Sharp Melting in DNA-Linked Nanostructure Systems: Thermodynamic

Sharp melting that has been found for DNA-linked nanostructure systems such as DNA-linked gold nanoparticles enhances the resolution of DNA sequence ...
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J. Phys. Chem. B 2007, 111, 8785-8791

8785

Sharp Melting in DNA-Linked Nanostructure Systems: Thermodynamic Models of DNA-Linked Polymers Sung Yong Park,† Julianne M. Gibbs-Davis, SonBinh T. Nguyen, and George C. Schatz* Department of Chemistry, Northwestern UniVersity, EVanston, Illinois 60208 ReceiVed: March 12, 2007; In Final Form: May 15, 2007

Sharp melting that has been found for DNA-linked nanostructure systems such as DNA-linked gold nanoparticles enhances the resolution of DNA sequence detection enough to distinguish between a perfect match and single base pair mismatches. One intriguing explanation of the sharp melting involves the cooperative dehybridization of DNA strands between the nanostructures. However, in the DNA-linked gold nanoparticle system, strong optical absorption by the gold nanoparticles hinders the direct observation of cooperativity. Here, with a combination of theory and experiment, we investigate a DNA-linked polymer system in which we can show that the optical profile of the system at 260 nm is directly related to the individual DNA dehybridization profile, providing a clear distinction from other possible mechanisms. We find that cooperativity plays a crucial role in determining both the value of the melting temperature and the shape of the melting profile well away from the melting temperature. Our analysis suggests that the dehybridization properties of DNA strands in confined or dense structures differ from DNA in solution.

1. Introduction DNA-linked nanostructure systems (Figure 1), in which multiple DNA strands of known sequence are attached to a central “core” like a nanoparticle, a dendrimer, a virus, or a polymer backbone, have attracted significant interest recently as a result of important advances in nanotechnology applications related to self-assembly and molecular recognition.1-10 The melting profile in particular has attracted interest, as it is found that this is usually considerable sharper than that for the corresponding dehybridization of DNA in solution.1,11-13 An interesting aspect of the melting process is that it can occur under conditions where aggregation is largely reversible,1 which is to be contrasted with the irreversible conditions of other nanoparticle aggregation processes.14,15 The melting behavior in DNA-linked nanoparticle aggregates has been the subject of several theoretical studies,12,16-24 but there is not yet agreement on the underlying mechanism. Two dominant explanations have been proposed, and there is a consensus that multiple DNA links between the nanoparticles play a significant role on the sharp melting. The first (more physical) explanation focuses on rapid morphological change of the nanoparticle aggregates during melting, including cluster melting and phase separation behavior,18-21,24 while the second (more chemical) explanation focuses on how the extreme chemical environment around the surfaces of the nanoparticles can lead to cooperative melting, a profound change in DNA dehybridization itself.12,23 In spite of some experimental indications of cooperative melting,12 the importance of this mechanism is very hard to establish because both explanations are not mutually exclusive. Experimentally, the sharp melting profile in the DNA-linked nanostructures is often measured by monitoring the change in the extinction spectra at 260 or 520 nm. In the case of the DNA-linked gold nanoparticles, the major * Corresponding author. E-mail: [email protected]. † Present address: Department of Biostatistics and Computational Biology, University of Rochester, Rochester, NY 14608.

Figure 1. DNA-linked nanostructure systems: (a) DNA-linked polymer system (b) DNA-linked nanoparticle system. Aggregates of DNA-linked nanostructures are formed in the presence of complementary oligonucleotide linkers.

contribution to the spectra comes from the gold nanoparticles rather than from the DNA,25 and so although the morphological change of the gold nanoparticle aggregates is clearly defined, it is uncertain whether cooperative melting induces the morphological change or not. A better system for studying DNA dehybridization behavior and to determine the importance of cooperativity involves DNA-linked polymer hybrids5,9 (Figure 1a), as the change at 260 nm mainly comes from hypochromism

