Coarse-grained dynamic RNA titration simulations

Abstract

Electrostatic interactions play a pivotal role in many biomolecular processes. The molecular organization and function in biological systems are largely determined by these interactions. Owing to the highly negative charge of RNA, the effect is expected to be more pronounced in this system. Moreover, RNA base pairing is dependent on the charge of the base, giving rise to alternative secondary and tertiary structures. The equilibrium between uncharged and charged bases is regulated by the solution pH, which is therefore a key environmental condition influencing the molecule’s structure and behaviour. By means of constant-pH Monte Carlo simulations based on a fast proton titration scheme, coupled with the coarse-grained model HiRE-RNA, molecular dynamic simulations of RNA molecules at constant pH enable us to explore the RNA conformational plasticity at different pH values as well as to compute electrostatic properties as local pK_a values for each nucleotide.

1. Introduction

RNA molecules are key players in a cell’s life as they are at the heart not only of the transcription process for protein synthesis but also of many regulatory mechanisms related to gene expression and enzymatic reactions [1–3]. Their function is intrinsically related to their three-dimensional structure [4], which in turns depends on the pairing of bases through hydrogen bonds (secondary structure). It is now recognized that RNA molecules have a significant structural plasticity due to possible alternative secondary structures with comparable free energies [5–8]. The environment plays a central role in shifting the equilibrium of different populations and influencing which structure the molecule adopts. This makes the conformational landscape of an RNA molecule rough, with possibly coexisting structures, that are each stabilized by external conditions. This is the case for riboswitches, among others, where two structures are known experimentally and where one or the other is stabilized depending on the presence or absence of a ligand [9,10].

pH is one of the environmental factors strongly influencing RNA structures, as ionized nucleotides change their coulombic electrostatic interactions and base pairing preferences. At low pH, adenosine can be protonated at its N1 atom and cytosine at its N3 atom and become positively charged, while at high pH, guanosine and uridine are deprotonated at their N1 and N3 atoms, respectively, becoming negatively charged [11]. If charged, the bases can no longer form the canonical Watson–Crick pairs A–U and G–C, since the charge lies on the Watson–Crick edge [12], but form alternative pairs. These charged pairs have been reported experimentally [13–15] as well as characterized by quantum mechanical calculations [16]. In particular, protonation of adenines has been found more commonly in experiments [17–21] and has been extensively investigated by different theoretical approaches [22–25].

Computational strategies to describe pH effects on RNA structure and activity often adopt an atomistic description of the system and involve solving the Poisson–Boltzmann (PB) equation [22,26] or performing different implementations of constant-pH molecular dynamics (MD) [23,27,28].

A diversity of tools is now available to deal with proton titration in numerical simulations of biomolecular systems, whose choice depends on the system studied, computational resources and sampling approaches among other factors (see [25,29] for detailed discussions). Though potentially accurate, adopting atomistic molecular descriptions can be extremely computationally expensive. Slow convergence issues have been reported by different research groups [30,31]. These methods are relatively apt for studying effect of the environment of one particular (small) structure, but they are not yet able to explore populations of alternative structures that molecules can adopt.

In particular, studies have been conducted on RNA using the standard nonlinear Poisson–Boltzmann (NLPB) method [22], a Gaussian-based dielectric function description for the NLPB (GNLPB) [26] and continuous constant-pH MD [23]. In the first two approaches, titration is computed for a given, rigid and kept static, molecular configuration. The method consists of a numerical solution of the second-order differential equation relating the electric field and the charge distribution of a system in a dielectric background. Different numerical methods are available for solving this equation. In the GNLPB implementation, the dielectric interface is a free parameter that can be tuned to fit experimental data, giving good agreement between experimental and computed pK_a values. Being a static method in which calculations are done on a given experimental structure with a constant electric potential, this approach cannot deal with charge fluctuations nor treat the response of the molecular conformation to changes in protonation states [26]. A recent implementation of constant-pH MD consists of running a MD simulation with an added artificial dynamical titration coordinate ‘λ’ regulating the transition between protonated and unprotonated states. The fully atomistic description of the system with explicit water allows studying mechanisms such as solvent-mediated proton transfers and following the structural rearrangements of the molecule subject to titration. However, this method suffers from severe sampling and convergence limitations, and typical simulations cover only a few dozen nanoseconds for systems of the size presented here.

Slow convergence and a limited ability of exploring large conformational rearrangements are the main drawbacks of current atomistic RNA simulations in general, even without considering the effects of pH. Indeed, at atomistic resolution, a wide exploration of the conformational space involving significant changes in the secondary structure is possible only for small systems of the size of a tetraloop, even when enhanced simulations are invoked [7,32–34]. To overcome this problem and study larger systems, in recent years, several coarse-grained models for RNA have been developed with the aim of addressing various structural and thermodynamics properties of these systems [35–38]. Among them, we developed the High Resolution Energy model for RNA (HiRE-RNA) [39,40] in which, by representing a nucleotide by six or seven beads—a higher resolution than most other coarse-grained RNA models—we are able to give an accurate description of base pairings, including both canonical and non-canonical pairs. Thanks to the significant reduction in degrees of freedom, with this model we have been able to explore large conformational changes, including complete folding of systems of several dozen nucleotides [5,40,41].

