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Nucleic acids are negatively charged biomolecules, and metal ions in solutions are important to their folding structures and thermodynamics, especially multivalent ions. However, it has been suggested that the binding of multivalent ions to nucleic acids cannot be quantitatively described by the well-established Poisson–Boltzmann (PB) theory. In this work, we made extensive calculations of ion distributions around various RNA-like macroions in divalent and trivalent salt solutions by PB theory and Monte Carlo (MC) simulations. Our calculations show that PB theory appears to underestimate multivalent ion distributions around RNA-like macroions while can reliably predict monovalent ion distributions. Our extensive comparisons between PB theory and MC simulations indicate that when an RNA-like macroion gets ion neutralization beyond a “critical” value, the multivalent ion distribution around that macroion can be approximately described by PB theory. Furthermore, an empirical formula was obtained to approximately quantify the critical ion neutralization for various RNA-like macroions in multivalent salt solutions, and this empirical formula was shown to work well for various real nucleic acids including RNAs and DNAs.
Nucleic acids are of biological significance in genetic storage, expression, and regulation.[1–6] Since nucleic acids are negatively charged molecules, metal ions in solutions could neutralize the negative charges on nucleic acids and favor their folding into native and functional structures.[1,7–15] Specifically, metal ions could enhance the stability of secondary and tertiary structures of nucleic acids[7,16,17] and could significantly influence nucleic acids helix assembly.[18–22] Importantly, multivalent ions such as Mg2+ and Cohex3+ play very efficient roles in DNA condensation[23–28] and RNA structural folding.[17,29–33]
To understand ion-nucleic acid interactions, several classic polyelectrolyte theories have been developed, including counterion condensation theory,[34–36] Poisson–Boltzmann (PB) theory,[37–42] modified PB theory,[43–48] and tightly bound ion (TBI) theory.[30,49] Among these, the PB theory is a well-defined theory with concise form and efficient solving algorithms, and has been widely applied to charged systems such as colloids, proteins, and nucleic acids.[50–57] However, due to the mean-field nature of the nonlinear PB theory, it ignores the discrete properties of ions, consequently underestimating the distributions of multivalent ions around highly charged polyelectrolytes such as nucleic acids,[27,58–62] and generally underestimates the efficient role of multivalent ions in the folding stability of nucleic acids.[63–68]
All-atom molecular dynamics and Monte Carlo (MC) simulations are widely used as important methods to complement theories and experiments in simulating charged systems and make predictions that agree well with experiments in a wide range of ion conditions.[23,69–76] Since multivalent ions are so important to nucleic acid structures and PB theory is concise and has efficient solving algorithms,[40,41,66] it is interesting to understand how much PB predictions deviate for multivalent ion distributions around nucleic acids and to quantify the extent to which ionic neutralization, multivalent ion distributions around nucleic acids can be approximately described by PB theory.
In the present work, we made extensive calculations on multivalent ion distributions around RNA-like macroions by PB theory and MC simulations, attempting to examine how much PB predictions deviate from the MC simulations. Furthermore, we attempted to obtain an empirical formula for quantifying the extent of ionic neutralization beyond which the multivalent ion distributions around RNA-like macroions can be approximately described by PB theory. Finally, this empirical formula was examined for various real RNAs and DNAs.
In this work, for simplicity, we used spherical macroions (RNA-like macroions) of various sizes and charge densities to model various RNAs structures, as shown in Fig.
