Phase behavior of DNA solutions in the presence of salts

Fluid Phase Equilibria 242 (2006) 103–109

Phase behavior of DNA solutions in the presence of salts Khaled A. Mahdi ∗ , Zuzana Drobiova Chemical Engineering Department, Kuwait University, P.O. Box 5969, Safat 13060, Kuwait Received 8 August 2005; received in revised form 18 January 2006; accepted 19 January 2006

Abstract The thermodynamic behavior of DNA solutions in the presence of salts is studied using a new phenomenological model of Helmholtz free energy of the system. The activity coefficient of the DNA–salt solution system is calculated and its asymptotic behavior suggests the formation of gel precipitate. Phase instability analysis of the derived free energy predicts multivalent salt concentration at which DNA precipitates. However the model is incapable to predict the redissolution of DNA at higher salt concentration. © 2006 Elsevier B.V. All rights reserved. Keywords: DNA precipitation; DNA–salt solution; Multivalent salt; DNA activity coefficient

1. Introduction Deoxyribonucleic acid, known as DNA, is a naturally existing biological polyelectrolyte. In solution, DNA dissociates into long anionic chain, which is neutralized by the presence of counterions. DNA chains exhibit stretched, rod-like, conformation in the presence of low concentration of multivalent salt (valence of two and higher), compacted form when ionic concentration is increased leading to DNA aggregation and precipitation. As more multivalent ions are added in the solution, the precipitated DNA chains become flexibly stretched and dissolve back [1–5,9]. The scope of this paper is to construct a thermodynamic model of DNA solutions in the presence of multivalent biological salts such as spermidine and spermine [4]. The model accounts for significant energetic interactions among constituents of the DNA–salt solution. The interaction of DNA with complex multivalent molecular ions is of considerable interest due to its importance in many biochemical applications and the significant role of biological ions interactions with nucleic acids, especially for DNA compaction [3–5]. When a DNA molecule, or any polyelectrolyte chain, compacts, several forces act against compaction such as bending and entropy of mixing, while the electrostatic attraction between positive ions and anionic chains favor compaction.

∗

Corresponding author. Tel.: +965 4985618; fax: +965 4839498. E-mail address: k [email protected] (K.A. Mahdi).

0378-3812/$ – see front matter © 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.fluid.2006.01.011

In the quest to understand the energetic contributions that favor compaction, experiments done by Delsanti on polystyrene sulfonate and Pelta on DNA [2–4], both polyelectrolyte solutions, proved that precipitation or aggregation is independent of the chain flexibility. Hence, the high charge density, a characteristic of polyelectrolyte system, is responsible for the observed phase behavior. Therefore, understanding the nature of the longand short-range electrostatic interactions among charged species in the solution should explain the thermodynamic of such system [4,7–9]. At different salt concentration levels, charged chains phase separate then redissolve to phase separate again from the solution. The first phase segregation is due to polyions–ion mediated attraction [9,14,15,18,20] and the second phase segregation is due to solvent–monomer interaction [12,13]. Several mechanisms to explain similarly charged polyions attraction have been proposed, most of them are based on the colloidal charge fluctuations, either charge fluctuations of counterions in the close vicinity of charged chain, or strong counterion correlations [5–7]. Theoretically, polyelectrolytes are treated within the scope of mean field theories such as Poisson–Boltzmann theory. This theory predicts that the effect of small ions in the solution is two-fold. First, the counterions renormalize the effective charge on the polyion due to ion condensation, as suggested by Manning [12]. Second the smaller free ions of negative and positive valences act to screen the charge of between polyions in the solution. Although the interaction between two similarly