Recently, we have extended to RNA a fast proton titration scheme (FPTS) computing titration by means of Monte Carlo simulations for a rigid macromolecule [42,43]. This method, thanks to a coarse-grained representation of the system, overcomes the convergence limitations of atomistic methods and allows to compute detailed electrostatic properties accounting for pH and salt concentration on a given RNA structure [24]. We have shown that pK_a values computed with this method on experimental structures, or on structures extracted from a MD trajectory, are comparable to values computed through the more expensive methods, at a much smaller computational cost. In this first study on RNA titration, we used HiRE-RNA MD simulations to generate structures that were then submitted to FPTS to compute titration properties. However, in that study, it is important to understand that the coarse-grained model was merely used as a means to generate distinct conformations that would constitute possible fluctuations of the experimental structure: the dynamic simulation was not affected by titration and pH, and the bases remained neutral at all times.

In this work, we present the proof of principle of the dynamic coupling of the FPTS to HiRE-RNA MD such that titration influences the structural and dynamical aspects while they also influence the acid–base equilibria. These constant-pH coarse-grained simulations (CpHCGMD) allow us to take into account both RNA structure and dynamics and their coupling with the protonation process, and therefore to study RNA molecules in different pH conditions, to extract local pK_a values and to study the predominant structures at the various pH values.

Following the landmark work of Baptista and co-authors [44] to describe the strong protonation-conformation coupling for proteins, we developed a new protocol for RNA simulations. By running a coarse-grained molecular dynamic (CGMD) simulation with external calls to the titration algorithm (here, the FPTS instead of the linear PB method used by Baptista and collaborators [27,44]), and by updating the base pairing schemes according to the protonation state returned by FPTS, we can explore the conformations of the molecule at a given pH and salt condition. By running simulations at multiple, independent, pH values, we can extract titration curves and obtain local pK_a values for any given titratable base. These may be compared with values obtained experimentally via NMR.

On several benchmark systems, our approach shows a net improvement in the agreement between computed and experimental pK_a values as compared to our previous work [24] (which did not include the dynamic coupling between titration, base pairing and conformational changes) and to other, more expensive, computational approaches.

This work is organized as follows. We first give a brief overview of the titration scheme and of the RNA coarse-grained model, followed by the description of how the two methods are combined to give a constant-pH CGMD simulation with the proper protonation-conformation coupling. We then introduce the systems on which the method has been tested and the results of applying CpHCGMD, compared with the FPTS (using a rigid RNA structure) and other methods.

2. Fast proton titration scheme

The FPTS is a numerical simulation protocol to efficiently describe acid–base equilibria in macromolecular systems where an effective Hamiltonian is derived from the modified version [45] of the classical Tanford–Kirkwood model [46] and solved by Metropolis Monte Carlo calculations. All titratable and nontitratable sites (amino acids or nucleotides) are represented at the coarse-grained level as a single Lennard-Jones like bead whose radius (R_i) is defined based on the molecular weights of each amino acid or nucleotide. Charges are assigned to the titratable groups according to the protonation/deprotonation process as a function of the solution pH and salt concentration. In short, the FPTS invokes an implicit solvent modelling where the molecule lies in a homogeneous dielectric medium and the effects of salt and ions are accounted for by a modified Debye–Hückel theory. The method has the advantage of being extremely fast (convergence can be reached in 10⁴–10⁵ Monte Carlo steps which take of the order of seconds in a common Intel i7 computer) and of being able to describe charge fluctuations, which is a key mechanism for macromolecular complexation. It has shown good accuracy in pK_a calculations for both proteins and RNA and was able to identify the main physical mechanisms involved in various electrostatic phenomena [24,47,48]. Despite these advantages, in the original FPTS, the macromolecular conformation is kept unaffected by the pH during the whole simulation resulting in a poor description of the protonation-conformation coupling. Moreover, given that the method performs a titration calculation on a given structure, it is only as accurate as the configurations that are used in the analysis and it cannot provide dynamical properties. A more in depth discussion of advantages and limitations of this approach can be found in [24,25,42,43,49]. Some details on the FPTS model, summarizing the main points of previous publications, are given in electronic supplementary material, section S1.

In our previous work, we have shown that the implementation of FPTS with a static conformation (for the sake of clarity, we will refer to it as FPTS_static from now on), where calculations are performed on an experimental structure or on snapshots extracted from MD simulations, are able to give an accurate description of some key electrostatic properties and phenomena of biomolecules such as the complexation of like-charged molecules [50,51], aggregation induced by pH variations [47], and give local pK_a values in a good range for both proteins and RNA molecules [24,25,43]. We have also shown that this approach gives results comparable to other constant pH methods at a much lower computational cost, since the evaluation for one structure (conformation) converges in a matter of seconds on a single processor desktop computer [24,43].

