 Research Article
 Open Access
 Published:
Prediction of solution properties and dynamics of RNAs by means of Brownian dynamics simulation of coarsegrained models: Ribosomal 5S RNA and phenylalanine transfer RNA
BMC Biophysics volume 8, Article number: 11 (2015)
Abstract
Background
The possibility of validating biological macromolecules with locally disordered domains like RNA against solution properties is helpful to understand their function. In this work, we present a computational scheme for predicting global properties and mimicking the internal dynamics of RNA molecules in solution. A simple coarsegrained model with one bead per nucleotide and two types of intramolecular interactions (elastic interactions and excluded volume interactions) is used to represent the RNA chain. The elastic interactions are modeled by a set of Hooke springs that form a minimalist elastic network. The Brownian dynamics technique is employed to simulate the time evolution of the RNA conformations.
Results
That scheme is applied to the 5S ribosomal RNA of E. Coli and the yeast phenylalanine transfer RNA. From the Brownian trajectory, several solution properties (radius of gyration, translational diffusion coefficient, and a rotational relaxation time) are calculated. For the case of yeast phenylalanine transfer RNA, the time evolution and the probability distribution of the interarm angle is also computed.
Conclusions
The general good agreement between our results and some experimental data indicates that the model is able to capture the tertiary structure of RNA in solution. Our simulation results also compare quite well with other numerical data. An advantage of the scheme described here is the possibility of visualizing the real time macromolecular dynamics.
Background
Ribonucleic acid, or RNA, is a key molecule for the synthesis of proteins in living cells. Transfer RNA (tRNA) carries an amino acid to the protein synthetic machinery of a cell, the ribosome, which is the particle responsible for translation of the genetic code. Ribosomes are mainly formed by the socalled ribosomal RNA (rRNA) along with a variety of protein subunits. As many biological macromolecules, the RNA functionality is related with its structure which, in turn, determines the RNA solution properties. Therefore, the latter are essential sources of information on RNA structure and dynamics [1]. However, the structural determination of even small RNA molecules like tRNAs (70 to 80 nucleotides long) or the 5S rRNA of E. Coli (120 nucleotides long [2]) is not an easy task, being most RNAs much larger molecules. For instance, besides 5S rRNA, the main classes of rRNA are 16S and 23S in prokaryotic cells, and 17S, 18S, 26S, and 28S in eukaryotic cells [2] (S stands for Svedberg, the unit of sedimentation coefficient; it is noteworthy that the notation for ribosomal RNAs, as for some other biomacromolecules, is made in terms of a hydrodynamic property).
It is wellknown that tRNA and rRNA are singlestranded chains that fold unto themselves giving rise to a secondary structure that contains both ordered (essentially rigid) doublehelical regions and disordered zones socalled loops (see Fig. 1). Double helices are formed when complementary base pairs belonging to two pieces of the strand interact with each other – after strand folding – through either WatsonCrick or Hoogsteen (nonWatsonCrick) interactions, whereas loops appear in regions with unpaired bases. The overall view of such structure can be described as some number of arms, or stems, joined by the loop regions, which may act as flexible hinges. Thus, RNA molecules have an intrinsic flexibility that allows them to adopt a wide range of threedimensional conformations. Although valuable insights about the tertiary structure can be obtained from atomic coordinates coming from xray crystallography measurements – available at the Protein Data Bank (PDB) (www.rcsb.org) – [3], experiments where the RNA flexibility can be revealed are most useful to characterize its conformational variability. In that sense, as above mentioned, dilute solution properties (hydrodynamic coefficients, scatteringrelated quantities, etc.) provide key information on the overall threedimensional structure of RNA when combined with theoretical predictive models. Clearly, those models should include flexibility in order to make adequate predictions.