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charged polyions reduces in magnitude, it remains repulsive. In other words, there is no strong attractive interaction between the molecules to cause precipitation and compaction that can be predicted by mean field theory [8]. Nevertheless, this contribution of long-range electrostatic interaction suggested by mean field theory must not be neglected in the free energy of the system where other strong attractive terms are included. Polyelectrolyte solution free energy consists of different contributions associated with specific intermolecular interactions, long- and short-range interactions. The long-range electrostatic contribution best described by Debye–H¨uckel theory [10,16]. This mean field theory treats ionic species in simple electrolyte solution as point particles. Mathematically, the structure function is expressed as delta function. In case of polyionic species, there exists a definite correlation function among the charges along the connected and flexible chain. The random phase approximation (RPA), a statistical mechanical method proposed by Doi and Edwards [11] characterizes the fluctuations of the system species and accommodate the structure function of the charged chain [12]. The free energy derived by RPA predicts phase segregation in salt free polyelectrolyte solution qualitatively but not quantitatively [17]. Strong attractive terms must be included in the free energy to better predict phase segregation [18]. One possible source of electrostatic attraction is induced dipole interactions between the fluctuating ions surrounding the charged chain [9]. This is conceptually similar to the London dispersion force arising from the mutual polarization of electron clouds. However, analysis of the balance between polyelectrolyte repulsive forces (mainly entropic) and attractive forces due to London dispersion interactions leads to the conclusion that dispersion forces would have to be two to five times larger to induce DNA precipitation [5]. Nagvekar et al. [13] describes the short-range interaction simply using a local composition model of the non-random two liquid (NRTL) to find counterion activity coefficients and the osmotic coefficients in polyelectrolyte solutions with and without added salts at finite concentrations. Although successful to explain the phase segregation at higher salt concentrations (i.e. segregation after redissolution), the model is incapable to describe the precipitation at lower salt concentration (i.e. initial precipitation prior to redissolution). However, the short-range attraction is expected to contribute to the precipitation but not significantly. A somewhat different attraction theory, which utilizes the concept of correlated ionic fluctuations has been developed by Rouzina and Bloomfield [21,22] and Solis and Olvera de la Cruz [17,23] independently. The attraction contribution emphasizes the pseudo-two-dimensional character of the counterion distribution very close to the highly charged DNA surface. When two DNA chains approach each other closely, the charges adjust themselves such that the positive charge is opposite to negative, leading to a net attraction. This form of attraction is related to forces explored in earlier work on charged planar surfaces, which suggested that, at low enough temperature, counterions should form a self-ordered two-dimensional Wigner crystal [18,20,23].

2. Thermodynamic modeling of DNA–salt solution DNA–salt solution system consists of the monomers of the flexible anionic chains (assumed Gaussian), counterions, salt cations and anions all species are dissolved in a solvent (a structure-less medium). The species densities are expressed as follows: ρm , ρp , ρc and ρa are monomer, counterion, salt cation and anion densities. Due to ion condensation [12], a fraction of the counterions and a fraction of the salt cations condense on the chain, in other words, they always exist in the close vicinity of the backbone of the charged chain. We define Γ p as the fraction of salt cations condensed per chain and similarly for counterions, Γ c . To distinguish the free and condensed species, we use the superscript f and c for free and condensed species, respectively. Hence, the total density of counterions is ρc = ρcf + ρcc and ρp = ρpf + ρpc for salt cations. The system Helmholtz free energy, referred to as the free energy in this paper, A, is the sum of the ideal (entropic) and the following contributions: (i) Hard core contribution, Ahc . (ii) Long-range electrostatic contribution of free ions, ADH . (iii) Long-range electrostatic contribution of ionic chains, ARPA . (iv) Short-range electrostatic contribution, ANRTL . (v) Electrostatic association contribution, Aassoc . The ideal mixing term results from the combinatorial arrangement of charged chains, free positive ions and negative ions is explicitly expressed as: Aid ρm ρm = ln + ρcf ln ρcf + ρpf ln ρpf + ρa ln ρa kB TV N N

(1)

where kB , T and V are Boltzmann constant, temperature and volume, respectively. The hard core contribution indicates that the particles have a finite size and are not actually point particles. This term is expressed as: ⎞ ⎛  

Ahc 3 bj3 ρj ⎠ − ρm ln 1 − am ρm = − ρj ln ⎝1 − kB TV j

j

(2) 3 where bj3 is the hard core size or cell volume of species j and am is cell volume of monomer [17,18]. The free energy of small free ions is found using the full Debye–H¨uckel expression [16], which accounts for the effect of particle size on the electrostatic charge density. This contribution describes the screening of long-range interactions resulted from the presence of free small ions in the system. It is given by:

ADH ln(1 + κb) − κb + 0.5(κb)2 = kB TV 4πb3

(3)

where κ is the inverse screeninglength or the square root of ionic strength defined as κ = 4πlB z2i ρi and b refers to the small ions diameter.