In the application of FPTS to RNA molecules, we consider that bases are the only titratable sites for a nucleotide in physiological conditions. In reality, RNA nucleotides have two titratable sites, the phosphate group and the nitrogenous base, but phospate groups are assumed to be negatively charged all the time due to the fact that their experimental intrinsic acid dissociation constant pK₀ is close to 1. On the contrary, the imino nitrogens of the bases can be protonated/deprotonated in close to physiological pH conditions due to their values of pK₀ (see electronic supplementary material, table S1, for the experimental values of pK₀ taken from [11]). Therefore, in our work, the whole nucleotide is associated with a single titratable group centred in its centre of mass (see electronic supplementary material, figure S1). The charge of a nucleotide can then take values of −1 and 0 for adenine and cytosine, which acquire a proton in acidic conditions, and of −2 and −1 for guanine and uracil that lose a proton in basic conditions.

3. Coarse-grained model

The coarse-grained model HiRE-RNA aims to simulate the behaviour of folding and assembly of RNA molecules of up to a few hundreds of nucleotides. Using MD and enhanced sampling simulations [52–56], this model has proven capable of folding non-coding RNAs of about 50 nucleotides from the sole knowledge of their sequence, i.e. in the absence of any structural information, and of larger systems when coupled to additional input such as the presence of a few well defined base pairs [39–41].

In HiRE-RNA each nucleotide is described by 6 or 7 beads: 1 particle for the phosphate group, 1 particle for the backbone oxygen, 3 particles for the sugar and 1 or 2 particles for the base (see figure S2 in the electronic supplementary material). Solvent and ions are taken into account implicitly. The force field follows a top-down approach with functional forms derived from the physical intuition of the system’s behaviour and parameters optimized based on structural knowledge. An important feature of the model, allowing a high geometrical accuracy in accounting for base stacking and base pairing, is the possibility of defining base planes thanks to the relatively high resolution of the coarse-graining [40]. In the electronic supplementary material, we briefly present the terms of the model most relevant for the coupling with protonation.

With respect to the original model, the new feature introduced in this work is that the energy of a base pair is evaluated according to the charged status of the base: charged or neutral. For neutral bases, the relative base pairing strengths ε_hb (see equation (S3) in the electronic supplementary material) have been parametrized starting from quantum mechanical calculations [57] and adjusted to correctly predict the folded structures of several small benchmark molecules including hairpins, H-pseudoknots, and double helices with and without non-canonical pairs. For the canonical base pairs A–U and G–C, our procedure based on assuring the stability of benchmark experimental structures [40] gives a ratio of the energies E_AU = 0.8 E_GC which is in perfect agreement with values of statistical potentials extracted from an extensive analysis of the NDB [36].

Initial values for the relative energies of charged base pairs were also obtained from quantum mechanical calculations [16] and adjusted by hand in order to maintain the secondary structure of the calibration system 1LDZ stable at experimental pH and ionic conditions. This procedure has not yet been optimized because of the scarcity of experimental data of charged base pairs, but we are currently investigating other strategies to obtain well-grounded energies for charged base pairs as well.

In our model, 24 neutral base pairs are currently taken into account, corresponding to the most frequent pairs reported in the NDB. In the work presented here, we also include three sets for protonated bases (A⁺–G, C⁺–G, A⁺–C), which are the most commonly occurring charged base pairs. Figure 1 shows the structure of protonated A⁺ and C⁺ as well as the three base pairs. All base pairs included in the model, labelled according to the Leontis–Westhof classification [12], are listed in table 1.

Figure 1.

Structures of neutral and protonated adenine (top) and cytosine (bottom) together with the pK_a values for isolated bases (left). For values of pH less than pK_a, the bases are protonated, while for values above they are in their neutral state. The three most common base pairs formed by these protonated bases are shown on the right (images of pairs from http://www.saha.ac.in/biop/www/db/local/BP/rnabasepair.html [16]). (Online version in colour.)

Table 1.

List of the base pairs included in the model classified according to the neighbouring edges as in the Leontis–Westhof nomenclature [12]. W: Watson–Crick edge, H: Hoogsteen edge, S: sugar edge. Charged base pairs are indicated with the symbol + (protonated A and C) following the edge on which the charge is present (W⁺ indicates a charged Watson–Crick edge).

base pair
A–A	trans WW	trans HH	trans HW
A–C	cis WS	trans HW	cis W+W
A–G	cis WW	trans HS	trans SS	cis W+H
A–U	cis WW	trans HW	cis HW
C–G	cis WW	trans WW	cis W+W
C–U	cis WW
G–G	cis HW	trans HW	trans SS
G–U	cis WW
U–U	cis WW	trans WW	trans WH

4. Coupling dynamics and titration

The idea of coupling coarse-grained dynamics and the FPTS stems from the fact that both HiRE-RNA and this titration scheme are defined at a coarse-grained resolution in the same implicit solvent framework.