10.1021/jp071985a CCC: $37.00 © 2007 American Chemical Society Published on Web 07/06/2007

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in the DNA, thus providing better sensitivity to the DNA dehybridization mechanism. In this paper, we use a combination of theory and experiment to show that cooperativity in DNA dehybridization is essential to melting in the DNA-linked polymer system and thus also plays a role in this process that goes beyond simple phaseseparation behavior. For this purpose, we will use simple thermodynamics and statistical mechanical arguments to construct a minimal model that captures the essence of the difference between melting behavior with and without cooperativity. Using this model to interpret the spectrum of DNAlinked polymer melting at 260 nm, we show that the theories based on cluster melting or phase transitions alone do not fully explain the observed results. 2. The Dissociation of a DNA-Linked Nanostructure Pair We begin by considering the dissociation of a DNA-linked nanostructure pair AB into nanostructures A and B, where the maximum number of DNA double strands between them is n. We defer giving a precise definition of what we mean by A or B, but one could imagine that each represents a polymer molecule that contains n DNA single strands. Using a simple two-state equilibrium model, the relative proportion of AB and A (or B) is determined by the chemical reaction

A + B h AB

(1)

We define the dissociation probability pdis(T) of the DNA-linked nanostructure pair AB as pdis(T) ) [A]/([AB] + [A]). This is related to the dehybridization of DNA double strands between the pair AB. To evaluate pdis(T), we need the dehybridization curve f(T) of each DNA double strand. Here we consider two different DNA dehybridization processes: ordinary (noncooperative) DNA dehybridization and cooperative DNA dehybridization. 2.1. Noncooperative DNA Dehybridization. For free DNA strands in solution, we can describe the dehybridization process of a double-stranded oligonucleotide using a two-state model, 26 where the relative proportion of D and S is determined by the chemical reaction

S+ShD

(2)

The chemical equilibrium condition corresponding to eq 2 is

fs(T)2CT [S][S] ) ) K(T) [D] 2(1 - fs(T))

(3)

where fs(T) is the fraction of single DNA strands at temperature T, i.e., fs(T) ) 2[S]/CT. Here, K(T) is the “conventional” equilibrium constant (with a units of concentration), and CT is the total molar concentration of single DNA strands in the sample when all DNA strands are dehybridized. The solution to eq 3 is

1 fs(T) ) (-K′ + xK′,2 + 4K′) 2

(4)

where K′ ) 2K(T)/CT. In our calculations, we have also assume simple van’t Hoff behavior K(T) ) exp[-∆Gs/kBT], with a Gibbs free energy of melting ∆Gs(T) ) ∆H - T∆S, where ∆H is the enthalpy of melting and ∆S is the entropy. Actually, ∆H and ∆S depend on the temperature, and thus we can expand ∆G in powers of T - TM using,

∆G ) c0 + c1(T - TM) + c2(T - TM)2 + c3(T - TM)3

(5)

where TM is the melting temperature and at T ) TM, f(TM) ) 1/ , so that -∆G(T )/k T ) -c /k T ) log(C /4). However, 2 M B M 0 B M T the coefficients of c2 and c3 are usually small, so that the contributions of these terms are not significant in the temperature region where the melting of the system takes place. Thus, the temperature variation of ∆H and ∆S is not significant, so that we can consider them as constant values ∆Ho and ∆So, respectively. In ref 27, the best fit can be found when we use the free energy at 37 °C. (A more accurate evaluation can be found in ref 24.) Hence, using a reasonable estimation of the free energy of dehybridization of double strands, we can obtain fs(T) in eq 4. The temperature variation of ln K′(T) depends mainly on the magnitude of ∆H, and as a consequence, when ∆H becomes bigger, the dehybridization profile fs(T) in eq 4 becomes narrower. One thing we emphasize is that the temperature profile of the DNA-linked polymer system at 260 nm is determined by the individual DNA’s rather than the breakup of larger units. Noncooperative DNA dehybridization (that does not involve a change in ∆H) makes the dissociation profile narrower due to multiple DNA double strands between each polymer pair, but the individual DNA dehybridization profiles for each DNA double strand are basically the same as in solution. As a result, we can use fs(T) in eq 4 to evaluate the dissociation probability pdis(T) of a nanoparticle pair that has n DNA double strand links in between. In this case, the probability that none of the single strands in the DNA-linked nanostructure pair forms a double strand can be simply approximated to

