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Ion-Mediated Nucleic Acid Helix-Helix Interactions

Department of Physics and Astronomy and Department of Biochemistry, University of Missouri, Columbia, Missouri

Correspondence: Address reprint requests to Shi-Jie Chen, E-mail: chenshi{at}missouri.edu.

	ABSTRACT

TOP ABSTRACT INTRODUCTION THEORY AND METHODS RESULTS AND DISCUSSION CONCLUSIONS AND DISCUSSION APPENDIX A: TREATMENT FOR... APPENDIX B: STRUCTURAL MODEL... APPENDIX C: PARAMETER SETS... APPENDIX D: MONTE CARLO... ACKNOWLEDGEMENTS REFERENCES

 
Salt ions are essential for the folding of nucleic acids. We use the tightly bound ion (TBI) model, which can account for the correlations and fluctuations for the ions bound to the nucleic acids, to investigate the electrostatic free-energy landscape for two parallel nucleic acid helices in the solution of added salt. The theory is based on realistic atomic structures of the helices. In monovalent salt, the helices are predicted to repel each other. For divalent salt, while the mean-field Poisson-Boltzmann theory predicts only the repulsion, the TBI theory predicts an effective attraction between the helices. The helices are predicted to be stabilized at an interhelix distance 26–36 Å, and the strength of the attractive force can reach –0.37 k_BT/bp for helix length in the range of 9–12 bp. Both the stable helix-helix distance and the strength of the attraction are strongly dependent on the salt concentration and ion size. With the increase of the salt concentration, the helix-helix attraction becomes stronger and the most stable helix-helix separation distance becomes smaller. For divalent ions, at very high ion concentration, further addition of ions leads to the weakening of the attraction. Smaller ion size causes stronger helix-helix attraction and stabilizes the helices at a shorter distance. In addition, the TBI model shows that a decrease in the solvent dielectric constant would enhance the ion-mediated attraction. The theoretical findings from the TBI theory agree with the experimental measurements on the osmotic pressure of DNA array as well as the results from the computer simulations.

	INTRODUCTION

TOP ABSTRACT INTRODUCTION THEORY AND METHODS RESULTS AND DISCUSSION CONCLUSIONS AND DISCUSSION APPENDIX A: TREATMENT FOR... APPENDIX B: STRUCTURAL MODEL... APPENDIX C: PARAMETER SETS... APPENDIX D: MONTE CARLO... ACKNOWLEDGEMENTS REFERENCES

 
Nucleic acids (DNAs and RNAs) are highly charged polyanionic chain molecules. The folding of nucleic acids requires the cations to neutralize the negative backbone charges. Therefore, the solvent ionic conditions, including salt concentration, ionic charge and size, solvent dielectric constant, and temperature play essential roles in the folding of nucleic acids (1–10).

Despite the extensive experimental and theoretical studies (11–19), our ability to make accurate quantitative predictions for the ion effects on the folding of complex secondary structures is quite limited, especially in multivalent metal ion solutions. Even if the ion-dependence of the secondary structure folding can be accurately predicted, we are still unable to quantitatively understand how ions assist the extended secondary structural segments to fold up into the compact tertiary structures. Experiments have shown that nucleic acid (RNA) folding undergoes a collapse process (20–28), where the secondary structural segments (e.g., helices) approach each other to form a compact state. A rudimentary process in chain collapse is the aggregation of helices. In this study, we investigate how the metal ions assist the folding (aggregation) of two finite-length helices and how the different ionic conditions affect the helix-helix electrostatic interactions. Since helix-helix recognition is a fundamental tertiary interaction in nucleic acids, this study may provide a paradigm for the ion-assisted nucleic acid tertiary interactions.

For RNAs, various experiments have shown that multivalent ions are efficient to cause chain compaction (20–28). Among other forces, the cation-mediated electrostatic helix-helix attraction can be a possible candidate for the driving force to cause the collapse of RNAs (20–22). Experiments have been designed to probe the driving force for RNA collapse in order to shed light on the mechanism of RNA compaction (28,29). In addition, DNAs, which are long polyanionic molecules, need to condense into compact particles in cells. DNA condensation, a process similar to RNA compaction, has been studied for over two decades (30–40). Experimental and theoretical studies suggest that a possible driving force for DNA helix-helix attraction may come from the correlated multivalent cations (30–40). In addition, for other polyelectrolyte molecules, such as F-actin, filamentous viruses fd and M13, and nucleosomes, experiments have shown the similar multivalent ion-induced collapse (aggregation) (41–47).

Parallel to the experimental development, different theories and computational models have been developed to treat the helix-helix electrostatic interaction. There have been primarily two types of polyelectrolyte theories used to treat helix-helix interactions: the Poisson-Boltzmann (PB) theory (48–55) and the counterion condensation (CC) theory (56). Both theories have been very useful in predicting many thermodynamic properties of nucleic acids and proteins in salt solutions. For the helix-helix interaction, however, the PB theory, which ignores the interion correlations, predicts only repulsive force (57). In contrast, the CC theory predicts attractive forces in both monovalent and multivalent ion solutions and the CC theory attributes the attraction to the increased ion entropy (58–61).