The RNA-like macroions and real RNAs (and DNAs) were immersed in monovalent, divalent, and trivalent salt solutions, and the solvent was modeled implicitly as a continuous medium of dielectric constant 78. The salts were considered as 1:1, 2:2, and 3:3 salts. The radii of monovalent, divalent, and trivalent cations were taken as 2.7 Å, 3.7 Å, and 4.2 Å,[77] respectively, and the radii of coions were taken as the same as those of the corresponding cations. The salt concentrations covered the typical wide ranges: [1 mM, 100 mM] for monovalent salt, [0.1 mM, 10 mM] for divalent salt, and [0.01 mM, 1 mM] for trivalent salt. For real RNAs and DNAs, the radii of the atoms were taken as their van der Waals radii.[30,51,77] Each phosphorus atom was assumed to carry an electronic charge −e, and the other atoms remained neutral.[8,9,17] This has been verified by previous studies as a robust simplified model for the all-atom force field of RNAs and DNAs.[8,9,17] In our systems, the interactions were simplified into Coulombic interaction and hard-core repulsion
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We employed PB theory and MC simulations to calculate ion distributions around RNA-like macroions or real RNAs and DNAs with different reduced charges
Monte Carlo simulations were employed to calculate ion distributions around RNA-like macroions, RNAs, and DNAs.[69,70,78] The simulation system was a canonical ensemble with a collection of ions and a macroion or an RNA (or DNA) in a cubic box at room temperature. In each MC simulation, the macroion or RNA was fixed at the center of the cubic box, and ions were diffusive and mobile in the simulated box. To diminish the boundary effect, the box size was kept at least eight times larger than the Debye–Hückel length of the ionic solution, and periodic boundary condition was employed. According to the Metropolis algorithm, the energy change
The nonlinear PB theory is a well-established mean-field theory, in which the solvent is treated as a continuous medium with a dielectric constant and ions obey Boltzmann distributions based on mean electrostatic potential ψ[40,44,47]
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Here, ρf is the charge density of fixed charges, and
In the present work, the three-dimensional algorithm developed in the TBI theory was used to numerically solve the nonlinear PB equation.[8,9,49] A thin layer of one ion radius was added to a macroion or RNA surface to take the ions’ excluded volume layer into account, and the three-step focusing process was employed to compute the detailed electrostatic potential near the macroion or RNA surface.[46,71] The grid size in the first run of the three-step focusing process depends on the salt concentration and was taken to be six times larger than the Debye–Hückel length from the macroion/RNA surface to effectively include the salt effect in solutions. The resolution for the first run varied with the grid size to make the iterative process computationally possible and was generally kept as 1 Å and 0.5 Å per grid for the second and the third runs, respectively. The iteration for a run was continued until the electrostatic potential change
In this work, ion distributions around RNA-like macroions or RNAs (and DNAs) were characterized by the net ion charge distribution Q(r) within a distance r from RNA-like macroions or RNAs (and DNAs)[29,80,81]
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Here,
In this work, by PB theory and MC simulations, we made extensive calculations of ion distributions for RNA-like macroions with typical surface charge density of real RNAs. We mainly focused on the deviations of ion distributions between PB theory and MC simulations, and attempted to derive an empirical formula for a “critical” ion neutralization, beyond which multivalent ion distributions around RNA-like macroions and real RNAs and DNAs can be approximately described by PB theory.
We examined monovalent ion distributions around RNA-like spherical macroions at different monovalent salt concentrations. As shown in Figs.
For a very high salt concentration, exclusion correlations between ions become strong and can cause the slightly lower predicted ion binding of PB theory since ion correlations can drive ions to self-organize to form low-energy states and consequently favor ion binding.[30,49]
Divalent ions such as Mg2+ are important for RNA structures and functions,[8,9,12] and we examined divalent ion distributions around RNA-like macroions. As shown in Figs.
The extensive comparison between PB theory and MC simulations shows that PB theory appears to underestimate the divalent ion distributions, compared to the MC simulations, and these underestimations of ion binding become slightly more apparent for higher salt concentration. This is because PB theory assumes fluid-like ion distribution obeying mean electrostatic field and ignores ion discrete properties such as ion–ion correlations. Such ion–ion correlations can drive ions to self-organize to low-energy micro-states and favor ion binding to polyelectrolytes such as RNAs and DNAs.[30,49] Thus, PB theory apparently underestimates the binding of divalent ions with strong correlations, compared with the MC simulations where ion–ion correlations are explicitly involved. Higher divalent concentration would cause stronger ion–ion correlations around RNA-like macroions and consequently cause more pronounced deviations between the mean-field PB theory and the MC simulations.