pdis(T) ) fs(T)n

(6)

A more elaborate consideration of noncooperative DNA hybridization includes entropic effects as found in refs 20 and 21. As n increases, pdis(T) shows a narrower dissociation profile and higher dissociation temperature. However, this behavior does not change the individual dehybridization profile because this behavior is not related to ∆H, a key parameter for broadness of DNA dehybridization profile. 2.2. Cooperative Dehybridization. Now we turn to cooperative dehybridization, where all n DNA double strands in the DNA-linked nanostructure pair melt together as a result of interactions between the neighboring duplexes. This can arise from enhanced salt concentration, as recently discussed in refs 12 and 23, but other factors can be involved such as strand confinement. In this case, the dehybridization curve of DNA double strands differs significantly from the individual dehybridization curve fs(T), as initially noted in ref 12. Here we introduce a phenomenological derivation of the cooperative DNA dehybridization profile to understand how cooperativity can change ∆H. First, we evaluate the Gibbs free energy difference ∆GAB for the dissociation of the DNA-linked nanostructure pair. This can be written as

∆GAB ) n∆Gs + ∆Gint

(7)

where ∆Gs is the Gibbs free energy difference between a DNA double strand and two separated single complementary strands, and ∆Gint is the Gibbs free energy change due to the interaction between DNA double strands in A and B. With a naive assumption that ∆Gint is much smaller than n∆Gs, we can approximate ∆GAB as n∆Gs. Hence, the equilibrium associated with reaction 1 can be written as

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[A][B] ) exp(-∆GAB/kBT) = exp(-n∆Gs/kBT) [AB]

(8)

Here, because [A] ) [S]/n and [AB] ) [D]/n, the chemical equation of the DNA strands that cooperatively melt can be written as

[S][S] = n exp(-n∆Gs/kBT) [D]

(9)

when the DNA double strands interact with each other but ∆Gint is negligible. Note that if ∆Gint has the same temperature dependence as ∆Gs, we can still use eq 9 but with an effective n. Using eq 9, the DNA dehybridization probability for the cooperative melting fc(T) ) 2[S]/CT, has the same solution as eq 4, but with

K′ ) 2n exp(-n∆Gs/kBT)/CT

(10)

