RNA interference, or RNAi, is a conserved mechanism whereby double-stranded RNAs (dsRNAs) silence protein-coding genes that contain sequence fragments complementary to those on the dsRNAs 123. In RNAi, the RNase III family enzyme, called Dicer, cleaves the dsRNA into small interfering RNAs (siRNAs) of ~21 nt 4. These small RNAs guide the silencing activities within the RNA-induced silencing complex (RISC). Proteins in the Argonaute family are critical to the RISC complex, playing essential roles in substrate selection and mRNA cleavage (56 and references therein).

Argonaute proteins are multi-domain proteins and contain both PAZ and PIWI modules 7. Structures of PAZ domains complexed with RNA demonstrated that siRNA recognition entails binding of the 3'-end by highly conserved aromatic residues 89.

Recently two atomic structures of intact, prokaryotic Argonaute proteins were obtained using X-ray crystallography, with the Pyrococcus furiosus (Pf-Ago) protein determined to 2.2 Å 10 and the Aquifex aeolicus (Aa-Ago) protein determined to 2.9 Å 11. Although the precise role of Argonaute proteins in prokaryotes is currently unknown, these crystal structures have yielded significant insight into mechanisms of eukaryotic Argonaute proteins; in addition, Aa-Ago has been shown to be capable of ssDNA- or ssRNA-directed cleavage of RNA 11.

The overall structure of Pf-Ago is composed of a 'crescent-shaped' base made up of the N-terminal, Mid- and PIWI- domains, with the PAZ domain mounted above the base by a 'stalk'-like region (Figure 1a) 10. A DDH motif (D558, D628, H745) with a divalent metal ion Mn²⁺ has been identified as the mRNA cleavage activity site in the PIWI domain from the Pf-Ago-Mn²⁺ complex 12. Structure-directed mutagenesis experiments have demonstrated that the PIWI domain of Argonaute is the catalytic unit of the RISC 13. Later crystal structures of RNA-associated PIWI proteins from diverse species support the similarity of PIWI to RNase H and confirm its mRNA-cleavage-related catalytic function 11121415.

The structure of the Aa-Ago protein is similar to that of the Pf-Ago protein in that it has the same domain ordering. However, its overall architecture is better described as a [2+2] bi-lobal protein with PAZ-containing (N-terminal and PAZ domains) and PIWI-containing (Mid and PIWI domains) lobes and the so-called "PIWI box" close to a hinge linking the two lobes (Figure 1b) 11. In contrast, the Pf-Ago protein's architecture has been described as a [1+3] structure with the PAZ domain connected to the N-PIWI-MID crescent base 10.

Insights from crystal structures of prokaryotic Argonaute proteins have led to recently proposed models of the RNAi reaction cycle in eukaryotes 61116. For the purposes of discussion, we refer to this cycle as the Patel Model, which includes guide RNA-directed mRNA loading, cleavage, and product release. In the Patel model, Argonaute assumes four different conformations (Figure 2). Large-scale motions of Argonaute are required to step through the different conformations of the model.

The conformational changes in the Patel Model suggest that large-scale flexibility and domain motions might play a critical role in the RISC machinery. Although the Debye-Waller factors (B-factors) provide insight into the Argonaute's flexibility within the crystalline environment, due to crystal packing constraints, the crystallographic data cannot provide direct observations of the large-scale motions. Therefore, to gain insight into the internal flexibility and accessible conformations of Argonaute, we have used an empirical, molecular-mechanics force field to perform all-atom normal mode analysis (NMA) 171819 on Pf-Ago and Aa-Ago. We wished to perform normal-modes analysis using a molecular-mechanics force field instead of an elastic network model 2021 because we were specifically interested in learning what local and global motions are favored within the context of an all-atom molecular mechanics model that includes explicit terms for electrostatic, van der Waals, and local geometry potentials; the analysis was performed using the CHARMM 31 molecular simulation program (see Methods). The normal modes show hinge motions, torsional motions, and breathing motions which are relevant to specific mechanisms proposed in the Patel model: the biological processes of mRNA/siRNA association and disassociation; and the shift of the 3'end of the guide strand RNA between the PAZ domain and the N/PAZ domain channel.