Computer simulations have shown that attraction occurs only in the presence of multivalent ions (62–72). Models based on the simplified structures of the helices (in an asymptotically dilute salt solution) suggest that the attraction may arise from the correlated ion configurations on the surface of the approaching polyelectrolytes (73–82). When two highly charged polyelectrolytes (helices) approach each other, the strong electric field drives the ions between the helices to self-organize in order to lower the total energy. In the low-temperature limit, the strongly correlated ions can form a Wigner crystal-like configuration and the attraction between two polyelectrolytes can be strong. As the temperature is increased, the Wigner crystal-like ion configuration would be disrupted gradually and the strength of attraction decreases. Both experiments (38,46) and computer simulations (63,66,69,79) have suggested the existence of the correlated states of the ions. However, the simplified models cannot treat detailed helix structure, which can be essential in the quantitative prediction of the interhelix force. In this study, we aim to obtain a detailed microscopic picture for the ion-mediated helix-helix interaction for realistic nucleic acid helix structures.

Recently, we developed a statistical mechanical theory (denoted the tightly bound ion theory, abbreviated as TBI theory) for nucleic acid molecules. An advantage of the theory is that it can explicitly account for the correlations and the fluctuations for bound ions near the molecule surface (83). As tested against experiments, computer simulations, and the PB and the CC theories (83), the TBI theory gives reliable ion-dependent predictions such as the ion dependence of the helix-coil transition for DNA in both NaCl and MgCl₂ solutions (84). In this study, we apply the TBI theory to investigate the ion-mediated nucleic acid helix-helix interactions. Compared with the previous simplified models, the present TBI theory (83,84) is based on realistic helical structure at the atomic level and can treat the dependence on the charge, the concentration, and the size of the added salt ions in the supporting solutions. Moreover, the present model can account for the different binding modes of the bound ions and the fluctuations of the binding modes and the electrostatic and excluded volume correlations between the bound ions. The fluctuations and correlations can play significant roles for multivalent ions such as Mg²⁺ ion solution.

In this work, based on the TBI theory, we calculate the electrostatic free energy for two parallel nucleic acid helices of finite length in the presence of monovalent or divalent salt solutions. We investigate how the helix-helix interaction depends on the ion valency, ion size, ion concentration, and solvent dielectric constant. We also analyze the driving force for the ion-mediated helix-helix attraction. In addition, we compare the predictions with the experimental measurements on osmotic pressure of DNA array as well as with Monte Carlo simulations.

	THEORY AND METHODS

TOP ABSTRACT INTRODUCTION THEORY AND METHODS RESULTS AND DISCUSSION CONCLUSIONS AND DISCUSSION APPENDIX A: TREATMENT FOR... APPENDIX B: STRUCTURAL MODEL... APPENDIX C: PARAMETER SETS... APPENDIX D: MONTE CARLO... ACKNOWLEDGEMENTS REFERENCES

 
Tightly bound ions and tightly bound region
The primary motivation to develop the tightly bound ion (TBI) theory is to go beyond the mean-field approach by treating the ion correlation and fluctuation effects for polyelectrolyte systems (83,84). The basic idea is to classify two types of ions according to ion-ion correlation strength: the (strongly correlated) tightly bound ions and the (weakly correlated) diffusively bound ones. Correspondingly, the solution can be divided into two regions: the tightly bound region and the diffusively bound region. The motivation to distinguish the two types of ions is to treat them separately. For the (weakly correlated) diffusively bound ions, we use PB; for the (strongly correlated) tightly bound ions, we use a separate treatment to explicitly account for the ion correlations and fluctuations.

The tightly/diffusively bound regions are determined by the ion-ion correlation condition. We consider 1), the electrostatic correlation, measured by the parameter (r),

(1)

and 2), the excluded volume correlation, measured by the ion radius r_c and the Wigner-Seitz radius a_ws(r) (83,84). In Eq. 1, r is the position vector, zq is the charge of cations, is the dielectric constant of solute, and a_ws(r) is the Wigner-Seitz radius, which is given by the cation concentration c(r) in excess of the bulk concentration c⁰ (73),

(2)

The tightly bound region is defined as position r such that either (r) is larger than a critical value _c so the Coulombic correlation is strong, or a_ws(r) is smaller than (r_c + r) so ions are so crowded that they can easily bump into each other (83,84):

(3)

Here, r is the mean displacement of ions deviating from their equilibrium positions, and 2(a_ws(r) – r) is the closest distance of approach between two ions before they overlap.

Equation 3 gives the criteria to characterize the strong Coulombic and excluded volume correlations, respectively. The value _c is chosen to be 2.6, the critical value for the gas-liquid transition point in ionic systems (85–87), and is used as the melting point for the correlated structure according to Lindemann's melting theory (88–90).

As an approximation, we compute c(r) using the nonlinear PB equation

(4)

(5)

where denotes the ion species, zq is the charge of the ion, and c⁰ is the bulk ion concentration. The value _f is the charge density for the fixed charges, ₀ is the permittivity of free space, and (r) is the electrostatic potential at r. From c(r) and Eq. 3, we can unambiguously define the tightly bound region (83,84). In Fig. 1 we show examples of the tightly bound regions around two parallel DNA helices separated at different distances.

