An Implicit Divalent Counterion Force Field for RNA Molecular Dynamics

Transcription

1 An Implicit Divalent Counterion Force Field for RNA Molecular Dynamics Paul S. Henke 1 1, 2, a and Chi H. Mak 1 Department of Chemistry and 2 Center of Applied Mathematical Sciences, University of Southern California, Los Angeles, California 90089, USA How to properly account for polyvalent counterions in a molecular dynamics simulation of polyelectrolytes such as nucleic acids remains an open question. Not only do counterions such as Mg 2+ screen electrostatic interactions, they also produce attractive intrachain interactions that stabilize secondary and tertiary structures. Here, we show how a simple force field derived from a recently reported implicit counterion model can be integrated into a molecular dynamics simulation for RNAs to realistically reproduce key structural details of both single-stranded and base-paired RNA constructs. This divalent counterion model is computationally efficient. It works with existing atomistic force fields, or coarse-grained models may be tuned to work with it. We provide optimized parameters for a coarsegrained RNA model that takes advantage of this new counterion force field. Using the new model we illustrate how the structural flexibility of RNA two-way junctions is modified under different salt conditions. a To whom correspondence may be addressed: Chi H. Mak, Department of Chemistry, University of Southern California, Los Angeles, CA Tel: , Fax: , cmak@usc.edu 1

2 I. INTRODUCTION RNA structure prediction has become an increasingly important problem as experimental results continue to reveal that RNA seems to play a much larger biochemical role than once thought. Functional, non-coding RNA molecules like ribozymes and riboswitches serve as important regulators of gene expression in vivo, 1 3 while studies on micro and small interfering RNAs have led to a deeper understanding of and possible treatments for genetic diseases such as cancer. 4,5 Other, less understood non-coding RNAs have also been shown to have significant influence over epigenetics and embryonic development. 6,7 To carry out this myriad of cellular roles, functional RNAs must fold into specific, compact tertiary structures, and improved RNA structure prediction is necessary to better understand the diversity of RNA functions. Molecular dynamics (MD) simulation of RNA has yielded many promising results toward this end, and is currently capable of generating physically accurate structures for small RNAs. 8,9 But while there is much promise in simulation as a tool to better understand RNA dynamics and ultimately predict full RNA tertiary structures, atomistic MD simulations are limited in the amount of real time that can be simulated for even small biomolecules. 10,11 These limitations have led to a desire for more computationally efficient methodologies and increased scientific interest in both implicit solute modeling and coarse-grained molecular modeling of nucleic acids Coarse-grained (CG) models of RNA and DNA abstract the important physical interactions from a fully atomistic model, capable of reducing the number of explicit polymer atoms needed in simulation by an order of magnitude or more. Similarly, implicit salt models allow the electrostatic interactions between the nucleic acid and solute ions to be treated globally, which eliminates the need for time-intensive ion equilibration, and further reduces the number of particles to be simulated. 22,23 RNA, a negatively charged polyelectrolyte, relies on positively charged counterions to fold into a compact, functional structure. Mean-field theories of electrolytes such as linearized Poisson-Boltzmann theory accurately predict the effects of monovalent salts on polyelectrolytes, but generally cannot account for the larger electrostatic correlations produced by higher valence ions. 24,25 Importantly, divalent ions like Mg 2+ have been found to have an especially powerful stabilizing effect on folded RNA structure Recent CG models have begun to incorporate divalent ions. Notably, Denesyuk et al. have found success including explicit Mg 2+ and Ca 2+ ions in a Gō-like 32 CG model that preserves native tertiary interactions. 33 Similarly, Hayes et al. have developed a modified Manning counterion-condensation model that is able to accurately predict the effects of excess Mg 2+ on RNAs in the presence of KCl and demonstrated these effects in a CG structure-based model

3 However, most RNA CG models proposed to date have either included only implicit monovalent ions via mean-field theories or have omitted the electrostatic effects of salts entirely. While these models have proven useful for studying RNA structure and thermodynamics, 17,37 large-scale conformational shifts are often reliant on the attractive forces arising from the presence of higher valence cations, and the absence of these interactions is detrimental to a full understanding of RNA conformational dynamics under physiological conditions. To study the effects of divalent counterions, we have previously proposed a tight-binding counterion model that can account for highly charged counterions implicitly 38 and a corresponding numerical approach for implementing this salt model in a Monte Carlo simulation. 39 In this paper, we present a divalent counterion force field for molecular dynamics (MD) simulations which has been built upon our implicit salt model. We also augment our previous algorithm 39 to provide a nearly 100-fold savings in computational time needed for calculating the ensemble forces arising from the RNA-counterion interaction, allowing for a computationally feasible integration of our model into a dynamics simulation. Applying this force field to atomistic MD simulations, we demonstrate that the effects of a large range of salt conditions can be reproduced in simulation by correlating to experimental results. Additionally, we parameterize a CG RNA model so that it is able to maintain stabilized secondary structural features under the influence of the attractive counterion-mediated free energy. We show that salt effects can be tuned so that it correctly mirrors experimentally determined dynamics of single-stranded RNA (ssrna) chains. We study the effects of divalent salt on the stabilization of RNA tertiary structure by simulating several RNA two-way junctions that include both ssrna and dsrna components, and we show that divalent ions play an important role in shifting the RNA conformational ensemble toward a more generally folded state. II. IMPLICIT DIVALENT COUNTERION FORCE FIELD The implicit counterion model has been presented in detail previously. 38 We will briefly review the most important aspects of it to motivate the discussions below. This model is used to compute the counterion mediated free energy, F CI, and its associated ensemble-averaged forces, which is the basis of the implicit counterion force field. The counterion model is illustrated in Fig. 1. Given an RNA conformation, the RNA molecule is solvated within a cloud of positive and negative ions. The ions are treated within a grand canonical ensemble. The full electrostatic free energy of this system includes U R, the self-interaction potential of 3

4 the RNA with coordinates {r}; U X, the self-interaction potential of the ions with coordinates {x}; and U RX, the interaction potential of the ions with the RNA: βf CI (r 1, r 2, ) βu R (r 1, r 2, ) ln Tr X e β(u X+U RX μ + N + μ N ) (1) where Tr X is the grand canonical trace over all ion coordinates. μ ± and N ± are the constant chemical potential and number of ions for both the positive and negative ions, respectively. β = (k B T) 1 for temperature T and Boltzmann constant k B. Figure 1 A given RNA molecule with a fixed conformation is coupled with a constant-volume ion reservoir with chemical potentials μ ± for positive and negative ions. The RNA is considered to have charged sites through which it interacts most strongly with the ions in solution. These interaction sites are shown here as open circles. The RNA is negatively charged, and so positive ions of charge z +, depicted as small filled circles, surround the RNA and may associate with the ion interaction sites. The positive ions in solution are most likely to associate with the RNA at regions where the negative charge is greatest. These electronegative regions are termed ion occupation sites. Any such site is defined to be occupied if a divalent ion is associated with it. For an RNA with N total ion occupation sites, any given site i is described by its occupation number: σ i = { 1, ion site i is occupied, 0, otherwise. (2) The negative charges on the RNA are concentrated at each phosphodiester bridge connecting the sequential nucleosides. Each bridge holds a single bound phosphate with a full e charge. Due to their high electronegativity, we have assumed that the ion-association sites are located at the phosphates: {r} = {r P }. 4