Thus we obtain the cooperative dehybridization profile fc(T), which becomes narrower as n increases. The probability pdis(T) for cooperative dissociation can be assumed to the same as fc(T), but in general, not all DNA double strands in a DNA-linked nanostructure pair may dissociate cooperatively, or there may be a distribution of possible n values. To describe this more general situation, we would need to combine the cooperative (including multiple n’s) and noncooperative solutions for pdis(T). However, for simplicity, we do not consider this general solution in this paper. 2.3. Comparison. Now let us compare the dissociation probability pdis(T) for both cooperative and noncooperative theories. When n ) 1, both dehybridization theories give the same result, which corresponds to the dehybridization of double strands in solution. Hereafter, we choose the parameters ∆H, ∆S, and CT for fs(T) in eq 4 in order to give a melting temperature TM ) 50.0 °C and full width at half-maximum (fwhm) ∆TM ) 9.2 °C. The resulting fs(T) resembles melting curves for 15 bp DNA, which we present later. If we set n ) 2, the melting temperature (at which pdis(TM) ) 1/2) increases to TM ) 51.7 °C (noncooperative) and TM ) 62.5 °C (cooperative), and the fwhm reduces to ∆TM ) 7 °C (noncooperative) and ∆TM ) 5 °C (cooperative). The larger increase in TM and narrower fwhm of pdis(T) in the cooperative theory is more consistent with the experimental results of DNA-linked polymer nanostructures (vide infra). Of course, higher melting temperatures and narrower transitions can be obtained in the noncooperative model by using a larger n value. For example, for n ) 30, the melting temperature TM for this model becomes 56.7 °C and the fwhm is 5 °C. This shows that it takes a very large number of noncooperative linkers to produce an effect that is comparable to a small number of cooperative linkers. Also note that this behavior is not related to the individual dehybridization profile. 3. Melting of a DNA-Linked Nanostructure Aggregate The theory to this point is incomplete because pdis(T) only describes the melting of two nanostructure pairs rather than an assembly of many such nanostructures as would be present in an aggregate. Hence, before making detailed comparisons with experiment, a more general theory is needed. In the case of DNA-linked polymers, due to the dominant contribution of DNA to the absorption at 260 nm, the melting profile M(T), compared with experimental data at 260 nm after applying a small correction,26 can be written as

M(T) ) 1 - Ntot(T)/Nmax

(11)

where Ntot(T) is the total number of DNA duplexes in the system at a given temperature T and Nmax is the maximum number of DNA double strands that can form. The total number of DNA duplexes in the system Ntot(T) strongly depends on the morphological change of DNA-linked nanostructures in the system because the morphology of DNA-linked nanostructures limits the number of possible DNA duplexes. Several structural models predict an abrupt morphological change such as a phase separation or sudden dissociation of aggregates (i.e., cluster melting).18-21,24 However, even in this circumstance, we can show that DNA dehybridization will dominantly contribute to the melting profile of the system for temperatures away from the onset of the abrupt morphological change. Let us assume that an aggregate consists of N DNA-linked nanostructures and that there are a total of Nlink connected pairs between all the nanostructures in the aggregate for a given temperature T, with each pair i involving ni DNA linkers. In this case, the total number of DNA duplexes Ntot can be estimated using Nlink(T)

Ntot(T) )

∑ i)1

ni(T)

(12)

where i denotes the index of each connected nanostructure pair and ni denotes the number of DNA double strands that connects the nanostructure pair i. If the maximum number of double strands in each connected pair is the same as n and if Nlink is so large that we can apply the central limit theorem to this problem, the summation in eq 12 can be replaced by an average. This leads to

Ntot(T) ) Nlink(T)〈n(T)〉

(13)

where 〈n(T)〉 is the average number of DNA double strands in a connected nanostructure pair. Also, the melting profile M(T) can be reduced to

M(T) ) 1 - Nlink(T)〈n(T)〉/Nmax

(14)

Thus, to describe the melting behavior of this system, we need to estimate the temperature dependence of Nlink(T) and 〈n(T)〉 in eq 14. Here, Nlink(T) is directly related to the morphological changes in the aggregates so that it can be evaluated from structural models, which we describe later. In the temperature region where Nlink(T) changes little, i.e., away from the abrupt morphological change, 〈n(T)〉 gives a dominant contribution to the melting profile. If we assume that each connected pair has the same maximum number of DNA double strands n and the dehybridization probability of all DNA double strands in the cluster is the same as f(T), the average number of DNA double strands in each connected pair will be

〈n〉 ) n(1 - f(T))

(15)

where f(T) corresponds to fs(T) for noncooperative DNA hybridization and to fc(T) for cooperative hybridization, and thus the melting profile M(T) of DNA-linked polymer systems is directly related to the individual DNA hybridization profile. Hence, even though Nlink(T) shows an abrupt transition, if the individual DNA hybridization profile shows broad melting that is away from the transition temperature of the abrupt transition, this broad hybridization profile still contributes to the temper-