The tRNA is a particularly simple case as it presents four short stems as sketched in Fig. 1 a. In usual conditions, those stems appear grouped forming two arms joined by a disordered region so that the crystal structure has the typical “Lletter” shape [4]. Thus, tRNA has been presented as a typical case among segmentally flexible macromolecules, composed by rigid parts connected by flexible hinges. Considerable effort has been devoted to the characterization of the structure and degree of flexibility of tRNA in dilute solution [5], by means of either measurements of solution properties [6–10] or atomistic molecular dynamics simulation [11] (particularly noteworthy are the pioneering works of Harvey, McCammon and coworkers [12–15]). Recent works and reviews on tRNA dynamics are those of Alexander et al. [16] and Agirrezabala and Valle [17]. Ribosomal RNAs are structurally quite more complex than tRNAs. For example, the secondary structure of the majority of the 5S rRNAs has five double helices, two hairpin loops, two internal loops, and a hinge region organized into three big domains (α, β, and γ) [18–20] as sketched in Fig. 1 b. Combining results from dilute solution experiments with numerical calculations, some models for the tertiary structure of 5S rRNA have been proposed [19, 20].
The level of detail, or coarsegraining, of a macromolecular model depends on the type of behavior that is intended to capture. Thus, a large variety of coarsegrained models and computational methods are used for biological macromolecules [21]. Since long, our group has developed procedures and devised computational tools to build coarsegrained models and predict solution properties of both rigid [22–24] and flexible macromolecules [25–27]. The latter are represented by beadandspring chains where, in the classical concept, each bead corresponds to a large piece of the macromolecular chain. Nonetheless, finer coarsegrained (beadandspring) models can be used whose elements represent, for instance, the repeating units – amino acid or nucleotide residues – of the biomacromolecules [28, 29]. Indeed, one of the present authors proposed forty years ago [30] to obtain hydrodynamic properties for doublehelical DNA using what is nowadays regarded as a coarsegrained model with one bead per nucleotide and the approximate Kirkwood formula for friction coefficients. The problem was revisited, in further work [28], using modern beadmodel hydrodynamics [22] for rigid DNA models. Subsequently, bending and twisting internal flexibility was included in the model by means of stiff, elastic springs [31]. The latter simulation of flexible, doublehelical DNA, which is precursor of the present work, is in the spirit of a popular family of coarsegrained models for biomacromolecules and supramacromolecular structures, the socalled “elastic network model” [32], which has been applied to RNA structures [33, 34].
Certainly, the preferred simulation method for short time scales has been, following the above mentioned pioneering works [11, 12, 15], the classical molecular dynamics technique. There are many commercial and public domain tools that implement that technique [21] which has been applied to a variety of RNA structures [35–37]. However, the large size of the macromolecules and the typical long time scales of their global dynamics are not usually accessible to conventional (atomiclevel) molecular dynamics simulations. An alternative is to apply the Brownian dynamics simulation technique to some type of coarsegrained model. Thus, in this work, we have attempted to devise a scheme for predicting global solution properties, that would also mimic the internal dynamics, of RNAs using the simple coarsegrained model with one bead per nucleotide, with intramolecular interactions represented by a minimalist elastic network, along with the Brownian dynamics technique.
Methods
Coarsegrained model
The RNA molecules (tRNA and rRNA) were modeled by beadandspring chains with one bead per nucleotide. In that model, nucleotides along the RNA strand form a string of touching beads all of them with the same hydrodynamic radius, σ=3.15 Å. That value is determined according to the features of the doublehelical regions which are modeled following previous works for doublestranded DNA [28, 31], taking into account that RNA double helices appear usually in the A form whereas DNA is commonly found in B (or Z) form. The A form is a right handed double helix with the following features [38]: major groove width 3.8 Å, minor groove width 10.9 Å, helix radius 10 Å, rise 2.8 Å, inclination 16.1°, number of base pairs per helical pitch 11. On the other hand, the phase angle, ϕ, which define the gap between the complementary helices and therefore the shape and size of the grooves [28], was set to ϕ= 200°.
Beads are the model elements where the friction and the intramolecular interactions take place. There are two types of intramolecular interactions in the model: elastic interactions (represented by Hooke springs) and excluded volume interactions. Hooke springs are used to keep the connectivity between beads along the strand as well as the secondary structure in the helical regions. Also, they introduce the required degree of flexibility in the model [25] and form a kind of elastic network (see Fig. 2). For the sake of minimizing the amount of interactions but keeping the doublehelical shape and the stiffness at short scale, we found it adequate [28] to connect each bead i (within every doublehelical region) to:

1.
its first neighbors along its helix piece (beads i±1), which keeps connectivity and bond equilibrium length.

