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J. Phys. Chem. B 2009, 113, 4698–4707
Physical Rationale Behind the Nonlinear Enthalpy-Entropy Compensation in DNA Duplex Stability E. B. Starikov*,† and B. Norde´n‡ Institute for Nanotechnology, Research Center Karlsruhe, Post Box 3640, D-76021 Karlsruhe, Germany and Department of Physical Chemistry, Chalmers UniVersity of Technology, SE-412 96 Gothenburg, Sweden ReceiVed: October 9, 2008; ReVised Manuscript ReceiVed: December 18, 2008
The physical-chemical sense of nonlinear entropy-enthalpy compensation based upon the standard thermodynamical parameters of high-temperature melting for doublet units in DNA duplexes has been considered. We are able to show that there are three, with no other constraints equally plausible, principal levels of DNA melting/hybridization description. First, DNA structure assembly/disassembly can be seen from the viewpoint of the conventional equilibrium thermodynamics without taking special care of the heat capacity ∆Cp value (by simply setting it equal to zero). Second, it is possible to assume that the ∆Cp is finite, but independent of temperature. At this approximation level the high-temperature DNA melting cannot be described, but only some special transition between metastable states of DNA duplexes in water solutions in the vicinity of ice melting point. Third, both the latter transition and the hightemperature DNA melting can be reproduced by one and the same approach, if the ∆Cp is assumed to be temperature dependent. These three approximation levels are equally justified from the nonlinear entropy-enthalpy compensation standpoint and by a generalized theory of temperature effects on themodynamical stability as is outlined here. Applicability of each of the approximation levels involved is discussed. 1. Introduction The problem of DNA melting/hybridization has a long history starting with the discovery of the double-helical structure in 1953. Revealing an ordered structure of a system entails questions about stability of such a structure with respect to changes in environmental parameters (humidity, temperature, etc.) and about conditions under which it can be formed. Very much has been done in this field during this appreciable time period, and a number of excellent surveys were published (see, e.g., refs 1a-c and references therein). Nevertheless, both experimental and theoretical studies on this very important topic continue even today,1d-f since separation of the two DNA strands from each othersas well as their reunionsare crucial stages of many important biological processes in cells, the details of which are still incompletely understood. Although DNA base-pair sequence dependence of melting and hybridization processes has been adressed extensively theoretically,1b,g-k the poser is still debatable (see, e.g., refs 1l, m and references therein). Specifically, the debates have focused around the problem of the correct functional forms of DNA-melting Hamiltonians and ways of their consistent parametrization. To compare the theoretical conclusions with the pertinent experimental data, the conventional statisticalthermodynamical treatment of these DNA-melting Hamiltonians is widely used, although some fundamental aspect may have escaped detailed theoretical consideration, namely, physically and chemically valid enthalpy-entropy compensation (EEC). Investigations of the latter phenomenon also have * To whom correspondence should be addressed. E-mail: starikow@ chemie.fu-berlin.de. † Institute for Nanotechnology. ‡ Chalmers University of Technology.