In the case of Pf-Ago, it was previously suggested that residues 317–320 might constitute a hinge region, enabling the PAZ domain to move towards and away from the crescent base 10. In this hinge motion, the PAZ domain, stalk, and linker region (i.e., residues 103–310) would move as a unit. It was suggested that this motion could allow RISC loading and mRNA binding between the PAZ/N-terminal cleft and the N-terminal/PIWI cleft. The normal mode analysis lends support to this suggestion, with a low-frequency mode involving a hinge motion between the PAZ and N-terminal domains. In this mode, both the PAZ and N-terminal domains flex, resulting in hinges through the stalk, linker and connecting regions of the PAZ and N-terminal domains. This motion is also consistent with RISC loading and mRNA binding in the PAZ/N and N/PIWI clefts.

In the case of Aa-Ago, the normal mode calculations show more collective motion between the PAZ-containing and PIWI-containing lobes, entailing the entire Argonaute protein, with hinging in both the PAZ and PIWI domains, including the PIWI box, as suggested by Patel and co-workers 11. Compared to Pf-Ago motions, Aa-Ago motions are more collective in nature, as reflected by the analysis of correlated motions in Figure 5.

Patel and co-workers described a 4-step model for RNAi target cleavage, which included (1) siRNA:mRNA duplex nucleation, where 2–8 base pairs are formed between the guide siRNA and the target mRNA; (2) siRNA:mRNA duplex propagation, where the duplex zippers to completion; (3) target cleavage, where the PIWI domain active site cleaves the phosphodiester bond between positions 10 and 11 of the mRNA target; and (4) mRNA release, where the cleaved mRNA dissociates from RISC 11. The normal mode analysis enhances steps (1) and (2) of the model in several interesting ways.

Step (1), nucleation of the siRNA:mRNA duplex, is complex and poorly understood, involving many substeps. In one substep, Argonaute must first bind to the mRNA. However, because target mRNAs often contain secondary and tertiary structure, it is likely that binding of Argonaute to its target requires helicase activity to melt mRNA structure. The low-frequency hinge modes may aid in exposing the guide siRNA to the mRNA target. Binding entails the nucleation between the guide siRNA strand and the target mRNA. It is not clear exactly how many base pairs form during the nucleation step; however, it is thought that nucleation begins at the 5' end of the siRNA 2627.

Once mRNA binding is achieved, another substep requires that Argonaute distinguish between correct and incorrect target sequences. This may occur by repeated binding and dissociation events, or by binding and scanning along the mRNA, in a manner similar to a polymerase, or even using mechanisms similar to those used by the ribosome, which detects the geometry of the codon-anticodon base pairs 2829. For example, Song et al.10 suggested that Pf-Ago is able to sense the minor groove width of the dsRNA in a manner similar to RNase H.

Squeezing of the base pairs by hinge modes or breathing modes might increase the fidelity of target recognition, enhancing the stability of cognate base pairs and destabilizing non-cognate base pairs. As described above, the breathing mode is the most effective mode for increasing and decreasing the volume available for duplex accommodation. Furthermore, in the double-stranded RNA association mechanism, breathing motions might facilitate sampling of low-energy states of the siRNA:mRNA duplex inside the Argonaute cavity, enabling tighter binding between the mRNA strand and the single-stranded siRNA.

Similarly, the hinge motion between the PAZ-containing and PIWI-containing lobes (in the case of Aa-Ago), or the hinge motion between the PAZ and N-terminal domains (in the case of Pf-Ago), might facilitate target recognition by enhancing thermal sampling of high-affinity binding-site conformations. We note that the Aa-Ago N-domain is closely related to the catalytic domain of the replication initiator protein 30.

