1610
J. Phys. Chem. B 2007, 111, 1610-1619
Sharp Melting of Polymer-DNA Hybrids: An Associative Phase Separation Approach Alexander Kudlay,* Julianne M. Gibbs, George C. Schatz, SonBinh T. Nguyen, and Monica Olvera de la Cruz Northwestern UniVersity, Department of Chemistry, 2145 Sheridan Road, EVanston, Illinois 60208-3113 ReceiVed: October 2, 2006; In Final Form: December 18, 2006
An associative equilibrium theory describing the sharp melting behavior of polymer-DNA hybrids is developed. The theory considers linear polymers with attached DNAs on each polymer that serve as “stickers” and with a two-state model governing the DNA melting equilibrium. For three or more oligonucleotides on each polymer, solutions of polymer-DNA hybrids are found to undergo phase separation at sufficiently low temperatures. The dense phase dissolves as temperature increases, which leads to a sharp increase in the fraction of nonhybridized DNA near the phase transition temperature, in agreement with experimental absorbance profiles at 260 nm. The melting temperature is predicted to have the same dependence on salt concentration as a solution of unattached DNAs and be weakly sensitive to the concentration of DNA in solution. The melting temperature is predicted to be higher than that of unattached DNA in solution, with the magnitude of the increase sensitive to the DNA hybridization cooperativity. The theoretical predictions are generally in good quantitative agreement with new experimental data (also presented here), which show the effect of the polymerDNA hybrid length and salt concentration on the melting profiles.
1. Introduction Nanoparticles that self-assemble via hybridization of complementary DNA oligonucleotides have attracted much interest recently.1-10 In particular, there has been a growing interest in using these systems as DNA detection probes, because they offer advantages in selectivity and sensitivity over conventional molecular probes, such as fluorophores.2,6 A specific feature in these systems that leads to their detection capabilities is the coupling of DNA hybridization to the meso- or macroscopic behavior of the nanoparticle solution. The system that has received the most attention is a solution of gold nanoparticles functionalized with DNA.2,4,6 The diameter of the gold nanoparticles in the solution has been varied between 13 and 50 nm; each nanoparticle carries 90 to 250 DNA strands, respectively.6 In these experiments, two species of nanoparticles, functionalized with different oligonucleotide fragments, are mixed in a solution. Upon the addition of DNA strands complementary to the DNA fragments on the nanoparticles, hybridization leads to the formation of large clusters or aggregates. The transition is conveniently observed in the visible light as a color change from red to purple.2,4,6 It was found that, once the clusters are formed at a low temperature, the dissolution of clusters upon heating, monitored by the absorbance change at 520 nm (gold surface plasmon) or 260 nm (hybridization of DNA plus nanoparticle absorption), is unusually sharp. The welldefined character of the melting transition enables the detection of single-base mismatches between the probe and target DNA as a shift in the melting temperature.2,6 This system has an unusually high sensitivity: concentrations of target DNA of tens of femtomoles can be detected colorimetrically, and even lower concentrations can be served using a variety of amplification schemes.2 * Corresponding author. E-mail:
[email protected]. Present address: Institute of Physical Science and Technology, University of Maryland, College Park, Maryland 20742.