a long history and still trigger hot debates as well (see, ref 1n and references therein). Nevertheless, systematically including the EEC into the theories of DNA melting/ hybridization is indispensable, for EEC is frequently observed in the relevant experimental studies. To this end, in their seminal work in ref 1o (it will be referred to as ref 1 from here on), Petruska and Goodman have observed a valid EEC for melting of nearest-neighboring doublets in DNA duplexes, which follows a nonlinear (rectangular-hyperbolic) functional relationship between the standard entropy (∆S) and enthalpy (∆H) changes in the states of the doublets during the melting process:
∆S )
a∆H aT0 + ∆H
(1)
where T0 and a are empirical parameters with the dimensions of temperature and entropy, respectively, implying that the measured melting temperature Tm is a linear function of ∆H:
Tm ) T0 +
∆H a
(2)
Fitting both parameters to the experimental data gives the following estimates: T0 ≈ 273 K and a ≈ 80 cal/(mol K). The model of the DNA melting1 underlying eqs 1 and 2 is very simple. Let us consider a DNA duplex consisting of n base pairs (BP) arranged as (n - 1) doublets of stacked BP with the stacking of the heterocyclic nucleobases contributing to duplex stability both hydrophobically and electrostatically. We may designate the doublet stack as MN and MN/M′N′ in terms of the both strands of DNA double helix, that is, some base
10.1021/jp8089424 CCC: $40.75 2009 American Chemical Society Published on Web 03/11/2009
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J. Phys. Chem. B, Vol. 113, No. 14, 2009 4699
sequence 5′-MN-3′ on one strand be paired with the complementary sequence 3′-M′N′-5′ on the opposite strand:
∂ (ln ∆) ∂β ∂ aT0 (ln ∆) ∂β ⇒ (y - x˜y˜′)(A - y˜′) + By˜′ ) 0
∂ R ln ∆ - β (ln ∆) ) ∂β
(
-a
)
-B + Bx + y ( √-4Bxy + (-B + Bx + y)2 , 2x T0 a A ≡ aT0, B ≡ , x˜ ≡ β, x ≡ , y ≡ ln ∆, R T ∂y ∂y (4) y˜ ≡ , y' ≡ ∂x˜ ∂x
⇒ y' )
Because the two strands are antiparallel, MN/M′N′ will always be equivalent to N′M′/NM, so that there can be 10 distinct canonical MN types arising from the 16 possible nearest neighbor base sequences in the conventional DNA, as well as different types of base pair mismatches. All the available body of DNA thermal denaturation studies indicates that each MN can be described by its own characteristic enthalpy change upon melting (∆HMN), thus including both hydrogen bonding within each of the neighboring base pairs and stacking interaction between them. Then, the average enthalpy and entropy changes per mole of doublets in a DNA duplex can be cast as follows:
j ) ∆H
∑ ∆HMN fMN,
∆Sj )
∑
∆HMN fMN Tm(MN)
(3)
where fMN is the mole fraction of the doublet type MN, the sum is carried over all the possible MN types, and we assume that melting temperatures of doublets ought to be significantly different from each other. With this in mind, one can otherwise consider ∆H and ∆S as functions of DNA sequence that can adopt some specific values, ∆HMN and ∆SMN, to be derived from systematical experimental data on DNA thermal denaturation. Then, plotting the ∆HMN vs ∆SMN reveals the nonlinear EEC given by eqs 1 and 2. In the literature (see, e.g., refs 2 and 3 and references therein), the nonlinear EEC embodied in eq 1 is honored as a useful empirical formula, but its physical reasons were never investigated in depth, to the best of our knowledge. To analyze the physics behind eq 1, the present communication uses the factor-analysis approach developed recently in ref 4. In what follows, we shall demonstrate that the type of nonlinear EEC described by eq 1 is indeed in accordance with the conventional laws of equilibrium thermodynamics. Besides, we shall elaborate the physical models delivering the best approximation to eq 1, and, in passing, discuss the difference between the linear and nonlinear EEC, because both the former and the latter are observed in DNA melting/hybridization experiments. We also discuss a number of seemingly disparate experimental findings that can be explained within the framework of our theory. 2. Nonlinear Enthalpy-Entropy Compensation Given by Equation 1 is a Direct Consequence of Equilibrium Thermodynamics Like in ref 4, we start with assuming that we are working with the distribution function ∆ in the isothermic-isobaric ensemble, so that enthalpy is expressed as H ) -(∂/∂β)ln ∆ and entropy as S ) R ln ∆ + (H/T), with R the universal gas constant, T temperature, and β ) (RT)-1. Substituting these expressions into eq 1, we get the following differential equation:
The “+” and “-” branches of eq 4 can be solved in closed form with respect to y ) y(x) to get the following solution branches, respectively.
