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DNA interaction with biologically active divalent metal ions: binding constants calculation
International Journal of Biological Macromolecules 34 (2004) 245–250
DNA interaction with biologically active divalent metal ions: binding constants calculation Elene V. Hackl∗ , Vladimir L. Galkin, Yurij P. Blagoi B.I. Verkin’ Institute for Low Temperature Physics and Engineering, National Academy of Sciences of Ukraine, 61164 Kharkov, Ukraine
Abstract In our previous work we have shown that under the action of Cu2+ , Mn2+ and Ca2+ ions DNA is able to transit into a compact state in aqueous solution. In the present work we carried out calculations of binding constants for divalent metal ions interacting with DNA in terms of the macromolecule statistical sum. The formula for calculation of the binding constants and cooperativity parameters was proposed. It was shown that on the “coil state”–“compact (globule) state” transition a single DNA molecule may undergo the first-order phase transition while the transition of the assembly of average DNA chains is of sigmoidal character typical of the cooperative and continuous transition. © 2004 Elsevier B.V. All rights reserved. Keywords: DNA; Metal ions; Binding constant; Phase transition.
As it is well known, in a solution DNA macromolecules exist in complexes with counterions, which together with hydration surrounding, determine its structure. Thus the question—how metal ions can effect the DNA structure—is very important both from the points of view of biophysics of nucleic acids and molecular biology. When characterising DNA-metal ion binding there are three general questions that usually rise: what places on DNA a metal ion interacts with (binding sites); how the interaction effects the DNA structure; and what is the affinity of the binding. In our previous work [1–3], we studied the interaction of Cu2+ , Mn2+ and Ca2+ ions with DNA macromolecule in aqueous and water–alcohol solutions at different metal ion concentrations. We have shown that all these ions interact both with the bases and with phosphate groups of DNA (at different [Met2+ ]/[DNA] ratios, see [1–3]). When interacting with DNA in an aqueous solution divalent metal (Met2+ ) ions studied induce DNA structural transitions similar to the DNA transition into compact state upon interaction with 3+ ions or polyamines—namely, a drastic decrease in the volume occupied by DNA molecules with reversible formation ∗ Corresponding author. Present address: Department of Biomolecular Science, UMIST, Manchester M601QD, UK. Tel.: +44 161 2008913. E-mail address: [email protected] (E.V. Hackl).
0141-8130/$ – see front matter © 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.ijbiomac.2004.08.003
of DNA dense particles of well-defined size and ordered morphology. Met2+ -induced DNA condensation in our experiments included both intra- and inter-molecular interactions. In the compact state DNA remained in the B-conformation limits [1,2]. A further study allowed us to suggest that the mechanism of DNA condensation under Met2+ ion action is not completely electrostatic but involves a partial destabilisation of DNA due to Met2+ ion interaction with DNA bases [3,4]. The present work is devoted to calculation of binding constants of divalent metal ions interacting with DNA in a condensation process. It should be noted that, as we have also shown in [1,2], at low concentrations Met2+ ions binding to DNA occurred while DNA condensation was not observed. Only at certain [Met2+ ]/[DNA] ratio (rather small in the case of Cu2+ ions and higher in the case of Mn2+ and Ca2+ ions) was DNA condensation started. Presence of the induction period (when no DNA condensation was observed despite condensing agent binding to DNA) was also shown for DNA inter-molecular condensation induced by common condensing agents (spermine, spermidine and cobalt(III) hexamine [5–8]). We suggested [2] that during the induction period, Met2+ ions are bound to DNA, but the degree of the binding remains below a threshold value and DNA condensation does not occur. As soon as the degree of Met2+ ion
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binding arrives at the threshold, DNA starts to condense. Therefore, to simplify the system, we divided all Met2+ ions interacting with DNA into two groups: (1) those that bind before condensation process starts; and, (2) those that bind in the condensation process (and that binding of these ions results in condensation). In the present work, as well as in our previous one [2], we consider only the second type of Met2+ ions—namely, those which bind in the condensation process. When calculating the binding constants we operate not with the value of the total Met2+ ion concentration but with the value of free Met2+ ion concentrations (see below). The main experimental method used by us in [1–4] was IR spectroscopy. We have found that upon DNA interaction with Met2+ ions the intensities of the absorption bands of DNA phosphate groups and bases increase significantly. We attribute the intensity increase observed to the DNA transition into compact state (see [1,2]). To detect the intensity increase the threshold concentration of Met2+ ions required; below this concentration binding of Met2+ ions to DNA did not result in the intensity increase (and, consequently, in DNA condensation, as was mentioned above). Dependencies of the relative change of intensity (R) of the absorption bands on the total Cu2+ (Mn2+ , Ca2+ ) ion concentration in solution are S-shaped curves with saturation. We define relative intensity R as R = Ii /I0 , where Ii is the intensity of an absorption band for DNA in the complex with i concentration of Met2+ ions, and I0 is the intensity of an absorption band for DNA without divalent ions. R value at saturation varied from ∼3.5 for Cu2+ ions to ∼1.5 for Ca2+ ions. To go from R([Met2+ ]) dependencies to r([Met2+ ]) ones (where r is the condensation degree, i.e. a part of DNA chain segments bound to Met2+ ions and formed a compact structure: r = DNA in compact state/total DNA), we assumed that in the Met2+ concentration interval, where no increase of the band intensities was observed (R = 1), r = 0 and DNA condensation is absent. When R value reaches saturation r = 1. As it was mentioned above, we consider only those divalent metal ions binding of which is coupled with DNA condensation. In this case the condensation degree will be directly proportional to the binding degree of Met2+ ions in the condensation process. Binding degree we define as ratio of bound Met2+ ions to the total number of Met2+ ions required to be bound to DNA to induce total DNA condensation. Binding degree as well as condensation degree varies from 0 to 1. Fig. 1 shows the dependencies of the binding degree on the total Met2+ concentration in solution. Dependencies were calculated using results given in our previous work [1,2]. Region 1 presents DNA in elongated coil state, while region 2 in a compact state. As is seen, metal binding (and, consequently, DNA condensation) occurs in a relatively narrow interval of Met2+ concentration that means that Met2+ induced DNA condensation is of high positive cooperativity.