Several theoretical approaches have been developed for describing the structural and thermal properties of the goldnanoparticle-DNA system. A simple thermodynamic model has been proposed by Jin et al.,6 in which cooperative melting arises because the neighboring gold nanoparticles are connected by multiple DNA strands. The effective salt concentration near these strands is increased, which leads to an increase in the DNA melting temperature relative to that of unattached DNA strands in bulk solvent. Once a connecting DNA chain melts, the effective salt concentration for nearby strands is lowered, and thus, the melting temperature for the remaining strands is decreased, which promotes their further melting. This leads to sharper melting of the multiple DNA linking strands and subsequent rapid dissolution of the nanoparticle cluster. Recent computer simulations corroborate the assumption of the melting temperature increase for arrays of DNA duplexes with small interduplex separations.11 The changes in the structure of the aggregates and their optical properties upon melting have been investigated by Park et al.,12-14 who assumed multiple linking between neighboring nanoparticles and used a simple probabilistic model to describe the cluster melting. The fractal structure of the clusters at different temperatures is investigated, and the scattering profiles are found to be similar to experimental findings. Lukatsky and Frenkel15 investigated the phase behavior of the gold-nanoparticle system using a simple mean-field theory of colloidal interactions. They concluded that the experimentally observed sharp melting is a manifestation of a phase transition. A general phase diagram of a colloidal system with short-range attraction and soft-core repulsion was studied by Tkachenko.16 Novel polymer-DNA hybrids,17 which are the subject of the present study, represent another DNA-hybridization driven selfassembling system. In this case, the hybrid molecule consists of a semirigid polymer backbone with a number of oligonucleotides chemically attached to it as side chains. Two different systems are experimentally investigated: a 17-mer polymer with
10.1021/jp0664667 CCC: $37.00 © 2007 American Chemical Society Published on Web 01/26/2007
Melting of Polymer-DNA Hybrids ≈5 oligonucleotides (each oligonucleotide consists of 15 base pairs attached through a T3 spacer) and a 37-mer polymer with ≈12 oligonucleotides (same sequence as for 17-mers). For each system, the solution contains two types of hybrid molecules: the first type carries n oligomers with the sequence 5′-(T3) ATC CTT ATC AAT ATT-3′, and the second type carries n complementary sequences (with the same T3 spacer). These polymer-DNA hybrids are typically present in a solution at micromolar concentrations. Similar to the gold-nanoparticle case, our polymer-DNA hybrids have been shown to exhibit a sharp melting transition (monitored by absorbance at 260 nm), which allowed the detection of single-base mismatches with the use of electrochemical labels.17 The DNA-linked polymer system is also of significant theoretical interest because it is not so complex as the gold-nanoparticle system: no charged nanoparticles are involved, the DNA loading is more controllable, and the optical measurements at 260 nm are controlled mostly by DNA (rather than nanoparticle optical properties), thus yielding a direct measure of DNA hybridization. As a result, polymer-DNA hybrids provide a good model system for elucidating the origin of the sharp melting. In the next section, we give a brief overview of the experimental methods used to make the polymer-DNA hybrids and to study their thermal properties. In section 3, a simple microscopic theory based on the mean-field Flory gelation model is presented. The solution of the polymer-DNA hybrids is shown to be phase separated at low temperatures when three or more DNAs per polymer are present. We argue that the origin of the sharp melting is a rapid dissolution of the dense phase at the phase transition temperature, which leads to sharp melting profiles in qualitative agreement with experiment. In section 4, we present new experimental results on the variation of the melting profiles with changing salt or polymer hybrid concentration. We show that, although the basic Flory theory correctly captures the increase in the melting temperature Tm (relative to a solution of unattached DNA) to achieve a quantitative agreement, we need to include the cooperativity in DNA hybridization on the polymers. This is done with the introduction of two cooperativity parameters. Conclusions are presented in Section 5. 