The function of DNA polymerase I from Thermus aquaticus (Taq polymerase) (see Fig. 1) requires coordinated domain and subdomain motions within this protein to generate a precise ligatable nick on a DNA duplex (11–13). Taq polymerase performs nucleotide replacement reactions in DNA repair and RNA primer removal in DNA replication (14). During such processes, Taq polymerase utilizes a DNA polymerase domain to catalyze the addition of dNTP to the 3′ hydroxyl terminus of an RNA primer and a 5′ nuclease domain to cleave the downstream, single-stranded 5′ nucleotide displaced by the growing upstream strand (11). Because the structure of Taq polymerase possesses an extended conformation with the polymerase and the 5′ nuclease active sites separated by ≈70 Å (15–17), the DNA needs to be shuttled between these two distant catalytic sites when switching from the DNA synthesis mode to the nucleotide cleavage mode. This scenario is similar to that which occurs when the DNA needs to be shifted from the polymerase active site to the 3′–5′ exonuclease catalytic center, which are ≈30 Å apart in the Klenow fragment (the polymerase domain plus the 3′–5′ exonuclease domain) domain of polymerase I, to cleave an incorrectly incorporated dNTP (18). In addition, the polymerase domain can communicate with the 5′ nuclease domain, as evidenced by biochemical experiments that show that the presence of the polymerase domain affects the activity of the 5′ nuclease domain (12, 19). These studies imply that dynamic coupling among protein domains plays a very significant (if not well appreciated) role in their biological functions.

NSE experiments were conducted at the Institut für Festkörperforschung (27). The wavelength was 8.6 Å. The path length of the sample cell was 4 mm. The data were collected over the range of 0.039 Å^–1 ≤ Q ≤ 0.260 Å^–1. NSE experiments were performed at 30°C.

The S(Q,t)/S(Q,0) spectra can be approximated by the first cumulant representation as where the decay rate of the dynamic form factor is and the effective diffusion coefficient is

DLS Experiments. DLS experiments were performed with a DynaPro (Wyatt Technology, Santa Barbara, CA), using a laser of wavelength of 824.7 nm at a fixed angle of 90°, corresponding to Q = 0.00108 Å^–1. The Taq polymerase concentration was 0.25 mg/ml in 70 mM NaCl/35 mM Tris·HCl, pH 8.0, in which the intermolecular interaction effect is eliminated. The DLS experiments were performed at 30.0°C in D₂O buffer. Because the size of Taq polymerase is much smaller than the wavelength of light used in a DLS experiment (QR_g ≈ 0.04, with R_g being the radius of gyration of Taq polymerase), DLS measures the center-of-mass translational diffusion constant of Taq polymerase. DLS experiments show that Taq polymerase does not aggregate in D₂O solution in which we conducted the NSE experiments (see Fig. 2).

Solution Small-Angle X-Ray Scattering (SAXS) Experiment. SAXS experiments were conducted with an in-house apparatus (28). The effective Q range covered was from 0.016 to 0.35 Å^–1. The Taq polymerase concentration was 5 mg/ml in 70 mM NaCl/35 mM Tris·HCl, pH 8.0/H₂O buffer. SAXS experiments were performed at 30.0°C. SAXS data reduction and data analysis procedures have been described in ref. 28. Inverse Fourier transformation of I(Q) gives the length distribution function P(r), which is the probability of finding two scattering points at a given distance r from each other in the measured macromolecule. Inverse Fourier transformation of I(Q) gives the length distribution function P(r), which is the probability of finding two scattering points at a given distance r from each other in the measured macromolecule (29):

We first examine the contributions of rigid-body translational and rotational diffusion to the oscillatory behavior of D_eff(Q) because the intramolecular interference in the static form factor of a rigid structure could, in principle, cause the oscillations in D_eff (32, 33). The D_eff(Q) calculated by the rigid-body model using Eq. 2 is shown in Fig. 4A. Although the experimental D_eff(Q) has maximums and minimums (see Fig. 4A) that correspond to the dip and rise in I(Q) (shown in Fig. 4B), respectively, the experimental D_eff(Q) shows much more significant oscillations than the calculated D_eff(Q) using Eq. 2 of the rigid-body model. Fig. 4A shows that the D_eff(Q) derived from Eq. 2 only agrees with the experimental data in the region of Q < 0.125 Å^–1, suggesting that Taq polymerase behaves as a rigid body only on length scales longer than 2π /Q > 50 Å. As Q > 0.125 Å^–1, the NSE-measured D_eff(Q) oscillates more markedly than that calculated by Eq. 2 of the rigid-body model. Thus, the dynamic behavior of Taq polymerase cannot be described by the rigid-body model when Q > 0.125 Å^–1.