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Figure 2. Estimated melting curve M(T) at 260 nm using the minimal cluster melting model. Here there is a sharp increase of M(T) at T ) T/M, where T/M is the melting temperature defined by eq 17. Also, the contributions to the dehybridization curve fs(T) or fc(T) occur below T/M.

ature dependence of the extinction spectra at 260 nm in the DNA-linked polymer system. This is to be contrasted with the situation for the DNA-linked gold nanoparticle system, where the morphological change dominates the observed spectra. 3.1. Minimal Model. Now we consider a minimal model for cluster melting. We assume that, below the melting temperature T/M of the cluster, no structural change takes place and the number of links in the cluster Nl is independent of temperature. At T ) T/M, all the links break at once so that a huge morphological change occurs. This leads to the following expression for the number of links as a function of temperature T

Nlink(T) )

{

Nl at T < T/M 0 at T > T/M

}

(16)

The melting temperature T/M is related to the dissociation probability of links,18,19 and thus we can define T/M as the temperature at which the system has a critical value pc of the dissociation probability pdis(T), i.e.,

pc ) pdis(T/M)

(17)

Now, plugging eqs 13, 15, and 16 into eq 14, we get the temperature dependence of the total number of DNA double strands in the cluster Ntot(T). From this, the melting curve M(T) can be expressed as:

M(T) ) 1 -

{

Ntot f(T) at T < T/M ) Nln at T > T/M 1

}

(18)

In Figure 2, we present the melting curve using this minimal cluster melting model. Below the melting temperature T/M, i.e., before the cluster melting transition occurs, the melting curve M(T) is proportional to fs(T) in the case of noncooperative melting. Moreover, as we increase n, the fs(T) branch becomes significant because T/M depends on n as we can see in eqs 6 and 17. Thus the contribution from cluster melting is less important: in the case of n ) 30, the resulting melting curve is almost the same as fs(T). On the contrary, for the cooperative case, this effect is less significant because the DNA dehybridization curve fc(T), the major contribution to the melting curve M(T) below T/M, is much sharper.

3.2. Reversible Model. Now we turn to a more realistic model to evaluate Nlink(T) and thus to directly compare the experimental results, which we will present later. The morphology at a given temperature can be obtained from the reversible model that is described in ref 19, with further details given in the Supporting Information. In this model, which is based on the idea that the equilibrium state at a given temperature is independent of the path that the system follows to get there, we split the dynamics of the DNA-linked nanostructures into the dissociation process of aggregates in the system due to DNA dehybridization and the association (or aggregation) process due to DNA hybridization. To mimic the association, we apply a reaction-limited cluster-cluster aggregation algorithm. This algorithm is used because, in the temperature region in which we are interested, diffusion takes place faster than DNA hydridization.28 To describe dissociation, we assume that DNA dehybridization takes place such that all DNA duplexes have the same probability to be dehybridized. As a result, all connected DNA-linked nanostructure pairs (or links) have the same probability to be cut. Hence, we randomly cut the links of a given aggregate with a given probability 1 - p, where we assume that p is the probability for each link to remain the same (The relationship between p and the DNA hybridization will be discussed later). After determining the connected nanostructure pairs “statically”, we check whether all nanoparticles are connected in the aggregate in the same way as in a bondpercolation model. If the aggregates are divided into several smaller connected aggregates or even into individual DNAlinked nanostructures, we consider that the smaller ones are free to move so that we can consider them as completely dissociated. The actual simulation procedure is as follows: First, we consider a large cluster on a simple cubic lattice that is generated by an RLCA association process. Initially, all two-particle pairs on nearest-neighbor sites are considered as linked so that nanoparticles in the large cluster are all connected. In each step of the simulation, we assume that the link between an adjacent pair of particles is broken with a probability 1 - p. After applying this probability to all particle pairs, we check the connectivity of each particle and use this to split the large cluster into a set of connected smaller clusters. This set is re-used as an initial distribution for the subsequent association process. We iterate the process until the simulation reaches equilibrium. For the equilibrium criteria, we monitored the radius of gyration of the largest cluster in the system, which eventually relaxes to a stationary value. (For more details, see, e.g., ref 19.) Moreover, it turns out that the probability p is related to the proportion of total single strands to double strands in the cluster at a given temperature,18,19 and we can simply consider it as the dissociation probability between a DNA-linked nanostructure pair pdis in the above discussion. For a given temperature T, we can then determine pdis(T) for noncooperative and cooperative melting. Thus, we find that, at a given temperature T, the simulation reaches a stationary state. We can further argue that the reversible model leads to a Gibbsian result because the model reaches a stationary state when we obtain detailed balance. A similar idea can be found in the theory of surface growth phenomena (see, e.g., refs 29 and 30.) Once the cluster morphology for a given temperature T is determined, we count the number of links Nlink(T), which can be written as Nlink(pdis(T)). In all the results we present herein, we have averaged over 20 different realizations of the morphology in determining average values of Nlink(pdis(T)). Figure 3 shows the estimated number of links Nlink(pdis(T))/Nl using the reversible cluster melting model. Here we use fs(T) as pdis(T).