2. Experimental Methods The 17- and 37-mer diphenyl acetylene homopolymer precursors to the polymer-DNA hybrids were synthesized following standard procedure, where the length of the polymers was equal to the catalyst-to-monomer ratio in the polymerization mixture.18 Polymer precursors were coupled to oligonucleotides also following previously reported methodology.18 The number of oligonucleotides per polymer was determined on the basis of the UV spectra of the polymer-DNA hybrids.17 For each hybridization, a solution of both complementary compounds (1.9 µM final concentration of each complementary strand, based on oligonucleotide concentration) was prepared in 1.0 mL of PBS buffer (10 mM phosphate, pH ) 7.0, [NaCl] ) 0.15 1.0 M) and allowed to equilibrate overnight. The thermal denaturation curves of these aggregates or duplexes were obtained by monitoring their absorbance at 260 nm as a function of temperature. For the salt variation studies, melting analyses of DNAcontaining materials were performed using a Hewlett-Packard (HP) 8453 diode-array spectrophotometer equipped with an HP 89090A Peltier temperature controller. The temperature was ramped up at 1 K/min, and the samples were stirred at 250
J. Phys. Chem. B, Vol. 111, No. 7, 2007 1611 rpm.19 For the polymer hybrid concentration variation studies, the melting was monitored using a Cary 500 instrument with constant heating at 0.25 K/min and a medium stirring rate. 3. Theory of Associative Phase Separation We start the theoretical discussion by presenting a simple Flory-type theory, which is the basis of our approach. We show that this simple approach yields the experimentally observed sharp melting of polymer-DNA hybrids arising from phase separation. The origin of the phase separation is the effective attraction between the polymer chains caused by hybridization of the complementary DNA oligomers on the polymer backbones. 3.1. Free Energy Model. To describe the experimental system we consider a solution of two types of polymer chains (p1 and p2) carrying complementary DNA oligomers, each chain carrying DNA of only one sequence type. Assuming a symmetric system, i.e., that the concentrations of the two types of polymer chains are equal, we denote the total number concentration of all polymers Fp ) 2Fp1 ) 2Fp2. The polymer chains are assumed to be monodisperse and comprised of N repeating units. For simplicity, each polymer chain is assumed to carry exactly n DNA oligomers. The distribution of attachment sites along the polymer backbone has no effect on the results of the mean-field theory developed in this section. (An approximate way of including a nonuniform distribution into the theory is discussed later in Section 4.) The fraction of monomers with attached DNA is f ) n/N, and the number concentration of DNA oligomers (of both types) in the solution is FDNA ) nFp. The two experimental systems, results for which are presented in the next section, are (i) N ) 17, n ) 5 ( 1 and (ii) N ) 37, n ) 12 ( 1. The fraction of monomers with attached DNA is the same (≈0.3) for both systems, as determined by experimental procedures. The melting profile measurements and salt variation results are presented for solutions with FDNA ) 3.8 µM (for both N ) 17 and 37 systems). To describe the contribution to the free energy due to thermally reversible association of complementary DNA oligomers, we employ a Flory-type gelation theory.20,21 The theory is based on two assumptions: (i) the same association constant for all associating groups () stickers), regardless of the states of neighboring stickers, and (ii) the theory takes into account only treelike aggregate structures, thus neglecting all cycles. The latter assumption can be shown to be equivalent to the mean-field assumption of the neglect of correlations between associating groups.22 With these two assumptions, the association of a given sticker pair occurs independently of the association of other sticker pairs in the system. As a result, within this mean-field theory, the distribution of stickers along the polymer chain and chain flexibility are not important.23 Consistent with these conclusions, the free energy of associating polymers in the Flory approach is equivalent to that of unconnected associating groups.23,24 The Flory approach to gelation is known to be adequate for description of phases rich in polymer (overlapping polymer chains regime) in which a given sticker is surrounded mostly by stickers from other polymers; thus, the effect of cyclization is generally insignificant.22,23,25 (For dilute solutions, the possibility of cyclization through intrachain association has been considered recently for flexible Gaussian chains.23,26) For the polymer-DNA hybrids considered in this work, the possibility of intrachain association is excluded because polymers carry only DNA of one type. However, due to the rigid nature of polymer backbones, the possibility of small cycles resulting from