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Mahdi and Olvera de la Cruz [17] demonstrated a scaling argument on the density limit of RPA applicability in polyelectrolyte solutions. The result of RPA derived for polyelectrolyte systems is also applied to DNA–salt systems accounting for the long-range electrostatic interactions between the chains and ions. The chains at a finite addition of salt resume a somewhat flexible Gaussian conformation. Therefore, the structure function of a Gaussian chain, approximated by the Debye function can be expressed in a Fourier representation as: ˆ mm (k) = D

Nρm 1 + (1/2)R2g k2

(4)

where R2g is the square of the radius of gyration given by a2 N/6, N is the number of monomers and k is the wavelength. Small ions have structure functions equal to their mean density ρi and the correlating function is the reciprocal of this mean density. Following the formulation of RPA outlined in the appendix, we construct neutral and charged correlating matrices. The neutral correlation matrix Sˆ ij0 (k) has four components each corresponds to a species in the system: 0 (k) = Sˆ 11

1 ˆ mm (k) D

(5)

1 0 (k) = Sˆ 22 ρc

(6)

1 0 (k) = Sˆ 33 ρp

(7)

1 0 (k) = Sˆ 44 ρa

(8)

The long-range electrostatic interactions among the four components in the system is simply the columbic potential, in our case the monomers, the counterions, the salt anions and cations. The elements of the electrostatic matrix Sˆ ije (k) are given by: zi z j Sˆ ije (k) = 4πlB 2 (9) k where zi and zj are the valence of species i and j. The matrix element Sˆ ij (k) is the sum of neutral Sˆ ij0 (k) and electrostatic Sˆ ije (k) contributions. The system matrix is a 4 × 4 matrix with its elements: zi z j (10) Sˆ ij (k) = Sˆ ij0 (k) + 4πlB 2 k Substituting Sˆ ij0 (k) and Sˆ ije (k) into Eq. (A.12), the RPA contribution to the free energy is obtained as:

 2 κi2 Nκm 1 ARPA ln 1 + 2 + 2 = kB TV k k (1 + s2 k2 ) 2(2π)3 −

2 κi2 Nκm − dk k2 (1 + s2 k2 ) k2

(11)

 2 f 2 = 4πl z2 ρ and κ2 = 4πl where κm B m m B i i zi ρi are the ionic strength of the monomers and all free ionic species, respectively, and s2 = 1/2R2g = a2 N/12. The last terms are subtracted from the logarithmic term to exclude self-interaction terms,

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which cause the divergence of the integration in Eq. (11) at long wavelengths. The electrostatic free energy by RPA upon the integration of (11) is: −1 ARPA = kB TV 48πs3 √  √ 2 × 2(X − Y )3/2 + 2(X + Y )3/2 + 6s2 κm −4 (12) 1 + s2 κi2

2 (1 − s2 κi2 )

and Y = where X = m . In the presence of salt in the system, the monomer contribution to the screening becomes minimal. The electrostatic contribution namely comes from the counterions and free ions. Since DNA can be treated as a polyelectrolyte, the NRTL equation derived by Nagvekar et al. [13] for polyelectrolyte solutions at finite concentration can account for the short-range electrostatic interactions of the charged species. NRTL is based on the local concentration of species in the solution that differ from the bulk concentration, therefore the first step in the development of the model is the establishment of some form of relationship between the local and overall concentrations. For electrolytes and polyelectrolytes, it is necessary to define an effective overall concentration before defining the local concentrations. An effective mole fraction for species i is defined as Xi = ρi Ci , where ρi is the overall density of species i, Ci is the absolute value of the valence, |zi |, for ionic species and unity for molecules. The effective local mole fractions Xji and Xii , of species j and i around a central species i are related by: Xj Xji = Fji Xii Xi

− 4s2 Nκ

(13)

where Fji = exp[−αji (fji − fii )/RT]. The symmetric energy parameter fji is a characteristic of the i–j interaction, and fij = fji . The non-randomness factor αji is an empirical constant that accounts for the fact that the different molecules are not fully statistically distributed through the liquid. τ is a binary interaction parameter. They found the symmetric excess Gibbs energy expression for the local composition model is given by: 