Fig. 1. Dependencies of the binding degree r on the total concentration of divalent metal ions in solution (D) for DNA phosphate groups interaction with Cu2+ (1), Mn2+ (2) and Ca2+ (3) ions in aqueous solution at 29 ◦ C (according to results of [1,2]).
1. Calculation of binding constants using the binding equation In our previous papers [1,2] calculations of the binding constants of Met2+ interacting with DNA in a condensation process were based on the binding equation in Scatchard form [9]. As in [1,2] we gave only the final equation, in the present work this method is described in more detail. We consider a model of “DNA—low molecular weight ligands” in which DNA macromolecule was taken as a chain of freely conjugated segments. To simplify we assumed that all segments have one negatively charged binding site (PO2 2− groups for DNA interaction with Met2+ ions at low concentration of Met2+ ions). At the first step of DNA–Met2+ complex formation binding of one divalent metal ion to two spatially separated segments occurs. If two segments belong to the same DNA chain, the Met2+ binding will result in a loop formation; loop size (radius) will be defined by the chain rigidity. It is clear that all factors that affect the chain rigidity (such as temperature, solvent, etc.) will effect the loop radius too. The complex formation will result in entropy loss: the higher the chain rigidity, the larger entropy loss will be. But because the ion binding constant is increased as a result of the complex formation the first bond can lead to decrease of enthalpy H. The binding constant of the first bond formation is determined as ∆G0 K0 = exp (1) R×T where G0 is the change in the Gibbs free energy upon binding: G0 = H0 − TS0 ,
(2)
where G0 , H0 and S0 are changes in thermodynamic parameters (free energy, enthalpy and entropy) upon ion binding to the macromolecule accompanied by a loop formation; R is
E.V. Hackl et al. / International Journal of Biological Macromolecules 34 (2004) 245–250
the gas constant, and T the temperature in K. The low probability of this process is because of high value of |S0 | (notice that S0 < 0 and H0 < 0), so that |TS| ∼ |H|. As a result |G0 | is low and the value of the binding constant К0 is low too. When forming the second bond between chain segments by second divalent metal ion, entropy will be decreased too. But in this case the entropy loss will be smaller than when forming the first bond because only a small part of the chain will be fixed and lose its mobility as a result of the second bond formation (i.e. main entropy changes proceed during the first bond formation). H will remain the same for the second bond formation so we can write down: G1 = 2H0 − T (S1 + S0 ),
|S1 | |S0 |
(3)
With the increase of the binding degree r, and the relative ordering of the complex structure, the loss in entropy will be decreased upon the attachment of every next ion. This means that G will be increased and K will rise exponentially. Upon the NTH bond formation GN = H0 − TS0 + (N − 1)(H0 − TS1av ) = G0 + (N − 1)(H0 − TS1av )
D = Cf + n × P × r
247
(8)
where P is the DNA molar concentration; D is the total Met2+ ion concentration in solution; Cf is the concentration of free Met2+ ions (that is the ions not taking part in the formation of the DNA compact structure); n is the coefficient reflecting the fact that not all potentially available binding sites need to be occupied by Met2+ ions to induce DNA condensation; r is the binding degree. D and P values were measured experimentally; r was calculated as it was described above; Cf was calculated according to (8); parameters K0 and were selected so as to minimise the deviations of the experimental points from the theoretical curve. Fig. 2A shows the theoretical r (Cf ) dependencies calculated according to (7). As it may be seen at ω < 8 binding isotherms are of a monotonous character corresponding to the continuous increase of the binding degree r with the increase of the free metal ion concentration Cf . The isotherm with ω = 8 having the intersection point with the vertical tangent, is critical for all values of the binding constants. At ω > 8 r (Cf ) dependencies acquire the non-monotonous S-like character with meta-stable and non-stable parts characterised with the reverse dependence of r on Cf . For a stable process
(4)
where S1av is an average change in entropy upon 2ND-NTH bonds formation between segments. Here N is the number of bonds formed between chain segments by Met2+ ions, therefore, N is proportional to the amount of segments formed bonds. Let Nt be the total amount of chain segments required to be bound with Met2+ to form a compact structure (i.e. the total amount of segments involved in a compact structure formation). Then N/Nt will be proportional to the degree of chain condensation r. For large N (N − 1) ≈ N, thus (4) taking into account (2) can be written as: GN ≈ H0 − TS0 +
N Nf (H0 − TS1av ) Nt
≈ G0 − r × W,
(5)
where W is a coefficient equal to −Nf (H0 − TS1av ). Thus the addition to the free energy is proportional to the binding degree r. Taking into account (5), let us write down (1) as: −G0 + W × r K = exp = K0 × exp(ω × r), (6) RT where K0 = exp(−G0 /RT ), ω = (W/(R × T )); K0 is the constant determining the critical (threshold) concentration of divalent metal ions corresponding to the beginning of DNA condensation, ω is the cooperativity parameter. Using (6) and binding equation in Scatchard’s form [9], one can obtain the dependence of r on Cf at different values of binding constants and cooperativity parameters: r = K0 × exp(ω × r) × Cf , (1 − r)
(7)
Fig. 2. Theoretical dependencies of the binding degree r on the free concentration of the divalent metal ions Cf calculated according to (7) (A) and (19) (B). Numbers at the curves-values of the parameters K0 (K0 ) and ω (ω ), respectively.