View larger version (45K): [in this window] [in a new window]   FIGURE 1  The tightly bound regions around two parallel 12-bp B-DNA helices in a divalent ion solution with different interaxis separations: (a) 50 Å; (b) 36 Å; and (c) 22 Å. The divalent salt concentration is 0.1 M and the cation radius is 3.5 Å. The red spheres represent the phosphate groups and the green dots represent the points at the boundaries of the tightly bound regions. The B-DNA helices are produced from the grooved primitive model (83,84,118).

(6)

where Z_M is the partition function for a given binding mode M (83,84),

(7)

Here, Z^(id) is the partition function for the uniform solution without the polyelectrolyte. The value c₊ denotes the total counterion concentration. The value N_b is the number of tightly bound ions for the mode M, and R_i denotes the position of the i^th bound ion. The configurational integral

provides a measure for the free accessible space for the N_b tightly bound ions.

Throughout the article, we use the following conventions for the notations:

• Superscripts b, d, and tot denote the tightly bound ions, the diffuse ions, and the total (bound and diffuse) ions, respectively;
  
• Subscripts E and S denote the energy (enthalpy) and the entropic free energy (–TS, where S is the entropy), respectively;
  
• Over-lines denote the statistical average over all the possible binding modes.
  

In Eq. 7, and are the energies of the bound ions, of the diffuse ions, and the entropy of the diffuse ions, respectively. In what follows, we present a brief account of the theory for the computation of and

The value in Eq. 7 is the mean Coulombic interaction energy between all the charge-charge pairs (including the phosphate groups and the tightly bound ions) in the tightly bound region (83,84),

(8)

where ₁(i) is the potential of mean force for the Coulomb interactions between charges within a tightly bound cell i, and ₂(i, j) is for the interactions between charges in different tightly bound cells i and j. For the two-helix system, i and j can be the tightly bound cells in the same helix or in different helices. Since the interactions between the cells in different helices is dependent on the interhelix separation x, the potential of mean force ₂(i, j) is x-dependent if i and j are for different helices.

₁(i) and ₂(i, j) are calculated as the average over all the possible positions R of the tightly bound ions in the respective cells (83,84),

(9)

where u_ii represents the Coulomb interactions between the charges in cell i, and u_ij the interactions between the charges in different cells i and j.

The value in Eq. 7 is the free energy for the electrostatic interactions between the diffusive ions and between the diffusive ions and the charges in the tightly bound region. The value in Eq. 7 is the entropic free energy of the diffusive ions. With the mean-field approximation for the diffusive ions (91,92), and can be calculated from these equations (83,84),

(10)

(11)

where '(r) is the electrostatic potential for system without the diffusive salt ions. The value '(r) is used here because (r)–'(r) gives the contribution to the electrostatic potential from the diffusive ions. The values (r) and '(r) are obtained from the nonlinear PB (Eq. 4, with salt) and the Poisson equation (salt-free), respectively.

From the above expressions, we can calculate the partition function Z, from which we obtain the electrostatic free energy G for the system:

(12)

Moreover, from the mode probability

(13)

we can calculate the mean electrostatic interaction

(14)

and the entropic free energy

(15)

Furthermore, we can decompose into contributions from the diffusive ions

(16)

and from the (tightly) bound ions

(17)

The above classification for the enthalpy and entropy is based on several approximations. First, the electrostatic energy and entropy of the ions in solution are coupled to the dielectric property of the solvent, which, in the continuum solvent model, is represented by the dielectric constant. The dielectric constant is intrinsically related to the solvent entropy. Therefore, the energy of the diffuse ions, implicitly contains the solvent entropy. Second, for each given ion-binding mode, the energy of the bound ions is computed as an average over the different configurations of the bound ions within the respective tightly bound cells. Therefore, contains the configurational entropy effect for the tightly bound ions. Besides the solvent entropy, the entropic free energy for the diffuse ions accounts for the translational entropy of the diffuse ions, and for the bound ions includes the combinatory entropy of the different binding modes as well as the configurational entropy of the tightly bound ions.

Numerical computation
The computation of the electrostatic free energy for two helices with the TBI theory can be summarized in the following steps (83,84).

First, we solve the nonlinear PB (Eq. 4) to obtain the ion distribution c(r) for the two-helix system (83,84), from which we determine the tightly bound region using Eq. 3.

Second, we compute the pairwise potential of mean force ₁(i) and ₂(i, j) for different tightly bound cells (i and j values) from Eq. 9 by averaging over all the possible positions of the tightly bound ions inside the respective tightly bound cells. In the calculations for ₁ and ₂, the excluded volume effects between ions and between ions and helices are accounted for by using a Lennard-Jones potential:

where r is the distance between the centers of ions and r₀ is the sum of the radii for the two spheres. Clearly the potentials of mean force are dependent on the detailed structure of the molecules (helices) and the size (and shape) of the ions. The calculated potentials of mean force are tabulated and stored for the calculations of partition function.

