Role of large thermal fluctuations and magnesium ions in t-RNA selectivity of the ribosome

Abstract

The fidelity of translation selection begins with the base pairing of codon-anticodon complex between the m-RNA and tRNAs. Binding of cognate and near-cognate tRNAs induces 30S subunit of the ribosome to wrap around the ternary complex, EF-Tu(GTP)aa-tRNA. We have proposed that large thermal fluctuations play a crucial role in the selection process. To test this conjecture, we have developed a theoretical technique to determine the probability that the ternary complex, as a result of large thermal fluctuations, forms contacts leading to stabilization of the GTPase activated state. We argue that the configurational searches for such processes are in the tail end of the probability distribution and show that the probability for this event is localized around the most likely configuration. Small variations in the repositioning of cognate relative to near-cognate complexes lead to rate enhancement of the cognate complex. The binding energies of over a dozen unique site-bound magnesium structural motifs are investigated and provide insights into the nature of interaction of divalent metal ions with the ribosome.

In 1944, Erwin Schrödinger made a prescient conjecture that biological machinery may be constructed in a manner “different from anything we’ve yet tested in the physical laboratory” (1). Human constructed machines are designed to minimize friction, whereas the molecular machinery of life is embedded in a viscous fluid where dissipation and the associated fluctuations are enormous (2).

Single molecule methods have allowed us to directly observe fluctuating transitions between intermediate states that help elucidate the interplay between structure, dynamics, and function. Through these observations, evidence is growing that biomolecular machinery may be coopting thermal fluctuations as a fundamental part of their operation. In our study of the minimal functional structure of the hairpin ribozyme, we suggested that the highly directional, enzymatic cleavage of a sequence specific strand of RNA was driven by reversible transitions between intermediate states driven by thermal fluctuations (3). Subsequent measurements of the transition rates between intermediate states in the full four-helix junction of the hairpin ribozyme reinforced this conjecture (4).

In single molecule studies of the ribosome (5), we proposed that thermal fluctuations between cognate and near-cognate t-RNA played a critical role in the selection fidelity of the ribosome (6–8). The single molecule studies indicate that base pairing of the codon-anticodon complex involve not only interactions with specific resides and ribosomal proteins, but also conformation fluctuations of the ribosome. These conformation changes bring about the small subunit to enfold around the ternary complex (9, 10). As a result, the cognate ternary complex, EF-Tu(GTP)aa-tRNA, swivels into an orientation that allows it to make stabilizing contacts with various functional groups such as GTP-activated center (GAC) and sarcin-ricin loop (SRL) near the GTPase activated state (5, 9, 10).

We infer that large and rare thermal fluctuations (Fig. 1) of the ribosome are essential to rotate the ternary complex into a position so as to facilitate stable contacts with GAC and the SRL in the large subunit (4, 5). This inference is based on the time scale for forming the stable contacts is between 65 ms and 100 ms at 15 mM and 5 mM Mg²⁺ concentrations respectively and the distance traversed by the ternary complex going to the GTPase activated state, the midFRET state, is over 70 Å. In this work, a framework to describe the probability of such rare events will be formulated.

Fig. 1.

The ternary complex undergoes large thermal fluctuation (denoted by x_c). These fluctuations involve configurational searches that are in the tail end of the probability distribution.

We also examine in detail various structural motifs of magnesium-binding sites to further elucidate the interplay between structure, dynamics, and function within the ribosome. Divalent metal ions serve an essential role in processes such as DNA bending (11, 12) and folding of RNA (13–15). In particular, magnesium ions play an important role in the subunits association, tRNA binding to the decoding site, and in general the structure and the stability of the ribosome (16–20). As shown in bacterial 70S ribosome, the divalent metal ions interact to hold the ribosomal subunits together (21). A comparative analysis of the crystal structures of the large subunits in Tetraopes thermophilus, Escherichia coli, and Haloarcula marismortuni ribosomes indicates that around 40% of the magnesium-binding sites are conserved not only across the domains of life, but also after the subunits associate (22, 23). In the folded conformation, various phosphate oxygen atoms in the ribosome are less than accessible to the solvent. These phosphate oxygens have an inclination for either inner-sphere or outer-sphere contacts with magnesium. In tertiary folding of ribosomal subunits, magnesium ions are thought to bury the phosphate oxygen atoms that are exposed in unfolded conformation (23).