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such isotherms have to be replaced with dependencies with a jump along r, that may evidence a phase transition. Thus at definite conditions (for example, in aqueous-ethanol solutions, where ω became more than 8 [2]) DNA transition into compact state under the action of divalent metal ions may be of phase transition character. At the same time our experimental data shows that DNA transition into a compact state occurs in a narrow but continuous interval of Met2+ ion concentrations (Fig. 1). Even in the case of Cu2+ -induced DNA condensation the “coilcompact state” transition was not discrete. In our experiments to record IR spectra of DNA–Met2+ complexes a high molecular weight DNA was used. Besides, DNA concentration in solution was relatively high (∼10−2 -5 × 10−2 M of phosphorus). Obviously, at such experimental conditions dispersion of DNA molecules on their molecular weight is present. But, when calculating binding isotherms, we did not take into account the distribution of values along the assembly of DNA molecules. To obtain a more sequential thermodynamic description for the system of ions interacting with the biopolymers, in the present work we carried out calculations in terms of the macromolecule statistical sum. It should be noted that in (7) and (8) we do not take into consideration the competition between cations of different valences interacting with DNA. As in our experiments [1–4] we used Na+ -salt DNA; therefore there are two types of cations in the solution—Met2+ and Na+ ions. In this case the binding constant of Met2+ ions will depend on concentration of monovalent ions. In the presence of competition between di- and mono-valent cations the total binding degree will be determined as rt = rMet 2+ + 0.5 × rNa + (coefficient 0.5 is due to the difference in cation valences), where rMet 2+ and rNa + are the binding degrees of Met2+ and Na+ ions, respectively. To calculate the parameters K0 and ω the system of two equations should be used instead of (7), in which the Na+ ions binding to DNA is taken into account as well. But a general view of binding isotherms shown in Fig. 2A was not changed when we considered two types of cations instead of one (not illustrated).
2. New model considering the statistical sum of the macromolecule As in our previous model let us consider a DNA macromolecule as a long chain of freely conjugated segments. Each DNA macromolecule usually used in our experiments consisted of 104 –105 nucleotides, therefore it may be presented as a macrosystem. A macromolecule is in a solution of low molecular weight ligands (metal ions). In a solution ligands can interact with a macromolecule. Ligands may interact only with definite parts of a macromolecule (with binding sites), thus we don’t consider DNA molecule as a uniformly charged cylinder. One ligand can form two bonds with two different binding sites on a macromolecule. One binding site of a macromolecule can form only one bond with a ligand, thus
each element of a macromolecule can be in one of two conditions: in condition a, if it does not interact with ligand, or in condition b, if it bound to a ligand. Let the energy of one element of the macromolecule chain in state a remain zero while the energy of the element in state b–W0 (W0 > 0). Ligands bound to the neighbouring elements can interact each other. Under interaction of bound ligands we imply not only their direct interaction, but indirect one as well. Let the energy of such interaction mean that W1 . W1 can be both positive and negative depending on the interaction forces (attractive or repulsive) between bound ligands. The statistical sum of a large canonical ensemble may be presented as: Z(µ) =
∞
[eβ µ N Z0 (N)]
(9)
N=1
where β = 1/T, N is the number of pairs (as our macromolecule is a double helix) of chain elements, Z0 (N) is the statistical sum of a chain consisting of N element pairs: g{σns }exp[−βEN {σns }] (10) Z0 (N) = {σns }
where EN {σns } is the total energy of a chain at a given configuration of spins {σ n s }, g{σ n s } is the statistical weight of the spin configuration given. Detailed calculation of the statistical sum for such system has been done in [10]. To calculate the statistical sum the authors used the standard matrix method. Making use of the results of [10], we can write down the statistical sum as a: Z(µ, µ1 ) =
∞ 2(N−1) N=1 K=0
exp{β[µ(N − 1) + µ1 K]}Z0 (N, K) (11)
where β = (kT)−1 , Z0 (N, K)—the statistical sum of a double helix consisting of N element pairs, containing K bound ligands, µ is the chemical potential of complementary pairs of chain elements, µ1 is the chemical potential of ligands bound to a polymer chain. Let η is the ligand concentration on a chain (that is the ratio of chain elements (segments) bound to ligands to the total quantity of macromolecule elements), than η=
1 ∂Z/∂µ1 × 2 ∂Z/∂µ
(12)
Authors of [10,11] showed that on the phase equilibrium curve ∂λ2 1 , (13) η= 2λ2 ∂ε where λ2 = ω2 χ2 eε+ε1 1/2
(14)
E.V. Hackl et al. / International Journal of Biological Macromolecules 34 (2004) 245–250
((ω2 )1/2 is a statistical weight of a non-bound element of a chain), 1/2
χ2 = ch(ε + ε1 ) + [sh2 (ε + ε1 ) + e−2ε1 ]
.
It is known that an equation of the phase transition curve may be found from the condition of equilibrium of chemical potentials of macromolecule different phases. In the method of large canonical ensemble [10] the chemical potential is determined by condition. N=
−∂Ω , ∂µ
(15)
Ω = −kT lnZ(µ, µ1 ), where N is the amount of complementary pairs of macromolecule elements. At N → ∞ Eq. (15) may be written as ∂Ω/∂µ = −∞. From formulae (13) and (14) one may get the dependence of µ1 on η: eε+ε1 =
ε=
[(2η − 1)e−ε1 + B] 2[η(1 − η)]1/2
1 β(µ1 + W0 ), 2
(16)
1 ε1 = − βW1 2 1/2
B = [4η(1 − η) + (2η − 1)2 e−2ε1 ]
.