Third, we enumerate the binding modes and for each mode, we calculate and from Eqs. 8–11. Summation over the binding modes gives the total partition function Z (Eq. 6), from which we can calculate the electrostatic free energy for the helices. For long helices, there are a large number of binding modes, therefore, a brute-force exhaustive enumeration for all the modes is practically impossible. So we have developed a special numerical method to efficiently treat long helices by using a coarse-grained approximation for the low-probability (i.e., small p_M) modes. The details of the method are given in Appendix A.

	RESULTS AND DISCUSSION

TOP ABSTRACT INTRODUCTION THEORY AND METHODS RESULTS AND DISCUSSION CONCLUSIONS AND DISCUSSION APPENDIX A: TREATMENT FOR... APPENDIX B: STRUCTURAL MODEL... APPENDIX C: PARAMETER SETS... APPENDIX D: MONTE CARLO... ACKNOWLEDGEMENTS REFERENCES

 
In this work, we focus on DNA helices. The same method and the physical mechanisms revealed here may also be applicable to RNA helices. To investigate the possible attraction between finite-length helices, we choose a model system with two parallel helices, each of length equal to one helix pitch. We compute the free energy G(x) of the system as a function of the interhelix distance x. We use G(x) to denote the free energy change of the system as the helices approach each other from far apart,

(18)

where G(x) and G() are determined from Eq. 12. We treat x = 50 Å as the outer reference distance (for separated DNA helices), i.e., we treat G(50 Å) as G(). Our control test shows that the results are not sensitive to the choice of the large reference distance 50 Å.

In this section, we use the TBI theory to analyze the general mechanism for the possible attraction between two helices in the presence of divalent salt ions. We then investigate how the ion valency, ion concentration, ion size, and solvent dielectric constant affect the helix-helix interactions. In the next section, we calculate the osmotic pressure of DNA aggregated array and compare our predictions with the experimental measurements.

Driving force for ion-mediated helix-helix attraction
The predicted free energy G(x) is shown as a function of the interhelix separation x in Fig. 2 for 0.01 M divalent salt. The free energy landscape G(x) shows that the two helices attract each other for x 28 Å and the most stable helix-helix configuration occurs at x 28 Å, where the free energy minimum is located. The results from the free energy landscape are in accordance with the previous findings that divalent ions can induce macroion-macroion attractions (62–82).

View larger version (15K): [in this window] [in a new window]   FIGURE 2  The free energy G(x), the electrostatic energy GE(x), and the entropic free energy GS(x), calculated from the TBI theory, as functions of the interhelix separation x for 0.01 M divalent salt concentration. The minimum contact distance for parallel helices is x 20 Å.

(19)

from Eqs. 14 and 15, respectively. Fig. 2 shows that, when the two helices approach each other from , G_E(x) decreases until x 22 Å. Thus, G_E(x) tends to give an attractive force for x 22 Å. On the other hand, as the two helices approach each other, G_S(x) increases and thus tends to give a repulsive force. Therefore, the helix-helix attraction comes from the electrostatic energy G_E(x); i.e., the attraction is enthalpically driven.

To understand the physical mechanism for the interhelix interaction, we examine the distance-dependence of the (bound) ion distribution and the free energy of the system. For large helix-helix separation, the helices and the associated bound ions are nearly independent with each other and the interhelix correlation is weak. The close approach between the two helices causes the following two effects.

First, in response to the enhanced electric field, more ions become tightly bound, causing a stronger charge neutralization (charge screening) for the polyanionic helices. This would weaken the average tendency of the repulsion between the two likely charged helices.

Second, the correlation between the tightly bound ions on the different helices becomes stronger. The (correlated) bound ions can self-organize to form the correlated low-energy states. Such correlated states can reach much lower energies than the (uncorrelated) mean-field states. So the interhelix correlation can cause a rapid decreasing of the electrostatic energy G_E(x) as x is decreased. This leads to an attraction between the helices.

In the correlated low energy states, the bound (divalent) ions tend to reside in the region of the lowest electric potential produced by all the bound and the diffusive ions as well as the phosphates on both helices. Shown in Fig. 3 are representative (most probable) low-energy binding modes and the mean charge neutralization for different phosphates for x = 24 Å and 50 Å. From the (most probable) bound ion distribution for x = 24 Å, we find that the strongest correlation occurs between ions bound to phosphates that are directly facing each other, such as phosphates P₁ and P₂ in Fig. 3. To more efficiently lower the energy, ions have higher tendency to bind to these phosphates and the ion-binding to these phosphates is highly correlated. The most probable (lowest energy) binding mode gives a strong attractive force. For the binding mode depicted in Fig. 3 a, phosphate P₁ bound by a divalent cation would attract the negatively charged unbound phosphate P₂. The statistical ensemble-averaged, mean-binding mode indicates that ions prefer to bind to the region between the two helices so as to interact with both helices strongly.

View larger version (25K): [in this window] [in a new window]   FIGURE 3  Illustrations for divalent ion binding to two parallel DNA helices for different interaxis separations: (a) x = 24 Å and (b) x = 50 Å. The circles represent the phosphates; the numbers in circles are the mean charge neutralization fraction of the tightly bound ions; the shaded circles represent the ion-binding phosphates for the most probable mode (with highest probability pM). Here, the divalent ion concentration is 0.01 M.