The implementation is based on three main assumptions:

• (1)The time for the charge exchange with the solvent (protonation and deprotonation of a base) is much faster than the coarse-grained dynamics, and the exchange can therefore be considered instantaneous.
• (2)To sample equilibrium structures the system requires a relaxation time after a titration event.
• (3)A hydrogen involved in base pairing is shielded from titration, whether the base is neutral or charged.

The first point constitutes the main difference with other CpHMD simulation where the proton exchange is performed gradually. It is based on the fact that the time step in a coarse-grained simulation does not correspond to the atomistic time, but it is longer. The third assumption is drawn from the experimental evidence that neutral bases involved in canonical base pairs exhibit local pK_a values shifted with respect to the isolated pK_a (lower values for A and C, higher values for G and U), and that protonated bases whose extra hydrogen is in a base pair exhibit the opposite behaviour, with pK_a shifts of 2 to 3 units.

In our CpHCGMD scheme, titration occurs at fixed time intervals during the MD run. Prior to titration, the dynamics gives to the titration algorithm (i.e. to FPTS_static) the information about the centre of mass of each nucleotide (instantaneous configuration), and about the total charge carried by each nucleotide. Nucleotides are allowed to attempt titration only if unpaired on the side of the base that can be titrated (the Watson–Crick edges), or if the absolute value of the energy of the base pair involving the titratable edge is below a chosen cutoff δ_bp, i.e. if the base pairing is weak. The choice of this value is discussed in the next section. Nucleotides that are tightly base paired on the Watson–Crick edge will contribute to the titration calculation because of the location of their net charge but are excluded from acquiring or losing a charge.

The titration algorithm returns the average charge of each nucleotide for the specific molecular configuration received from the MD run. This value is expressed in a fraction of a charge and can be interpreted as the probability of finding the nucleotide in the uncharged or charged state. Since in the dynamics bases must have integer charges, following titration, the charge of the bases is updated according to the probability returned by titration (i.e. the average charge of each base). For a fixed structure, the average charge recovered after a statistically significant number of titration events and probabilistic charge assignment will be the same as the charge returned by the FPTS_static alone.

In the subsequent dynamics, before a new titration event, the new charge of the base is taken into account by considering the electrostatic interaction given by the presence/absence of a charge on the base and by selecting the appropriate set of base pairing parameters, for the charged or neutral base. On the one hand, the Debye–Hückel potential gives rise to new attractive or repulsive interactions with other charged bases and with the negatively charged phosphate groups. On the other hand, if a base becomes charged, the energetic minima corresponding to base pair geometries for the Watson–Crick edge of neutral base are no longer present but new minima, with different distance and angles between the bases ($r_0,{\alpha }_{1_0},{\alpha }_{2_0}$ in equation (S3) in the electronic supplementary material), appear for adenines and cytosines (see table 1 and figure 2). The opposite occurs if the base goes from charged to neutral. The strength of the interaction (ε_hb in equation (S2) in the electronic supplementary material) is also changed for the charged base pairs with respect to the strength of neutral base pairs. These changes in base pairing properties are key for the correct representation of the behaviour of RNA and constitute the main new feature of the method presented in this work.

Figure 2.

Two alternative pairings for bases A and G: the canonical WC base pair between G (red) and A (orange) and the non-canonical base pair between the Hoogsteen edge of G and the protonated WC edge of A⁺ (blue). Two separate sets of variables, indicated here by the coloured dashed lines and arcs, are used in the potential as references for the position of the base pairing energy minimum. For the neutral base A, the orange set is present and the blue set is absent, while for the charged base A⁺ the opposite occurs. (Online version in colour.)

To give an example, let us assume that a neutral base A and a base G are paired through their Watson–Crick edges. At a first titration attempt, their base pairing is strong and both bases will not be allowed to change their charged state. The titration will return a charge state of 0 for both and the probability of the bases to be assigned a different charge will be zero as well. At a second titration attempt, the structure has changed because of the dynamics and the two bases are either unbound or loosely bound. They will be allowed to titrate and will be assigned an average charge by the titration algorithm, according to the assigned pH and salt conditions. The base A is assigned a charge of z, with 0 ≤ z ≤ 1, similarly for G. A random number k between 0 and 1 is drawn. If k ≥ z the charge assigned to A will be 0, if k ≤ z the charge will be +1 and the base will be protonated. Let us assume the second case occurs. The base A has now become A⁺ and will be able to form the base pairs A⁺–G, A⁺–C, but none of the other pairs involving its Watson–Crick side. It will still be able to form base pairs occurring on its Hoogsteeen or sugar edge. If an A⁺–G or an A⁺–C pair is formed, at the next titration event, the stability of this pair will be evaluated in the same way as for neutral bases: if the pair is strong, the base will not be allowed to change charged state (because the extra hydrogen is involved in the pair with the other base) and the base will remain A⁺ in the following dynamics; if the pair is weak, the base will be allowed to protonate and it could lose the extra proton converting back to A.