In the limit case η 1, µ1 = kT ln η − W0 At W1 = 0 for any concentration we obtain η µ1 = kT ln − W0 1−η In a solution of free low molecular weight ligands with concentration c (c 1) µ1 = kT ln c + f
(17)
where f is standard chemical potential. At this we suppose that ligands on a polymer chain and in a solution are in equilibrium. From the equality of ligand chemical potentials µ1 on a polymer chain determined by (16) and in solution (17) the dependence of η on c follows: 1 η= 2
249
a macromolecule (W1 ) corresponds to the cooperativity parameter ω (as we defined this value in [1,2]). We mentioned above that W1 can be both positive and negative depending on the interaction forces (attractive or repulsive) between bound ligands. In it’s turn cooperativity parameter ω can be both positive and negative too. As we indicated above, W0 is the energy of the chain segments bound to a ligand (metal ion); therefore, this value has a physical meaning of the binding constant (K0 ). It should be noted that this binding constant K0 might be not equal to the binding constant K0 calculated according to (7). Value c was defined as ligand concentration in solution that corresponds to the concentration of free ligands Cf . Taking into account all above the Eq. (18) may be re-written as ω 1 C f K0 e − 1 r = 1 + (19) 2 2 ω (Cf K0 e − 1) + 4Cf K0 Theoretical dependencies r (Cf ) calculated according to (19) are shown in Fig. 2B. As it is seen, in this case binding isotherms are of monotonous character at any values of binding constant K0 and cooperativity parameter ω . Thus we have shown that use of the exact expression for the statistical sum leads to the disappearance of bimodality on the r (Cf ) dependencies (Fig. 2). This means that the bimodal character of r (Cf ) dependencies is typical for a case when the distribution of values in ensemble of DNA macromolecules is not taken into account. In other words for a single DNA chain the Met2+ -induced transition into a compact state can be of phase transition character. But when we take into account the statistical sum of macromolecules (i.e. we take into account the ensemble of macromolecules) the isotherms describing the Met2+ -induced DNA transition into a compact state are of monotonous character. Thus upon the “coil state”–“compact (globule) state” transition, single DNA molecules may undergo the first-type phase transition; while the transition of the ensemble of DNA chains is of sigmoidal character typical for the cooperative and continuous process. This explains why in experiments, even in the case of DNA interaction with Cu2+ ions, DNA condensation (as a consequence of this interaction) occurred in a narrow but detectible interval of Cu2+ ion concentrations [1,2].
×1+
c × exp(β(W2 −W1 ) − 1
,
(c × exp(β(W2 −W1 ) − 1) +4c × exp(βW2 ) (18) 2
where W2 = f + W0 . As we defined η as a ratio of chain elements (segments) bound to ligands to the total quantity of chain elements, the physical meaning of η and r (where r is the binding degree defined in [1,2] as a part of chain segments bound to metal ions and formed a compact structure) values are the same. This allows us to transit from η to r value in (18). Besides, the interaction energy of ligands bound to the neighbour parts of
3. Comparison of both models with experimental results To compare two Eqs. (7) and (19) and to find out how both our models can fit the experimental results we did the following: for each K0 (K0 ) we chose ω (ω ) so that to minimise the root-mean-square deviation (ϑ) of a theoretical curve from experimental results. Parameters K0 and ω were calculated according to (7) while parameters K0 and ω —according to (19), experimental r and Cf values were taken from [1,2]. Dependencies of deviation ϑ on the binding constant K0 (K0 ) for DNA interaction with Mn2+ (A) and Ca2+ (B) ions are
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Fig. 3. Dependencies of root-mean-square deviation (ϑ) of theoretical curves from experimental results on binding constant K0 (K0 ) for DNA interaction with Mn2+ (A) and Ca2+ (B) ions. Parameters K0 and ω were calculated according to (7), parameters K0 and ω – according to (19), experimental D, r and Cf values were taken from [1,2]. Table 1 Binding constants K0 (K0 ) and cooperativity parameters ω (ω ) calculated according to (7) (K and ω) or (19) (K and ω ) for DNA interaction with Cu2+ , Mn2+ and Ca2+ ions in aqueous solution (K0 and ω parameters were chosen to minimise the deviation of experimental results from a theoretical curve) Met2+
K0
ω
K0
ω
Cu2+
4.5 1.7 0.5
11 7 7.2
2.5 0.8 0.08
8 3.3 3.8
Mn2+ Ca2+
shown in Fig. 3. As is seen from the figure, ϑ dependencies on K0 and K0 have similar character with well-expressed minima both for Mn2+ and Ca2+ ions interacting with DNA. The minimum corresponds to the pair of K0 (K0 ) and ω (ω ) parameters, which gives the lowest deviation of the experimental data from theoretical curve and, therefore, describes the binding of the divalent metal ions studied to DNA in the best way. These pairs of parameters are given in Table 1. The comparison of the two models shows that they both allow de-
termination of the binding parameters (binding constant and cooperativity) sufficiently well. As can be also seen from the data presented in the Table 1, both formulae give the same order of the binding constants for Met2+ ions interaction with DNA in a condensation process (Cu2+ > Mn2+ > Ca2+ ) and correlation between binding parameters (binding constants and parameters of cooperativity). Thus we can conclude that both formulae can be used to calculate the binding parameters for divalent metal ions interacting with DNA. In the present model considering the ligand binding to DNA elements we didn’t take into consideration all other kinds of interactions between macromolecules including interactions occurring by means of solvent molecules. As an example of such interactions one may mention the electrostatic forces between macromolecule charges. It is known that electrostatic interaction essentially influences the properties of strongly charged polyelectrolytes in solution. Due to such macromolecular interactions the permissive chain configurations in the statistic sum stop being equivalent (as we considered in the present work). Another assumption was using the model of a homogenous medium with a constant macroscopic value of dielectric permeability neglecting the change of this value in the vicinity of DNA surface.