Trivalent ions can very efficiently play a role in RNA folding[1,2,4,5] and DNA condensation,[25,26] so we examined trivalent ion distributions around RNA-like macroions. As shown in Figs.
Furthermore, figure
As discussed in the above section, PB theory significantly underestimates multivalent ion distribution (e.g., Q(r)) around RNA-like macroions because PB theory ignores ion–ion correlations, and correlations between binding ions near macroions are very strong for divalent and trivalent ions. Since such strong correlations between binding ions are induced by the highly charged RNA-like macroions, the correlations between binding ions can become weak when the highly charged macroions get ionic neutralization by binding ions. As shown in Figs.
First, we used
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The parameters a and b are 0.07852 and 0.00017 for divalent salt, and 0.066826 and 0.00025 for trivalent salt, respectively; z is the valence of multivalent ions, [c] is the bulk salt concentration of multivalent ions in mol/L, and S is the macroion surface area in Å2. The presence of σc in Eq. (
Beyond the above described RNA-like spherical macroions, we examined the multivalent ion distributions around real RNAs and DNAs, including 8-bp and 12-bp RNA duplexes, 12-bp and 24-bp DNA duplexes, 28-nt BWYV pseudoknot, 58-nt rRNA fragment, 76-nt tRNAPhe, and 158-nt P4-P6 domain of the tetrahymena ribozyme. Here, we also used Eq. (
As discussed above, the higher ion charge leads to stronger Coulomb attraction between ions and RNAs (or DNAs) and causes stronger ion binding, which is consequently more enthalpically dependent and less ion-concentration dependent. The more pronounced deviation between MC simulations and PB theory for ions with higher valence and for higher ion concentration is attributed to the stronger ion–ion correlations. As described above, higher salt concentration and ion valence would cause stronger ion–ion correlations, which can drive ions to self-organize to low energy microstates and favor ion binding.[30,49] Such ion–ion correlations are ignored in PB theory, causing greater apparent deviations of PB theory for higher ion valence and higher concentrations.
In analogy to the above described RNA-like macroions, the value
In the above, we discussed obtaining a series of
To answer this, we first examined the values of
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![](cpb_27_1_018203/cpb_27_1_018203_ieqn59.gif)
![](cpb_27_1_018203/cpb_27_1_018203_ieqn60.gif)
![](cpb_27_1_018203/cpb_27_1_018203_ieqn61.gif)
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Finally, an ellipsoidal shell with the same
From the above discussions, the empirical formula Eq. (
In this work, we employed PB theory and MC simulations to calculate the ion distributions around RNA-like macroions and various RNAs and DNAs, and our calculations covered wide ranges of ion conditions including monovalent, divalent, and trivalent salts with different bulk concentrations. Through the extensive calculations and detailed comparisons between PB theory and MC simulations, we have obtained the following major conclusions.
In this work, for simplicity, the radii of coions were kept the same as those of cations. Different radii of coions might bring slightly different ion distributions, which is beyond the scope of the present work and deserves to be discussed elsewhere. The present work has also employed some approximations and simplifications. First, the dielectric discontinuity at the boundary between solvent and RNA-like macroions/RNAs was ignored, which may slightly affect the ion binding at the boundary. Second, RNAs were modeled as spherical macroions, which would ignore some details of structures. Finally, the employment of the empirical formula from the analyses for RNA-like spherical macroions to real RNAs and DNAs also ignored structural details of RNAs and DNAs. Nevertheless, the present work provides extensive calculations and analyses on multivalent ion distributions around various RNA-like macroions and nucleic acids over a wide range of salt conditions, and an empirical formula was derived to measure the “critical” extent of ion neutralization beyond which the multivalent ion distributions around RNAs and DNAs can be described by PB theory. This empirical formula would be very useful for future two-phase polyelectrolyte models for quantifying multivalent ion-nucleic acid interactions.[79–81]
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