like interference between particles.
In the formulation by Zernike and Prins 1,2 , the SAXS intensity scattered with liquid-like interference between particles is written as,    (1) where q is the scattering vector length, I e (q) is the intensity scattered by one electron, N , the average number of particles in the irradiated volume, v 1 , the average volume per particle in the irradiated volume, f(q), the spherically averaged scattering amplitude of a particle, and P(r) is the probability distribution function of particles with an inter-particle separation, r. In the formulation, a square of the ensemble-average for the scattering amplitude of a particle, <f(q)> 2 is put equal to an ensemble-average for a square of the scattering amplitudes, <f 2 (q)> and is written by f 2 (q). We used the experimentally-obtained in place of f 2 (q). P(r) for a system of hard spherical particles is employed, where P(r) = 0 for r < 2R (= a diameter) and P(r) = 1 for r > 2R. An interference peak calculated using this model is too broad to fit the experimental profiles. This is because the model has a sharp transition of P(r) from zero to one in real space (see Figure 1). To fit the broad interference peak, we introduce a new parameter allowing for a gentle transition of P(r). Here, the following P(r) for the system of soft spherical particles is used, where r 0 is the closest approach distance between particles in the medium, and  is an adjustable parameter (see Figure 1). Using the Sommerfeld expansion for the integration, equation (1) where the term I(q)/c| c=0 [1 -f 1 ] corresponds to an interference function for the system of hard spheres, which was derived by Debye 3 . The term I(q)/c| c=0 [f 2 ] is a correction factor for the contact between soft particles. Due to this term, the interference peak becomes sharper.

Description of time dependent changes of the average molecular weight in a nucleation-controlled polymerization
We employ the kinetic model of polymerization developed by Ferrone 4 . The principal assumption of this model is that all polymerization processes are sufficiently slow to allow equilibrium between nuclei and monomers. A typical diagram of free energy change (G) of formation of various oligomers relative to the monomer state is as shown in Figure 2. When the concentration of nuclei is effectively low, they form a barrier to further growth. The rate of the formation of polymers is determined by the population of nuclei and the rate of elongation of the nuclei (the rate on crossing this barrier) J * (= k + * c *k -* ; k + * and k -* are the rate constants of elongation and dissociation of the nucleus, respectively). If c * is the concentration of nuclei, polymers (the concentration, c p ) are then formed at the rate ** p dc Jc dt  (4) where J* is the rate of elongation of the nucleus. Once polymers are formed, they add mass by accretion to their ends. If the monomer addition is done with the same rate J (= k + ck -: c is the concentration of monomers), independently of the polymer size, n, then the rate of the total concentration, of monomersthat are incorporated into the polymers is written as If all molecules are classed as either polymers or monomers, then putting the original concentration as c 0 , we can write at t, (6) By the use of a perturbation approach, the first-order solution to the set of equations has the form J 0 is a J at the concentration, c 0 . A and B are parameters (see the paper of Ferrone 4 ).
The weight-average molecular weights, M ave , which are obtained by SAXS is written as, where M is the molecular weight of a monomer, and M i , c i and n i are the molecular weight, the concentration and the number of the oligomer i, respectively, and n c and n tot are a size of the nucleus and the total number of molecules, respectively. The sums of in i and i 2 n i in equation (9) at t are written approximately as 2 00 0 Using these equations, the time-dependence of the weight-average molecular weight is finally, (11) in which we put  = B 2 A/n tot and  = J 0 . When n c is equal to 4, equation (11) is written down by Since the experimental data have some errors, we cannot decide whether the singular value,  j 2 , is real zero or near zero. Thus, we cannot deduce explicitly the number of the kinds of complexes, n, during polymerization. To estimate n, we employed the following indicator (IND) function which has been proposed by Mallnowski 5,6 .

Isoscattering points
Assuming that all oligomer species scatter independently, the SAXS intensity profiles for a mixture of m different actin oligomers including monomers are described by 8 , where c l is a weight concentration of the oligomer l, and I l (q) is a scattering intensity of the oligomer l per weight. I l (q) is written by 8 , where R jk is a distance between the subunit j with a form factor of F j (q)exp(i j ) and the subunit k with F k (q) exp(i k ) which are included in the same oligomer l. < > denotes an orientational average. The first term corresponds to the scattering from individual actin subunits in each oligomer, and hence it has the same intensity profiles for all species. The second term accounts for interference between actin subunits in each oligomer. When the second terms for all species appearing during polymerization are zero at a fixed q, the profiles have an isoscattering point. In other words, all species must share a common periodicity in order to have an isoscattering point. When an actin subunit is approximated as a spherical molecule with a radius of R, the above equation 8 is, where F(q) is the form factor of a sphere with a radius of R. To confirm the above consideration, we calculated the second term of equation (17) for a series of linear oligomers with the axial displacement of 5.52 nm and a series of helical oligomers having an F-actin type feature with the diameter of 16 nm, a 13 subunits/7 turns helical symmetry and an axial shift of 2.76 nm (see Figure 3). In the case of helical oligomers, the scattering profiles of the oligomers larger than the trimer give rise to a distinct isoscattering point. From the position of isoscattering point, the average nearest-neighbor distance of the subunits in the oligomers can be estimated. In the mixture of oligomers including monomers, dimers and trimers, the crossing point becomes blurred.