5 Using the ion occupation representation defined by Eq. (2) in tandem with a mean field treatment of the surrounding non-associated ion cloud in the solution, the free energy of the polymer with the counterions as defined in Eq. (1) is then reduced to: βf CI = ln(tr σ e β(h σ μ + N σ ) ) (3) where Tr σ is the trace over all possible sets of ion occupation numbers and H σ is the effective Hamiltonian, which depends on the total number of associated ions, N σ, and the spatial conformation of the RNA, R: This effective Hamiltonian includes an electrostatic two-body interaction term between occupied sites, χ ij, a one-body term arising from the interaction of occupied site with the surrounding field, ξ i, and a self-energy term, h σ, accounting for the interaction of non-occupied sites. Use of H σ to evaluate the ensemble average for the total occupation number, N σ, reveals that the number of ions associated with the RNA at physiological salt concentrations in the range from 0.05 mm to 5 mm MgCl 2 is effectively constant, such that the total ion charge neutralizes the backbone charge of the RNA. That is, N σ N/z + for the relevant salt concentrations. As a simplification, we can therefore consider N σ to be constant, and treat the associated ions within an effective canonical ensemble and strictly as a function of the RNA conformation, R. Within this effective canonical ensemble, all ion configurations are charge neutral, with q i = 0 i (or +1 or -1 if the number of charge sites is odd), and the set of all such ion configurations is denoted S. For divalent ions (z + = +2) the effective Hamiltonian reduces further to: H σ = 1 2 χ ij(n σ, R)σ i σ j + ξ i (N σ, R)σ i + h σ (N σ, R) (4) ij i e2 1 H 2+ = (4πε 0 ) 2ε q iq j ij r ij (5) where ε 0 is the vacuum permittivity, ε is the dielectric constant and r ij is the distance between ion sites i and j. Each site is either occupied by a divalent ion, which gives it a net +1 charge, or it is unoccupied, which gives it a 1 charge. Therehfore, the ion association site charges in H 2+ are restricted to q i = {+1, 1}. To evaluate F CI in practice, we have extended the previously described boundary search alrogithm. 39 Boundary search takes a static RNA conformation as input, and defines a discrete-space mapping of the canonical ion-configuration space, S, about that conformation. Once the ion 5

6 configuration space has been constructed, boundary search applies a uniform-cost search method to efficiently find the energy states most relevant to the ensemble. For a well-formed RNA, the ground state and other low-lying energy states dominate the canonical ensemble and can be used to generate a very good approximation for F CI. In our previous study, boundary search was incorporated into an atomistic Monte Carlo simulation of RNA with constrained base pairs and shown to work well for compact RNA structures up to 400nt in length. However, it has not been integrated into a dynamics simulation which requires the calculation of forces. Because boundary search seeks out a full ensemble of relevant ion configuration states, calculating the forces arising from the free energy is in principle straightforward. Generally, these forces are given by the gradients of the counterion ion free energy: for charge-site coordinates {r P }. For a given ion configuration s S, the force on site i is f si = E s, where E s is the energy for configuration s given by Eq. (5). Evaluation of Eq. (6) shows that the ensemble average force on site i is given by: F = F CI r P (6) f i = f sie βe s s e βe s s (7) The direct application of Eq. (7) to compute forces turns out to be somewhat problematic in practice. While the original boundary search algorithm can be directly incorporated into a molecular dynamics simulation according to Eq. (7), there are significant computational efficiency issues that arise in doing so. In particular, the full boundary search algorithm runs in O(N 3 ) time for number of nucleotides N. In practice, this run time makes the calculation of F CI and F a major computational bottleneck. We work around this bottleneck by modifying boundary search so that it does not run a complete search for each dynamic time step. This modification can be justified when considering the following two points. First, the small time steps of a dynamical simulation necessarily prohibit large motions of particles between time steps, and so the relevant ion configurations are slow to change over the course of a simulation. Using our coarse-grained model for RNA as described in Section IV.A, we measured this behavior and found that the average duration within which a particular ion configuration ground state persisted was 750 time steps for an ssrna loop 20 nt in length, with a persistence length of 1.5 nm. Second, boundary search does a good job of finding many different low-lying energy configurations 6

7 beyond just the ground state. Unlike a stochastic search such as Monte Carlo which may only locate a subset of relevant states, boundary search is guaranteed to find a new state on each iteration. We take advantage of this feature by keeping track of the low energy ensemble once the search has completed, and then utilizing that ensemble to generate the ensemble forces for several MD iterations before running boundary search again. The number of charge configurations to be identified in the search is defined heuristically by a parameter max_nodes. max_nodes determines how many search iterations are performed and therefore how many ion configurations are evaluated. Our previous report on boundary search found that a value of max_nodes = 10 was able to account for counterion-induced stability for RNA structures up to 400nt in length. 39 Full pseudocode for the modified boundary search is shown in the following: Let P be a persistent set of explored energy nodes Let R be the set of nodes with connected edges {u,v}, where u P and v P If a full search is to be performed Initialize P to be empty Initialize R with node s S While P max_nodes Select the node u R that has the lowest total energy value P = P {u} Foreach node v P; R with connected edge {u,v} R = R {v} End foreach Endwhile Else Recalculate energy values in P for current charge site conformation Use all ion configurations in P to calculate F CI and F In practice, a full refreshed boundary search is performed periodically (every 200 time steps in a coarse-grained simulation, and 0.05 ps in an atomistic simulation), and we use a value of max_nodes = 20. These values were chosen to be conservative considering the bounds described above. Use of this modified boundary search provides an approximate speedup proportional to the number of searches skipped. In our case where we use a time step 1 fs for atomistic simulation, we found an order of 100-fold savings in computation time spent calculating F CI and the associated forces F when compared to using a full search each time step. With this modification, boundary search can be incorporated into MD simulations without seriously impacting the run time. Note that F CI, when calculated this way, provides free energy and force estimates for divalent ion conditions under which the polymer with its associated counterions is approximately neutral. Under such conditions, the chemical potential of the ionic solution is high enough to saturate approximately 7

8 half the ion association sites on the RNA. If Mg 2+ ions are forced onto an RNA beyond saturation, the overall charge on the chain will turn positive, producing an overcharged state that would be highly thermodynamically unfavorable. On the other hand, in some cases we are interested in how the RNA behaves under ionic concentrations that are less than saturated, and in these cases where we have an undercharged chain, F CI must be replaced by a simple partial-charge electrostatic potential U PC, which is defined as: U PC = (4πε 0) 1 2 (q P )2 r phosphate pairs (8) The partial charge q P may be considered to be an adjustable parameter, tuned to correlate with known experimental results under different undercharging ionic conditions. When q P tends to zero, the limiting effect is similar to saturating the system with monovalent ions. For divalent counterions, we expect that at a critical experimental ion concentration the electrostatics will be fully screened without the counterions producing any attractive or repulsive electrostatic interactions. This critical ion concentration would therefore correspond to q P = 0 in U PC and simultaneously ε = in F CI. Above this critical concentration, F CI should be used. Below this critical concentration, U PC should be used. (Note that because there is no screening term applied to F CI, U PC is also not screened so that it may be compared more directly against F CI.) While this implicit counterion model has been developed to describe the free energy due to divalent ions such as Mg 2+, the effects of monovalent ions such as Na + are experimentally similar to Mg 2+ but at concentrations roughly 100- to 1000-fold lower. For example, structural characteristics such as radius of gyration, small-angle scattering intensities and persistence lengths observed for ssdnas in solutions with NaCl concentrations of ~ 0.1 to 1 M are similar to those with ~ 1 mm MgCl 2. 40,41 The dielectric constant ε is used an adjustable parameter in F CI and it can be calibrated against experimental Mg 2+ or Na + concentrations to reproduce experimental structural properties. On the other hand, below the critical counterion concentration, the electrostatic energy for undercharged chains U PC in Eq. (9) should be used, and the partial charge q P may be considered as the adjustable parameter, which can similarly be calibrated against experiments. III. ATOMISTIC SIMULATIONS OF SINGLE-STRANDED RNA WITH COUNTERIONS When included in MD simulations, the counterion free energy F CI is expeceted to produce effective attractive intrachain interactions. We tested our divalent counterion force field by 8