The protein functional groups, the structural positioning of the RNA, and the inner-sphere coordination of the protein atoms to them, create several unique motifs for the binding of Mg²⁺ ions in the large subunit of H. marismortuni ribosome as well as in E. coli (22, 23). Klein et al. have identified and structurally classified 116 magnesium-binding sites from a high-resolution crystal structure of H. marismortuni (23). The structural classifications are labeled by the number of inner-sphere contacts (SI Text) they form and on the three-dimensional arrangement of the protein atoms or the RNA (23).

Here we present a theoretical framework to determine the probability that as a result of large and rare thermal fluctuations (Fig. 1), the cognate ternary complex forms stabilization contacts with the GTPase activated state. We also investigate the binding energies of several unique site-binding Mg²⁺ structural motifs that are a result of the protein functional groups, the structural positioning of the RNA, and the inner-sphere coordination of the protein atoms to them.

Results

Large Thermal Fluctuations Play a Crucial Role in the Selection Process.

The probability of forming contacts to stabilize the GTPase activated state involves configurational searches that are in the tail end of probability distribution (Fig. 2). To determine the probability of such an event, let us assume that the thermodynamic state of the 30S subunit and the codon-anticodon complex is described by a set of extensive macroscopic variables {ψ_l; l = 1,…,n}. The state variables are assumed to be independent, and can be expressed as sums of molecular variables.

Fig. 2.

The figure is a plot of the probability density (y-axis) vs. coordinate of the ternary complex (x-axis). x denotes the departure of the ternary complex from its equilibrium position. As thermal fluctuations reach a critical value, a near-cognate ternary complex will reach distance x_c that allows it to be in spatial contacts with GAC/SRC sites in the ribosome. Small variations in the repositioning of cognate relative to near-cognate (mismatched) complexes lead to significant rate enhancement.

We assume, following Onsager and Machlup (24), that the macroscopic variables defining a state are Gaussian random variables (25). The probability P{ψ_l} of a given fluctuation is proportional to exp(S(ψ₁,…ψ_n)/k_B), where S(ψ₁,…,ψ_n) is the entropy, which is a function of the macroscopic variables {ψ}. On expanding the entropy about the equilibrium value, , where the sums over i and j run over the phase variables, and χ^-1 is a covariance matrix.

Now consider a fluctuation in component i = 0 for which Δψ_o≥a, where a is a positive quantity, and the parameter Δ is small. Because Δ is small, the fluctuations defined by ψ_o are large and rare events. The probability for the rare event occurs in the tail end of the distribution and is given by

[1]

For simplicity, we have considered a one-dimension probability distribution.

The dominant contribution to the probability is expected to come from those configurations for which Δψ - a is small. By carrying out the integral for Δ ≪ 1, one obtains

[2]

The quantity in the square brackets in the above equation dictates the size of the fluctuation. The exponential term governs the decay of probability of the rare event.

Let I{Φ}/Δ² = Ψ/2Δ², where Φ_i = Δψ_i and . The rare event is defined by Φ_o≥a. Now minimize Ψ subject to the constraint Φ_o≥a. Analysis shows that the minimum of I evaluated at the extremum Φ^∗ is

[3]

Observe that I{Φ^∗}/Δ² is equal to negative of the exponent that appears in the second term of [2].