Xa ANRTL j=c Xj Fjc,ac τjc,ac   = Xc kB TV a X a k=c Xk Fkc,ac c a 

Xc j=a Xj Fja,ca τja,ca   + Xa (14) c X c k=a Xk Fka,ca a c This expression is for the completely generalized case of a multicomponent system with any number of polyelectrolytes and salts. Here, c refers to cationic and a to anionic species regardless of their source (polyelectrolyte or the added electrolytes). The subscripts j and k refer to any of the species in the system unless otherwise indicated. Primes are used to distinguish different species of the same type [13,19]. When two charged monomers are associated to the same cationic species through ion condensation, a cluster of charged particles is formed. If the two charged monomers are in the same chain, distanced apart, the attraction will cause a collapse

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of charges is lB z2eff Ne2 ln Ne , where zeff is the effective charge of monomers along the line. The total number of lines in the system is zp ρpc . Following the above steps, the total free energy of all charged lines in the system is: z2 ρ 2 Aassoc ρm line = lB eff mc ln kB TV zp ρp zp ρpc

Fig. 1. A description of the association or ionic bridge model that describes the short-range attraction of multivalent ions and monomers.

or compaction of the chain. On the other hand, if the two charged monomers are in different chains, the association leads to aggregation of the chains. The late type of association is referred to as ion-bridge [4]. Regardless of the type of the association, the energy of the cluster is described similarly. The cross links are formed from the attraction of multivalent cation and a number of monomers equivalent to the valence of this cation, as shown in Fig. 1. The cluster or the cross links composed of charged monomers and cationic species is described as Wigner crystal. The energy of formation of such cluster can be described by the columbic electrostatic energy as: Aassoc jun kB TV

= lB


zi zj i=j

rij

(15)

where rij is the ionic distance between neighboring ions and lB is Bjerrum length. The total energy of the cluster is the sum of the attractive energy between non-similar charges and the repulsive term between the monomers. The total charge of the cluster is neutral, as every multivalent ion is attracted to equal numbers of monomers as its valence. The system’s free energy of all links is:


lB zm z2j Aassoc lB z2m zj (zj − 1) c jun = (16) + ρj kB TV a+b 4a j

The ratio of the small ion to the monomer size b/a will be designated as β, and the ratio of Bjerrum length to monomer size as ξ = lB /a. Eq. (16) can be rewritten as: Aassoc jun kB TV

=−




ξωj ρjc

(17)

where ωj is the cluster parameter given by: ωj =

zm z2j 1+β

+

z2m zj (zj − 1) 4

(18)

As mentioned earlier, there exists a line of monomers carrying charges between the cross links, see Fig. 1. To find the free energy associated with all the charged lines in the system, it is divided into a number of regions equivalent to the number of cross links. The total energy of all charged lines in the system is calculated as follows. The system is divided into regions that equal the numbers of the condensed multivalent ions ρpc . The total number of monomers in each region is ρm /ρpc . The total number of charges in each line is Ne = ρm /zp ρpc . The electrostatic energy of line

(19)

where zeff is the effective charge. In the presence of monovalent and multivalent counterions, the polyelectrolyte chain, due to ion condensation, consists of three types of monomers: neutralized monomers, due to monovalent counterions, charged monomers, monomers attracted to multivalent ions with charge zm + zp . The monomer effective charge becomes: zeff = (zc + zm )Γc + zm (1 − Γp − Γc ) + (zm + zp )Γp = z m + z c Γ c + z p Γp

(20)

where Γ c and Γ p are fractions of monomers neutralized by monovalent and multivalent ions per chain, respectively [12]. The total free energy of the system, which is the sum of all contributions outlined in Eqs. (1)–(3), (12), (14), (17) and (19) is written as a A(ρm , ρc , ρp , ρa , Γ c , Γ p ) where ρm , ρc , ρp and ρa are monomer, counterion, multivalent cation and anion densities, and Γ c , Γ p are fractions condensed of counterions and multivalent cations. This function is minimized with respect to Γ c and Γ p , at different set of parameters. The minimization is constrained by the conservation of mass and electroneutrality condition expressed as: zm ρ m + z c ρ c + z p ρ p + z a ρ a = 0

(21)