For each trajectory, we recorded the assigned charge of 0 or 1 for each base at every titration event, whose average gives the partial charge associated with the whole population sampled by the dynamics for each base. To obtain local pK_a values, we fit data of partial charge versus pH with a sinusoidal function using the Hill equation and extract the pH for which the function has a value of 0.5. The spread of the points at mid-height gives us an estimate of the error.

4.1. Simulation protocol and analysis

The current CpHCGMD protocol has been chosen having explored a wide spectrum of possibilities in terms of how often to exchange between dynamics and titration, what cutoff to use to assess whether a base pair is sufficiently stable to shield its bases from titration, the balance between convergence of the FPTS and speed, and the length of the simulation. The criterion to make these choices was that of being able to reproduce at best the pK_a values reported in the literature for the adenines of a well characterized test system.

We now discuss the values of the parameters chosen through calibration and then used for the simulation of all systems.

• (i)5000 steps of MD are taken between each titration event as relaxation time. With this interval, the system has enough time to break or form base pairs and to readjust the overall conformation accordingly. Results are not very sensitive to this parameter that can be changed up to 20–30% without much effect on the molecule’s behaviour.
• (ii)10 000 steps are taken for the Monte Carlo titration algorithm. As shown in previous studies [24], this is a value for which the convergence of FPTS_static has been reached up to a few per cent difference with longer simulation times. It is therefore a good compromise between speed and accuracy. Results of titration are stable if this value is lowered by 5–10%, and are substantially unchanged if this value is increased.
• (iii)Simulations times are between 200 and 300 ns. In this range, the values of 〈z〉 for each base have converged and the system has explored various conformations. This has been verified computing 〈z〉 for the first half and the second half of the simulation separately and observing that the value remains unchanged. However, it is a well known problem that conversion times between alternative conformations can be long even with a coarse-grained model where simple MD can be trapped in local minima and not evolve further (indeed complete folding simulations are typically done using extensively enhanced sampling techniques [52,53,58]). To overcome the biases of a particular structure being trapped in a local minimum, we run several independent replicas to estimate titration curves. In this work, we have chosen to consider five replicas. A higher number of replicas would increase the precision in the calculation of pK_a values, since our main source of error is the dispersion of 〈z〉 from one replica to another.
• (iv)The energetic cutoff δ_bp for the base pair to be shielded from titration is taken as about 1/3 of the maximal base pairing energy (that of the canonical Watson–Crick GC pair). This term is intrinsically related to the temperature of the simulations, since what matters for the stability of the pair are the fluctuations of the interactions. This is the most sensitive parameter of our method. Shifting the cutoff of a few per cent can significantly alter the results. The value chosen was selected in the calibration procedure where we ran simulations for one specific system (1LDZ) at various values of δ_bp and retained the one giving the best agreement between computed and experimental pK_a values.

For all systems under investigation, we performed CpHCGMD simulations starting from one of the experimental structures with fixed pH ranging from 1.0 to 10.0 at 0.2 intervals, for a total of 230 independent trajectories for each system, considering the five replicas. Each full trajectory took about 48 h computing time on a single Intel core (Intel X7560, Bull Curie supercomputer in TGCC) for molecules ranging in size between 20 and 40 nucleotides. The computing effort to obtain a complete titration curve, as those shown in the Results section, was therefore of 11k cpu hours (about 2 days of computation). This time, even though three orders of magnitude larger than titration calculations done with the static FPTS, is still two to three orders of magnitude smaller than computing times of methods based on atomistic simulations. For comparison, an atomisic simulation with constant charges (i.e. not coupled with a titration scheme) of 200 ns on the same systems considered here requires 10k cpu hours. Therefore, the time necessary to run one constant pH all-atom molecular dynamics simulation, which takes longer than a regular atomistic MD, is comparable to the one used to compute CpHCGMD simulations for a 100 different pH values. Hence our method is at least 100 times faster than all-atom molecular dynamics approach.

5. Systems

In our work, we consider three RNA molecules, whose experimental secondary structures are reported in figure 3.

Figure 3.

Base pairing of the three systems extracted from the experimental structures. Solid lines represent canonical pairs, dashed lines represent non-canonical pairs. The green lines for the stem-loop branch-point helix (17RA) highlight that two alternative pairs are proposed by the experiment. (Online version in colour.)

The reference system used for calibration of the method is the lead-dependent ribozyme (LDZ) that has been thoroughly characterized experimentally via NMR.

The structure of the LDZ (PDB ID: 1LDZ) consists of a 30-nucleotide hairpin including a bulge and one A⁺–C pair for A25 [17]. Experimental pK_a values were determined for all adenines at a salt concentration of 100 mM and room temperature. The system exhibits a pK_a shift toward lower values for bases paired through canonical Watson–Crick pairs, values close to those of isolated nucleotides for unpaired bases in the apical loop, and a significantly enhanced value for the charged base pair. Given the abundance of experimental information, this RNA system has become the reference for titration computations, with computed values published for PB solvers, constant-pH MD methods, and the FPTS_static [22,24,25].