During Step (2), siRNA:mRNA duplex propagation, zippering of the full siRNA:mRNA target duplex occurs. If the target is cognate, the process culminates with the transfer of the 3'-end of the guide siRNA strand from its PAZ binding site to the N-terminal domain. Even though the present simulations were performed without the duplex, the primary torsion mode bears an interesting relationship to zippering activity: the axis of the N-terminal/PIWI domain torsional mode is approximately parallel to the model hybrid duplex axis. If significant in the presence of the duplex, this torsional motion might act to test the stability of siRNA:mRNA base pairs in a more precise fashion than the hinge or breathing modes. In addition, the hinge motion between the PAZ and N-terminal domains in the case of Pf-Ago might facilitate the transfer of the 3'-end of the guide siRNA from the PAZ domain to the N-terminal domain described in the Patel model.

Finally, we note that although target recognition might naively be expected to be similar during RNAi, RNA-dependent RNA polymerase (RDRP) activity, and transcription, there are dramatic differences in structure and dynamics among Argonaute, RDRPs, and RNA polymerases. These differences are almost certainly driven by differences in detailed functional requirements. For example, during RNAi, once the target is recognized, there is no need for translocation along the mRNA. During RDRP activity and transcription, however, the polymerase must move along the template one nucleotide at a time. In addition, the biological role of Argonaute in prokaryotes is currently unclear. As more is learned about the relations among the structure, dynamics, mechanisms, and biological functions of these proteins, it will be interesting to consider the implications for their evolutionary history, which will not only tell a rich and interesting story in its own right, but will also yield general insight into how complex and exquisitely scripted behaviors evolve in macromolecular systems.

NMA was performed using the CHARMM simulation program 31. First, the crystal structure was relaxed to a configuration x at a local minimum in the potential energy U(x), and the integrity of the minimized structure was confirmed 32. To gain insight into large-scale motions, we excluded the crystalline environment. Next, the Hessian ∂²U/∂x_i ∂x_j was calculated at the minimum; normal modes and frequencies were then obtained by solving an Eigenvalue problem. Finally, as in previous NMA studies of proteins 171819, low-frequency modes were interpreted in light of the biological function of the protein complexes.

The atomic model of Aa-Ago was taken from the 2.9 Å X-ray structure (Protein Data Bank (PDB) entry 1YVU 11). Hydrogen atoms were added to the protein using the HBUILD module of CHARMM 31 and waters in the x-ray structure were removed. The extended-atom model TOPH19 and the polar-hydrogen parameter set for proteins, PARAM19, were used for the energy minimization and normal mode analysis 33. To avoid large charge-charge interactions, the distance-dependent dielectric constant ε = 2r was used to include screening effects due to bulk solvent. The non-bonded electrostatic interaction was shifted to zero at 7.5 Å and the corresponding cutoff distance for the non-bonded neighbor list update was 8 Å.

Prior to calculating the normal modes of the Aa-Ago protein, the structure was relaxed by energy minimization, including 200 steepest descent minimization steps followed by 200 steps of adapted basis Newton-Ralphson minimization (ABNR) 31 with constant harmonic constraints, used to remove excess strain. This was followed by 5000 steps of ABNR minimization with gradually decreasing harmonic constraints imposed. Finally, 2410 steps of ABNR minimization were carried out without constraints, using a gradient threshold of 0.01 Kcal/mol/Ų for termination of the minimization. As in previous normal-mode analyses of large systems 3435, use of this gradient threshold resulted in a "minimized" structure that was similar to the crystal structure (the C_α RMS deviation between the X-ray and minimized structure is 1.1 Å), but at the cost of yielding a significant number of negative-frequency modes in the normal-mode analysis. By comparison, we found that a smaller threshold of 0.001 Kcal/mol/Ų yielded no negative-frequency modes, but resulted in a 2.3 Å RMSD with respect to the crystal structure. As in previous studies 3435 we chose to increase the fidelity of the "minimized" structure to the crystal structure at the cost of creating some negative modes in the normal-mode analysis.

To calculate the normal modes of the structure, the Hessian was calculated using the VIBRAN module of CHARMM and was diagonalized using the DIAG module. Eighteen modes with negative frequency were found; detailed examinations of these modes revealed that they only involve local side-chain displacements and not global conformational changes.