9 incorporating it into full atomistic MD simulation of an ssrna with implicit ions but no explicit solvent. Using the TINKER molecular modeling package 42 and Amber ff94 force field parameters, 43 we ran simulations of ssrnas of lengths N = 10, 20, 40, 60 nt. ssrna chains were generated by equilibrating an 88 nt RNA (PDB ID: 4LVV) 44 in MD for 1 ns under a purely steric Weeks-Chandler-Andersen (WCA) replacement for the Lennard-Jones (LJ) potentials, with chains of the desired lengths extracted from the 88 nt sequence as initial conformations in a number of parallel simulations. We have used WCA instead of LJ because in the absence of explicit solvent molecules, employing unmodified LJ would have produced unrealistically large intrachain attractions, causing overcompaction of the chain. A previous study 45 has also suggested that employing a repulsive-only modification to LJ in a ssdna MD simulation produced radius of gyration in agreement with experimentally observed scaling behaviors. Twelve simulations of 10 ns each were run at T = 310 K within the canonical ensemble for each value of N, with forces from the gradients of F CI. The runs were repeated using different values of ε = 8, 10, 12 and 14 as well as for a neutral system (ε = ). The time required to compute forces due to F CI is a function of N, but we found that inclusion of F CI resulted in only a 6% slowdown per MD time step for the largest simulations we ran, which was N = 60. In all the simulations, the only non-bonded interactions included were WCA and F CI, and in this way any observed effects can be conclusively attributed to F CI, independent of effects arising from partial charges, van der Waals or base-base interactions. We analyzed the ssrna simulation trajectories by calculating the scaling exponent, ν, for the radius of gyration, the persistence length. We also ran an additional set of simulations to measure the small-angle x-ray scattering (SAXS) intensities. The radius of gyration, R g, is expected to decrease as attractive forces arise with increasing salt concentration. Polyelectrolyte size scales as a power law with the number of nucleotides, N, like R g = cn ν, where c and ν are constants for a given salt concentration, and ν is the scaling exponent. For any neutral, self-avoiding random coil polymer, the scaling exponent is expected to be ν = 0.588, 46 and a similar value was found experimentally for denatured proteins. 47 With respect to nucleic acids, scaling has been studied experimentally for ssdna, 48,49 but not for ssrna. Of note, Sim et al. used SAXS to study scaling of poly(dt) under varying monovalent salt concentration, 41 and reported a range of ssdna scaling exponents of ν = 0.58 for M NaCl to ν = 0.67 for M NaCl. (Note that M Na + is a relatively high monovalent salt concentration compared to the physiological concentration of ~ 0.15 M.) 9

10 Figure 2 Radius of gyration, R g, as a function of ssrna length, N, derived from atomistic MD simulation using F CI with varying values for ε. Simulation results (unfilled points) are compared with experimental R g values of poly(dt) ssdna in M NaCl and M from Sim et al. (filled triangles and squares, respectively). 41 RNA sizes in simulation with no electrostatics (unfilled circles) as well as with F CI using ε = 12,14 are about the same as ssdna under high monovalent salt concentration. Alternatively, for ε < 12, divalent ion effects become stronger and R g decreases below values attained experimentally with monovalent ions. Also shown are scaling exponents, ν, for the different electrolyte conditions. While experimental scaling data for R g have only been reported for ssdna, the scaling exponent is likely very similar between the two types of nucleic acids. Scaling behavior for MD simulated ssrna is illustrated in Fig. 2 wherein the average R g values are shown as a function of sequence length with implicit ions via F CI, using different values of ε = 8, 10, 12 and 14 and for a neutral chain with no counterions (ε = ). Experimental values for R g measured for ssdnas are also displayed in Fig. 2 for two different NaCl concentrations, M and M. The MD simulated ssrnas under neutral conditions and ε = 14, 12 have values of R g that are approximately the same as that found experimentally for ssdna at M. But further lowering ε to between 12 to 8 produced increasingly compact chains with smaller R g as a result of the effective attractive interactions coming from the counterion free energy F CI. 10

11 Figure 3 A) Scaling exponent, ν, as a function of dielectric constant, ε. As ε increases, ν approaches neutral ssrna chain scaling, which was found to be ν = 0.64 ± B) Persistence length, l p, as a function of dielectric constant, ε. As with scaling, increasing ε results in in l p approaching neutral ssrna chain persistence length, which was found to be l p = 1.4 ± 0.1 nm. The scaling exponents ν derived from the sequence length dependence of R g in Fig. 2 are shown as a function of ε in Fig. 3a. Without counterions, we found ν = 0.64 ± When F CI was applied with ε = 14 or 12, the exponents changed only slightly, both yielding ν This suggests that scaling is not affected by F CI at larger values of ε beyond about 14. But decreasing ε to 10 produced a much more pronounced change in the scaling exponent, lowering it to ν = 0.53 ± At ε = 8, ν decreased further to 0.43 ± Both of these exponents are smaller than the value expected for a self-avoiding chain, indicating that the electrostatic influence of divalent ions is dominating the entropically-driven conformational ensemble of a sterically self-avoiding chain when ε is in the range around 10 to 12. In addition to scaling, we have also measured the persistence length, l p. There have been many more experimental studies of persistence length for ssdna 40,41,50 56 than for ssrna, 40,50,51 but it is typically found that ssrna is only marginally more rigid than ssdna because of the extra hydroxyl group on the ribose, which reduces the flexibility of the sugar-phosphate backbone through steric exclusions. Importantly, experimental estimates for the persistence length of ssdna vary tremendously with cation concentration and valence, and range from l p = 0.6 nm under very concentrated salt conditions of 50 mm MgCl 50 2 to l p = 2.1 nm at 0.1 mm NaCl 40 to l p = 4 nm at 8 mm urea. 57. Similarly, ssrna persistence lengths have been shown to vary widely, from l p = 0.8 nm in 120 mm MgCl 2, 50 to l p = 2.5 nm in 0. 1 mm NaCl. 40 Furthermore, experimental results for l p may span a large range even 11

12 within similar salt conditions. For example, experiments studying ssdna in 100 mm NaCl have reported values of l p that range from 1.3 nm 58 to 2.5 nm. 55 Given this variance within current experimental results, we present the range of l p values that our divalent counterion force field is capable of producing as a function of ε rather than conflate values of ε directly with specific salt concentrations. In this way, future simulations done using this force field may use ε as a tuning parameter that modifies the RNA stiffness to match the effective divalent salt concentration. The MD simulations found that a neutral RNA chain had persistence length l p = 1.4 ± 0.1 nm, which is similar to what was found experimentally for ssrna in 0.5 M NaCl. 40 When F CI is included, we found that the persistence length l p decreased from 1.3 ± 0.1 nm for ε at 14 to 0.6 ± 0.1 nm for ε at 8. These results are shown in Fig. 3b. A value ε in the range 10 to 14 correlates closely with the range of persistence lengths found experimentally for divalent ions, and using the correlations in Fig. 2 and Fig. 3, ε can be adjusted to match a divalent salt concentration of interest using the radius of gyration or the persistence length. While experimental results have found ssrna to be much stiffer than the range presented here in low concentration monovalent salt solutions, we do not explicitly investigate those conditions here. Rather, MD studies of atomistic RNA using a Poisson-Boltzmann treatment of monovalent ions have been shown to accurately account for the effects of monovalent salts found experimentally. 22 To further assess the structural details of MD simulated ssrnas in the presence of F CI, we have computed scattering intensities to be compared against experimentally measured SAXS data. To most effectively match experimental data, a separate set of MD simulations was run under an NPT ensemble using the Amber94 ff99 force fields with a tapered Lennard-Jones potential in place of the sterics-only WCA potential. 59 The LJ potential used was tapered off at 0.3 nm and switched off completely at 0.4 nm, in a similar fashion to previous MD studies of ssdna. 60 MD trajectories were generated at 310K until equilibrated, and scattering intensity curves from these simulations with N = 40 are shown in Fig. 4 for F CI using dielectric ε = 10, 12, and 14 (triangles, crosses, and asterisks) as well as neutral conditions (ε = ) (circles). SAXS results obtained by Chen et al. for U 40 in 100mM NaCl are shown in Fig. 4 as the solid line. 40 We find that our results match experiment well in the range of ε = 12 to 14. These results suggest that a value of ε in this range approximate physiological monovalent salt conditions, which is effectively similar to low-concentration divalent counterion conditions. At ε = 10, the scattering profile reveals a more compact structure as the intensity peak rises in the low-q range and tapers off more quickly at high-q. This indicates that for values ε = 10 and below, F CI begins to exhibit attractive behavior due to Mg 2+ beyond that produced by monovalent salts alone. 12