Two other systems were then studied using the method without further calibration: the stem-loop branch-point helix, the domain of the hairpin ribozyme and the adv5.

The stem-loop branch-point helix (BPH) is a 21-nucleotide structure determined by NMR (PDB ID: 17RA) [18]. One A base in the central portion of the stem is unpaired. Depending on the NMR model this is either A6 or A7, while the other base is bound to U16 via a canonical pairing. At 10 mM salt concentration, pK_a values are below 5.0 for three of the five A bases, while A7 exhibits a significant shift toward higher values (pK_a(A7) = 6.1) and A13 a moderate shift toward higher values (pK_a(A13) = 5.5). None of the structures deposited in the NDB suggests formation of pairs involving protonated bases.

The B domain of the hairpin ribozyme consists of 38 nucleotides, including an adenine-rich internal loop composed of 16 nucleotides with 8 adenines involved in non-canonical pairs (PDB ID: 1B36) [19]. At 50 mM salt concentration, all of the adenines of the internal loop exhibit a raise in pK_a values ranging from 4.8 to 5.8 suggesting the formation of stabilizing interactions in the loop that could include additional stacking and hydrogen bonding among all bases in the loop.

6. Results

The main result of our work is the proof of principle that the coupled CGMD/FPTS can lead to good results for the protonation mechanism, provided that a reasonable choice for the energies of protonated base pairs with respect to neutral base pairs is made and a sensible protocol is selected, as detailed above. This was achieved by working on a calibration system by manually adjusting the different simulation parameters to obtain a good agreement between calculated and experimental pK_a values.

After the calibration procedure, we applied the coupled dynamic/titration to two other systems, without further changes in the simulation parameters and protocols, and compared pK_a values obtained with the coupled dynamics to results obtained from applying alone the FPTS_static and with published results obtained by other CpH simulation methods. We point out that, once the calibration is performed, in the framework of our study, there is no free parameter that can be fitted to obtain specific pK_a values. The good agreement between computed and experimental data therefore demonstrates that our scheme is capable of capturing the basic physics of the process.

6.1. Computed pK_a values

In the electronic supplementary material, tables summarize computed pK_a values from CGMD/FPTS and establish a comparison with experimental values and with values obtained by other methods, when available. The comparison between experimental pK_a values and those computed through CGMD/FPTS and through FPTS_static is reported in figure 4.

Figure 4.

Comparison of the computed pK_a values from the FPTS_static and CGMD/FPTS with the experimental values. Horizontal error bars correspond to experimental uncertainties, while vertical error bars indicate computational uncertainties. The shaded area shows the values for which the experimental value is only known to be below a certain pK_a. Dashed lines represent linear regressions of the simulation data. The values of the slope of the regression (s) and of the Pearson correlation coefficient (rc) and the p-value of the correlation (pv) are reported for each kind of simulation. For 17RA alone we do not compute the correlation because there are only two points for which experimental values are known precisely. We do, however, include all systems in the overall correlation (bottom right plot). (Online version in colour.)

For 1LDZ, through calibration of the relative base pairing energies of charged and uncharged bases and of the energetic cutoff shielding a formed base pair, we were able to achieve results comparable to the best calculations from the far more expensive (in cpu times) PB method and without the need to tune any parameter to best reproduce the experimental pK_a shifts.

For non-calibrated systems, pK_a values obtained through the coupled dynamic/titration scheme represent a clear improvement over results obtained using static evaluations, exhibiting a higher correlation with experiments and a greater sensitivity in discriminating bases at low effective pK_a values and those at high values. FPTS is particularly sensitive to salt concentration and global conformation and we have shown in our previous work that it is indeed able to find the correct range of pK_a values for a given system. However, static calculations were lacking sensitivity to the variations in a given system from one base to another and were unable to account for the details of the local environment. For all systems, dynamic calculations are now able to make that distinction, exhibiting a wide range of computed values. This can be observed for 1B36 in particular. Values from the static calculations are already a good estimate for experimental measures, but the range of values for the different bases is limited to 1 pH unit. The dynamic calculation is more sensitive to the local environment and exhibits a wider range of pK_a values, correlating better with the trend of the experiment.

6.2. Structural analysis

The coarse-grained dynamics are able to sample large conformational changes and allow one to investigate possible different structural populations. We have analysed the trajectories at various pH values to understand what were the main structural features and how these correlate with the computed pK_a values.