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formation of multiple DNA duplexes between polymer backbones is increased. In this paper, we first develop a basic Flory theory (in which all cycles are neglected) and compare its result with experiment. In Section 4, we modify this simple approach by assuming that the DNA oligomers hybridize in blocks, which roughly estimates the importance of the corrections due to cyclization (fluctuations). Due to the effective attraction between polymer strands that arises from DNA hybridization at sufficiently low temperatures, the solution phase-separates into dilute and concentrated phases, with respect to polymer (and thus DNA). For simplicity, we assume that the concentrated phase is stabilized by the balance of associative and steric (hardcore) interactions; the van der Waals interactions are assumed to be less important and are neglected. The dilute phase concentration is determined primarily by the balance between translational entropy of the polymers and the free energy of the polymer in the concentrated phase. In accordance with this model, the free energy density of a homogeneous solution has the form
Fp F ) Fp ln + FDNA[(1 - Γ) ln (1 - Γ) + Γ ln Γ] VkBT 2e FDNAΓ FDNAΓ 1 ln K + (1 - FpNVm - FDNAνDNA) ln (1 2 2e Vm FpNVm - FDNAνDNA) (1) with V being the total volume of the system and kBT the thermal energy. The first term is the translational energy of the polymer chains. The second and third terms are due to the reversible association of DNAs on the polymer chains. In accordance with the Flory theory of gelation, the second term describes the entropy of mixing and the third one the enthalpic gain from association. (For a detailed derivation and discussion, see refs 23,24, and 26.) In eq 1, we have introduced the conversion Γ, which is defined as the fraction of hybridized (associated) DNA (associated) oligomers in the system: Γ ≡ FDNA /FDNA. We have also introduced the DNA association constant K ≡ e-∆G/(RT), where ∆G ) ∆H - T∆S is the melting free energy of a DNA duplex, which can be decomposed into the enthalpy ∆H and entropy ∆S of melting. The fourth term in eq 1 is the steric contribution, which is written using a simple lattice model.27 Specifically, Vm and νDNA are the steric volumes of the monomer unit and the DNA oligomer, respectively, so that FpNVm and FDNAνDNA are the volume fractions occupied in the system by monomers and the oligomers. We assume Vm is the volume of an elementary cell. The conversion Γ is determined from minimization of the free energy (given by eq 1) according to ∂F/∂Γ ) 0. This minimization yields the usual law of mass action:
FDNA Γ K ) 2 2 (1 - Γ)
(2)
K ) e-∆G/(RT)
(3)
Using this result to eliminate K from eq 1, one finds that the free energy takes the following simple form:
Fp 1 Γ F ) Fp ln + FDNA + ln(1 - Γ) + (1 - FpNVm VkBT 2e 2 Vm FDNAνDNA) ln(1 - FpNVm - FDNAνDNA) (4)
[
]
It is convenient to express eq 3 through a single volume fraction,
e.g., the volume fraction of monomeric units φm ≡ FpNVm, which gives the following dimensionless free energy:
[
]
φm φ m F Γ ln + φm + ln(1 - Γ) + ) kBTV/Vm N 2e 2 VDNA VDNA 1 - φm 1 + f ln 1 - φm 1 + f Vm Vm
( [
]) ( [
])
(5)
Note that here the conversion Γ is not a minimization parameter as in eq 1. Rather, its value is determined by the mass action law for a given concentration and temperature. Because we assumed that each polymer chain carries the same number n of DNA oligomers, the polymer and DNA concentrations are trivially related: FDNA ) nFp. Therefore, at a given temperature, the free energy of eq 4 is a function of only one concentration variable, φm, which can be used to characterize the separation of the system into the two phases: dilute and concentrated in terms of φm. In our formulation of the free energy model, we only implicitly consider electrostatic effects and the entropy of counter- and salt ions. This is done in two ways. The influence of electrostatic interactions and ion entropies on DNA hybridization thermodynamics is characterized with a phenomenological dependence of ∆G on NaCl concentration, which is introduced below. Highly screened (effectively short-range) repulsions between DNA oligomers are modeled with an effective steric radius. This simple approach is a good approximation, because the experiments are typically conducted at rather high salt concentrations of 0.1 to 1.0 M, corresponding to the Debye-Hu¨ckel screening radius of 10 to 3 Å. Comparing it with the length of the polymer (≈ 82 Å), we conclude that we can neglect all long-range electrostatic effects between DNA strands on different polymer chains. Such effects could give rise to collective contributions to the free energy at lower salt concentrations;28,29 however, they are not significant in our case. This simple treatment of electrostatics is confirmed by the experimental results, as discussed below. 3.1.1. Phase Separation and Sharp Melting. The effective attraction arising from the hybridization of complementary DNAs on the polymer chains leads to phase separation at sufficiently low temperatures.30 To obtain the monomer volume (c) fractions of the coexisting dilute (φ(d) m ) and concentrated (φm ) phases we equate the pressure and the chemical potential in the two phases. Using the dimensionless free energy density F ≡ F/(kBTV/Vm) given by eq 4, we numerically solve the equations