The values of the fractions condensed are used to evaluate the monomer activity coefficient, determined by the derivative of the free energy equation with respect to monomer density. Plots of monomer activity coefficient versus salt concentrations are calculated at specific values of the parameters β, ξ, z and ρm . 3. Results and discussion The monomer activity coefficient is found to diverge at certain values of salt and monomer concentrations. This divergence, or asymptotic behavior, indicates that there exist a highly crosslinked phase, possibly a gel phase. Theoretically, the association term in the total free energy is dominant and significant to induce the formation of a cross linked phase and predict segregation. Analysis by Olvera de la Cruz and Ermoshkin [20], proves the presence of gel in DNA/multivalent salt solutions in the precipitated phase. The proposed model captured this phenomenon as proven by the activity coefficients. The model predicts the precipitation transition in DNA/multivalent salt solution as observed experimentally within reasonable accuracy. At the transition, the predicted ratio of multivalent salt to DNA concentrations are 0.211 and 0.147 for salt valence, Zp , equals 3 and 4, respectively, where the size ratio is taken as 0.4. Table 1 summarizes these predictions. The transition value depends on the valence and cluster parameter,

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Table 1 Experimental and predicted values for the ratio of salt concentration to DNA concentration at different salt valences Zp

Experimental [4]

Perdicted

3 4

0.23 0.12

0.211 0.147

Fig. 4. The monomer activity coefficient vs. multivalent salt concentration at different dimensionless Bjerrum lengths ([dashed line] ξ = 2, [solid line] ξ = 2.5); the following parameters are fixed z = 4, ρm = 10−2 mM, β = 0.5.

Fig. 3. The monomer activity coefficient vs. multivalent salt concentration at different salt ions to monomers ratio ([dashed line] β = 0.5, [solid line] β = 0.6); the following parameters are fixed ξ = 2, z = 4, ρm = 10−2 mM.

which is a function of valence and the ratio of salt to monomer ionic sizes. At higher valences, the salt to monomer ratio at the transition decreases significantly, which means a lower salt concentration is required to precipitate the polyelectrolyte chains.

Fig. 2. The monomer activity coefficient vs. multivalent salt concentration at different salt valences ([solid line] z = 3, [dashed line] z = 4); the following parameters are fixed ξ = 2, ρm = 10−2 mM, β = 0.5.

The model also predicts that the transition ratio is dependent only on the multivalent ion valence and ionic ratio of ion to monomer size β. For instance at similar ions valence, the larger the ions are, the smaller is the electrostatic energy of the cluster and more salt is required to precipitate the chains. This result is physically realistic because large ions have small charge density as compared to some ions. Plots 2 thru 5 describe the monomer activity coefficient variation with the multivalent salt concentration in several cases. Fig. 2 suggests that lower values of multivalent valence increase the values of monomer activity. Fig. 3 shows noticeable effect

Fig. 5. The monomer activity coefficient vs. multivalent salt concentration at different monomer concentrations, ρm ([dashed line] ρm = 10−1 mM, [solid line] ρm = 10−2 mM); the following parameters are fixed z = 4, ξ = 2, β = 0.5.

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of size ratio on the monomer activity coefficient. The larger the ionic size, the less the charge density and the less salt activity is expected. The decrease in the salt activity must be compensated by an increase in monomer activity according to Gibbs–Duhem relation. Fig. 4 is expected thermodynamically, the activity coefficient decreases with decreasing temperature; i.e. higher values of dimensionless Bjerrum length. Fig. 5 illustrates expected thermodynamic result of the proportionality of monomer activity coefficient and monomer concentration. 4. Conclusion The model predicts phase segregation in DNA–multivalent salt solutions as observed in several experiments. It suggests that the precipitated phase might have the morphology of a gel due to the divergence of the monomer activity coefficient at certain monomer and salt concentrations. Although the model predicts the first transition (precipitation), the transition at higher salt concentration, re-dissolution, is not predicted. A stronger charge screening contribution to the free energy is needed. It is possible that a very strong screening term must be added to the free energy to possibly predict the second transition. The contribution of NRTL to the system free energy to describe the short-range is small when the concentration of salt is low. NRTL term becomes significant when the long-range electrostatic is screened with the presence of smaller ions. This explains its significance when used at more concentrated system such as one studied by Nagvekar et al [13]. At higher concentrations, the species become much closer to each other making the effect of short-range interactions as described by NRTL is significant. The monomer activity coefficient is greatly influenced with temperature, monomer concentration and ionic size. The effects of temperature and monomer concentration are expected thermodynamically for any substance. The ionic size effect is related to the strength of charge density, the high charge density in smaller ions results in higher chemical activity of these ions. The main drawback of the model is the complicated form of the monomer activity coefficient derived by the free energy and the need of mathematical optimization to evaluate the fractions of multivalent and counterions condensed on the charged chain. List of symbols a Monomer size A Helmholtz free energy b Small ion size D Structure function of Gaussian chain H Hamiltonian k Wave vector kB Boltzmann constant lB Bjerrum length N Total number of charges p Momentum Rg Radius of gyration S Components of correlation matrix T Temperature U Potential energy