A similar protocol was used to calculate the normal modes of the Pf-Ago protein (PDB entry 1U04 10). The RMS deviation between the x-ray and the minimized structure is 1.3 Å for Pf-Ago. Six negative modes were found for Pf-Ago protein.

Analysis of two-point correlations of atomic fluctuations provides an overall picture of the collectivity of the internal motions within the protein structures. We analyzed two-point correlations using the covariance matrix of the atomic motions: the correlation between the motions of the ith atom and the jth atom is calculated using the normal modes as

〈 Δ r i ⋅ Δ r j 〉 = k B T m i m j ∑ k u i k ⋅ u j k λ k MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGacaGaaiaabeqaaeqabiWaaaGcbaWaaaWaaeaacqqHuoarcqWHYbGCdaWgaaWcbaGaemyAaKgabeaakiabgwSixlabfs5aejabhkhaYnaaBaaaleaacqWGQbGAaeqaaaGccaGLPmIaayPkJaGaeyypa0tcfa4aaSaaaeaacqWGRbWAdaWgaaqaaiabdkeacbqabaGaemivaqfabaWaaOaaaeaacqWGTbqBdaWgaaqaaiabdMgaPbqabaGaemyBa02aaSbaaeaacqWGQbGAaeqaaaqabaaaaOWaaabuaKqbagaadaWcaaqaaiabhwha1naaBaaabaGaemyAaKMaem4AaSgabeaacqGHflY1cqWH1bqDdaWgaaqaaiabdQgaQjabdUgaRbqabaaabaacciGae83UdW2aaSbaaeaacqWGRbWAaeqaaaaaaSqaaiabdUgaRbqab0GaeyyeIuoaaaa@5694@

where r_i is the displacement vector of the ith atom, m_i is the mass of the atom, k_B is the Boltzmann's constant, and T is the temperature. u_ik is the (dimensionless) mass-weighted displacement vector of the ith atom within the kth eigenvector u_k and λ_k is the corresponding eigenvalue (units Kcal/g/Ų). The sum runs over positive modes. Here we are more interested in analyzing the overall correlations between residues, as opposed to individual atoms. Thus, the displacement u_ik in Equation (1) was replaced by u˜ik MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGacaGaaiaabeqaaeqabiWaaaGcbaGafCyDauNbaGaadaWgaaWcbaGaemyAaKMaem4AaSgabeaaaaa@303F@, the mass-weighted displacement of the center of mass of the ith residue in the kth eigenvector:

u ˜ i k = 1 M i ∑ n i m n i u n i k m n i = 1 M i ∑ n i m n i u n i k MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=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@5F73@

where M_i is the total mass of the ith residue and n_i is the index for all the atoms belonging to the ith residue. The correlation between the motions of residue i and j is thus defined by:

D i j = k B T M i M j ∑ k u ˜ i k ⋅ u ˜ j k ω k 2 MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGacaGaaiaabeqaaeqabiWaaaGcbaGaemiraq0aaSbaaSqaaiabdMgaPjabdQgaQbqabaGccqGH9aqpjuaGdaWcaaqaaiabdUgaRnaaBaaabaGaemOqaieabeaacqWGubavaeaadaGcaaqaaiabd2eannaaBaaabaGaemyAaKgabeaacqWGnbqtdaWgaaqaaiabdQgaQbqabaaabeaaaaGcdaaeqbqcfayaamaalaaabaGafCyDauNbaGaadaWgaaqaaiabdMgaPjabdUgaRbqabaGaeyyXICTafCyDauNbaGaadaWgaaqaaiabdQgaQjabdUgaRbqabaaabaacciGae8xYdC3aa0baaeaacqWGRbWAaeaacqaIYaGmaaaaaaWcbaGaem4AaSgabeqdcqGHris5aaaa@4E51@

The atomic RMSF was used to describe the flexibility of the proteins, which is the self-correlation and the diagonal term of the covariance matrix 36. Following Equation (1),