Sharp Melting in DNA-Linked Nanostructure Systems

Figure 3. Estimated number of links Nlink(pdis(T))/Nl using the reversible cluster melting model. The error bars are smaller or comparable to the point size and thus are not visible.

Figure 4. Estimated melting curve M(T) at 260 nm using the reversible model. The resulting melting curves are governed by a DNA dehybridization curve fs(T) for the individual melting (fc(T) for the cooperative melting).

Similar to Nlink(T) in eq 16 of the minimal cluster melting model, at the melting transition, Nlink sharply changes. However, unlike the minimal model, it shows the temperature dependence of Nlink(T) away from the melting transition, especially above the melting transition. The long tail above the melting transition reflects the contribution of small clusters, which can exist even in temperatures above the melting transition. In Figure 4, we present M(T) using the aforementioned reversible model and eq 14. The left-most curve is based on the dehybridization curve fs(T) with n ) 1. This shows behavior similar to the analogous curve in Figure 2 for temperatures below T /M, and then it smoothly rises near T /M to an upper branch that looks like fs(T) but shifted down in temperature by about 5 °C. (The resemblance of the melting curve for temperatures above T /M to shifted versions of fs is more than superficial, as we discuss later.) For larger n, the noncooperative expression for T/M gets shifted to the right, and the melting curve broadens somewhat. For the cooperative melting case, the melting curve shifts to much higher temperatures, even for n ) 2. The fc(T) curve governs the melting curve for temperatures below T/M. Above T/M, the melting curve is smooth but shifted down compared to fc(T) by about 2 °C. As a result, the cooperative case leads not only to a higher melting temperature but also to a narrower temperature range. For n > 2, the

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Figure 5. Estimated melting curve M(T) at 260 nm using the reversible model. The resulting melting curves are bounded by two DNA dehybridization curves fs(T) and f ′s(T) for the individual melting (fc(T) and f c′ (T) for the cooperative melting).