[exp(2C) ( 6B][exp(2C) ( 2B(1 + 2x)] 4[exp(2C) ( 2B] B[2 exp(2C) ( 3][2 exp(2C) ( (1 + 2x)] (5) y2(x) ) 2 ( 4 exp(2C)
y1(x) )
where C is an integration constant. As both y1 and y2 in eq 5 are linear functions of x, it is straightforward to use them in deriving the expression for the change in the Gibbs energy ∆G of the “hidden process” behind the melting of nearest-neighboring doublets in DNA duplexes, just like for the linear EEC:4
∆G ≡ RTy1(x) )
aT0[exp(2C) ( 6B] + exp(2C) ( 2B
RT [6B ( exp(2C)] ≡ ∆H - T∆S 4 aT0[2 exp(2C) ( 3] ∆G ≡ RTy2(x) ) + 2 exp(2C) ( 1 3 aT ( exp(2C) ≡ ∆H - T∆S 2
[
(6)
]
so that y1 and y2 in eq 6 are, in fact, identical expressions for the Gibbs free energy difference, because C is an arbitrary constant. It should also be noted that only the “-” branches of y1 and y2 have physical sense (in that they would in effect describe the Gibbs free energy difference). Equation 6 shows that Equation 1, in addition to being an empirical formula, is indeed homomorphic with the statisticalthermodynamical theory, like the conventional linear EEC. Now, the question arises as to what is the physical sense of the hidden process behind eq 1. More specifically, it is important to decide which kind of thermochemical model we should use for adequately describing the process of DNA duplexes melting at the doublet level. 3. Toward Thermochemical Models of DNA Duplexes Melting at the Doublet Level Usually, any discussion of biopolymers’ thermal stability assumes that, in some temperature range near the melting transition, the heat capacity change at a constant pressure, ∆Cp, is approximately independent of temperature and is greater than zero. Under the additional assumption that biopolymer melting might be effectively considered a two-state process, one may then use the Gibbs-Helmholtz equation to arrive at the
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Starikov and Norde´n
Figure 1. Schematic plot of the second expression in eq 8a (here plotted in the equivalent form, ∆Cp ) (ax)/(x - 1) - (a)/(ln(x)), where x ) T0/T). We can see that ∆Cp exhibits quite a weak temperature dependence on T0/T in the 0.7-1.5 span, which covers 410-190 K, if the nominal T0 is taken to be equal to 273 K, that is, the temperature value found in ref 1.
following expression for the temperature-dependent Gibbs free energy (for a more detailed discussion see ref 5):
∆H ) ∆H 0 + ∆Cp(T - T 0) ∆S ) ∆S 0 + ∆Cp ln(T/T 0)
(7)
∆G ) ∆H - T∆S + ∆Cp[T - T - T ln(T/T )] 0
0
0
0
where T 0 is any reference temperature, whereas ∆H 0 and ∆S 0 are changes in partial molal enthalpy and entropy at that temperature, respectively. To check whether eq 7 is applicable to the nonlinear EEC given by eq 1, we assume that T0 ) T 0 and substitute the expressions for (∆H s ∆H 0) and (∆S s ∆S 0) from eq 7 into eq 1, just to see how strong would be the resulting temperature variation of ∆Cp. After some trivial algebra, we get the following equation for the ∆Cp:
∆Cp ln(T/T0) )
a∆Cp(T - T0) aT0 + ∆Cp(T - T0)
(8)
aT0 a + -T + T0 ln(T/T0)
temperature dependence of the DNA thermodynamic parameters under study, we shall adopt the thermochemical model of eq 7 from here on. Besides, to illuminate the difference between the linear and nonlinear EEC, we may apply an analogous approach to the general linear EEC in the form of ∆H ) Tc∆S + b, where Tc is the “compensation temperature” (see ref 4). We get the following result:
∆Cp(T - Tc) ) Tc∆Cp ln(T/Tc) + b ⇒ b ∆Cp ) T - Tc - Tc ln(T/Tc)
(9)
which leaves the ∆Cp(T) with a strong singularity (discontinuity) exactly at the T ) Tc and appreciable temperature dependence around this temperature point (see Figure 2). However, this is completely at odds with our initial assumption. Thus, neither the zero approximation of thermochemistry nor the model given by eq 7 are applicable when the EEC is linear, and one has to assume that ∆Cp is noticeably temperature-dependent in such a case. 4. Approximate Solutions to Equation 4 s Two Physical Models of DNA Structure
which has the two solutions:
∆Cp ) 0 and/or ∆Cp )
Figure 2. Schematic plot of eq 9. Here, the temperature dependence of ∆Cp turns out to be non-negligible, especially in the vicinity of T0.