〈 Δ r i 2 〉 = k B T m i ∑ k | u ik | 2 ω k 2 MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGacaGaaiaabeqaaeqabiWaaaGcbaWaaaWaaeaacqqHuoarcqWHYbGCdaqhaaWcbaGaemyAaKgabaGaeGOmaidaaaGccaGLPmIaayPkJaGaeyypa0ZaaSaaaeaacqWGRbWAdaWgaaWcbaGaemOqaieabeaakiabdsfaubqaaiabd2gaTnaaBaaaleaacqWGPbqAaeqaaaaakmaaqafajuaGbaWaaSaaaeaadaabdaqaaiabhwha1naaBaaabaGaeeyAaKMaee4AaSgabeaaaiaawEa7caGLiWoadaahaaqabeaacqqGYaGmaaaabaacciGae8xYdC3aa0baaeaacqWGRbWAaeaacqaIYaGmaaaaaaWcbaGaem4AaSgabeqdcqGHris5aaaa@4C0D@

To assess the overlap between a specific conformational change and a normal mode, we denote the C_α conformational change by Δx = (Δx₁, Δx₂, ⋯, Δx_N), where Δx_i is the displacement of the ith C_α atom. The overlap of a normal mode with this conformational change is evaluated as:

where ukα MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGacaGaaiaabeqaaeqabiWaaaGcbaGaeCyDau3aa0baaSqaaiabdUgaRbqaaGGaciab=f7aHbaaaaa@307C@ is the normalized vector consisting of the C_α subspace of the k^th eigenvector 3738. From the decomposition, one may identify which normal modes contribute to specific conformation changes. (Note that because ukα MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGacaGaaiaabeqaaeqabiWaaaGcbaGaeCyDau3aa0baaSqaaiabdUgaRbqaaGGaciab=f7aHbaaaaa@307C@ is normalized, the sum of ck2 MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGacaGaaiaabeqaaeqabiWaaaGcbaGaem4yam2aa0baaSqaaiabdUgaRbqaaiabikdaYaaaaaa@2FA0@ over all modes is greater than 1).

To investigate the implications of the normal mode analysis for the proposed conformational change upon

c k 2 = ( Δ x ⋅ u k α ) 2 | Δ x | 2 , MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGacaGaaiaabeqaaeqabiWaaaGcbaGaem4yam2aa0baaSqaaiabdUgaRbqaaiabikdaYaaakiabg2da9KqbaoaalaaabaWaaeWaaeaacqqHuoarcqWH4baEcqGHflY1cqWH1bqDdaqhaaqaaiabdUgaRbqaaGGaciab=f7aHbaaaiaawIcacaGLPaaadaahaaqabeaacqaIYaGmaaaabaWaaqWaaeaacqqHuoarcqWH4baEaiaawEa7caGLiWoadaahaaqabeaacqaIYaGmaaaaaOGaeiilaWcaaa@4607@

duplex formation, we used the hypothetical model of Patel and co-workers 11. In particular, to study the variation of the volume occupied by DNA-RNA duplex with normal-mode motion, the model was used to locate C-alpha protein atoms within 5Å of the duplex. This set of atoms was then used to analyze deformations of the crystal structure in terms of duplex accommodation. The selected C-alpha atoms form the shape of cylinder, with its axis aligned with that of the duplex. We defined the radius r of the cylinder as the largest distance between any C-alpha atom and the axis – the radius for the unaltered crystal structure is denoted by r₀. The volume of the cylinder, defined as V (r) = π × z × r², where z is the length of the duplex, was used as a measure of the space available to accommodate the duplex. We then imposed a conformational change to the protein according to each normal-mode vector, using an amplitude corresponding to a RMSD of 0.05Ų per atom. (We also considered a frequency-dependent amplitude as in Eq. (1), which led to identical conclusions.) We found that r takes different values when imposing different normal-mode motions. The volume changes associated with normal-mode motions were calculated as Δ V = V (r) - V (r₀).

DM carried out the calculations, performed analysis, made the figures and wrote the paper. MEW helped formulate problem, helped perform analysis and helped write the paper. KYS formulated problem helped perform analysis, helped make figures, and helped write the paper.