In addition, as the helices approach each other, more ions become tightly bound and hence the ion entropy would decrease and the entropic free energy G_S(x) would increase. The competition between the decreasing (attractive) G_E(x) and the increasing (repulsive) G_S(x) results in an overall net decreasing (attractive) G(x).

In the following sections, we show how the ion valency, ion concentration, ion size, and solvent dielectric constant affect the energy component G_E(x) and the entropy component G_S(x) and the shape of the free energy landscape for the interhelix interactions.

Ion valency effect on helix-helix interaction
Monovalent and divalent ions have contrasting effects on the helix-helix interaction. The values G(x) of the systems with monovalent and divalent salts are shown in Fig. 4, a and b, as functions of the interhelix distance x. As a comparison, the predictions from the TBI theory and from the PB theory are both shown in the figures. For monovalent salt, the TBI theory and the PB give nearly identical results for G(x), and both PB and TBI theories predict only repulsive helix-helix interaction. For divalent salt, however, the TBI theory predicts an effective attraction between the two helices, while PB predicts a repulsive force. The different results for monovalent and divalent ions clearly show the role of the ion valency in helix-helix interaction. The findings from the TBI theory agree with the findings from the previous experiments (30–34), computer simulations (62–72), and theoretic analysis (73–82).

View larger version (38K): [in this window] [in a new window]   FIGURE 4  (a) The free energy G(x) as a function of interhelix separation x for different monovalent salt concentrations: 0.01 M, 0.1 M, and 0.6 M (from top to bottom). (Dotted lines) Poisson-Boltzmann theory. (Solid lines) TBI theory. Some parts of the dotted lines are not visible because they are underneath the solid lines. (b) The electrostatic free energy G(x) as a function of interhelix separation x for different divalent salt concentrations: 0.001 M, 0.01 M, and 0.1 M (from top to bottom). (Dotted lines) Poisson-Boltzmann theory. (Solid lines) TBI theory. The inset shows G(x) at high divalent salt concentration. (c) The electrostatic energy GE(x) (solid lines) and entropic free energy GS(x) (shaded lines) for different monovalent salt concentrations. (d) The electrostatic energy GE(x) and entropic free energy GS(x) for different divalent salt concentrations. The inset shows GE(x) and GS(x) at high divalent salt concentrations. The minimum contact distance for parallel helices is x 20 Å.

To gain more detailed insights into the ion valency effect on helix-helix interaction, we analyze the components of the free energy G(x): the electrostatic energy G_E(x) and the entropic free energy G_S(x) (see Eq. 19). Fig. 4, c and d, show G_E(x) and G_S(x) as functions of the interhelix separation x for monovalent and divalent salts, respectively.

For the entropic free energy G_S(x), as the two helices approach each other, in general, the stronger electric field (for smaller x) around the polyanionic helices drives more cations to bind to the helix (especially between the two helices), which results in a decrease in the entropy of the ions and thus an increase in the entropic free energy G_S(x). In particular, when the helices are very close to each other (x 20–30 Å), the strong electric potential between the polyanionic helices would significantly restrict the freedom of the bound ions and thus cause a rapid decrease of ion entropy and hence a rapid rise of G_S(x).

For the enthalpic free energy G_E(x), as discussed above, the (correlated) bound cations between the helices would self-organize and interact more strongly with the two negatively charged helices, which lead to a decrease in the electrostatic energy G_E(x). For smaller x, G_E(x) decreases more rapidly with x due to the large number of the bound ions between the helices and the smaller distance between the cations and the negatively charged phosphates. However, for very small helix-helix distance (x 23 Å) when helices are tightly packed, the volume exclusion between the bound ions and helices causes a slight increase in G_E(x). In fact, when the helices are very closely packed, a fraction of bound ions may be pushed out from the strongly correlated interhelix region due to the volume exclusion and the ion-ion electrostatic repulsion.

From Fig. 4, c and d, we find that the TBI theory gives good agreement with the Monte Carlo simulations for G_E(x) (see Appendix D for the details on the Monte Carlo simulations). The competition between G_E(x) and G_S(x) results in the free energy landscape G(x) = G_E(x) + G_S(x). From Fig. 4, we find the following distinctive features for the free energies of the system in monovalent and divalent ion solutions:

1. Because the (bound) divalent ions carry higher charges, for small x, the decrease in the electrostatic energy G_E(x) for divalent ions is more pronounced than for monovalent ions, i.e., as two helices approach each other, G_E(x) decreases faster in divalent ion solution than in monovalent ion solution.
   
2. The divalent ions, which carry higher charges, are more efficient in charge neutralization than monovalent ions. As a result, G_S(x) for divalent ions is much smaller than for monovalent ions, i.e., charge neutralization of the helix requires less divalent ions than monovalent ions. Consequently, as the helices approach each other, divalent ions have a smaller entropic decrease (i.e., smaller free energy G_S(x) increase) than monovalent ions.
   