Figure 3 Intensity profiles of the interference term between the subunits in oligomers.
Each arrow indicates the zero point of the interference term.

A kinetic model for actin polymerization
Our kinetic model for actin polymerization including the formation of nonpolymerizable dimer is formulated as follows.

Figure 4 A sequential reaction scheme for actin polymerization including a nonpolymerizable dimer.
A 1 , A s , A i and F denote G-actin, nonpolymerizable dimer, intermediates comprising i actin molecules (i-mer) and F-actin, respectively. k s+ and k s-are the rate constants of nonpolymerizable dimer formation and deformation, respectively, and k +i and k -i are the rate constants of the association and dissociation of a monomer for i-mer, respectively.
In the model, F-actin is defined as the complexes larger than an 8-mer (see Supplementary Figure 3), and the time-dependent SAXS intensities were calculated using the fractions of species present at a given time. In the growth phase of F-actin, k +i = 5.5 M -1 s -1 and k -i = 1.8 s -1 were assumed. A 1 , A s and A i are the number concentration of G-actin, nonpolymerizable dimer and i-mer, respectively, and w i is the weight concentration of i-mer. Then the changes of A s , A 2 and A i with time and w i are expressed as follows, ( 1) where M is the molecular mass of a monomeric actin. We calculated the number concentration of i-mer up to 100-mer. Even if it is calculated up to 1000-mer, the result was the same. Using A 7 , A 8 and A 9 thus obtained, the number concentration of F-actin (N) and the weight concentration of F-actin (F) together with that of monomers (A 1 ) were calculated as follows.
The change of A 1 with time is These differential equations were solved using the ode command in the Scilab Package 7 .

Explanation of a nucleus size in actin polymerization by the simple geometry of F-actin.
F-actin is constituted by two kinds of contacts between actin subunits with a longitudinal configuration along each strand (longitudinal contact) and a diagonal one between the two strands of F-actin (diagonal contact) 9 . By either one of the two kinds, a longitudinal dimer or a diagonal dimer is assumed to be formed. Upon the formation of helical trimer containing one turn of the basic helix of F-actin, the association of a monomer to the diagonal dimer simultaneously generates both the longitudinal and diagonal contacts likewise in the case of F-actin. As shown in Figure 5a, the diagram of the free energy change G 0 in the pathway via the diagonal dimer would alter from the small slope (dG 0 /di) to the large one at the dimer, and thus the nucleus in the downhill diagram is taken as the dimer by definition 10 . In contrast, in the pathway via the longitudinal dimer, two diagonal contacts are simultaneously generated. The diagram of the free energy change G 0 would alter as shown in Figure 5b, and the nucleus is a trimer. Collectively, the formation of dimer or trimer as a nucleus has been deduced from the structural geometry of F-actin dependently on the strength of the two contacts 11 . In the normal condition, the longitudinal contact in F-actin is stronger than the diagonal contact, and a trimer is expected to be a nucleus.

Figure 5 Schematic diagrams of free energy change versus oligomer size (i) via the diagonal and longitudinal dimers.
The free energy changes for the diagonal contact and the longitudinal contact are denoted by -E d and -E l , respectively (E d , E l >0) . a, The path through the formation of diagonal dimer. The diagonal dimer is dominantly formed when the diagonal contact is stronger than the longitudinal one (E d >E l ). In this diagram, the magnitude of free energy change for monomer addition upon the formation of each species is expressed: the diagonal dimer = -E d , trimer = -(E l + E d ), tetramer = -(E l + E d ), pentamer = -(E l + E d ). b, The path through the formation of longitudinal dimer. The longitudinal dimer is formed at the condition of E l > E d . The free energy change for monomer addition upon the formation of each species is expressed: the longitudinal dimer = -E l , trimer = -2 × E d (>-(E l + E d )), tetramer = -(E l + E d ), pentamer = -(E l + E d ). Dashed lines in both graphs are the lines extrapolated from the line at the prior step.

Supplementary Figures
Supplementary Figure 1   a, A parallel dimer. The dimer was made from the subunit arrangement along the crystal contacts of PDB code: 2FXU 12 . Theoretical SAXS intensity profile is indistinguishable from that of the dimer along the long-pitched helical strand of F-actin in the measured q range b, An anti-parallel dimer. The dimer was made from the arrangement of the anti-parallel dimer in the actin crystal of PDB code: 1LCU 13 . A similar dimer can be made from contacts in the actin crystal of PDB code: 2OAN 14 .

Supplementary Figure 3 SAXS intensity profiles calculated from G-actin, dimers, various intermediates and F-actin models.
The SAXS curves of oligomers larger than heptamer (7-mer) are indistinguishable from that of F-actin in the measured q range. The scattering profiles from oligomers larger than the trimer exhibit a distinct crossing point at q = 0.65 nm -1 .