∂F ∂F | ) | ∂φm φm(d) ∂φm φm(c)
(
-F +
)
(
(6)
)
∂F ∂F | ) -F + | ∂φm φm(d) ∂φm φm(c)
(7)
(c) to obtain φ(d) m and φm at a given temperature. Variation of the temperature then yields the phase diagram. To compare the theoretical modeling results with experimental data, we need to specify ∆G explicitly. For the sequence 5′(T3) ATC CTT ATC AAT ATT-3′ used in our experiments, the nearest-neighbor model31,32 with the dangling T end correction33 yields ∆H ) -116.8 kcal/mol and ∆S ) -330.6 cal/ (mol K) for the standard NaCl salt concentration of 1.0 M. We use these values to calculate the melting temperature (R is the gas constant)
Melting of Polymer-DNA Hybrids
Tm(1.0 M) )
J. Phys. Chem. B, Vol. 111, No. 7, 2007 1613
∆H ≈ 326.1 K ∆S + R ln(FDNA/4)
(8)
for a solution of unattached oligomers with the total DNA concentration FDNA ) 3.8 µM. The experimentally obtained value is Tm ≈ 329.8 K. To bring the theoretical value in agreement with experiment we change ∆S to ∆S ) -326.5 cal/ (mol K). This small correction can be attributed to the influence of the dangling T3 segment, because the dangling end correction33 taken into account in ∆H and ∆S is only applicable to a single dangling T base. Note that we modify only ∆S because the dangling end corrections are ≈3 times larger for ∆S (see ref 33) than for ∆H. The enthalpy ∆H is well-modeled as being independent of the salt concentration.32,34 To obtain the dependence of ∆S on [NaCl], we utilize the empirical formula from Owczarzy et al.34 The formula fits the dependence of T-1 m with a linear and quadratic in ln[NaCl] terms and takes into account the fraction of GC base pairs in the sequence, which gives a better precision than other empirical fits available in the literature:34
1 1 ) + (4.29f(GC) - 3.95)10-5 ln Tm([NaCl]) Tm(1 M) [NaCl] + 9.40 × 10-6(ln[NaCl])2 (9) where [NaCl] is in mol/L, and f(GC) is the fraction of GC base pairs. For our sequence, f(GC) ) 0.2, which gives a linear coefficient of - 3.09 × 10-5. We measured the dependence Tm([NaCl]) experimentally for our sequence using a solution of unattached DNA chainssthe results are plotted in Figure 3b with open circles. To achieve the best fit using an expression of the type of eq 7, we found it necessary to correct the linear coefficient in it to
1 1 ) -5.41 × 10-5 ln[NaCl] + 9.40 × Tm([NaCl]) Tm(1 M) 10-6(ln[NaCl])2 (10) Such a change can be attributed to the dangling end effect, because eq 7 was obtained for oligomers without any dangling ends; other effects are also possible. Because ∆H is assumed not to depend on [NaCl], we can use eq 6 to obtain ∆S([NaCl])
∆S([NaCl]) ) ∆S(1 M) + 6.32 ln[NaCl] - 1.10(ln[NaCl]) (11) 2
Summarizing, for our oligomer sequence with the T3 spacer, the association constant K is given by eq 3 with ∆G ) ∆H T∆S, where ∆H ) -116.8 kcal/mol is independent of salt, and ∆S is given by eq 9 ([NaCl] is in mol/L) with ∆S(1 M) ) -326.5 cal/(mol K). The free energy given by eq 4 is formulated in terms of the monomer volume fraction φm and uses the monomer and DNA excluded volumes, Vm and VDNA, respectively. As such, we also need to specify Vm and VDNA. Assuming the length of each backbone polymer repeating unit to be ≈0.5 nm,35 the width to be ≈1.8 nm, and the third dimension to also be ≈1.8 nm (to take into account thermal rotations of the side group on the backbone), we arrive at Vm ≈ 2 nm3. This gives the following relation: φm ) N(1.2 × 10-6)Fp[µM] between the monomer volume fraction φm and the concentration of polymers Fp (expressed in µM). For a single-stranded DNA (ssDNA), we assume a base length of 0.5 nm (see refs 36 and 37), which for an 18-base oligomer (15 probe bases plus a T3 spacer) gives the contour length of ≈9 nm. The bare steric radius is assumed