cooperative melting curve is shifted to even higher temperatures, is even sharper, and the contribution of cluster melting is less important. 3.3. Relation of Structural Model to Phase Separation. The resemblance of the melting curve for temperatures above T /M to shifted versions of fs is more than superficial. In the Appendix, we show that the upper branch has the same mathematical form as fs but for a lower effective concentration that produces the shift. This result reflects that, above T /M, only nanostructure pairs or small clusters with a few nanostructures can exist and DNA hybridization can occur only when two nanostructures meet together. As a result, the DNA hybridization time is increased. Under this circumstance, as we discuss in the Appendix, we can consider the effective concentration to be reduced due to the increased hybridization time, and the best fit for this model is f s′(T) with a concentration C ′T ) CT × 0.08. Hence, as shown in Figure 5, the melting transition takes place between fs(T) and f ′s(T). In the case of noncooperative melting, even when n > 1, the melting curves remain between these two curves, so that fs(T) or f ′s(T) govern a large portion of the temperature dependence of the melting curve. In contrast to this, for cooperative melting, the melting curves are bounded by fc(T) and f ′c(T), where the curve f ′c(T) has the same concentration C T′ as f ′s(T), and thus it has not only a higher melting temperature but also a narrower temperature range. In both cases, the DNA dehybridization curves dominate away from the cluster melting temperature. In the above reversible model, we found that the melting transition takes place between two different DNA dehybridization curves that have different concentrations. This result resembles phase separation behavior that has recently been found in mean-field modeling of the DNA-linked polymer aggregates.24 Here we treat the separation behavior phenomenologically: below a critical point (pdis(T) < pc), two phases (a dense aggregate phase and a very dilute phase) exist together, and above the critical point (pdis(T) > pc), there is only one homogeneous phase. When phase separation occurs, most of the DNA double strands are in the dense aggregate phase, and the concentration of DNA strands in this phase is much higher than that in the dilute phase. Hence, we assume that the concentration of DNA strands in this dense phase is effectively much (∼factor of 100 times) higher than the total concentration CT. In Figure 6, we present the resulting melting curves of the phenomenological phase separation model. We assume nonco-

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Park et al. TABLE 1: Summary of Structural Models in This Work 1.

2. 3.

Figure 6. Melting curve M(T) using the phase separation model. We present three different results depending on the number of DNA strands between a DNA-linked structure pair (n ) 1, 2, and 12). We also present two single DNA melting curves for different concentrations (single (low) and (high)).

Figure 7. Normalized extinction at 260 nm for an annealed DNAlinked polymer aggregate solution (blue circles). For comparison, we present the melting curve M(T) using the reversible model and cooperative DNA hybridization (blue line). Unlike the melting curve for unmodified DNA duplexes in solution (pink squares), the melting curve of DNA-linked polymers exhibits a sharp melting transition with a large melting temperature increase, but does not show the contribution of individual DNA dehybridization in the temperature away from the sharp melting.

Minimal model. Microscopic thermodynamic model. No structural change below the melting temperature. At the melting temperature, all clusters are dissociated abruptly. Reversible model. Microscopic thermodynamic model. Structural change in all temperature takes into account. Phase separation model. Mean-field method. The system is described by macroscopic variables.

ties. The concentration of the polymer-DNA hybrids is 3.2 µM in 0.3 M NaCl, 10 mM phosphate buffer (pH ) 7.0). Melting of the system was observed by monitoring the UV absorbance at 260 nm with stirring at 250 rpm while heating the solution in 1 °C intervals with a hold time of 1 min at each interval. Melting curves of the corresponding DNA strands without polymer backbones at the same DNA and salt concentration were also measured for comparison. In Figure 7, we present the normalized corrected absorbance at 260 nm. The measured absorbance A260 is corrected for background by subtracting the average absorbance for the interval 350-400 nm. Compared with the dehybridization curve for DNA strands in solution, the DNA-linked polymer aggregates show a much higher melting temperature and a sharper melting transition. The curve also does not follow the individual DNA dehybridization curve away from the melting transition and is in excellent agreement with our cooperative theory based on n ) 2. Although it may be possible to generate comparable agreement with experiment using the noncooperative theory, as a result of a fine-tuned choice of concentration CT and using a large value of n, the value of n required would be unphysical, and it is unlikely that this choice of parameters would be robust to changes in concentration of nanostructures, salt concentration, the number of DNA strands attached to one polymer, and so on. Indeed, exhaustive experiments have similarly shown sharp melting behaviors over a wide range of experimental conditions,24,31 and thus we can conclude with confidence that the experimental results are more consistent with a model where cluster melting theory is combined with cooperative melting. The value n ) 2 suggests that, on average, there are two DNA strands that interact strongly enough with each other to exhibit cooperative melting behavior. This value is similar to a value estimated in analyzing DNA-linked gold nanoparticle aggregates,12 but the connection of n to the structure of either of these aggregate systems remains to be established. 5. Summary