References [1] Hackl EV, Kornilova SV, Kapinos LE, et al. J Mol Struct 1997;408–409:229. [2] Kornilova S, Hackl E, Kapinos L, et al. Acta Biochim Pol 1998;45:107. [3] Hackl EV, Blagoi YuP. Met Ions Biol Med 2000;7:361. [4] Hackl EV, Blagoi YuP. Met Ions Biol Med 2000;7:771. [5] Porschke D. J Mol Biol 1991;222:423. [6] Burak Y, Ariel G, Andelman D. Biophys J 2003;85:2100. [7] Porschke D. Biochemistry 1984;23:4821. [8] He S, Arscott PG, Bloomfield VA. Biopolymers 2000;53:329. [9] Scatchard G, Ann NY. Acad Sci 1949;51:660. [10] Kosevich AM, Galkin VL. J Exp Theor Phys 1971;60:821. [11] Kosevich AM, Galkin VL, Nechiporenko IN. J Exp Theor Phys 1974;66:261.
DNA interaction with biologically active divalent metal ions: binding constants calculation Elene V. Hackl∗ , Vladimir L. Galkin, Yurij P. Blagoi B.I. Verkin’ Institute for Low Temperature Physics and Engineering, National Academy of Sciences of Ukraine, 61164 Kharkov, Ukraine
Abstract In our previous work we have shown that under the action of Cu2+ , Mn2+ and Ca2+ ions DNA is able to transit into a compact state in aqueous solution. In the present work we carried out calculations of binding constants for divalent metal ions interacting with DNA in terms of the macromolecule statistical sum. The formula for calculation of the binding constants and cooperativity parameters was proposed. It was shown that on the “coil state”–“compact (globule) state” transition a single DNA molecule may undergo the first-order phase transition while the transition of the assembly of average DNA chains is of sigmoidal character typical of the cooperative and continuous transition. © 2004 Elsevier B.V. All rights reserved. Keywords: DNA; Metal ions; Binding constant; Phase transition.
As it is well known, in a solution DNA macromolecules exist in complexes with counterions, which together with hydration surrounding, determine its structure. Thus the question—how metal ions can effect the DNA structure—is very important both from the points of view of biophysics of nucleic acids and molecular biology. When characterising DNA-metal ion binding there are three general questions that usually rise: what places on DNA a metal ion interacts with (binding sites); how the interaction effects the DNA structure; and what is the affinity of the binding. In our previous work [1–3], we studied the interaction of Cu2+ , Mn2+ and Ca2+ ions with DNA macromolecule in aqueous and water–alcohol solutions at different metal ion concentrations. We have shown that all these ions interact both with the bases and with phosphate groups of DNA (at different [Met2+ ]/[DNA] ratios, see [1–3]). When interacting with DNA in an aqueous solution divalent metal (Met2+ ) ions studied induce DNA structural transitions similar to the DNA transition into compact state upon interaction with 3+ ions or polyamines—namely, a drastic decrease in the volume occupied by DNA molecules with reversible formation ∗ Corresponding author. Present address: Department of Biomolecular Science, UMIST, Manchester M601QD, UK. Tel.: +44 161 2008913. E-mail address: [email protected] (E.V. Hackl).
0141-8130/$ – see front matter © 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.ijbiomac.2004.08.003
of DNA dense particles of well-defined size and ordered morphology. Met2+ -induced DNA condensation in our experiments included both intra- and inter-molecular interactions. In the compact state DNA remained in the B-conformation limits [1,2]. A further study allowed us to suggest that the mechanism of DNA condensation under Met2+ ion action is not completely electrostatic but involves a partial destabilisation of DNA due to Met2+ ion interaction with DNA bases [3,4]. The present work is devoted to calculation of binding constants of divalent metal ions interacting with DNA in a condensation process. It should be noted that, as we have also shown in [1,2], at low concentrations Met2+ ions binding to DNA occurred while DNA condensation was not observed. Only at certain [Met2+ ]/[DNA] ratio (rather small in the case of Cu2+ ions and higher in the case of Mn2+ and Ca2+ ions) was DNA condensation started. Presence of the induction period (when no DNA condensation was observed despite condensing agent binding to DNA) was also shown for DNA inter-molecular condensation induced by common condensing agents (spermine, spermidine and cobalt(III) hexamine [5–8]). We suggested [2] that during the induction period, Met2+ ions are bound to DNA, but the degree of the binding remains below a threshold value and DNA condensation does not occur. As soon as the degree of Met2+ ion
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binding arrives at the threshold, DNA starts to condense. Therefore, to simplify the system, we divided all Met2+ ions interacting with DNA into two groups: (1) those that bind before condensation process starts; and, (2) those that bind in the condensation process (and that binding of these ions results in condensation). In the present work, as well as in our previous one [2], we consider only the second type of Met2+ ions—namely, those which bind in the condensation process. When calculating the binding constants we operate not with the value of the total Met2+ ion concentration but with the value of free Met2+ ion concentrations (see below). The main experimental method used by us in [1–4] was IR spectroscopy. We have found that upon DNA interaction with Met2+ ions the intensities of the absorption bands of DNA phosphate groups and bases increase significantly. We attribute the intensity increase observed to the DNA transition into compact state (see [1,2]). To detect the intensity increase the threshold concentration of Met2+ ions required; below this concentration binding of Met2+ ions to DNA did not result in the intensity increase (and, consequently, in DNA condensation, as was mentioned above). Dependencies of the relative change of intensity (R) of the absorption bands on the total Cu2+ (Mn2+ , Ca2+ ) ion concentration in solution are S-shaped curves with saturation. We define relative intensity R as R = Ii /I0 , where Ii is the intensity of an absorption band for DNA in the complex with i concentration of Met2+ ions, and I0 is the intensity of an absorption band for DNA without divalent ions. R value at saturation varied from ∼3.5 for Cu2+ ions to ∼1.5 for Ca2+ ions. To go from R([Met2+ ]) dependencies to r([Met2+ ]) ones (where r is the condensation degree, i.e. a part of DNA chain segments bound to Met2+ ions and formed a compact structure: r = DNA in compact state/total DNA), we assumed that in the Met2+ concentration interval, where no increase of the band intensities was observed (R = 1), r = 0 and DNA condensation is absent. When R value reaches saturation r = 1. As it was mentioned above, we consider only those divalent metal ions binding of which is coupled with DNA condensation. In this case the condensation degree will be directly proportional to the binding degree of Met2+ ions in the condensation process. Binding degree we define as ratio of bound Met2+ ions to the total number of Met2+ ions required to be bound to DNA to induce total DNA condensation. Binding degree as well as condensation degree varies from 0 to 1. Fig. 1 shows the dependencies of the binding degree on the total Met2+ concentration in solution. Dependencies were calculated using results given in our previous work [1,2]. Region 1 presents DNA in elongated coil state, while region 2 in a compact state. As is seen, metal binding (and, consequently, DNA condensation) occurs in a relatively narrow interval of Met2+ concentration that means that Met2+ induced DNA condensation is of high positive cooperativity.