to be ≈0.5 nm. Additionally, the effective electrostatic radius is taken to be 0.5 nm regardless of the salt concentration (corresponds to the Debye-Hu¨ckel screening radius at [NaCl] ≈ 0.3 M). Assuming that the ssDNA is essentially a rod,37 the excluded volume VDNA is ≈30 nm3. It should be also noted that in the formulation of the free energy we neglected the differences between the excluded volumes of two ssDNAs and one duplex (the volume, calculated with similar assumptions, is ≈45 nm3). Results of the theory that take that feature into account show that the corrections are negligible. The excluded volumes Vm and VDNA are certainly approximate; however, we directly checked how the variation in Vm and VDNA affects the resulting phase diagrams. Changing Vm from 1 to 5 nm3 corresponds to a shift of the coexistence curve by ≈1 K; variation of Vm/VDNA from 1 to 30 (maximum physically reasonable range) corresponds to ∆T ≈ 4 K. Therefore, variation of Vm and VDNA within physically reasonable limits does not lead to significant changes of the phase diagrams. Figure 1a presents a theoretically calculated phase diagram in terms of log Fp versus temperature T, where Fp is the total concentration of polymer chains in µM. In the phase diagram, the salt concentration [NaCl] ) 0.3 M. We present coexistence lines (solid curves) for the two experimental systems: polymers with n ) 5 DNA strands on the chain (N ) 17, f ≈ 0.3) and longer chains with n ) 12 (N ) 37, f ≈ 0.32). A solution of the polymer-DNA hybrids is phase separated into a dilute and concentrated phases in the area below the coexistence line; the solution is homogeneous in the area of parameters above the coexistence line. (c) The densities of the dilute F(d) p and concentrated Fp phases can be determined from the coexistence lines by the usual lever rule. As is clear from Figure 1a, for a given system, F(c) p changes relatively little with increasing T; at the same time, the variation of F(d) p is significant (note the log scale). For the n ) 12 case, the coexistence curve lies above (higher temperatures) that of the n ) 5 system (for all, except very high concentrations). For the n ) 5 system, we have also plotted two additional curves in Figure 1a. The dashed curve below the solid coexistence line is the spinodal,27 which is calculated from the usual condition:
∂2F )0 ∂φ2m
(12)
In the area between the coexistence line and the spinodal, the homogeneous solution can exist as a metastable state, even though the temperate is lower than the equilibrium phase transition temperature. The spinodal is physically important in that it shows the range around the transition temperature in which the system can be found in metastable states, thus where thermal equilibration is expected to be slow. For Fp ∼ 1 µM (typical experimental concentration), such a range is only ∼3 K, but it can be seen to rapidly increase with decreasing Fp. The dotted line above the solid coexistence line is the solgel boundary for the n ) 5 system,28 which is obtained in the homogeneous state from the usual condition,23,22 Γ ) 1/(n 1). The gelation line separates the sol (small Fp) and gel (large Fp) states of the homogeneous solution. From the topology of the phase diagram, we can infer that the concentrated phase exists in the gel state. The variation of the fraction of non-hybridized DNA (γ ≡ 1 - Γ) with temperature is plotted in Figure 1b. The conversion Γ is calculated using conversions in the dilute and concentrated
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Figure 1. (a) Phase coexistence lines (solid lines) for two different systems n ) 5 (N ) 17) and n ) 12 (N ) 37). For both, [NaCl] ) 0.3 M. For the n ) 5 case, the spinodal (dashed curve) and gelation line (dotted curve) are shown. (b) Solid line: total fraction of non-hybridized DNA γ for n ) 5 at FDNA ) 3.8 µM as a function of temperature. Upper and lower dashed curves: fractions of non-hybridized DNA in the dilute and dense phases, respectively.
phases: Γd and Γc, which are determined from eqs 2 and 3 using the solutions of eqs 6-7. Weighted with the fractions of polymer chains in the two phases, ξd and ξc, the total conversion of this two-phase system is given by
Γ ) ξdΓd + ξcΓc ξd )
(d) F(c) p - Fp Fp F(c) - F(d) Fp p
p
ξc )
(13) (c) Fp - F(d) p Fp F(c) - F(d) Fp p
(14)
p
(In the homogeneous solution, Γ is given simply by eq 2.) Experimentally, the variation of A260, the optical absorbance at 260 nm of the polymer-DNA hybrid solutions, is used to determine changes in γ.39 If we assume that the maximum absorbance at high temperatures A(max) 260 is entirely due to nonhybridized DNA, the lowest absorbance A(min) 260 is due to fully hybridized DNA; additionally, if we assume that there is no variation of absorbance coefficients with temperature, γ is given (max) (min) simply by γ ) (A260 - A(min) 260 )/(A260 - A260 ). Note that all absorbance measurements are corrected for background by subtracting the average absorbance in the 350-400 nm interval. The solid curve in Figure 1b depicts the fraction of nonhybridized DNA γ as a function of T, calculated for the n ) 5 case. The total concentration of polymer chains in the system is Fp ≈ 0.77 µM (FDNA ) 3.8 µM for n ) 5, N ) 17), [NaCl] ) 0.3 M. The two dotted curves are the fractions of nonhybridized DNA in the two phases: γd ≡ 1 - Γd (upper curve) and γc ≡ 1 - Γc (lower curve); they go up to Tm only (above Tm, the solution is homogeneous). In agreement