13 Figure 4 SAXS curves for ssrna simulations with N = 40 for several values of ε as well as neutral electrostatic conditions compared to experimental data for U 40 at 100mM NaCl found by Chen et al. 40 Values of ε = 12 and 14 show close agreement with the experimental results as F CI with these valuesof ε approximates low-concentration Mg 2+, having a similar effect as highconcentration Na +. For ε = 10 the scattering profile shifts, indicating a more compact structure. IV. PARAMETRIZATION OF A COARSE-GRAINED RNA MODEL In this section, we formulate a coarse-grained model of RNA that meaningfully incorporates our implicit divalent ion model. When building a CG model, the attractive nature of F CI must be accounted for. As described in the previous section, divalent ions have an outsized influence over ssrna because ssrnas are inherently quite flexible, but counterions have a smaller effect on secondary structures in RNAs. Because of the disparity in counterion influence over ssrna versus RNA in duplex form, we have parameterized our model on the basis that individual nucleotides must be treated with different potentials depending on their secondary structure, and toggling between these potentials is then necessary in simulation when the secondary structure of the molecule changes. We validate our model by comparing both ssrna and double-stranded RNA (dsrna) chains in atomistic simulations against experiments. Recent CG models of nucleic acids have shown much promise in meaningfully incorporating explicit multi-valent ions like Mg 2+ into structural MD simulations. 33,36,61 However, until these recent developments, most previously proposed CG models of RNA and DNA either omit electrostatics entirely or utilize a linearized Poisson-Boltzmann potential to account for the effects of counterions. 23 This results in a Debye-Hückel screening factor for the negatively charged monomers: 13

14 U DH = (4πε 0) 1 2ε charged sites i,j q iq j r ij e r ij/κ D (9) where q i is the charge on site i, r ij is the distance between sites i and j, ε is the relative dielectric, and κ D is the Debye length. For a nucleic acid, the charge on each phosphate site is negative, q P = e, and U DH is thus purely repulsive for RNA. U DH provides a reasonable approximation for understanding polymeric behavior in most monovalent salt concentrations, but fails to encapsulate the actual salt conditions found in vivo. The cellular ionic milieu contains physiologically essential divalent and trivalent ions, which not only help to neutralize the inherent charge on the RNA, but also actively stabilize a compact conformation via a net attractive free energy of interaction. Divalent ions and Mg 2+ in particular are known to have a universally stabilizing effect on folded RNA conformations arising from a net attractive electrostatic potential, 27 and in that sense the free energy of interaction due to divalent ions is inherently different from U DH. A. Model Design The counterion-mediated free energy, F CI, is a function of the conformation of the RNA backbone. Specifically, the modified boundary search algorithm works on the assumption that the RNA phosphate sites are generally well separated in such a way that the neighboring phosphate sites along the backbone are usually the closest to each other in space. The principal secondary structure of RNA, the A-form helix, is structurally very stiff, with a persistence length l p nm. 62,63 Conversely, single-stranded loop regions of RNA are much more flexible with a persistence length l p 1 2 nm. 40,41 These flexible loops allow for large-scale RNA motions, and it is therefore important that our CG model is able to stabilize helical dsrna structures while simultaneously preserving the flexibility of ssrna loops regions. On the other hand the model must also account for an overall attractive electrostatic free energy coming from the counterions. Unlike other CG potentials, F CI is not defined around equilibrium positions in space, and the electrostatics alone could easily collapse any RNA structure if not appropriately counterbalanced by the other structural forces. We implement a 3-bead per nucleotide CG model wherein each nucleotide is encapsulated by three bound and sterically-interacting beads, which represent the phosphate, sugar, and nucleobase. Similar models have been used for many previous CG nucleic acid simulations 13,15,16,18 because they represent a balance between the speed increases gained from coarse-graining while preserving the 14

15 most important physical attributes of RNAs, such as base-pairing, base-stacking, and electrostatic interactions. Figure 5 Each coarse-grained nucleotide is treated differently depending on its internal state. A) if the nucleotide is unpaired, a stiffness force is applied such that angle α is extended. B) If the nucleotide is base-paired, then a dihedral force is applied about each torsional angle φ. Additionally, the base-pair is kept structurally rigid by applying a force to extend angle γ between bases, as well as a distance harmonic force to keep the bases separated by a distance d 0. In all cases, bond lengths are set at a distance r 0 and bond angles are set to θ, depending on the type of beads involved. Compared to most 3-bead models, our model incorporates two important modifications. Because of the vast difference in dynamics between ssrna and dsrna sections, it is necessary to treat each nucleotide differently depending on whether it is base-paired within a helix or if it is part of a single-stranded segment. We apply a different set of potentials to each nucleotide depending on its current internal state, according to a state parameter s i : s i = { 0, nucleotide i is unpaired 1, nucleotide i is base paired (10) If the nucleotide is base paired, then a dihedral potential is imposed upon it, forcing consecutive base-paired nucleotides into an A-form helical structure. Conversely, if the nucleotide is unpaired then no dihedral potential is applied, and the torsion angles can rotate freely, which allows for far more flexibility in a segment with multiple unpaired nucleotides. This separation between paired and 15

16 unpaired sections of RNA has successfully been used previously for structure prediction 64,65 and for the study of the topology of two-way junctions. 66 Although unpaired nucleotides are free to rotate, a stiffness potential that allows for control over the flexibility of the single-stranded sections is imposed to prevent collapse due to electrostatic attraction. The differences between paired and unpaired nucleotides are illustrated in Fig. 5. Additionally, we have re-parameterized our CG potentials around the presence of an overall attractive electrostatic free energy. It is standard for coarse-grained models to strictly control all of the bond distances, angles, and dihedral angles through the use of harmonic, quartic, and Lennard-Jones potentials. In the case of a polyelectrolyte, a repulsive electrostatic potential like U DH in Eq. (10) has a decay term that is strong enough to prevent the electrostatics from overwhelming the other forces, breaking apart the polymer, or disturbing its overall desired structure. But the opposite problem arises in the case of an attractive electrostatic free energy like F CI. Rather than the electrostatics repelling the structure, the electrostatics must be prevented from collapsing the polymer. The lowest-energy distance for two opposite charges is zero, and so all other potentials must be appropriately balanced to prevent this collapse from occurring. Our CG model is comprised of nine terms in addition to the electrostatic potentials U PC and F CI described in Section II. Each of the other potential terms is discussed individually below. Parameterization of these potentials was carried out to specifically reproduce A-form helix structure in double-stranded sections and to match experimentally determined scaling and dynamics in the singlestranded sections. The complete energy function of the CG model is: U = U WCA + U FENE + U stiffness + U stack + U angle + U BP + U paired distance + U paired angle + U dihedral + [F CI, U PC ] (11) The general polymeric potentials, U WCA and U FENE are defined as: U WCA = { 4ε ((σ σ r )12 ( r )6 ), r/σ < 2 1/6 all pairs ε, otherwise. (12) 16