Even though large thermal fluctuations take place in the movement of the ternary complex, our results indicate that the probability of such a rare process is localized around the most likely configuration. This feature is a unique attribute of Varadhan’s formulation of stochastic processes of rare events, and in which I{Φ^∗} is the action functional (26–28). The results can be generalized to Gaussian Markov and non-Markovian processes.

Site-Bound Magnesium Ions Contribute to Structural Stability of the Ribosome.

Which of the magnesium ions observed in the crystal structure of the large subunit of H. marismortuni and E. coli are essential for its structural stability are far from well understood. To provide insights into how site-bound metal ions contribute to structural stability, we have investigated the binding energies of several unique Mg²⁺ binding sites in the ribosome. One such motif (geometrical class IIIa; SI Text) is shown in Fig. 3 (top row, left). This Mg²⁺ ion (labeled Mg 10) links via inner-sphere the phosphate oxygen atoms in helix 35 and helix 90, to an interface between protein L3 and domains V and II of 23S RNA 23S RNA subunit of H. marismortuni (23). Mg 10 stabilizes the quaternary as well as the tertiary structures of the 23S subunit of the ribosome (23).

Fig. 3.

Magnesium-binding motifs extracted from high-resolution crystal structure of the 23S RNA subunit of H. marismortuni and from the 70S bacterial ribosome E. coli. Mg 10 (top row, left), Mg 7 (top row, right), Mg 8 (second row left), Mg 4 (second row, right), and Mg 26 (third row, left) are from the large ribosomal subunit of H. marismortui. A magnesium-binding motif along with its interactions in the intersubunit bridge B3 from the large subunit of E. coli 70S ribosome is depicted in the third row (right). The region encircled denotes the quantum mechanical region. The remaining atoms in the motif define the molecular mechanics region. The color code is such that white denotes the hydrogen atoms, red denotes the oxygen atoms, blue denotes the nitrogen atoms, green denotes the carbon atoms, and purple denotes the phosphate group.

Magnesium ions are known to bind to small structural motifs in 23S RNA. These ions stabilize the RNA structure by neutralization of part of charge on the phosphate groups. One such case we studied (geometrical class IIa; SI Text) consists of a Mg²⁺ ion, (labeled Mg 27) from the 23S RNA subunit of H. marismortuni that facilitates interactions (Fig. 4; second row, left) between the phosphate group and the 2-amino of G nucleotide with pro-S_P phosphate oxygen of nucleotide 5′ and the 6-oxo group of guanosine base (23). Here, the magnesium ion forms two inner-sphere coordination with protein residues and RNA bases.

Fig. 4.

Magnesium-binding motifs extracted from high-resolution crystal structure of the 23S RNA subunit of H. marismortuni (PDB:1s72). Mg 9 (top row, left), Mg 15 (top row, right), Mg 27 (second row, left), Mg 29 (second row, right), Mg 30 (third row, left), Mg 39 (third row, right), and Mg 18 (fourth row, left). The region encircled denotes the QM region. The remaining atoms in the motif denote the MM region. The color code is such that white denotes the hydrogen atoms, red denotes the oxygen atoms, blue denotes the nitrogen atoms, green denotes the carbon atoms, and purple denotes the phosphate groups.

Another unique Mg²⁺ ion binding motif occurs in large subunit of E. coli 70S ribosome. Most of the observed magnesium-binding sites in 30S within 70S are identical to those in the isolated 30S subunit. An exception is a binding site near bridge B3; the latter has been proposed to serve as a pivot point for ratchet-like motion of the small subunit (22). In the 30S subunit, a magnesium ion that adjoins (Fig. 3; third row, right) tandem G-A base pairs in helix 44 of 16S rRNA interacts by hydrogen bonding with G-C base pairs in helix 71 of 23S rRNA (22). Upon association of the 30S and the 50S subunits to form the 70S ribosome, crystal structures reveal that this magnesium ion, instead, forms inner-sphere coordination with nucleotide G1417 (22).