with experiments, γ exhibits a sharp increase upon approaching the phase transition temperature Tm, where it shows a kink. For convenience, we will define the theoretical melting temperature as the phase transition temperature, which is consistent within an error of ≈1 K with the usual experimental definition of Tm as the position of the peak in the first derivative. However, other definitions, such as γ(Tm) ) 1/2 are in principle possible. As we can see from Figure 1b, the origin of the sharp increase of γ near Tm is the existence of the two phases with greatly varying concentrations and therefore conversions between them. This sharp increase in γ is mostly due to the dissolution of the concentrated phase (with low values of γ); the T in the values of γd and γc themselves is not so significant. Most of the
variation of γ takes place in the phase-separated state of the system, whereas in the homogeneous state (T > Tm) the variation is much smaller. We further investigate the origin of sharp melting by calculating the melting curves γ(T) for different numbers of DNAs attached to the polymer while keeping the total concentration of DNA strands in the solution FDNA constant. Such melting curves are plotted in Figure 2a. For all curves, FDNA ) 3.8 µM, and [NaCl] ) 0.3M. The solid curves are for varying n, and Fp ) FDNA/n: n ) 3 (N ) 17, f ≈ 0.18); n ) 5 (N ) 17, f ≈ 0.3); and n ) 12 (N ) 37, f ≈ 0.32). The cases n ) 5 and 12 correspond to the experimental 17- and 37-mer systems discussed in the next section. The dashed curve is a reference case for a solution of unattached DNA strands, which exhibits broad melting behavior. For n ) 2, we do not find phase separation: the melting curve is smooth and coincides with that of free DNA, which is in agreement with experiments on DNA dimers.40 As we can see from Figure 2a, the melting sharpness is related to an increased Tm relative to the case of free DNAs in solution. This increase is due to the fact that the hybrids are mostly in the dense aggregate phase below the melting temperature (due to the specific shape of the coexistence curve). The DNA concentration in the dense phase is much higher than the total concentration of DNA in solution, which strongly shifts the equilibrium toward hybridization in the mass action law (eq 2). As T rises, the dense phase dissolves, manifested by a sharp increase in γ. At T > Tm, when the solution becomes homogeneous, the variation of γ is slow and is the same as in the case of free DNA strands, which is a consequence of the mean-field assumptions of the Flory gelation approach.22,23,25 Our theory predicts that with increasing n the melting temperature Tm as well as the sharpness of the transition should increase. The increase of Tm corresponds in Figure 1a to a shift of the coexistence line to higher temperatures. The comparison of melting profiles for solutions with different total concentrations of polymer chains is presented in Figure 2b. The profiles correspond to the case in Figure 1a where we fix Fp at 0.1,1,10, and 100 µM and increase T. For all curves, [NaCl] ) 0.3 M, n ) 5, and N ) 17. Overall, the variation of Tm with Fp is rather small (e.g., Tm(1µM) - Tm(0.1µM) ≈ 1.8 K). The sharpness of the phase transition is predicted to decrease slightly for increasing Fp. This effect is due to the increase in (c) the concentration of the dilute phase F(d) p ; Fp remains nearly
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Figure 2. (a) Fraction of non-hybridized DNA γ as a function of temperature for n ) 3 (N ) 17), n ) 5 (N ) 17), and n ) 12 (N ) 37). For all cases, FDNA ) 3.8 µM, and [NaCl] ) 0.3 M. The dotted curve is the reference system of free DNA in solution. (b) Effect of varying the polymer concentration Fp (indicated in the plot) on the melting profiles. [NaCl] ) 0.3 M, n ) 5 (N ) 17).
Figure 3. (a) Experimental normalized corrected absorbance profiles at 260 nm as a function of temperature at different [NaCl] as indicated in the plot. FDNA ) 3.8 µM, n ) 5 (N ) 17). (b) Comparison of theoretical and experimental dependences of the melting temperature Tm on [NaCl]. Solid lines: theory for n ) 5 (N ) 17) and n ) 12 (N ) 37). Experimental data: n ) 5 (open triangles); n ) 12 (solid squares). The solution of free DNA is shown by open circles. In all cases, FDNA ) 3.8 µM.
constant. As a result, there is less difference between γd and γc and thus a less pronounced increase of γ when the concentrated phase dissolves. However, the decrease in sharpness is predicted to be significant only for rather large values of Fp. Experimentally, for the 17-mer system at 0.3 M NaCl, we find negligible (within the experimental error of 1 K) variation of Tm and sharpness upon the increase of Fp from 1 to 5 µM. These experimental observations are consistent with the theory, which predicts a