V z Z

Volume Ion valence Partition function

Greek letters α Non-randomness factor β Ratio of small ion to monomer size κ Ionic strength or inverse screened length ξ Ratio of Bjerrum length to monomer size ρ Density τ Energy interaction parameter Γ Fraction of condensed ions ω Cluster parameter Subscripts a Anion c Counterion i,j,k Indices for species m Monomer p Multivalent ion Superscripts 0 Neutral c Condensed e Electrostatic hc Hard core id Ideal NRTL Non-random two liquid RPA Random phase approximation T Transpose Appendix A. Random phase approximation Gaussian approximation, linear mean field approximation, linear response theory or random phase approximation all refer to the same technique, a powerful tool in statistical mechanic application to characterize system’s fluctuation. In polymer physics it was first proposed by Doi and Edwards [11]. According to statistical mechanics, the Helmholtz energy A is related to the partition function Z as: A = −ln Z kB TV

(A.1)

where kB is Boltzmann constant. The partition function is an integral over all positions r = {r1 , r2 , . . . , rn } and momenta p = {p1 , p2 , . . . , pn } of all particles exist in the system:   H(r, p) Z= exp − dpdr (A.2) kB T where H is the Hamiltonian of the system given as the sum of kinetic  and potential energy of all particles in the system H(r, p) = j p2j /2mj + U(r). If all the momenta are integrated out, the potential energy in the integral is left out:   U(r) Z ∝ exp − dr (A.3) kB T

K.A. Mahdi, Z. Drobiova / Fluid Phase Equilibria 242 (2006) 103–109

The partition function has to be represented in terms of concentration fluctuations (number density) instead of particle posi˜ is expressed mathetions. The number density of particles ρ(r) matically as:
˜ = ρ(r) δ(r − Rj ) (A.4) j

where δ(r) is three-dimensional Dirac delta function. It is preferred to work with the Fourier representation of particle density. The Fourier representation is expressed as follows: ˆ ρ(k) =

1
exp(−ikRj ) V

(A.5)

j

where k is the wave vector and V is the volume. Note that we are defining Fourier transforms as follows: 1 ˆ ˜ exp(ikr) dr ρ(k) = ρ(r) (A.6) V V ˜ = ˆ exp(−ikr) dr ρ(r) ρ(k) (A.7) (2π)3 When Eq. (A.3) is inserted into Eq. (A.5) using Dirac delta functions, the free energy takes the following form:    A − A0 1
ˆ exp − = δ(ρ(k)) − exp(−ikRj ) kB TV V j k>0    U({Rj })  ˆ dRj dρ(k) × exp − kB T j k>0 (A.8) If we integrate out {Rj }, Eq. (A.8) becomes an integral over the density fluctuation only:     ˆ A − A0 U[ρ(k)] ˆ exp − = exp dρ(k) (A.9) kB TV kB T k>0

The potential energy of the system has to be expressed in terms of the density fluctuations. In the linear limit the potential energy ˆ U[ρ(k)] is formally expressed as:
ˆ ρ(−k) ˆ ˆ S(k) ˆ U[ρ(k)] = ρ(k) + O(ρˆ 2 ) (A.10) ˆ is the inverse of the structure function defined as G ˆ = where S(k) ˆ ρ(−k)) ˆ (ρ(k) averaged over the distribution function. Eq. (A.10) can be easily extended to include several components. Instead ˆ of ρ(k), which represents one component, vector representation is used to accommodate all the components of the system ρˆ i (k). ˆ S(k) becomes an n × n matrix with its elements Sˆ ij (k) tensorial notation is used to represent matrices:

U[{ρˆ i (k)}] = ρˆ i (k)Sˆ ij (k)ρˆ jT (−k) (A.11) where T represents the transpose. Next the free energy of the system is evaluated by substituting Eq. (A.11) into (A.9) and performing the integration over all components and wave vec-