3. The competition between G_E(x) and G_S(x) results in an apparent attractive force that tends to bring the two helices together in divalent ion solution in order to lower the free energy. However, in monovalent ion solution, no such attractive force exists, even for high salt concentrations. We note that the analysis here is also in qualitative accordance with the previous simulations for two like-charged spheres in monovalent and divalent salts (64,65).
   

Ion concentration effect on helix-helix interaction
Fig. 4, a and b, shows that the ion concentration has great influence on the helix-helix electrostatic interactions. We use c to denote the ion concentration. For monovalent ions, the increase of c significantly weakens the repulsion between DNA helices (see Fig. 4 a). This is because higher c gives lower entropic cost for ion-binding and hence more bound ions. A larger number of bounds ions means stronger ionic screening (neutralization) for the helices and thus weaker helix-helix repulsion.

For divalent ions, the relationship between G(x) and c is more complicated (see Fig. 4 b). As the divalent ion concentration c is increased, the attraction between the helices becomes stronger. However, as c continues to increase and exceeds a certain critical value c*, further addition of salt ions would weaken the attraction. For two helices of 10-bp in divalent ion solution, we find that c* 0.1 M. The salt-enhanced attraction at low c and the salt-weakened attraction at high c may correspond to the salt-induced bundle formation and bundle resolubilization for DNA helices (36,37,93) and for other polyelectrolyte molecules (43–47).

Also shown in Fig. 5 are the minimum free energy G_min, the stable interhelix separation x_min, and the mean equilibrium helix-helix distance computed as

as functions of divalent salt concentration c. We find that G_min, x_min, and all show nonmonotonic behavior: to decrease with the increase of c (<c*), and to slightly increase with the addition of salt when c exceeds a critical value c*. The predicted stable helix-helix separation x_min shown in Fig. 5 as well as the salt dependence of x_min agree with the experimentally measured results for the interhelical spacing of an hexagonal DNA array (36,93,97) (details in Comparisons with Experimental Measurements).

View larger version (9K): [in this window] [in a new window]   FIGURE 5  (a) The minimum free energy Gmin as a function of divalent salt concentration for variant cation radii: 4.5, 3.5, and 2.5 Å (from top to bottom). (b) The equilibrium interhelix separation (solid lines) and xmin (dotted lines) corresponding to minimum free energy Gmin as functions of divalent salt concentration for different cation radii: 4.5, 3.5, and 2.5 Å (from top to bottom). (c) The mean charge neutralization fraction fb of the tightly bound ions per phosphate as a function of divalent salt concentration for different cation radii: 4.5, 3.5, and 2.5 Å (from the top to bottom). The interaxis separations are 50 Å (solid lines) and 26 Å (dotted lines), respectively.

The entropic free energy G_S(x)
For higher ion concentration c, because of the lower entropic cost for ion-binding, more ions are bound to the helices and the charge neutralization is stronger. Therefore, as the two helices approach each other (i.e., x becomes smaller), the increase in the strength of the negative electric field (around the helices) and hence, the increase in the number of bound ions, are less pronounced for higher c. Consequently, the decrease of the entropy (and the increase of the entropic free energy G_S(x)) for smaller x would be less significant for higher c,

(20)

See Fig. 4 d for the divalent ion concentration from 0.001 M to 0.1 M.

For very high divalent ion concentration c0.1 M, the ionic entropy decrease upon binding is small and thus more ions would bind to the helices, resulting in possible full-neutralization or over-neutralization (charge inversion); see Figs. 5 and 6. In such cases, as the helices approach each other, the strong Coulomb repulsion between the large number of bound ions causes the ions to avoid each other, especially for the bound ions that reside between helices. Such effect can cause a saturation in the c-dependence of the number of the bound ions and of the free energy G_S(x); see Fig. 4 d (inset). In fact, for the strongly overneutralized case, we find that the amount of the tightly bound ions (especially the tightly bound ions in the interhelix region) slightly decreases when the two helices become closer. Such a prediction is consistent with the results in a recent computer simulation (71).

View larger version (17K): [in this window] [in a new window]   FIGURE 6  (a) The entropic free energies of diffusive ions (black lines) and the entropic free energy of the tightly bound ions (shaded lines) as functions of interhelix separation x for different divalent salt concentrations. (b) The mean charge neutralization fraction fb of the tightly bound ions per phosphate as a function of interhelix separation x for variant divalent salt concentrations. (c) The mean volumes vb (in Å3) of the tightly bound region per phosphate versus interhelix separation x for different divalent salt concentrations. The minimum contact distance for parallel helices is x 20 Å.

From Fig. 4 d (inset), we also find that for and x 30–40 Å, G_S(x) decreases slightly when the two helices are closer, indicating a (slight) entropic attractive force:

(21)

The total entropic free energy G_S(x) is the net result from the two different types of ions (the tightly bound ions and the diffusive ions). We would like to understand how the different types of ions play different roles in the helix-helix folding process and what results in the slight entropic attraction. In what follows, we examine the component entropic free energies (Eq. 16) of the diffusive ions and (Eq. 17) of the tightly bound ions separately.