The molecular fragments that define the magnesium-binding motifs are extracted from high-resolution crystal structure of the 23S RNA subunit of H. marismortuni (PDB:1s72) (23) and from the crystal structure of h44 of 16S rRNA (PDB:2QB9) and fh71 of 23S rRNA (PDB:2QBA) for the 70S bacterial ribosome E. coli (22). Each complex is partitioned (SI Text) into an inner region and an outer region (Figs. 3 and 4) (29, 30). The inner region consisting of between 24–56 atoms is treated by ab initio electronic structure calculations (QM). All calculations in the QM region are single point and were performed using density functional techniques (DFT) at the DFT-M06-2X/6-31++G** level of approximation (SI Text) (29). The remainder of the atoms (outer region) in the magnesium-binding motifs are described by molecular mechanics (MM) and based on the OPLS-2005 force field (29).

The binding energy, which is the sum of the interaction and the solvation energies, of Mg 10 (type IIIb) is -88.745 kcal/mol while that of Mg 4 (type IIa) is -73.367 kcal/mol. In contrast, the binding energy of Mg²⁺ ion from E. coli 70S ribosome is -47.168 kcal/mol. Other type IIa magnesium ions in the 23S RNA subunit of H. marismortuni that we have studied (Figs. 3, 4 and Table S1) have binding energies -72.983 kcal/mol (Mg 9), -76.625 kcal/mol (Mg 18), and -81.452 kcal/mol (Mg 30). Binding energies of Mg 7 (type IIIb), Mg 26 (type IIIa), and Mg 30 (type I) are comparable. The former two magnesium ions interact with three water molecules, while the latter with five water molecules (Table S1).

Discussion

The ternary complex, as a result of rare and large thermal fluctuations (Fig. 1), forms contacts with SRC/GAC leading to stabilization of the GTPase activated state. Our theoretical findings establish that configurations, which make significant contribution to the probability of the rare process, have the inherent property that they are localized tightly around the most likely configuration. We show that this configuration is obtained by minimizing an action functional with an appropriate constraint. The minimized action gives the probability of the rare event.

Small variations in the repositioning of cognate relative to near-cognate complexes lead to significant rate enhancement (Fig. 2). Let x be the departure of the ternary complex from its equilibrium position. As thermal fluctuations reach a critical value, a near-cognate ternary complex will reach distance a, that allows it to make spatial contacts with GAC/SRC at elongation factor-Tu. In contrast, the critical fluctuation will allow a cognate ternary complex to reach, instead, a distance a - |γ|, where γ is a constant. From [2], the ratio of the rate of docking of cognate to the near-cognate ternary complex is approximately

[4]

where the subscripts c and nc stand for cognate and near-cognate complexes, ν is the docking frequency, and we have assumed that χ_oo;c ≈ χ_oo;nc. Observe that the model predicts that rate enhancement is dictated in part by χ_oo;c, the second derivate of the entropy with respect to macroscopic variables.

The large fluctuation theory proposed here is reminiscent of a multidimensional threshold model in chemical kinetics by Slater (31) and in the quantum theory that describes barrier crossing in condensed phases by Wolynes (32). There are also processes in biological systems where fluctuations in the environment are slower than events that lead to barrier crossings (33). In these systems, classical kinetic laws are inapplicable because the time dependence of correlation functions display nonexponential behavior. Wang and Wolynes have studied such a system by considering a reaction that occurs when a threshold fluctuation of an environmental variable takes place (33, 34). These authors have shown that rare events make a significant contribution to the statistical time path of an intermittent process (33, 34).

Whitford et al. have recently shown that large thermal fluctuations play a vital function in the process of accommodation in the ribosome (35). This work supports the mechanism proposed in this paper, namely that small structural changes in the cognate tRNA probablity distribution can make the rare events for the cognate ternary complex more likely to take place than for the noncognate ones. There is an amplification effect because the events occur at the tail of the distribution.