17 U FENE = 1 2 k Fr 2 0 ln (1 ( r 2 ) ) r 0 bound pairs (13) U WCA is the Weeks-Chandler-Andersen purely repulsive potential which is included to prevent steric overlap of the beads. We have used ε = 0.24 kcal/mol, and σ = nm for all bead pairs except for phosphate-phosphate pairs, for which σ PP = nm was used because phosphates are chargecarrying. The larger value of σ PP more closely approximates the true van der Waals radius of a phosphate molecule (~0.5 nm), and more importantly puts a hard limit on the minimum distance between point charges. U FENE is a bonding potential describing finite extensible polymers. 67 This potential is applied to all consecutive pairs of beads along the RNA backbone. U FENE is used specifically to prevent the polymer from crossing itself, with r 0 = 2σ and k F = 7ε/σ 2. 68,69 (In these and all other potential terms, r is the distance between the two beads being evaluated.) Single-stranded structure is enforced by the potentials U stiffness, U stack, and U angle : U stiffness = k α (1 + cos α i ) 2 ( 1 s i ) conseutive sugar beads (14) U stack = 4ε stack (( σ stack r neighbor bases ) 12 ( σ stack ) 6 ) r (15) U angle = 1 2 k θ (θ i θ 0 ) 2 consecutive beads (16) U stiffness is applied across neighboring sugar bead triplets, where α i is the angle between three consecutive sugars about a central sugar i as depicted in Fig. 5A and k α = 10ϵ. The minimum of this potential is at an angle of π, and U stiffness therefore confers an inherent stiffness to the RNA backbone that can compete with and counterbalance an attractive electrostatic free energy. Moreover, U stiffness effectively allows for direct parameterization of the flexibility of the single-stranded sections of the RNA through the parameter k α. 70 Since it is specifically applied to single-stranded segments, U stiffness is active only for sugars in unpaired nucleotides (s i = 0). 17

18 U stack is a base-stacking potential which takes a general Lennard-Jones form. This term is incorporated so that the equilibrium distance between bases is representative of an A-form helix, with ε stack = 4ε and σ stack = 1.07σ. U stack helps maintain helical structure and facilitate base-pairing by keeping neighboring bases together, which then allows consecutive base pairs to dynamically form more easily following base-pair nucleation. U angle is a harmonic potential that enforces a given bond angle between all consecutively bound beads where θ i is the internal angle within a bead triplet about the center bead i, and k θ = 400ε. Table I gives values for θ 0 which have been parameterized in combination with U FENE and U dihedral to help preserve the appropriate distance between nearest-neighbor phosphate beads, as well as to enforce an A-form helical geometry in base-paired sections of an RNA. Table I Bond angle values for U angle. Each bond angle is defined between a triplet of beads bound by U FENE. Bead Triplet θ 0 value (degrees) P-S-B P-S-P S-P-S The base-pairing potentials U BP, U paired distance, and U paired stiffness are defined as: U BP = 4ε bp (5 ( σ r )12 6 ( σ r )10 ) base beads (17) U paired distance = [k bp1 (r d 0 ) 2 + k bp2 (r d 0 ) 4 ]s i paired base beads (18) U paired stiffness k γ (1 + cos γ i ) 2 s i paired base beads and sugars (19) U BP is a general base-base interaction. U BP is applied only to Watson-Crick base pairs (A-U, G-C) and G- U wobbles with ε AU = ε GU = 2.5ε and ε CG = 4ε, and provides an attractive potential which allows those bases to hydrogen-bond. But without a priori specifying base pairs, this potential may give rise to clusters of three or more bases due to its general radial symmetry. U paired distance is a harmonic 18

19 distance force, where d 0 = 0.8 nm, k bp1 = 1ε, and k bp2 = 400ε. This potential keeps paired nucleotides at an appropriate distance from each other, and restricts the electrostatics between paired phosphate sites from overwhelming the helical structure. Note that the distance between bases here does not accurately mirror the true distance between bases in an A-form helix. This is not particularly important because we are more interested in the backbone structure, and so the more relevant physical distance is the distance between phosphates in base-paired nucleotides. In order to keep the RNA duplex structure appropriately rigid, U paired stiffness is applied where k γ = 400ε and γ i is the angle about each base, its paired base, and its bound sugar as illustrated in Fig. 2B. U paired stiffness is applied symmetrically to each nucleotide in a particular pair, and forces the nucleotides apart orthogonally with respect to the helical axis. In addition to the potentials applied to the bases themselves, nucleotides in paired state also have the potential U dihedral applied to the beads: U dihedral = ζ(t) 1 2 k φ [cos(φ i ) cos(φ 0 )]2 s i conseutive beads (20) where φ i is the torsional angle between four consecutive beads, and φ 0 is given in Table II, with k φ = 140ε. This potential allows neighboring base-pairs to arrange into an A-form helix. Table II Equilibrium dihedral angle values for U dihedral where the bead sequence is read in the 5 to 3 direction. Bead Triplet θ 0 value (degrees) P-S-P-S S-P-S-P B-S-P-S 33.6 S-P-S-B 54.0 Note that many MD and CGMD simulations apply a standard harmonic potential about the desired dihedral angle, U φ = 1 2 k φ (φ i φ 0 ) 2. However, the related force term U φ has a singularity and is unstable for some values of φ i. 71 Because U dihedral is toggled dynamically in this model, the initial value of φ i is not known before it is applied, and it cannot be guaranteed that this singularity will not arise. Because of this there is an intrinsic symmetry in U dihedral and the helices that form in our model may be left- or right-handed depending on the initial conditions of base-pairing. Additionally, although an explicit singularity is avoided by using U dihedral instead of U φ, some initial angles of φ i were 19

20 observed to create forces strong enough to break the polymer if simply toggled on in a single time step. To avoid this, a time-dependent scaling factor ζ(t) is applied to U dihedral, which linearly increases the force over a 2000 time step interval after the pair has been toggled: t, t < 2000 time steps ζ(t) = { 20 ; where t = 0 at base pairing (21) 1, t 2000 time steps B. Validation of CG Model We have validated the new CG model by independently analyzing the conformational statistics of both ssrna unpaired loops and dsrna hairpins. Typically some combination of ss and ds structures will exist in a given RNA simulation, and it is important to verify that each type of structure is correctly modeled by the CG dynamics. Figure 6 Initial conformation of the RNA structure for all simulations is a fully extended polymer conformation with bond angles and bond length equilibrated. On top is the full structure with all beads explicitly depicted. Below is the backbone structure, which illustrates the overall RNA conformation more simply. Purple tail sections are inert and orange sections have non-interacting bases. Seen here is an initial conformation for a hairpin loop with secondary structure found in Fig. 9, with N BP = 15. For the simulations performed, we applied a standard velocity Verlet integration in the context of Langevin dynamics, with a thermal bath temperature of 310K. Time steps of dt = 0.01 ps were used in relative units, however it is unlikely that this time step corresponds directly to real time due to interaction modifications inherent in coarse graining, and so any dynamical data is presented in units of time steps. All simulations were initialized with a fully extended single-stranded RNA structure with bond angles minimized and torsion angles set to an A-form helix as seen in Fig. 6. Velocities were randomly initialized with a Gaussian distribution about the desired temperature. The base pairing parameter s i, initially set to 0 for all nucleotides, was toggled to 1 whenever two pairing bases were 20