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tors. The resultant free energy expression takes the following form:

 det[Sˆ ij (k)] 1 ARPA ln = dk (A.12) kB TV det[Sˆ ij0 (k)] (2π)3 where Sˆ ij0 (k) is the reference state, usually taken as the state of no interaction. References [1] Y. Burak, G. Ariel, D. Andelman, Onset of DNA aggregation in presence of monovalent and multivalent counterions, Biophys. J. 85 (2003) 2100–2110. [2] M. Delsanti, J.P. Dalbeiz, O. Spalla, L. Belloni, M. Drifford, Phase diagram of polyelectrolyte solutions in presence of multivalent salts, ACS Symp. Ser.: Macro-ion Charact. (1994) 381. [3] I. Sabbagh, M. Delsanti, Solubility of highly charged anionic polyelectrolytes in presence of multivalent cations: specific interaction effect, Eur. Phys. J. E (2000) 1–75. [4] E. Raspaud, M. Olvera de la Cruz, J.L. Sikarov, F. Livolant, Precipitation of DNA by polyamines: a polyelectrolyte behavior, Biophys. J. 74 (1998) 381–393. [5] W.M. Gilbart, R.F. Bruinsma, P.A. Pincus, V.A. Parsegian, DNA inspired electrostatics, Phys. Today 53 (2000) 38. [6] J.L. Barrat, J.F. Joanny, Persistence length of polyelectrolyte chains, Europhys. Lett. 24 (5) (1993) 333. [7] J.F. Joanny, L. Leibler, Weakly charged polyelectrolytes in a poor solvent, J. Phys. France 51 (6) (1990) 545. [8] M.O. Khan, B. J¨onsson, Electrostatic correlations fold DNA, Biopolymers 49 (1999) 121–125. [9] V.A. Bloomfield, R.W. Wilson, D.C. Rau, Polyelectrolyte effects in DNA condensation by polyamines, Biophys. Chem. 11 (1980) 339–343. [10] Y. Levin, M.E. Fisher, Criticality in the hard-sphere ionic fluid, Physica A 225 (1996) 164–220. [11] M. Doi, S.F. Edwards, The Theory of Polymer Dynamics, Clarendon Press, 1986. [12] G. Manning, Limiting laws and counterion condensation in polyelectrolyte solutions. I. Colligative properties, J. Chem. Phys. 51 (1969) 924. [13] M. Nagvekar, F. Tihminlioglu, R.P. Danner, Colligative properties of polyelectrolyte solutions, Fluid Phase Equilibr. 145 (1998) 15–41. [14] P.S. Kuhn, Y. Levin, M. Barbosa, Polyelectrolyte solutions with multivalent salts, Physica A 266 (1999) 413–419. [15] Y. Levin, Theory of counterion association in rod-like polyelectrolytes, Europhys. Lett. 34 (6) (1996) 405–410. [16] D.A. McQuarrie, Statistical Mechanics, Harper and Row, pp. 326–340. [17] K. Mahdi, M. Olvera de la Cruz, Phase diagrams of salt free polyelectrolyte semidilute solutions, Macromolecules 33 (2000) 7649–7654. [18] F.J. Solis, M. Olvera de la Cruz, Attractive interactions between rod-like polyelectrolytes: polarization, crystallization and packing, Phys. Rev. E 60 (1999) 4496. [19] G.H. Van Bochove, Two and Three Liquid Phase Equilibria in Industrial Mixed-Solvent Electrolyte Solutions, Delft University Press, 2003. [20] M. Olvera de la Cruz, A.V. Ermoshkin, Polyelectrolytes in the presence of multivalent ions: gelation vs. segregation, Phys. Rev. Lett. 90 (2003) 125504 (1–4). [21] P.G. Arscott, A.Z. Li, V.A. Bloomfield, Condensation of DNA by trivalent cations. I. Effects of DNA length and topology on the size and shape of condensed particles, Biopolymers 30 (1990) 619– 630. [22] I. Rouzina, V.A. Bloomfield, Macroion attraction due to electrostatic correlation between screening counterions. 1. Mobile surface-adsorbed ions and diffuse ion cloud, J. Phys. Chem. 100 (1996) 9977–9989. [23] F.J. Solis, M. Olvera de la Cruz, Collapse of flexible polyelectrolytes in multivalent salt solutions, J. Chem. Phys. 112 (2000) 2030–2035.