2.
its second neighbors along its helix piece (beads i±2), which accounts for bending interactions.

3.
its third neighbors along its helix piece (beads i±3), which accounts for torsional interactions.

4.
its counterpart in the complementary helix piece (bead i+N _{ bp }), which accounts for interactions between nucleotides forming the base pair.

5.
the first neighbors of its counterpart in the complementary helix piece (beads i+(N _{ bp }±1)), which is necessary in order to keep the double helical structure.
N _{ bp } is the number of base pairs within a given doublehelical region. In the above description, we use the term “helix piece” instead of “strand” because every nucleotide in the RNA molecule actually belongs to a unique strand. Figure 2 a shows the model for a double helix (in the straight, equilibrium conformation) displaying all the connectors involving one of the innermost beads (the rest of connectors was removed for the sake of clarity). On the other hand, each bead in a loop is only connected to their two first neighbors (beads i±1).
The Hooke spring potential reads
where l is the instantaneous spring length, l _{ e } is the equilibrium spring length, and H is the spring constant. Following previous works [31], we set \(H=200k_{B}T/l_{e,1}^{2}\) for all connectors regardless the type of interaction represented. In that expression, k _{ B } T is the Boltzmann factor and l _{ e,1} is the equilibrium length of the connector binding bead i to its first neighbor (l _{ e,1}=2σ=6.30 Å, in our touchingbeads model). For the chosen value of H, the rootmeansquare fluctuation in the spring length, \(\sqrt {\langle l^{2} \rangle  \langle {l_{e}^{2}} \rangle }\), is 5 % of l _{ e }.
In addition to the elastic interactions, nonconnected beads interact with each other through the following excluded volume (EV) LennardJones potential that prevents beads from overlap and mimics good solvent conditions:
where r _{ ij } is the instantaneous distance between the interacting beads, ε _{ LJ }=0.1k _{ B } T, and σ _{ LJ }=0.8l _{ e,1} [39].
Simulation, hydrodynamics, and solution properties
We performed Brownian dynamics (BD) simulations of beadandspring models of tRNA and 5S rRNA in order to obtain their solution properties in a solvent with viscosity η _{ s }=0.01 poise at T=293 K. The deterministic forces in the BD algorithm are the derivatives of the potentials defined by Eqs. (1) and (2). Both simulation and analysis were carried out with the software package SIMUFLEX [27] that implements a procedure based on the ErmakMcCammon algorithm [40]. That BD algorithm takes into account the hydrodynamic interaction (HI) effect through the RotnePragerYamakawa tensor [41]. Although the rigorous inclusion of the HI effect is necessary in order to generate the right macromolecular dynamics, BD simulations without HI (computationally less expensive) can be used as a “smart Monte Carlo” procedure [42] because they sample correctly the equilibrium conformational space. In such an approach, an ensemble of equilibrium conformations is generated and properties are calculated by means of the socalled rigid body treatment [43]. In the rigid body treatment, conformations are considered as instantaneously rigid and the hydrodynamic theory for rigid bead models is applied [23]. According to that procedure, conformational properties (like the radius of gyration) can be rigorously calculated whereas some overall hydrodynamic entities (like the translational diffusion and the intrinsic viscosity) can be acceptably estimated [44].
The initial conformations of the RNA models (including helices and loops) were built using an inhouse program. After an equilibration period, long BD simulations with a duration about 100 times the relaxation time of the molecules were carried out. The resulting trajectories were divided into five pieces in order to calculate averages and standard deviations. We computed three properties: the radius of gyration R _{ g }, the translational diffusion coefficient D _{ t }, and the harmonic relaxation time τ _{ h }.