21 observed to be within a distance r < 3σ for a consecutive 10 time steps. Additionally, because the forces that control structure are applied asymmetrically to the last two nucleotides on both the 3 and 5 ends of the RNA, those nucleotides (colored in all backbone figures as purple) were excluded from base pairing and electrostatic interactions. This exclusion prevented poorly defined end-effects from interfering with the RNA dynamics. All statistics gathered also ignored these four end nucleotides. 1. Coarse-Grained Single-Stranded RNAs We first show that our model is capable of correctly reproducing experimental results for singlestranded segments of RNA. Specifically, we aimed to simulate the dynamics of true ssrna by maintaining a persistence length, l p, similar to that found experimentally. l p is a good measure of local flexibility on the nucleotide scale, and so it is important to accurately reproduce that flexibility in CG simulations. In general, when F CI is used as the electrostatic potential, an attractive force is generated which mirrors a chain saturated with counterions, and l p decreases as the dielectric constant ε decreases. On the other hand, when U PC is used as the electrostatic potential, the system is undercharged and l p decreases as q P decreases toward an electrically fully-screened neutral state. As with our atomistic simulations, we note that there is no generally accepted value of l p that corresponds to any particular Mg 2+ concentration. Therefore we present our results as a function of the parameter ε and show that our CG model is capable of accurately reproducing the range of ssrna flexibility found experimentally. The persistence length of ssrnas was determined by simulating a 64 nt RNA which has base pairing interactions turned off. Similar to the ssrna MD calculations described above, l p was calculated using the orientational correlation length definition of persistence length, 72 where the vectors between adjacent sugar beads were used as the correlated vectors. Results of these simulations are shown in Fig. 7. In the case of a neutral chain when electrostatics is turned off, we found a persistence length of 3.3 nucleotides, which when multiplied by the average distance between sugar beads gives a value of l p = 1.5 ±.1 nm, which is equivalent to the value of l p found in our neutral atomistic simulations. When U PC was used as the electrostatic potential, the persistence length ranged from l p = 1.63 ± 0.02 at q P = to l p = 3.3 ± 0.4 nm at q P = For values of q P > 0.02, the persistence length became too large to effectively measure with an RNA of the length we simulated. The range allowed by U PC accounts for most relevant monovalent salt concentrations, and l p increased as expected with q P. 21

22 Figure 7 Persistence length, l p, under the influence of different electrostatic potentials. On the left hand side, results for saturated counterion conditions with F CI are shown as a function of the dielectric constant, ε. On the right hand side, results for undercharged conditions with U PC are shown as a function of the partial charge, q P. Both potentials converge to an electrically neutral value of l p = 1.5 ± 0.1 nm. As ε is decreased in F CI, l p approaches experimental estimates for RNA in the presence of high-concentration divalent counterions, while increasing q P in U PC leads to values of l p > 3nm, which is what is found experimentally for RNA in dilute monovalent salt conditions. When F CI was used as the electrostatic potential, we found that ε 12 resulted in a persistence length almost indistinguishable from that found with no electrostatics. Using F CI with ε = 10 resulted in a value l p = 1.4 ± 0.1 nm and ε = 8 gave a value l p = 1.3 ± 0.1 nm. Below ε = 8, the magnitude of F CI became too attractive to be physically reasonable for physiological salt conditions, as the electrostatics overwhelmed the other energy terms. Compared to neutral chains, these values of l p reflect overall attractive intrachain interactions coming from the counterion-induced free energy, and they are therefore relevant for divalent counterions. Rivetti et al. found a value of l p = 1.3 nm for ssdna in 10 mm NaCl and 2 mm MgCl 2, 54 which is close to physiological divalent salt concentration. We therefore postulate that ε = 8 provides a good estimate for the value of l p for ssrna under physiological salt conditions or a higher concentration of purely divalent counterions. ssrna scaling was determined by simulating ssrnas of lengths N = 24, 34, 44, 54, 64 nt under the effects of U PC or F CI. Results for the scaling properties of are displayed in Fig. 8. When no electrostatics was applied, the scaling of R g gave ν = 0.59 ± 0.02, which closely matches theoretical 46 and experimental 47 results for a self-avoiding coil. Experimental studies of poly-dt ssdna found scaling values ranging from ν 0.59 to 0.72 in NaCl concentrations of 0.01 to 1.0 M respectively. 41 Use of 22

23 U PC resulted in values of ν = 0.61 ± 0.03 for q P = 0.003, and ν = 0.77 ± 0.02 for q P = 0.02 which mirrors the experimental scaling observed in monovalent salt. Figure 8 Scaling exponent, ν, under the influence of different electrostatic potentials. On the left hand side, results for saturated counterion conditions with F CI are shown as a function of the dielectric constant, ε. On the right hand side, results for undercharged conditions with U PC are shown as a function of the partial charge, q P. Both potentials converge to an electrically neutral value of ν = 0.59 ± As ε is decreased in F CI, ν also decreases, which demonstrates the attractive forces arising from F CI allow the ssrna to approach ideal polymer scaling of ν = 0.5. On the other hand, as q P is increased in U PC, the ssrna approaches linear scaling, indicating that the electrostatics overwhelm all other potentials and force the ssrna into a fully extended conformation. 2. Coarse-Grained RNA Hairpin Loops Next, we show that the RNA helices generated in the CG simulations correctly reproduce the structural properties of experimentally determined RNA A-form helices. The magnitude of F CI is especially responsive to the backbone structure of the RNA conformation, and small changes in structure show up prominently in the value of F CI. Taking advantage of this property, and because our model is primarily constructed around the inclusion of F CI, we used it as a means of validating the dsrna structure generated by our CG model To analyze the structure of an RNA hairpin loop structure, base pairs were specifically enforced and not permitted to break once formed. Explicitly enforcing base pairing allowed us to verify that the structures formed and behaved as expected. Base pairing was defined to maintain a tetraloop structure with N BP being the number of C-G base pairs in the stem with overall secondary structure illustrated in Fig

24 Figure 9 Illustration of a standard dsrna hairpin loop structure with a tetraloop bend at one end. The 5 and 3 ends, shown here in purple, are non-interacting. The gray section is completely base-paired, with s i = 1 for all nucleotides in gray. The orange tetraloop is not base-paired, and nucleotides in the loop have s i = 0, allowing for free dihedral rotation. Dynamically generated hairpin loops were compared to crystal structures for N BP = 4 (12 nt) and N BP = 15 (34 nt). As a baseline, we simulated ssrna chains of equivalent lengths using atomistic simulation with the same parameterization as in Section III and no electrostatic interactions. These simulations provided an entropic conformational ensemble, for which values of F CI were generated using ε = 8. For a hairpin loop with N BP = 4, a standard crystal structure (PDB ID: 1GID) 73 was found to have a counterion free energy ΔF CI = 9 kcal/mol relative to the baseline ssrna. At equilibrium, a CG hairpin loop of equivalent size was also found to have an average counterion free energy ΔF CI = 9 ± 1 kcal/mol. Similarly, a crystal structure with N BP = 15 (PDB ID: 4FNJ) 74 had a counterion free energy ΔF CI = 44 kcal/mol relative to ssrna, and an equivalent CG structure had an average free energy difference of ΔF CI = 44 ± 3 kcal/mol at equilibrium. The crystal structures and example dynamic structures used in this calculation are shown in Fig. 10 alongside traces of ΔF CI over time steps compared to the constant crystal structure value represented by a dashed line. These results confirm that our model was able to maintain the appropriate backbone structure of a dsrna A-form helix even in the presence of both a heat bath and an attractive electrostatic free energy. In the CG RNA helix, the distance between adjacent phosphate beads was found to be r PP = 0.60 ± 0.06 nm. We compare this to the average value of r PP measured across 15 crystal structures of different RNAs, which was found to be 0.6 ± 0.2 nm. The average distance is the same, although the variance in the CG model is smaller than the variance found experimentally. 24