Diffusive ions
As shown in Fig. 6 a, for low ion concentration c 0.1 M, as the two helices become closer, of the diffusive ions decreases, resulting in an apparent attractive force between the helices. This is because as the helices approach each other, the aggregation of the phosphate charges causes a stronger cation-attracting field and hence more bound ions (see Fig. 6 b) and stronger charge neutralization. As a result, the diffusive ions around the helices are less confined and have larger translational entropy (i.e., lower free energy ) for smaller x:

(22)

For high salt concentration, the helices are fully or over-neutralized. For smaller x, the strong repulsion between the bound ions would inhibit further increase of ion-binding. As a result, the c-dependence of 22 becomes saturated; see the curves for c = 0.1 M and 0.2 M divalent ion concentrations.

Bound ions
The apparent attractive entropic force shown in Eq. 21 can be understood from the entropic free energy for the tightly bound ions. The entropy of the tightly bound ions has two components: the entropy for the partitioning of the different binding modes and the entropy N_b(x)S_b(x) of the bound ions in tightly bound cells. Here N_b(x) is the number of the bound ions and S_b(x) < 0 is the entropy change for an ion to transfer from the diffusive region into the tightly bound region. The value S_b(x) is determined by the bulk ion concentration c and the volume v_b of the tightly bound region: S_b(x) k_B ln(v_b(x)c).

Fig. 6 c shows that as the helices approach each other, the volume of a tightly bound cell v_b(x) varies nonmonotonically: v_b(x) increases for x 30 Å and decreases for x 30 Å. We note that the variation of v_b(x) with x is similar to that of the volume of the condensed ions in CC theory (58–61). The increase of v_b(x) for x 30 Å tends to cause a smaller |S_b|. In contrast, as the helices approach each other, N_b(x) increases, which tends to cause a larger |S_b| (see Fig. 6 c).

For low (divalent) ion concentrations (c 0.001–0.01 M in Fig. 6), the N_b(x)-effect dominates over the v_b(x)-effect, resulting in a net increase of |S_b| (i.e., an increase in the entropic free energy ) as x decreases. Therefore, for low ion concentrations, the profile gives no attractive force.

For high (divalent) ion concentrations (c 0.1–0.2 M in Fig. 6), as x ( 30 Å) decreases, the increase in N_b is small (see Fig. 6 b) and the v_b(x)-effect dominates. As a result, |S_b| decreases (i.e., decreases) with x, causing an apparent effective entropic attractive force (58); see Fig. 4 d, inset, and Eq. 21. However, we note that, as we show in the following section, the above entropic effect is weaker than the effect from the electrostatic energy G_E(x).

The electrostatic energy G_E(x)
As shown in Fig. 4, b and d, G_E(x) increases with increased ion concentration c:

(23)

The salt dependence of electrostatic energy G_E(x) is a result of the competition between the ion-helix attraction and the ion-ion repulsion. For higher ion concentration, the helices are strongly neutralized (due to more bound ions). For example, the tightly bound ions neutralize 97% and 70% phosphates in 0.1 M and 0.01 M divalent solutions, respectively, for two separated helices. The strong ion-helix attraction dominates the electrostatic energy G_E(x).

In a high-c solution, the helices are already significantly neutralized before they approach each other. Therefore, as the two helices become closer, the increase in N_b in a low-concentration c solution is much more pronounced than that in a high-c solution. Consequently, the enhancement in the ion-helix attraction is much more significant for a low c solution. Moreover, in a high-c solution, when the helices, bound by a large number of tightly bound ions, approach each other (e.g., for x 20–30 Å), the Coulomb repulsion between the bound ions would further damp the interhelix attraction. Such effect is particularly strong for the overneutralized case. As a result, G_E(x) is higher for higher ion concentration.

The total free energy G(x)
The combination of G_E(x) and G_S(x) gives the total free energy G(x) = G_E(x) + G_S(x). For the divalent ions, from the folding energy landscape G(x) shown in Fig. 4 c, we find that there exists a critical ion concentration c* (0.1 M for the two-helix system), which is related to the charge over-neutralization as shown in Fig. 5. A higher ion concentration c would strengthen (weaken) the interhelix attraction to stabilize (destabilize) the two-helix system for c < c* (c < c*). The TBI theory shows that the full-neutralization by the multivalent ions is the starting point of the weakening of the helix-helix attraction. This is in accordance with the previous experimental measurements (43,93), theoretical predictions (94), and computer simulations (71).

In our TBI theory, the ions are assumed to have the charges of bare ions (i.e., completely dissociated ions in solution). Based on this assumption, the charge inversion occurs (83) and may be responsible for the weakening of ion-mediated helix-helix attraction at high concentration. As a caveat, we note that the charge inversion for a polyanion may be accompanied by the association of the co-ions (chloride ions) around the bound cations. Thus, a clearcut boundary for the charge inversion may not exist (37,47). Recently, a different possible mechanism for the resolubilization of DNA bundles has been proposed (93,95): at high concentration, multivalent ions cannot dissociate completely from the co-ions (chloride ions), thus the clusters of the multivalent ions with the associated chloride ions would carry less charges than bare ions. As a result, ions would become less effective to mediate the helix-helix attraction and thus helix-helix attraction is weaker.