The macroscopic translational diffusion of a Brownian particle (consequence of its Brownian motion) is characterized by the time correlation function of the center of mass or Einstein equation 〈d ^{2}〉=6D _{ t } Δ t, where 〈d ^{2}〉 is the mean squared displacement of the center of mass, Δ t is the displacement duration, and D _{ t } is the translational diffusion coefficient of the particle. Long BDHI simulations were carried out to obtain D _{ t } using that equation. On the other hand, BDnoHI simulations were used as a “smart Monte Carlo” procedure to generate equilibrium conformations and apply the rigid body treatment to obtain τ _{ h } and R _{ g }. The harmonic relaxation time τ _{ h } characterizes the overall tumbling of the molecule and can be compared to experimental rotational relaxation times. In case of a rigid body with arbitrary shape, the harmonic mean of its five characteristic rotational relaxation times is calculated as τ _{ h }=1/(6D _{ r }) [23], where D _{ r } is a rotational diffusion coefficient obtained from the eigenvalues of the rotational diffusion tensor. Although the rigid body hydrodynamics is not strictly applicable to the reorientational dynamics of flexible structures (like tRNA and rRNA), it allows for the calculation of an ensemble average 〈D _{ r }〉 from which to calculate a value τ _{ h }=1/(6〈D _{ r }〉). Finally, since R _{ g } is a conformational property, it is correctly obtained from a conformational average over a BDnoHI trajectory.
Results and discussion
A beadandspring model for the yeast phenylalanine tRNA, yeast tRNA ^{phe} (PDB code 1EHZ), was built according to the structure sketched in Fig. 1 a [45]. After a BDnoHI simulation for equilibrating the initial conformation, BD simulations with and without HI were performed. Figure 3 a shows two of the many conformations adopted by that molecule along the Brownian trajectory once the equilibrium has been reached (the “Additional file 1” in the supplementary material displays the movie of the trajectory). Flexibility is noticed in the variation of the angle between arms (interarm angle θ) as better appreciated in Fig. 4 a. There, it is depicted the fluctuation along the Brownian trajectory of the angle θ subtended by the two arm vectors, one defined along the “acceptor stem” and the other one defined along the “anticodon stem”. As observed, fluctuations are quick and the value of the angle varies in a relatively wide range, in agreement with some molecular dynamics simulations [11]. Figure 4 b is an histogram that shows the frequency of occurrence of the different θ values as taken at intervals of 10°. Its broadness is indicative of the flexibility of the structure. Clearly, there is a most probable equilibrium value around θ∼ 90° (the precise average value being θ= 88°), which is coincident with that found in the crystal structure. Therefore, overall (timeaveraged) hydrodynamic properties can be reasonably estimated using rigid bead models built upon the atomic coordinates coming from PDB files [46, 47].
Table 1 collects our BD results for R _{ g }, D _{ t } and τ _{ h }, along with values calculated previously using rigid body hydrodynamics [47] and experimental results. The rigid body hydrodynamic calculations were carried out with the program HYDROPRO [24] which is able to build a bead model from the atomic (PDB) coordinates. There is a general good agreement among HYDROPRO calculations, BD simulations and experimental results. The advantage of our BD scheme is the possibility of evaluating, in addition to global solution properties, the internal dynamics of the molecule as illustrated by Fig. 4.
On the other hand, a beadandspring model of 5S rRNA of E. Coli (PDB code IC2X:C) was built according to the structure sketched in Fig. 1 b [20]. Again, after equilibrating the initial conformation, BD simulations with and without HI were performed. Figure 3 b illustrates two of the many conformations adopted by our 5S rRNA model along the equilibrium Brownian trajectory (the “Additional file 2” in the supplementary material displays the movie of the trajectory). The flexibility is revealed by the variation of the angle between the domains (α, β, and γ) as well as the expansion of the loops whereas the helical regions keep a rigid structure.
Table 2 shows the values for R _{ g }, D _{ t } and τ _{ h } coming from our BD simulations along with some values selected from Skibinska et al. [20]. In that work, authors collect experimental data from several sources and present their own experimental data. Also, they make some rigid body hydrodynamic calculations (partly carried out with the program HYDROPRO [24]) and propose a likely tertiary structure for the 5S rRNA.