25 Figure 10 A) A hairpin loop with N BP = 4 is simulated and compared to a crystal structure of an equivalent RNA hairpin loop, shown at left. An example of the backbone structure determined in simulation is shown at right. B) A longer hairpin loop with N BP = 15. A crystal structure is shown at left compared to a dynamically generated structure shown at right. At bottom is a 2ns trace of ΔF CI (ε = 8) for each RNA shown below in black, compared to the value of the experimentally determined structures, shown as an orange dashed line. All ΔF CI values are in relation to F CI values of unfolded ssrna of equivalent length. These analyses confirmed that our CG RNA model was capable of reproducing experimentally realistic behavior in the presence of either monovalent or divalent counterions. ssrna segments are especially attuned to the different electrostatic potentials applied, and our results show that the tuning parameters for either counterion potential can be easily adjusted to mirror any observed experimental structures. Similarly, helical stems of RNA are more stable experimentally, and our model preserves this stability by applying conformation-preserving forces to nucleotides within a stable secondary structure. 3. Singly connected RNA Hairpins 25

26 While we have independently demonstrated that our CG RNA model effectively reproduces the most important backbone qualities of both ssrna and dsrna in for relevant monovalent and divalent salt conditions, it is important to show that these two states can meaningfully coexist in simulation. To that end, we performed several MD simulations of two RNA hairpins singly-linked by an ssrna linker of length M Loop, with secondary structure depicted in Fig. 11. Each hairpin was comprised of a base-paired RNA helix N BP in length capped by a tetraloop. Figure 11 Secondary structure of the singly-linked RNA hairpins being simulated. Each hairpin, shown in gray has N BP base pairs and is capped by an ssrna tetraloop. The two hairpins are singly connected by a short ssrna loop of length M loop. The 5 and 3 ends, shown here in purple, are non-interacting. Previous SAXS experiments by Bai et al. have been done on similarly formed DNA constructs. 75 Specifically, these experiments used DNA structures composed of two 12-bp DNA helices connected by a single (CH 2 -CH 2 -O-) 3 linker. For our CG RNA model to most closely mimic the length of the DNA helices and the flexibility of the linker used experimentally, we used values N BP = 10 and M Loop = 2. Scattering profiles from our simulations can be seen in Fig. 12 for simulations using F CI with ε = 8 (crosses), U PC with q P = 0.1 (circles), and no electrostatics (triangles). Experimental results from Bai et al. are shown for 20 mm Na + (dashed line) and 1.2 M Na + (solid line). In 20 mm Na +, the DNA structure is highly repulsive to itself which leads to a fully extended structure. There are two characteristic peaks in the scattering profile found experimentally for this extended structure. Likewise the simulated RNA structures become extended when applying U PC, and we find that the scattering profile of our simulated RNA construct with repulsive electrostatics largely matches the scattering profile found experimentally. Conversely, the experimental curve for 1.2 M Na + corresponds to a structure that fluctuates randomly between a fully extended and fully closed state. As expected, we find that our simulations with neutral electrostatics closely matches the experimental curve closely because the simulated RNA structure is entropically driven and has no preferred conformation. Lastly, simulations run with F CI show a more well defined peak in the low-q range, suggesting an overall more compact structure beyond what was 26

27 found experimentally for DNA, but as we expect should be the case with RNA in the presence of divalent ions. Figure 12 SAXS curves for simulations of the singly-linked RNA hairpins depicted in Fig. 11. Simulations were run with F CI (ε = 8), U PC (q P = 0.1), and electrostatically neutral conditions in order to reproduce attractive, repulsive, and non-specific electrostatics respectively. Experimental SAXS profiles for DNA from Bai et al. are shown for both 20 mm Na+ (dashed line) and 1.2M Na+. Simulation results for U PC closely matches the experimental low monovalent salt curve while neutral electrostatics curve closely matches the experimental high-monovalent salt curve. V. APPLICATION TO BENDING FLEXIBILITY OF RNA TWO-WAY JUNCTIONS Previous theoretical studies have predicted that two helices or two like-charged rods or nucleic acid helices will attract each other in the presence of divalent ions Follow-up experimental studies showed that DNA duplexes may not demonstrate this predicted attraction under the influence of divalent ions, 75 but that RNA duplexes do enter into an attractive regime at concentrations of ~6 mm Mg 2+ and above. 79 In our previous studies, 38 we also found that F CI is predicted to be negative in the presence of Mg 2+ at similar concentrations, suggesting a net attractive average force between RNA helices. To test if our CG model correctly reproduces this behavior, we performed MD simulations of two RNA helices linked by a 2-way junction, designed to study the attractive nature of RNA helices, and we indeed observed bending effects similar to those found experimentally. Simulations were run of RNAs composed of two helical stems with N BP base pairs each, joined by a symmetric bulge consisting of two unpaired single-stranded segments each M Loop nt in length. The secondary structure of this construct is depicted in Fig. 13. The base pairs were constrained to stay paired, maintaining a helical stem structure on either side of the bulge. Two sets of twelve simulations were run for each value of N BP. The first set of runs was performed with no electrostatics, for which the 27

28 single-stranded sections were found to have a persistence length l p = 1.5 ±.1 nm. The second set was run with effective counterions included via F CI with ε = 8, for which the loop section had persistence length l p = 1.3 ±.1 nm. A value of ε = 8 was chosen because it is most closely emulates true salt conditions found in vivo, as described in Section IV.B.1. Because the helical sections are base-paired, the portions of the structure that are most relevant to the structural fluctuations of this two-way junction are the connecting ssrna bulges. Figure 13 Illustration of the secondary structure of the RNAs being simulated. Two equal-length helical stems with N BP base pairs each are connected by two ssrna sections, each of length M Loop. The orange tetraloop and inner loop sections have no base interactions, and the final two nucleotides on the 5 and 3 end are non-interacting and colored purple. Simulations were performed for N BP = 3, 6, 9, 12, 15. In all cases M Loop = 6, which at over three times the persistence length of unfolded ssrna, was long enough to allow the two helices to interact meaningfully without interfering in the overall conformational ensemble. We measured the bend angle θ between the two helices, determined by taking the center point of two base pairs on each helix symmetrically about the loop region, and then using the vector connecting those two points as the helical axis to calculate θ through the inner product between the helical axes. Examples of two N BP = 12 structures, one closed and one open, with their associated axis vectors are shown in Fig. 14. Figure 14 The bend angle, θ, is calculated by taking the vector along each helical axis. Here, two conformations generated in simulation for N BP = 12 are shown. A) An open conformation with θ = and B) a closed conformation with θ =

29 Figure 15 Normalized distributions of the bend angle, θ, for N BP = 3, 9, and 15 in plots A, B, and C respectively, for simulations with no electrostatics (Neutral), shown as a dashed blue line, and those with implicit divalent ions via F CI with ε = 8, shown as a solid black line. Distributions shown here are deconvoluted with sin θ to correctly reflect the conformational shift. In all cases, as well as for N BP = 6, 12 (not shown), the conformational distribution is shifted toward a more closed state in the presence of divalent counterions, as depicted in Fig. 14B. The distribution of θ for the neutral simulations is nearly identical for all values of N BP, with θ = 110 ± 30 in all cases. The distribution of θ with divalent ions is slightly more varied, but indicates that the distribution is not necessarily a function of N BP. 29