(8a)
The first solution is known as the zero approximation of thermochemistry, rendering ∆H and ∆S temperature-independent quantities,5 whereas the second solution exhibits quite a weak temperature dependence of ∆Cp for the T/T0 in the 0.7-1.5 span (see Figure 1), which covers 190-410 K for the nominal T0 ) 273 K. Although at first glance eq 8a is possessed of a seemingly essential singularity at T ) T0, it is quite easy to check that, in accordance with eq 8a, limxf1((ax)/(x - 1) (a)/(ln x)) ) 1/2, where x ) T0/T. Therefore, we may assume that ∆Cp is practically temperature-independent and nonzero in the conventional temperature range, in accordance with our initial assumption. Interestingly, both of these approximations were shown to be applicable to DNA melting and to deliver consistent results.3 However, since we are interested in the
With the proper thermochemical model at hand, we revert to the solutions of eq 4. We shall employ the Bakhshali approximation to the square roots, expanding them in terms of a quantity d that is small in comparison to N2:
√N2 + d ≈ N +
d d2 2N 8N3 + 4Nd
(10)
Applying the first two terms of this expansion to the “+” and “-” branches of eq 4, we obtain two approximate differential equations:
2(-B + Bx + y+) - 4Bxy+ /2(-B + Bx + y+) 2x 4Bxy /2(-B + Bx + y -) y- = 2x (11) y+ =
Nonlinear Enthalpy-Entropy Compensation
J. Phys. Chem. B, Vol. 113, No. 14, 2009 4701
Figure 3. Schematic plot of two eq 11 solution branches given by eqs 12a and 12b. The blue (lower) curve is y ) B + Bx - 2Bx, whereas the violet (upper) curve is y ) B + Bx + 2Bx. The area between these two curves shows the physical inapplicability area for eq 11, according to eq 13.
which have the following solutions:
y+ ) B(x - 1) ( √B2(2x + x2) + x2C + 2B2x2 ln(x) (12a) y- )
B(x - 1) LambertW[B(x - 1)exp(BC)]
(12b)
with C being an integration constant and LambertW is the “Lambert W function” or product logarithm.6 It should be noted that the approximations given by eq 12a will be valid under the following condition:
Here we shall follow the approach of7 in trying to set up our own phenomenological metastability theory in the vicinity of some particular reference temperature (which should not necessarily be very low). Our consideration starts with the GibbsHelmholtz expression for the temperature-dependent free energy given by eq 7, which is an adequate model in our case, as we have seen in the preceding section. Unlike in the work,7 we do not actually need to separately reconstruct the special form of the functional dependence of enthalpy on entropysit is represented by eq 1 in our case. Therefore, we can assume that, in eq 7, the reference temperature is T0 ) T0, T0 ) ∆H0/∆S0, and ∆Cp is just equal to some entropy change ∆S obeying eq 1. Remarkably, the latter ∆S is not necessarily equal to ∆S0, but the ∆H in eq 1 must be equal to ∆H0. In most of the processes where biomacromolecules participate, ∆Cp is usually connected with changes of biopolymer solvent-accessible surface areas, differences in electrostatic interactions, and, anyway, with the presence of EEC (see, e.g., refs 9-11 and references therein). The surface area changes are closely related to the hydrophobicity-hydrophilicity interplay and as such can be viewed as entropic effects. Furthermore, it is doubtless that, especially for DNA polyanions, electrostatic contributions to ∆Cp must be very important. Remarkably, the latter contribution may translate into a special kind of molecular dynamics (cf. the concept of DNA plasmon12-15). Therefore, the idea that ∆Cp is determined by entropic effects, and using it in this form to study EEC phenomena seems to be plausible. Moreover, we can even anticipate that ∆Cp is simply proportional to entropy change in the temperature range shown on Figure 1.16 Then, assuming that ∆S ) ∆Cp in eq 1, the expression for ∆G in eq 7 can be recast as follows:
∆G ) ∆H0(1 - 1/x) +