Fig. 1. Dependencies of the binding degree r on the total concentration of divalent metal ions in solution (D) for DNA phosphate groups interaction with Cu2+ (1), Mn2+ (2) and Ca2+ (3) ions in aqueous solution at 29 ◦ C (according to results of [1,2]).
1. Calculation of binding constants using the binding equation In our previous papers [1,2] calculations of the binding constants of Met2+ interacting with DNA in a condensation process were based on the binding equation in Scatchard form [9]. As in [1,2] we gave only the final equation, in the present work this method is described in more detail. We consider a model of “DNA—low molecular weight ligands” in which DNA macromolecule was taken as a chain of freely conjugated segments. To simplify we assumed that all segments have one negatively charged binding site (PO2 2− groups for DNA interaction with Met2+ ions at low concentration of Met2+ ions). At the first step of DNA–Met2+ complex formation binding of one divalent metal ion to two spatially separated segments occurs. If two segments belong to the same DNA chain, the Met2+ binding will result in a loop formation; loop size (radius) will be defined by the chain rigidity. It is clear that all factors that affect the chain rigidity (such as temperature, solvent, etc.) will effect the loop radius too. The complex formation will result in entropy loss: the higher the chain rigidity, the larger entropy loss will be. But because the ion binding constant is increased as a result of the complex formation the first bond can lead to decrease of enthalpy H. The binding constant of the first bond formation is determined as ∆G0 K0 = exp (1) R×T where G0 is the change in the Gibbs free energy upon binding: G0 = H0 − TS0 ,
(2)
where G0 , H0 and S0 are changes in thermodynamic parameters (free energy, enthalpy and entropy) upon ion binding to the macromolecule accompanied by a loop formation; R is
E.V. Hackl et al. / International Journal of Biological Macromolecules 34 (2004) 245–250
the gas constant, and T the temperature in K. The low probability of this process is because of high value of |S0 | (notice that S0 < 0 and H0 < 0), so that |TS| ∼ |H|. As a result |G0 | is low and the value of the binding constant К0 is low too. When forming the second bond between chain segments by second divalent metal ion, entropy will be decreased too. But in this case the entropy loss will be smaller than when forming the first bond because only a small part of the chain will be fixed and lose its mobility as a result of the second bond formation (i.e. main entropy changes proceed during the first bond formation). H will remain the same for the second bond formation so we can write down: G1 = 2H0 − T (S1 + S0 ),
|S1 | |S0 |
(3)
With the increase of the binding degree r, and the relative ordering of the complex structure, the loss in entropy will be decreased upon the attachment of every next ion. This means that G will be increased and K will rise exponentially. Upon the NTH bond formation GN = H0 − TS0 + (N − 1)(H0 − TS1av ) = G0 + (N − 1)(H0 − TS1av )
D = Cf + n × P × r
247
(8)
where P is the DNA molar concentration; D is the total Met2+ ion concentration in solution; Cf is the concentration of free Met2+ ions (that is the ions not taking part in the formation of the DNA compact structure); n is the coefficient reflecting the fact that not all potentially available binding sites need to be occupied by Met2+ ions to induce DNA condensation; r is the binding degree. D and P values were measured experimentally; r was calculated as it was described above; Cf was calculated according to (8); parameters K0 and were selected so as to minimise the deviations of the experimental points from the theoretical curve. Fig. 2A shows the theoretical r (Cf ) dependencies calculated according to (7). As it may be seen at ω < 8 binding isotherms are of a monotonous character corresponding to the continuous increase of the binding degree r with the increase of the free metal ion concentration Cf . The isotherm with ω = 8 having the intersection point with the vertical tangent, is critical for all values of the binding constants. At ω > 8 r (Cf ) dependencies acquire the non-monotonous S-like character with meta-stable and non-stable parts characterised with the reverse dependence of r on Cf . For a stable process
(4)
where S1av is an average change in entropy upon 2ND-NTH bonds formation between segments. Here N is the number of bonds formed between chain segments by Met2+ ions, therefore, N is proportional to the amount of segments formed bonds. Let Nt be the total amount of chain segments required to be bound with Met2+ to form a compact structure (i.e. the total amount of segments involved in a compact structure formation). Then N/Nt will be proportional to the degree of chain condensation r. For large N (N − 1) ≈ N, thus (4) taking into account (2) can be written as: GN ≈ H0 − TS0 +
N Nf (H0 − TS1av ) Nt
≈ G0 − r × W,
(5)
where W is a coefficient equal to −Nf (H0 − TS1av ). Thus the addition to the free energy is proportional to the binding degree r. Taking into account (5), let us write down (1) as: −G0 + W × r K = exp = K0 × exp(ω × r), (6) RT where K0 = exp(−G0 /RT ), ω = (W/(R × T )); K0 is the constant determining the critical (threshold) concentration of divalent metal ions corresponding to the beginning of DNA condensation, ω is the cooperativity parameter. Using (6) and binding equation in Scatchard’s form [9], one can obtain the dependence of r on Cf at different values of binding constants and cooperativity parameters: r = K0 × exp(ω × r) × Cf , (1 − r)
(7)
Fig. 2. Theoretical dependencies of the binding degree r on the free concentration of the divalent metal ions Cf calculated according to (7) (A) and (19) (B). Numbers at the curves-values of the parameters K0 (K0 ) and ω (ω ), respectively.