Next, we compare the structure of Taq polymerase in solution by SAXS with the crystal structure to examine whether the existence of multiple static structures in solution could possibly cause the deviations of the NSE-measured D_eff(Q) from the rigid-body behavior. The static form factor I(Q) calculated from the crystal structure coordinates is shown in Fig. 4B, together with that measured by solution SAXS, which is an ensemble average of all possible structures that can be adopted by Taq polymerase in solution. The length distribution functions P(r) calculated from the crystal structure and solution SAXS data are also plotted in Fig. 4C. As shown in Fig. 4 B and C, both I(Q) and P(r) from solution SAXS are very similar to those calculated from the crystal structure. The ensemble-averaged global structure of Taq polymerase in solution by SAXS is thus very close to the crystal structure. If there were distinct multiple structures, we would expect the SAXS results to be significantly different from the crystal structure. Thus, the oscillatory behavior of D_eff(Q) measured by NSE shown in Fig. 4A must arise from the internal motions of Taq polymerase. In the following, we analyze the NSE results from Taq polymerase by using a normal mode framework and statistical mechanics to show how the oscillation in D_eff(Q) can be explained by internal motion.

Normal Mode Analysis (NMA) Suggests That the Lowest Frequency Normal Modes of Internal Motions in Taq Polymerase Are Interdomain Motions. We have carried out NMA on Taq polymerase by using the program elnémo (34) to identify the type and direction of internal motion (35). In NMA, the first six modes of lowest frequency are the translational motion and rotational motion of the protein molecule as a whole. NMA suggests that the lowest frequency modes of internal motion, which are modes 7 and 8 (Fig. 1), are the en bloc relative motion between the 5′ nuclease and the Klentaq domains. The direction of motion of modes 7 and 8, shown in Fig. 1, is consistent with our previous structural study by small-angle neutron scattering that finds the 5′ nuclease domain to be in closer contact with the thumb subdomain when Taq polymerase binds to a structural specific overlap flap DNA (17). In higher normal modes, subdomain motions appear. Specifically, in normal modes 9 and 10, the relative motions of the 5′ nuclease, the 3′–5′ exonuclease, and the polymerase domains are apparent (see Fig. 1).

Thus, NMA predicts that the lowest frequency mode of internal motion in Taq polymerase involves the relative motions of two domains, the polymerase plus the 3′–5′ exonuclease domain (together called the Klentaq domain) and the 5′ nuclease domain, which are connected by a spring-like linker (see Fig. 5B). Higher normal modes display relative motion of three rigid domains, with the Klentaq domain split into its 3′–5′ exonuclease and polymerase components (see Fig. 5C).

NSE Can Determine the Domain Mobility Tensor That Defines the Degree of Dynamical Coupling Between Domains. The above analysis shows that a rigid-body analysis of Taq polymerase is inadequate and that the ensemble-averaged solution structure is very close to the crystal structure. We therefore generate a progression of models that systematically include internal normal modes by considering (i) the lowest frequency internal modes in which the 5′ nuclease and the Klentaq domains are treated as two oscillating lobes and (ii) the two lowest frequency modes that include the 5′ nuclease domain but in which the Klentaq domain is now further subdivided into its polymerase and 3′–5′ exonuclease domain components.

First, we treat the Klentaq domain and the 5′ nuclease domains as separate rigid objects whose coordinates are assumed to vary little from the crystal structures (see Fig. 5B). The time evolution of the coordinates can be described by the Langevin equation for the two domains at center-of-mass coordinates that comprise the protein (36):



where is the domain mobility tensor and is defined by in terms of the velocities and forces for each domain. Here, U is the potential of mean force between the two domains and f is the usual random thermal force with ensemble averages



where k_B is Boltzmann's constant and T is the temperature. Eq. 3 indicates that protein motion and thus its normal modes arise from a convolution of the static structural forces (U) and the dynamical and hydrodynamical effects (H).

The first cumulant (defined in Methods) of the dynamic form factor can be calculated, because effects that occur on different timescales can be separated (37–40). There are rapidly fluctuating “Brownian” forces due to collisions with solvent molecules, and, in addition, there are longer timescale forces on domains due to the influence of other domains. The first cumulant of S(Q, t)/S(Q, 0) for the two-domain model is thus



where {b_j} are atomic neutron scattering lengths. The D_eff(Q) of Eqs. 3–5 reveals the dynamic events that occur on the timescales of internal modes (9) that are much longer than the Brownian timescale τ_B (40).