Ion size effect on helix-helix attraction
Several experiments have demonstrated the important role of ion size in RNA collapse transition (22), DNA condensation (39), and the compaction of other polyelectrolyte molecules such as anionic rodlike M13 and fd viruses (44,45). In this section, we focus on the physical mechanism and quantitative prediction for the ion size effect. To be specific, we use different ion radii 2.5 Å, 3.5 Å, and 4.5 Å. As shown in Fig. 7 a, for small ion radius, the TBI theory predicts a smaller distance (x_min and ) between the helices, a lower minimum free energy (G_min), and a stronger helix-helix attraction. In what follows, we discuss the ion size-dependence of the free energy landscape G(x).

View larger version (28K): [in this window] [in a new window]   FIGURE 7  (a) The free energy G(x) as a function of interhelix separation x for different divalent ion radii: 4.5 Å, 3.5 Å, and 2.5 Å (from top to bottom). The divalent salt concentration is 0.01 M. (Dotted lines) Poisson-Boltzmann theory. (Solid lines) TBI theory. (b) The electrostatic energy GE(x) (solid lines) and entropic free energy GS(x) (shaded lines) for different divalent ion radii. The divalent salt concentration is 0.01 M. (Solid lines) TBI theory. (Symbols) Monte Carlo simulations. (c) The mean charge neutralization fraction fb of the tightly bound ions per phosphate as a function of interhelix separation x for different cation sizes. (d) The mean volume vb (in Å3) of the tightly bound regions per phosphate as a function of interhelix separation x for different divalent cation sizes. The minimum contact distance for parallel helices is x 20 Å.

(24)

In addition, Fig. 7 b shows that for small x (23 Å), G_E(x) increases rapidly with a decreasing x. This is because the bound ions are pushed out from the strongly correlated interhelix region when the helices are very close. Such effect is stronger for more bulky ions because of the larger excluded volume, as shown in Fig. 7 b.

In Eq. 24, the ion size effect is stronger for the entropic G_S(x) than for the enthalpic G_E(x), and the net free energy G(x) = G_S(x) + G_E(x) gives a stronger decrease for smaller ions (see Fig. 7 a):

(25)

As a result, the helices are more stabilized by smaller ions.

Moreover, because smaller ions can make closer contact with the helices, the helices can be more closely packed at a smaller stable distance, causing a smaller interhelix distance and lower free-energy minimum (see Figs. 7 a and 8, a and b):

View larger version (11K): [in this window] [in a new window]   FIGURE 8  (a) The minimum free energy Gmin versus divalent ion radius for different divalent salt concentrations: 1 mM, 10 mM, and 100 mM (from top to bottom); (b) The equilibrium interhelix separation (solid lines) and xmin (dotted lines) corresponding to minimum free energy Gmin as a function of cation radius for different divalent salt concentrations: 1 mM, 10 mM, and 100 mM (from top to bottom).

The above results are in accordance with the recent experiments (22,39,44,45). The sensitivity of G_m and x_min (or ) on the ionic radius might also (partially) account for the experimental finding that some divalent ions can condense DNA molecules whereas other divalent ions cannot. For example, at room temperature and in aqueous solution, Mg²⁺ ions cannot condense DNA, while Cu²⁺ ions can induce DNA condensation (30,34,96–99). Mg²⁺ ions have large hydrated radii (4.5 Å) (83,84,100), thus the ion-mediated attraction is not strong and the stable helix-helix distance is not small enough to form a compact helix-helix collapsed structure. Another possible reason for the different abilities of different divalent ions in condensing DNA is the site-specific ion binding. For example, Mg²⁺ ions bind to phosphate groups, while Mn²⁺, Cu²⁺ ions bind to both phosphate groups and bases (1,98,99). The binding of ions to bases may favor condensed DNA because of the stronger helix-helix attraction due to the specific charge distribution on the helix surface (77,101) and the stronger flexibility for helix bending (98,99,102,103). The TBI model allows the bound ions to move around the phosphates and inside the grooves (83,84). However, the model does not treat the site-specific ion-binding.

The present TBI model does not treat the possible ion desolvation upon binding. In the folding of nucleic acid molecules, the polyanionic chain can fold into compact structures, some of which can form electrostatically attractive pockets for ion binding (6,10). In these pocket regions and in regions with ions bound to specific charged groups, ions can be desolvated. A dehydrated ion can have a much smaller size than a hydrated one. For example, the radius of a Mg²⁺ ion can change from 4.5 Å (hydrated) to 0.65 Å (fully dehydrated) in the dehydration process (6,83,84,100). The (small) dehydrated cations can approach the surrounding negatively charged groups at a much smaller distance and the interaction is much stronger.

In addition, the dielectric effects (e.g., the dielectric discontinuity (83,102) and the dielectric saturation (104)) result in a stronger charge-charge interaction owing to the decreased effective dielectric constant (83,102,104,105) (as compared to the dielectric constant bulk solvent). To account for the dielectric effect, the distance-dependent dielectric constant (r) has been introduced (104,105). The use of (r) can cause a stronger charge-charge interaction, especially between the nearby charges, and a stronger ion-ion correlation