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such isotherms have to be replaced with dependencies with a jump along r, that may evidence a phase transition. Thus at definite conditions (for example, in aqueous-ethanol solutions, where ω became more than 8 [2]) DNA transition into compact state under the action of divalent metal ions may be of phase transition character. At the same time our experimental data shows that DNA transition into a compact state occurs in a narrow but continuous interval of Met2+ ion concentrations (Fig. 1). Even in the case of Cu2+ -induced DNA condensation the “coilcompact state” transition was not discrete. In our experiments to record IR spectra of DNA–Met2+ complexes a high molecular weight DNA was used. Besides, DNA concentration in solution was relatively high (∼10−2 -5 × 10−2 M of phosphorus). Obviously, at such experimental conditions dispersion of DNA molecules on their molecular weight is present. But, when calculating binding isotherms, we did not take into account the distribution of values along the assembly of DNA molecules. To obtain a more sequential thermodynamic description for the system of ions interacting with the biopolymers, in the present work we carried out calculations in terms of the macromolecule statistical sum. It should be noted that in (7) and (8) we do not take into consideration the competition between cations of different valences interacting with DNA. As in our experiments [1–4] we used Na+ -salt DNA; therefore there are two types of cations in the solution—Met2+ and Na+ ions. In this case the binding constant of Met2+ ions will depend on concentration of monovalent ions. In the presence of competition between di- and mono-valent cations the total binding degree will be determined as rt = rMet 2+ + 0.5 × rNa + (coefficient 0.5 is due to the difference in cation valences), where rMet 2+ and rNa + are the binding degrees of Met2+ and Na+ ions, respectively. To calculate the parameters K0 and ω the system of two equations should be used instead of (7), in which the Na+ ions binding to DNA is taken into account as well. But a general view of binding isotherms shown in Fig. 2A was not changed when we considered two types of cations instead of one (not illustrated).
2. New model considering the statistical sum of the macromolecule As in our previous model let us consider a DNA macromolecule as a long chain of freely conjugated segments. Each DNA macromolecule usually used in our experiments consisted of 104 –105 nucleotides, therefore it may be presented as a macrosystem. A macromolecule is in a solution of low molecular weight ligands (metal ions). In a solution ligands can interact with a macromolecule. Ligands may interact only with definite parts of a macromolecule (with binding sites), thus we don’t consider DNA molecule as a uniformly charged cylinder. One ligand can form two bonds with two different binding sites on a macromolecule. One binding site of a macromolecule can form only one bond with a ligand, thus
each element of a macromolecule can be in one of two conditions: in condition a, if it does not interact with ligand, or in condition b, if it bound to a ligand. Let the energy of one element of the macromolecule chain in state a remain zero while the energy of the element in state b–W0 (W0 > 0). Ligands bound to the neighbouring elements can interact each other. Under interaction of bound ligands we imply not only their direct interaction, but indirect one as well. Let the energy of such interaction mean that W1 . W1 can be both positive and negative depending on the interaction forces (attractive or repulsive) between bound ligands. The statistical sum of a large canonical ensemble may be presented as: Z(µ) =
∞
[eβ µ N Z0 (N)]
(9)
N=1
where β = 1/T, N is the number of pairs (as our macromolecule is a double helix) of chain elements, Z0 (N) is the statistical sum of a chain consisting of N element pairs: g{σns }exp[−βEN {σns }] (10) Z0 (N) = {σns }
where EN {σns } is the total energy of a chain at a given configuration of spins {σ n s }, g{σ n s } is the statistical weight of the spin configuration given. Detailed calculation of the statistical sum for such system has been done in [10]. To calculate the statistical sum the authors used the standard matrix method. Making use of the results of [10], we can write down the statistical sum as a: Z(µ, µ1 ) =
∞ 2(N−1) N=1 K=0
exp{β[µ(N − 1) + µ1 K]}Z0 (N, K) (11)
where β = (kT)−1 , Z0 (N, K)—the statistical sum of a double helix consisting of N element pairs, containing K bound ligands, µ is the chemical potential of complementary pairs of chain elements, µ1 is the chemical potential of ligands bound to a polymer chain. Let η is the ligand concentration on a chain (that is the ratio of chain elements (segments) bound to ligands to the total quantity of macromolecule elements), than η=
1 ∂Z/∂µ1 × 2 ∂Z/∂µ
(12)
Authors of [10,11] showed that on the phase equilibrium curve ∂λ2 1 , (13) η= 2λ2 ∂ε where λ2 = ω2 χ2 eε+ε1 1/2
(14)
E.V. Hackl et al. / International Journal of Biological Macromolecules 34 (2004) 245–250
((ω2 )1/2 is a statistical weight of a non-bound element of a chain), 1/2
χ2 = ch(ε + ε1 ) + [sh2 (ε + ε1 ) + e−2ε1 ]
.
It is known that an equation of the phase transition curve may be found from the condition of equilibrium of chemical potentials of macromolecule different phases. In the method of large canonical ensemble [10] the chemical potential is determined by condition. N=
−∂Ω , ∂µ
(15)
Ω = −kT lnZ(µ, µ1 ), where N is the amount of complementary pairs of macromolecule elements. At N → ∞ Eq. (15) may be written as ∂Ω/∂µ = −∞. From formulae (13) and (14) one may get the dependence of µ1 on η: eε+ε1 =
ε=
[(2η − 1)e−ε1 + B] 2[η(1 − η)]1/2
1 β(µ1 + W0 ), 2
(16)
1 ε1 = − βW1 2 1/2
B = [4η(1 − η) + (2η − 1)2 e−2ε1 ]
.