The nature of interaction of magnesium ions with RNA is generally classified either via site binding or diffuse binding (36, 37). According to the counterion condensation model, the magnesium ions form a delocalized cloud that is spread over the phosphate backbone (36, 37). These nonspecific interactions, i.e., diffuse binding by the magnesium ions lead to substantial neutralization of the negative phosphate charges and thermodynamic stabilization of the RNA (37). In contrast, in site binding, magnesium ions with its partial layer of water molecules can directly coordinate with RNA (13, 14, 38). Which of the site-bound magnesium ions observed in the crystal structure of the large subunit of H. marismortuni and E. coli are essential for its structural stability are not well understood.

The binding energy of various magnesium structural binding motifs in the large subunit of H. marismortuni is listed in Table 1 (SI Text). In all cases, the total solvation energy is destabilizing while the interaction energy is stabilizing. A comparison of the gas phase energy and the solvation phase energy of the ligand, i.e., the magnesium ion with its associated water molecules, indicates that many energy values were consistent and predictable for each classified type of binding site (Tables S2 and S3). For type IIa motifs investigated, the gas phase energy of the ligand is around -505.15 Hartree, for example. There were other binding motifs, such as Mg 7 (type IIIb) and Mg 8 (type IIIb), whose ligand gas phase energy as well as solvation phase energy (Tables S2 and S3) displayed characteristics of other types, namely Mg 26 (type IIIa) and Mg 16 (type IIIa). Remarkably, the solvation energy also does not vary much within a given type. These trends are also observed at the LMP2 level of approximation for the QM region.

In summary, we have developed a theoretical technique that allows us to determine the probability that the ternary complex forms contacts leading to stabilization of the GTPase activated state. The configurational search for such process is in the tail end of the probability distribution, and is a rare event. The probability distribution of the rare event is localized around the most likely configuration. The binding energies of over a dozen site-bound magnesium structural motifs were investigated and provided insights into the nature of interaction of divalent metal ions with the ribosome and its structural stability.

Materials and Methods

The magnesium-binding motifs were extracted from the crystal structures (22, 23) of H. marismortuni (PDB:1s72) and E. coli (PDB:2QB9; PDB:2QBA) and capped with hydrogen atoms. The optimization of all hydrogen atoms is implemented by freezing the coordinates of all heavy atoms. All minimizations were carried out in Maestro (29) using a conjugate gradient method in which the energy change and the gradient criterion are taken to be 1.0 × 10^-7 kcal/mol and 0.01 kcal/ mol Å, respectively.

Because the number of atoms in the motifs is large, we partitioned the complex into inner and outer regions (30) (see SI Text) using Qsite suite of programs (29). Quantum mechanical approximations at the DFT-MO6-2X/6-31++G** level and OPLS-2005 force field (29) were employed in the inner and the outer regions, respectively, to obtain the interaction energies in the gas phase (SI Text). The number of atoms in the inner region varied between 24 and 56 depending on the total number of atoms in the magnesium ion structural motif. The interaction energy is, by definition, the difference between single-point energies of the molecular fragments that define the binding motif and the monomers.

We have determined the effects of solvation on the interaction energy by solving the nonlinear Poisson-Boltzmann equation using finite element method (39). The displacement threshold and nonbonded cutoffs were set to 0.1 Å and 12.0 Å. This former signifies how far a given atom moves from the coordinates used in the previous step of the finite element grid calculation of the nonlinear Poisson-Boltzmann equation. Convergence criterion is based on energy change (1.0 × 10^-7 kcal/mol) and gradient (0.01 kcal/mol Å). The nonbonded interactions (electrostatic and van der Waals) cutoff is set to 30 Å while the dielectric constant ϵ in the gas phase is set to unity. Because water is the solvent, the dielectric constant was taken to be a constant, namely 80.4. The temperature and pressure were fixed at 298.15 K and 1 atm respectively. The binding energy is the sum of the interaction and the solvation energies.