As observed, the results coming from our BD simulations agree relatively well with both experimental data and rigid body hydrodynamic predictions. In particular, our predictions for D _{ t } and R _{ g } are coincident with some of the experimental measurements collected in Table 2 [48, 49]. If we use our predicted D _{ t } value (6.2×10^{−7} cm ^{2}/s) and the Svedberg equation (\(s/D_{t}=M(1\bar {v}\rho)/RT\)), we obtain a value for the sedimentation coefficient s≃5 S what further supports the validity of the scheme employed in this work. In order to obtain that value of s, we assume for the 5S rRNA a molecular weight M=40000 Da [50] and a partial specific volume \(\bar {v}=0.53\) cm ^{3}/g [51] as well as a solvent (water) density ρ≃1 g/cm ^{3}.
On the other hand our calculated τ _{ h } is smaller than the experimental value obtained by Skibinska et al. [20], although close to the value predicted by their rigid body hydrodynamic calculations. As already mentioned, the rigid body hydrodynamic treatment does not capture rigorously the reorientational dynamics of flexible molecules and it is not the most adequate procedure to calculate their rotational relaxation times. For the case of 5S rRNA, the rigid body treatment seems to underestimate the rotational relaxation time. In any case, theoretical predictions are reasonably good if one considers the existing discrepancy among the experimental data themselves. In the “Additional file 1” (yeast tRNA ^{phe}) and the “Additional file 2” (5S rRNA) of the supplementary material, one can appreciate the internal and rotational dynamics of the RNA molecules modeled in this work.
Conclusions
We have devised a computational scheme to model RNA molecules, which present a partially disordered structure, and calculate their solution properties. According to the model, each nucleotide is replaced by a bead so that the secondary structure is represented in detail. The beads are connected by Hooke springs that form a type of elastic network. The model dynamics can be simulated by any Brownian dynamics algorithm. The algorithm and the analysis procedures employed in this work are implemented in our publicdomain software SIMUFLEX available at http://leonardo.inf.um.es/macromol. The previously mentioned program HYDROPRO is also available at that web site.
The good agreement of our simulation results with some experimental data as well as with other theoretical results obtained by alternative numerical approaches ensures the validity of our scheme for the prediction of solution properties of RNAs. It is also indicative that the model is able to capture the tertiary folds of RNA molecules in solution. The advantage of the Brownian dynamics simulation technique (including hydrodynamic interaction) respect to other procedures is the possibility of simulating the long time internal dynamics of big macromolecules in real time.
In a previous work [52], we tested the adequacy of a computational scheme based on a residue level model and the Brownian dynamics simulation to predict the internal dynamics of partially disordered proteins. The present work on the more complex RNA molecule further confirms the success of that type of approach which is expected to be appropriate for the study of other disordered biological macromolecules.
Availability of supporting data
All the supporting data are included as additional files.
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Acknowledgments
Financial support was provided by Ministerio de Economía y Competitividad within project CTQ201233717 including FEDER funds and Fundación Séneca de la Región de Murcia (grant 19353/PI/14).
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Authors’ contributions
AAB carried out the Brownian dynamics simulations and data analysis. JGHC participated in the data analysis and wrote the manuscript. FGB designed the simulations and the analysis. JGT conceived of the study. The authors read and approved the final manuscript.
Additional files
Additional file 1
Movie of the BDHI equilibrium trajectory of our yeast tRNA ^{phe} model. (AVI 26726 kb)
Additional file 2
Movie of the BDHI trajectory of our 5S rRNA model from the initial conformation. (AVI 32972 kb)
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Benítez, A.A., Hernández Cifre, J.G., Díaz Baños, F.G. et al. Prediction of solution properties and dynamics of RNAs by means of Brownian dynamics simulation of coarsegrained models: Ribosomal 5S RNA and phenylalanine transfer RNA. BMC Biophys 8, 11 (2015). https://doi.org/10.1186/s1362801500257
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Keywords
 Brownian dynamics
 Coarsegrained model
 Hydrodynamics
 Transfer RNA
 Ribosomal RNA
 Diffusion coefficients
 Internal dynamics