(-B + Bx + y)2 . 4Bxy ⇒ y , B + Bx - 2B√x or y . B + Bx + 2B√x (13) The regions of Gibbs free energy values where the approximation eq 12a is valid or invalid are shown on Figure 3. Now we are ready to investigate the physical impact of eq 12a in detail. 5. DNA Metastability at Both Low and High Temperatures? Let us consider first eq 12a. The solution y+ describes two branches of the Gibbs free energy, with the additive branching term having square-root-like dependence on temperature. This mathematical structure is reminiscent of that appearing in the phenomenological metastability theory advocated by Kornyushin.7 In this work, the low temperature region was considered, assuming that the entropy can be regarded as entirely configurational entropy (with practically no thermal contributions7) and, mathematically, as an independent variable (a propos, this is in full accordance with the general theory of thermodynamics by Lieb and Yngvason8). Then, the EEC concept is invoked in explicit form, by recalling the fact that enthalpy, being one of the thermodynamic potentials, must be somehow a function of entropy. The latter functional dependence at low temperatures is approximated by the third degree polynomial with some physically reasonable coefficients. Kornyushin minimizes the Gibbs free energy with respect to entropy and in this way determines an expression for the thermodynamic barrier dividing metastable and more stable states of the system.7
aT0∆H0 [1/x - 1 + ln(x)/x] aT0 + ∆H0 (14)
so that, zeroing the first derivative of ∆G with respect to ∆H0, we immediately see that ∆G has a minimum at
∆Hmin )
aT0(x - 1) - aT0√(1 - x)[1 - x + ln(x)] x-1 (15a)
and a maximum at
∆Hmax )
aT0(x - 1) + aT0√(1 - x)[1 - x + ln(x)] x-1 (15b)
Hence, it is straightforward to show that the temperature dependence of the Gibbs free energy barrier (which contributes to kinetical stabilization of the metastable state) can be expressed as follows:
∆∆GT 1 (T < T0) the system is in a metastable state that is destroyed at T ) T0, so that the latter is in effect a kind of critical temperature (Tc). But what happens to the system at T > T0? This question can be answered with the help of eq 12a. Indeed, if we assume that this equation describes the minimum and maximum Gibbs free energy of the one and the same metastable state, we arrive at the following approximate expression for temperature dependence of the same Gibbs free energy barrier as in eq 16:
∆∆Gappr ) 2B√2x + x2(1 + C) + 2x2 ln(x)
(17)
Plotting eqs 16 and 17 together (cf. Figure 4), we can see that the latter can be rendered a good approximation for the former at T < T0, if we assume that ∆∆Gappr ) 2∆∆GT T0. Indeed, the barrier grows again from its zero value. Therefore, the temperature T0 determined in ref 1 can be considered the temperature of a nonequilibrium phase transition. It should be noted that such a transition cannot be detected calorimetrically, since the heat capacity is not changing significantly in the relevant temperature range, as we have seen in the previous section. Besides, this transition ought to be reversible, unlike other known transitions between metastable states (for example, crystallization of amorphous materials or diamond-graphite transformations of the solid carbon), where not the derivatives of the Gibbs free energy but the latter itself experiences a discontinuity versus temperature (zero-order phase transition). With all this in mind, we might interpret the strange phase transition under study in terms of the so-called rare-region effects.17 Specifically, the latter are understood as a result of a certain amount of quenched (intrinsic, independent of time and other external parameters, or frozen-in) disorder, like impurities, dislocations, grain boundaries, etc. To this end, the nonhomogeneous, irregular nucleotide sequence of nucleic acid polymers ought to represent a bright example of the quenched disorder. The presence of the quenched disorder is always connected with some rare, but large disorder fluctuations that occur in rare spatial regions. Such phenomena are known to have powerful
effects on the nature of phase transitions that range from inducing strong power-law sigularities in the free energy to smearing, or even a complete destruction, of phase transitions (for more details see the review article ref 17 and references therein). Moreover, we also see no special singularities or discontinuities in the temperature dependence of the Gibbs free energy barrier in the DNA melting temperature range. This means that the metastable phase transition model in question cannot describe DNA melting, which is, of course, a clear restriction of the model given by eq 12a. Still, an introduction of this model poses some fundamental questions as to the physical nature of the DNA double helix, which we shall elaborate below. Indeed, even at temperatures higher than T0 and up to their melting, DNA duplexes might persist in a kind of a metastable state, which is quite different from the one at T < T0 (the latter ought to correspond to a true glassy state). Such a conclusion is directly supported by the recent experiments demonstrating multiple metastable intermediates in DNA unzipping under a constant force18 and by a