6.2.1. Lead-dependent ribozyme

The experimental structure of 1LDZ has three adenines bound through canonical base pairings: A4, A8 and A12. For A4 and A12, these pairs are present in most of the structures at physiological pH and start opening only for low pH, protecting the bases from protonation. Indeed, the experimental local pK_a values for these two bases are estimated below 3, and our computed values are in agreement. In our simulations, the canonical pairing of A8 is less stable allowing adenine protonation more easily at low pH values. If protonated, this base can form an A⁺–G base, stabilizing the protonation and justifying the medium elevation in effective pK_a that is found experimentally and that we also predict through simulations. In the experimental structure, A25 is protonated and forms an A⁺–C base pair. In our simulations, this pairing is present up to pH values close to physiological conditions. The pK_a values for A16, A17 and A18 computed from the CGMD/FPTS methods show an improvement with respect to the calculations with a single rigid conformation, but they are still overestimated with respect to experiment, in particular for A18. These three bases are in the apical loop of the stem, unbound and exposed to the solvent. In the simulations, for all structures that are still folded into a hairpin, the three bases remain exposed to the solvent and do not form base pairs, even though their positions can be stabilized by stacking. Since no base pairing is involved, the discrepancy between experimental and computed values had to be found outside the dynamic coupling scheme. A possible cause is the coarse-graining of the FPTS that reduces the nucleotides to a single bead and does not take into account possible rotamers therefore not distinguishing between a base pointing inward inside the loop and one pointing outward, isolated into the solvent.

6.2.2. Domain B of the hairpin ribozyme

The experimental structure of 1B36 is a hairpin including a large internal loop. One adenine (A36) is part of the terminal stem, while all others are found in the internal loop. In our dynamic calculations, low pK_a value is predicted for A36 which forms a canonical Watson–Crick pair in the helical stem, while elevated values are predicted for all the adenines in the internal loop, reproducing the experimental trend. For folded structures, A36 remains stable even at low pH values.

NMR studies revealed that the internal loop is structured with several non-canonical pairs. In the dynamic simulations at physiological pH, we observe a rapidly fluctuating charge for A6 and the base is mainly unbound. A8 and A9 can both be found bound to U31. When A8 is bound to U31, A9 is bound to U30 through two canonical pairs, but this configuration can exchange with a second one where A9 is bound to U31 and A8 is either free or bound to A29. A10 and A11 are predominantly bound to U28 and U26 respectively, but the pairs still allow for some charge fluctuations, indicating a relatively low stability. A27 and A29 can both temporarily bind to C11 through an A⁺–C pair; however, A29 can also be found forming a triplet with A10–U28 pair. In this second case, the base uses its Watson–Crick edge for the triplet formation. A32 can pair to G7 through an A⁺–G pair.

These observations provide us a structural justification to the computed pK_a values. A8, A9, A10 and A12 have a rather low effective pK_a due to the shielding given by the possible canonical pairs. These values are not as low as for a base whose pairing is stable, but they are lower than for free bases, such as A6, and lower than for bases forming pairs while protonated. A27 and A32 have a raised pK_a possibly due to their ability to form pairs while protonated. The only clear discrepancy between simulations and experiment is observed for A29, probably due to the possible triplet formation.

As the pH is lowered, the internal loop loses any structure and becomes an open bulge with only the Watson–Crick stems formed. For even lower pH, the structure opens with the melting of the terminal stem including A36.

The general picture that emerges for this system is that the internal loop is only marginally stable with interactions that can break and interchange. In experiments, these alternative conformations might coexist or one of them may be stabilized by the external conditions that we cannot take into account in the coarse-grained model, such as the presence of explicit ions, that would rigidify some structures making them sufficiently stable to be observed by NMR experiments.

6.2.3. Branch-point helix

The structure of 17RA is a hairpin including a single base bulge. Two adenines are part of the helical stem, two are part of the loop and one is the unpaired base of the bulge. In the structures proposed by NMR, the identity of the base in the bulge can change between adenine A6 and adenine A7. In some structures, A6 is bound to U16 through canonical base pairing and A7 is unpaired; in other structures, the opposite occurs. In our simulations, both conformations are explored, with a preference for the second. The computed pK_a values are consistent with the presence of one stable canonical base pair formed by A17 and U5 and with another canonical pair formed by A7 and U16. The average pK_a value and the rapidly fluctuating charge dynamic for A6 suggest that this base is mainly unpaired; however partial charge values 〈z〉 exhibit a large variability, suggesting the presence of two families of structures, one with A6 canonically paired, and one with A6 unpaired (see figure 5). These results capture the general feature of two coexisting alternative structures; however, the relative equilibrium appears to be shifted in the simulations with respect to the experiment where pK_a values were measured, which would suggest a paired A6 and unpaired A7. This discrepancy could be due to several reasons including the fact that in the experiment, one of the conformations has been stabilized by environmental conditions that we cannot take into account.

Figure 5.

Titration curves for the weakly paired adenine A6 of 17RA (left) and for the strongly Watson–Crick paired base A17 (right). The effective pK_a is read off the intersection between the fitting curve and the 〈z〉 = 0.5 axes (dashed line). The spread of the points at 〈z〉 = 0.5 gives the error in our estimates. (Online version in colour.)

In the simulations, A10 and A13 remain unbound in the loop for all folded structures. As in the case of 1LDZ, the pK_a values of the bases in the loop are overestimated. In particular for A10, the model probably fails to capture some of the interactions that stabilize the position of the base in the loop, that typically involves some hydrogen bond network among the nucleotides in the loop.