operative DNA dehybridization, and for pdis(T), we simply use eq 6 and treat it in the same way as in the above minimal model. The result is very similar to the melting curve, which has been obtained from a mean-field model based on Flory-Huggins theory.24 4. Experimental Evidence Now let us compare our theoretical results with experimental data that we have generated for organic polymers that have been modified with multiple copies of the same DNA sequence (5-6 strands per polymer chain).5,9 Because of the multiple DNA strands for each polymer-DNA hybrid, when complementary polymer-DNA hybrids are combined, they form large aggregate structures linked through DNA duplexes. For this experiment, we annealed these aggregates at 55 °C overnight in order to avoid forming fractal structures, which could rearrange during the melting experiments and possibly affect the melting proper-

In this work, with a combination of theory and experiment, we investigated a DNA-linked polymer system in which we can show that the optical profile of the system at 260 nm is directly related to the individual DNA dehybridization profile, providing a clear distinction from other possible mechanisms. We considered three different structural models that are summarized in Table 1, but we found that, without the contribution of cooperative dehybridization, the results from the models could not explain the experiment. We concluded that cooperativity plays a crucial role in producing sharp melting, which suggests that the dehybridization properties of DNA strands in a confined or dense structure differ from DNA in solution. Our work strongly suggests that cooperativity in the DNA dehybridization process is an essential part of the observed sharp melting. The underlying mechanism is not fully understood, but recent work suggests that counterion cloud overlap provides a plausible mechanism.23

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[SI][SI] [DI]

)

fIs(T)2CT 2(1 - fIs(T))

) KI(T) ) e-∆GI/kBT

) e-∆GII/kBT e∆/kBT ) KII e∆/kBT

(24)

and thus if we set the effective concentration Ceff to CT e-∆/kBT, we can describe case I with an equilibrium constant KII and an effective concentration Ceff that is smaller than the actual concentration CT by e-∆/kBT. Acknowledgment. We gratefully acknowledge support from the NSF through the NSEC, the NIH through the CCNE, and the AFOSR through the DARPA DURINT. S.Y.P. thanks to S. Ryu, H. Hwang, and A. Kudlay for useful discussions. Supporting Information Available: Reversible model for the formation of DNA-linked nanostructure aggregates. This material is available free of charge via the Internet at http:// pubs.acs.org. Figure 8. (a) DNA-linked polymer aggregate; (b) free DNA strands with equivalent concentrations.

Appendix Effective Concentration Shift in Dense Aggregates of DNA-Linked Nanostructures. We consider two different situations: (I) DNA-linked polymer aggregates (Figure 8a) and (II) free DNA strands with equivalent concentrations (Figure 8b). We assume that the temperature of these systems is well above the melting temperature, so that in the systems of interest, there exist only a small number of DNA double strands. In this situation, we compare the rates of DNA hybridization between above two cases. DNA hybridization in case I takes place only when two nanostructures meet together, and these nanostructures diffuse slowly compared with the free DNA single strands in case II. Hence, we can compare the rate of hybridization between two cases so that

kIhyb < kIIhyb

(19)

On the other hand, the rates of DNA dehybridization are all the same because, for DNA dehybridization, thermal fluctuation plays most important role, so that

kIdehyb = kIIdehyb

(20)

Hence, the equilibrium constants Ki ) kidehyb/kihyb (i ) I or II) should satisfy

KI > KII

(21)

Using the van’t Hoff relation, we can write

∆GI < ∆GII

(22)

∆GI ) ∆GII - ∆

(23)

or, equivalently,

where ∆ > 0. In case I, the equilibrium condition of eq 3 is

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