It is important to point out that the formula Eq. 5 arises as the result of a delicate limiting process. A central feature of this process is the order in which two important limits are taken. These limits are (i) the limit in which a stiff spring becomes perfectly rigid and (ii) the limit of zero time in the first cumulant of the effective diffusion constant. If the second limit is taken first, very fast underdamped modes can appear (32). In this case, it is no longer reasonable to neglect inertial modes, and the usual derivations of Eq. 5 are invalid. These difficulties can be avoided by taking limit (i) first (32). Thus, such relations are correct for perfectly rigid bodies (Eq. 2) or for rigid bodies connected by soft spring linkers (Eq. 5), as we consider in this study. The existence of underdamped motion requires that the spring constant for a linker connecting domains of mass m and friction constant ζ be >ζ²/4m (9); explicit calculations as well as measurements (23) indicate that proteins are well within the overdamped soft spring regime.

Eqs. 2, 4, and 5 explicitly show that, given the structural coordinates of a protein, the NSE experiment essentially tests models of the domain mobility tensor , which defines the velocity of domain j given the force applied to domain k.We construct the simplest possible model of a domain mobility tensor for internal motion with friction constants ζ_j, j = 1, 2 appropriate for each domain and evaluate D_eff(Q) using Eq. 5 for the case in which Taq polymerase is separated into two domains, a 5′ nuclease domain (j = 1) and a Klentaq domain (j = 2). The D_eff(Q) for the two-domain model (see Fig. 5B) is then where D_j = (k_BT)/ζ_j with D₁ = D_5′-nuc and D₂ = D_Klentaq, and are the rotationally averaged static form factors, which can be approximated by the crystal structure coordinates as we show by SAXS (see Unusual Internal Dynamic Behavior in Taq Polymerase Revealed by NSE) and previous small-angle neutron scattering studies (17). In Eq. 7b, the sum is taken over the N_j atoms in domain j, and N₁ and N₂ are the number of atoms in domain 1 and domain 2, respectively. The cross-term 1–2 in the numerator of Eq. 7a disappears because the mobility tensor is diagonal. Subtleties in using Eq. 7 arising when rigid constraints are applied (41–44) are avoided by explicitly separating the center-of-mass coordinate of each domain before performing the ensemble average and then subsequently folding in domain form factors. Eq. 7 was also verified by explicit calculation using Eq. 2 with U taken as a harmonic oscillator potential. As per Eq. 7, the first cumulant is explicitly independent of the interdomain spring constant.

The calculated D_eff(Q) using Eq. 7a, shown in Fig. 4A, also has peaks and dips in the intermediate Q values as the NSE-measured D_eff(Q). When using Eq. 7a to calculate the curves shown in Fig. 4A, we find that the friction constants ζ₁ and ζ₂ of both domains in Taq polymerase increase by a factor of ≈2 compared with those of the separated individual domains as calculated by the Kirkwood–Riseman formula (45). As the domains are in close proximity, the friction constant for each domain is increased because of the fluid displaced between them by their motion. Quantitatively similar phenomena have been found when two circular disks or two spheres approach one another in viscous media or when a sphere approaches a wall (9, 46–48).

We then extend Eq. 7a to the case of a three-domain model (see Fig. 5C). In this model, the Klentaq domain is subdivided into the 3′–5′ exonuclease and polymerase domains. We then repeat the analysis of Eqs. 7 to show The result calculated by Eq. 8 appears in Fig. 4A and demonstrates that the systematic inclusion of higher normal modes consistently improves the agreement with the NSE experiment. Thus, our mobility tensor analysis shows that NSE data reveal coupled correlated domain motion within Taq polymerase. Moreover, we show that the internal motion can be systematically analyzed by reducing the data within a normal mode framework.

The rms amplitude x^{2 1/2} and the spring constant of interdomain motion can be estimated from the equipartition theorem, which states where k is the spring constant, k_B is Boltzmann's constant, D is the Stokes–Einstein diffusion constant, τ is the relaxation time that can be estimated from the NSE results, and T is the temperature. Thus, for relaxation times of the order of 10 ns, the estimated amplitude x^{2 1/2} of the normal mode is ≈1–10 Å. For x^{2 1/2} = 7 Å, the spring constant k for the linker region is ≈8.5 × 10^–3 N/m. This value is less than one-third of the spring constant of myoglobin (23) but ≈5.6 times larger than the reported spring constant of cross-linked polystyrene (49).