In the limit case η 1, µ1 = kT ln η − W0 At W1 = 0 for any concentration we obtain η µ1 = kT ln − W0 1−η In a solution of free low molecular weight ligands with concentration c (c 1) µ1 = kT ln c + f
(17)
where f is standard chemical potential. At this we suppose that ligands on a polymer chain and in a solution are in equilibrium. From the equality of ligand chemical potentials µ1 on a polymer chain determined by (16) and in solution (17) the dependence of η on c follows: 1 η= 2
249
a macromolecule (W1 ) corresponds to the cooperativity parameter ω (as we defined this value in [1,2]). We mentioned above that W1 can be both positive and negative depending on the interaction forces (attractive or repulsive) between bound ligands. In it’s turn cooperativity parameter ω can be both positive and negative too. As we indicated above, W0 is the energy of the chain segments bound to a ligand (metal ion); therefore, this value has a physical meaning of the binding constant (K0 ). It should be noted that this binding constant K0 might be not equal to the binding constant K0 calculated according to (7). Value c was defined as ligand concentration in solution that corresponds to the concentration of free ligands Cf . Taking into account all above the Eq. (18) may be re-written as ω 1 C f K0 e − 1 r = 1 + (19) 2 2 ω (Cf K0 e − 1) + 4Cf K0 Theoretical dependencies r (Cf ) calculated according to (19) are shown in Fig. 2B. As it is seen, in this case binding isotherms are of monotonous character at any values of binding constant K0 and cooperativity parameter ω . Thus we have shown that use of the exact expression for the statistical sum leads to the disappearance of bimodality on the r (Cf ) dependencies (Fig. 2). This means that the bimodal character of r (Cf ) dependencies is typical for a case when the distribution of values in ensemble of DNA macromolecules is not taken into account. In other words for a single DNA chain the Met2+ -induced transition into a compact state can be of phase transition character. But when we take into account the statistical sum of macromolecules (i.e. we take into account the ensemble of macromolecules) the isotherms describing the Met2+ -induced DNA transition into a compact state are of monotonous character. Thus upon the “coil state”–“compact (globule) state” transition, single DNA molecules may undergo the first-type phase transition; while the transition of the ensemble of DNA chains is of sigmoidal character typical for the cooperative and continuous process. This explains why in experiments, even in the case of DNA interaction with Cu2+ ions, DNA condensation (as a consequence of this interaction) occurred in a narrow but detectible interval of Cu2+ ion concentrations [1,2].
×1+
c × exp(β(W2 −W1 ) − 1
,
(c × exp(β(W2 −W1 ) − 1) +4c × exp(βW2 ) (18) 2
where W2 = f + W0 . As we defined η as a ratio of chain elements (segments) bound to ligands to the total quantity of chain elements, the physical meaning of η and r (where r is the binding degree defined in [1,2] as a part of chain segments bound to metal ions and formed a compact structure) values are the same. This allows us to transit from η to r value in (18). Besides, the interaction energy of ligands bound to the neighbour parts of
3. Comparison of both models with experimental results To compare two Eqs. (7) and (19) and to find out how both our models can fit the experimental results we did the following: for each K0 (K0 ) we chose ω (ω ) so that to minimise the root-mean-square deviation (ϑ) of a theoretical curve from experimental results. Parameters K0 and ω were calculated according to (7) while parameters K0 and ω —according to (19), experimental r and Cf values were taken from [1,2]. Dependencies of deviation ϑ on the binding constant K0 (K0 ) for DNA interaction with Mn2+ (A) and Ca2+ (B) ions are
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E.V. Hackl et al. / International Journal of Biological Macromolecules 34 (2004) 245–250
Fig. 3. Dependencies of root-mean-square deviation (ϑ) of theoretical curves from experimental results on binding constant K0 (K0 ) for DNA interaction with Mn2+ (A) and Ca2+ (B) ions. Parameters K0 and ω were calculated according to (7), parameters K0 and ω – according to (19), experimental D, r and Cf values were taken from [1,2]. Table 1 Binding constants K0 (K0 ) and cooperativity parameters ω (ω ) calculated according to (7) (K and ω) or (19) (K and ω ) for DNA interaction with Cu2+ , Mn2+ and Ca2+ ions in aqueous solution (K0 and ω parameters were chosen to minimise the deviation of experimental results from a theoretical curve) Met2+
K0
ω
K0
ω
Cu2+
4.5 1.7 0.5
11 7 7.2
2.5 0.8 0.08
8 3.3 3.8
Mn2+ Ca2+
shown in Fig. 3. As is seen from the figure, ϑ dependencies on K0 and K0 have similar character with well-expressed minima both for Mn2+ and Ca2+ ions interacting with DNA. The minimum corresponds to the pair of K0 (K0 ) and ω (ω ) parameters, which gives the lowest deviation of the experimental data from theoretical curve and, therefore, describes the binding of the divalent metal ions studied to DNA in the best way. These pairs of parameters are given in Table 1. The comparison of the two models shows that they both allow de-
termination of the binding parameters (binding constant and cooperativity) sufficiently well. As can be also seen from the data presented in the Table 1, both formulae give the same order of the binding constants for Met2+ ions interaction with DNA in a condensation process (Cu2+ > Mn2+ > Ca2+ ) and correlation between binding parameters (binding constants and parameters of cooperativity). Thus we can conclude that both formulae can be used to calculate the binding parameters for divalent metal ions interacting with DNA. In the present model considering the ligand binding to DNA elements we didn’t take into consideration all other kinds of interactions between macromolecules including interactions occurring by means of solvent molecules. As an example of such interactions one may mention the electrostatic forces between macromolecule charges. It is known that electrostatic interaction essentially influences the properties of strongly charged polyelectrolytes in solution. Due to such macromolecular interactions the permissive chain configurations in the statistic sum stop being equivalent (as we considered in the present work). Another assumption was using the model of a homogenous medium with a constant macroscopic value of dielectric permeability neglecting the change of this value in the vicinity of DNA surface.
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