30 Normalized histograms of the bend angles for N BP = 3, 9, and 15 are shown in Fig. 15. Note that the statistics for θ are biased by a Jacobian, which is sin θ. Removing the Jacobian from the distributions, the histograms show that that with counterions via F CI the probability that the RNA was found in a closed conformation is enhanced compared to a neutral molecule. With no counterions the bend angle statistics stayed roughly the same as N BP was varied, with an average value of θ = 115 ± 34 for all values of N BP. But with counterions included via F CI, the average value of θ is shifted down in all cases, from θ = 103 ± 40 for N BP = 12 to θ = 108 ± 34 for N BP = 6. This shift does not vary monotonically with N BP, and it a direct result of counterion effects. The observed effects on the conformational ensemble due to divalent counterions was relatively small, as the average bending angle θ showed a deviation from neutral chains of no more than 10 across all values of N BP. In spite of this seemingly small effect, folded RNAs are typically stabilized by many other types of tertiary interactions, and counterion-mediated free energy is one of many factors determining RNA folding dynamics. But even minor counterion-induced distortions in the bendability of one junction can produce large-scale cumulative effects, since a tertiary RNA fold is invariably made up of a network of many flexible joints. Also, the fact that divalent counterions allow the RNA to more easily reach a closed conformation indicates that tertiary interactions that only engage at close range, such as those between tetraloop and tetraloop-receptors, would be more likely to form in the presence of divalent ions, suggesting that the role divalent ions play could be quite significant when the flexibility of the entire folded tertiary structure is considered. Table III Free energy differences between open and closed states for two-way junctions with and without divalent counterions (all free energy values in kcal/mol). N BP ΔF folding (neutral) ΔF folding (F CI ; ε = 8) Using the bend angle distributions, we can derive a quantitative estimate for the free energy of folding for the two-way junctions we have studied. Defining the free energy of folding as the free energy difference between the open and closed conformations: 30

31 ΔF folding = k B Tln ( P open P closed ) (22) where P open is the probability of the RNA existing in an open state and P closed is the probability of the RNA existing in a closed state, and using an arbitrary bend angle cutoff θ 50 to select out closed states from the open states, values for ΔF folding obtained are given in Table III. The free energy of folding is ~0.5 kcal/mol smaller with divalent counterions present compared to a neutral system in all cases. At our simulation temperature of 310 K, this difference is of the order of k B T. In the presence of counterions, ΔF folding seems to decrease with the length of the helices, presumably because of increasing counterion-mediated attractive interactions developing between the two stems. However, this trend also appears to be broken between N BP = 12 and 15. While our approach here is somewhat arbitrary, there is no clear method of defining the difference between RNA states that are open and closed. We provide these values of ΔF folding to show that the effects of divalent ions are not particularly large, but nevertheless do impact the conformational ensemble of these large RNAs. Figure 16 Decay of the correlation function of folding, C(t)/C(0), with N BP = 6,9,12 for both neutral simulations and those with counterions via F CI. C(t)/C(0) reaches stable exponential decay for which rate constants, k fold are measured by fitting (black lines) and given in Table IV. The correlation function shows a slower decay with counterions present than without. While the folding free energy shows a weak dependence on helix length, a kinetic analysis shows that the folding rate is strictly a function of stem length and is higher in the presence of counterions. We have determined the rate of folding, k fold, in the simulations by calculating the autocorrelation function of the closed state over the course of the simulation: 31

32 C(t) = δh(0) δh(t) (23) where h(t) is the Heaviside function 1, θ 50 h(t) = { 0, otherwise (24) and δh(t) = h(t) h. The correlation function C(t) is expected to follow the phenomenological firstorder rate law: C(t)/C(0) = exp ( k fold t) (25) Fig. 16 shows that decay of the correlation functions becomes exponential after a short transient, and k fold reaches its plateau value 80 for all simulations after about 1,000 to 2,000 time steps. k fold values were determined by linear fits to the curves in Fig. 16 and they are given in Table IV. Table IV Rate constants as a function of N BP for two-way junctions with and without divalent counterions (values in units of (time steps) 1 ). N BP k fold ( neutral) k fold (F CI ; ε = 8) The rate data in Table IV show that the mean residence time τ = k 1 fold in the closed state under neutral conditions is 5.5e4, 1e5, 1.3e5, and 1.7e5 time steps for 6, 9, 12, and 15 bp, respectively, which scales approximately linearly with helix length with an average τ of approximately 1.1e4 time steps/bp. On the other hand, the residence time in the presence of counterions for the closed state is 6.6e4, 1.2e5, 1.9e5, and 2.5e5 time steps for the same helix lengths, which correspond to average τ of approximately 1.1e4, 1.3e4, 1.5e4, and 1.7e4 time steps/bp for 6, 9, 12, and 15 bp, respectively. In contrast to the thermodynamic analysis above which shows only weak counterion dependence, kinetics analysis is more sensitive and reveals an unambiguous salt effect which stabilizes and prolongs the lifetime of the closed state. Results of these simulations on two-way junctions demonstrate that our CG model is capable of reproducing the attractive influence of high concentrations of Mg 2+ that has also been seen experimentally. 79 The overall effect on the RNA conformations due to F CI is small and non-specific, but 32

33 nevertheless important when considering overall RNA folding dynamics. The presence of divalent counterions allows for more flexibility in single-stranded loop portions of RNA and shifts the conformational ensemble toward a folded state. VI. CONCLUSION We have developed a force field for implicitly calculating the effects of divalent ions on RNAs in MD simulation. Incorporating this force field into atomistic MD simulation, we showed that our force field replicates many experimental properties of ssrna in the presence of high salt concentration. This force field can therefore be tuned to match a range of counterion conditions not accessible with a simple Debye-Hückel implicit electrolyte potential. Additionally, we have constructed a new coarse-grained RNA model for use with our implicit divalent counterion force field that is primarily designed around an attractive counterion-induced intrachain free energy. We have verified and benchmarked our CG model against experimental analyses of ssrna, demonstrating that it is fully capable of matching true ssrna dynamics under a range of salt conditions, including NaCl and MgCl 2. Furthermore, it is able to dynamically form stable A-form helical structures in the presence of both repulsive and attractive electrostatics. Finally, from extensive simulations using our CG model, we have shown that divalent counterions play an important role in allowing RNAs in a two-way junction to form folded conformations more often and for longer durations, which suggests that other tertiary interactions may work cooperatively with divalent ions to form stable, folded RNA structures. ACKNOWLEDGEMENTS This material is based upon work supported by the National Science Foundation under Grant No. CHE

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37 Solution V Ion Resevoir μ +, μ

38 N R g (nm) 7 ν = 0.67 ν = ν = 0.58 ν = 0.53 ν = Sim et. al M NaCl Sim et. al M NaCl Neutral RNA F CI ε = 8 F CI ε = 10 F CI ε = 12 F CI ε = 14

39 Scaling l p (nm) exponent,ν A B ε

40 q I(q) (a.u.) U 40, 100mM NaCl; Chen et al. Neutral (310K) F CI ε = 10 (310K) F CI ε = 12 (310K) F CI ε = 14 (310K) q(nm -1 )

41 A B α θ r 0 d 0 γ φ

42

43 F CI U PC l p (nm) Fully Saturated Counterions Undercharged Conditions ε q P 0.02

44 Scaling exponent, F CI Fully Saturated Counterions ε Undercharged Conditions U PC q P

45 NBP

46 A B ΔF CI (kcal/mol) N BP = 4 N BP = Time Steps

47 N BP =10 M Loop =2 N BP =10

48 q I(q) (a.u.) PEG mm Na+; Bai et al. 12 PEG M Na+; Bai et al. Neutral (310K) F CI ε = 8 (310K) U PC q P = 0.1 (310K) q(nm )

49 N BP M Loop N BP

50 A B θ