Overall, for this system, we can highlight the existence of two competing structures, which are sensitive to the pH not by forming stabilizing pairs with the charged bases, like for the previous systems, but by the stabilization of the canonical base pairs and the exposure to the solvent of a titratable group.

6.3. pH melting curves

Dynamical simulations at different pH values allow to compute unfolding curves for the whole molecule. In figure 6, we report the percentage of folded structures present in the simulations as a function of pH. Using a crude definition, we have identified a structure as folded if its root-mean-square deviation (RMSD) with respect to the experimental structure is less than 7 Å. Above this value, the structure starts adopting an open conformation. For systems having a simple two-state dynamics such as 1LDZ and 17RA, we obtain a sigmoidal curve allowing to establish at what pH one would expect the molecule to melt. For 1B36 the multitude of conformations adopted by the inner loop does not allow to use a simple RMSD definition to identify folded and unfolded structure. Melting curves are consistent with the measurement of pK_a values as low as 3 for 1LDZ, with a significant fraction of the population still folded, and with values above 5 for 17RA, since the fraction of population still folded drops rapidly below this value.

Figure 6.

Fraction of folded structures as a function of pH exhibiting the expected sigmoidal behaviour. (Online version in colour.)

7. Conclusion

In this work, we have presented the proof of principle of a simulation scheme for RNA molecules accounting for different pH values based on coupling two coarse-grained models: HiRE-RNA to simulate the dynamical behaviour, with a representation of a nucleotide by six or seven beads, and FPTS to compute the electrostatic properties of a given structure of the molecule, with the representation of a nucleotide by one bead. By incorporating the titration calculation into the coarse-grained MD, we developed a new CpHCGMD scheme able to study the conformational properties of a given system at different pH values as well as to compute properties linked to the electrostatics of the configuration such as local pK_a values for different bases. We have focused on adenine protonation as it is the most commonly studied nucleobase for protonation effects. We were able to show that, through an initial reasonable calibration of the coarse-grained model parameters and set-up of the coupling between dynamics and protonation, we obtain pK_a values in good agreement with experimental data with limited computational effort.

Having at our disposal the information from all trajectories at all sampled pH values, we can investigate the structural behaviour of the molecule and give a microscopic justification of local pK_a values in terms of how the base is paired or unpaired and thus exposed to the solvent. This ability is what makes our calculations particularly interesting as they can complement ensemble experiments where pK_a values are measured, but where the specific behaviour of a given base might be hard to determine because of the coexistence of several conformers, as is often the case for RNA molecules. Thanks to the gain in speed from the coarse-grained representation, our simulation can investigate large conformational changes and rearrangements of secondary structures. This is infeasible with current simulation methods based on an atomistic representation due to the prohibitive computational costs.

We have shown that through an accurate calibration of the energetics of the different base pairs, neutral or charged, and of the coupling with protonation, we are able to recover an excellent agreement with experiment. Results for non-calibrated systems are also good but show a margin for improvement, which indicates that a more complete parametrization is needed, focusing on a larger set of benchmark systems. Balancing the relative energies of the different base pairs is particularly challenging because of the lack of experimental data on the relative stability of all possible base pairs. To overcome this problem, we are investigating the possibility of obtaining these data from quantum mechanical calculations of base pairs in solvation. A better assessment of relative base pairing energies will also allow us to expand the number of different base pairs included in the model and better address the titration of cytosine, guanine and uracil as well. Another source of error comes from the cutoff used to determine the strength of the base pair relative to protonation. Despite the generally good results, we are investigating whether a smoother transition, coupled to a probabilistic assignment dependent on the temperature, would improve the accuracy.

The coupling CGMD/FPTS is promising for the study of RNA structure and dynamics. As was shown in other studies using coarse-grained modelling [41,59,60], complementing coarse-grained dynamics with atomistic studies is a successful strategy to address the complexity of RNA folding. A new direction of multi-scale modelling will now be to complement coarse-grained titration with more detailed atomistic models in which the coarse-grained dynamics can provide families of structures with sensible physical properties at a given pH for further investigation.

Data accessibility

Simulations trajectories and titration data are available for download: doi:10.5281/zenodo.2562864.

Authors' contributions

F.B. and S.P. equally contributed envisioning the work and carrying out its implementation for RNA, performing all the tests related to the coupling CGMD-HiRE-RNA/FPTS and writing the manuscript. E.F. refined the parametrization of HiRE-RNA to re-equilibrate the coarse-grained model after the introduction of charge bases and titration scheme.

Competing interests

We declare we have no competing interests.

Funding

F.B. and S.P. were supported by the IDEX Brasil of Sorbonne Paris Cite University and University of São Paulo for funding the international collaboration. F.B. was also supported by the Fundação de Amparo á Pesquisa do Estado de São Paulo (Fapesp 2015/16116-3), the Conselho Nacional de Desenvolvimento Cientifico e Tecnológico (CNPq), and the USP through the NAP-CatSinQ (Research Core in Catalysis and Chemical Synthesis).