big number of earlier as well as later findings (see, e.g., refs 19, 20 and references therein). The structural basis of the DNA metastability has been systematically analyzed in refs 21-24, which revealed that DNA duplex structure at the dinucleotide and tetranucleotide level results from an intricate competition between the modalities of basestacking interactions and phosphodiester backbone conformational preferences. Due to this competition, neither of the two systems can adopt its genuinely optimum conformation. With the latter idea in mind, metastability-related physical concepts were invoked in trying to explain the well-known, but still poorly understood, phenomen of DNA A-tracts bendability.25-28 Finally, the development of a metastability when increasing temperature with respect to T0 can also be considered in terms of the “freezing-upon-heating” concept (see, e.g., refs 29-31 and references therein), so that DNA duplex structure might possibly be viewed as a result of the “inverse freezing” (reversible glass transition upon heating). All these topics definitely deserve much more special interest and will be dwelt on elsewhere. 6. Combined Prediction of DNA Melting Transition and the Corresponding Kauzmann Temperature Using Equation 12b Finally, let us consider the second approximate solution to eq 4, namely, the one given by eq 12b. Unlike eq 12a, it contains only one branch where the conventional Gibbs free energy expression is divided by the Lambert W function containing an arbitrary integration constant C in its argument. Thus, we might consider this a kind of approximation for the Gibbs-Helmholtz equation, where ∆Cp is now taken to be somehow temperature dependent, unlike in the model described by eq 7. With this in mind, it is possible to find explicit formulas for ∆S(T) and ∆Cp(T) in a very simple way, just by properly differentiating ∆G(T) ) RTy- in eq 12b. The corresponding algebraic and numerical results can easily be obtained using the MATHEMATICA software package (by Wolfram Research Inc.), but the resulting formulas are tedious, so we will restrict ourselves to only discussing a plot of ∆S(T), ∆Cp(T), and ∆G(T), see Figure 5. Interestingly, the presence of the arbitrary integration constant allows to adjust the functions ∆S(T), ∆Cp(T), and ∆G(T) in such a way that (a) ∆G(T) will at some special temperature Tm have a branching point from where on it ceases to be a real function
Nonlinear Enthalpy-Entropy Compensation
Figure 5. Schematic plot of the Gibbs free energy change (∆G), entropy change (∆S), and heat capacity change (∆Cp) versus x ) T/T0, based upon detailed investigations of eq 12b. There are two special temperature points on this graph: T ) T0, where ∆Cp ) 0 and ∆S has minumum; and T ) Tm, where ∆G, ∆S, and ∆Cp start to be complex functions of temperature, with ∆S and ∆Cp exhibiting here a significant discontinuity as well.
of temperature. This branching point does not contain any discontinuity, because only an infinitely small imaginary part will appear at T ) Tm and increase with temperature in the T > Tm range, leaving the real part basically unchanged; (b) both ∆Cp(T) and ∆S(T) will at the same temperature T ) Tm also have a branching point similar to that of ∆G(T), but both real and imaginary parts of them will experience a strong discontinuity at T ) Tm; (c) ∆Cp(T) will also be equal to zero at some temperature T0 < Tm, and ∆S(T) will have a minimum at this same temperature point, whereas ∆G(T) will have no further special points in the temperature range involved; (d) by fitting the integration constant C in eq 12b, the abovementioned T0 can be made correspondent to the T0 found inref 1, with Tm being situated in the range of typical DNA melting temperatures. To sum up, eq 12b can be used as a viable approximation to DNA temperature stability curves in a rather broad temperature rangesfrom the water freezing point up to the DNA melting point. Further, the model given by eq 12b clearly demonstrates that DNA melting process strongly resembles a phase transition of the first order. To this end, we recall a recent fierce debate about the order of DNA melting transition in the literature (see refs 32-38 and references therein). Interestingly, systematic studies on self-complementary (DNA beacons) and nonselfcomplementary oligonucleotides show that single-stranded selftransitions should play a very important role in the thermodynamics and kinetics of the DNA duplex formation.39 Such self-transitions may most probably be connected with the sequence-dependent intrastrand stacking propensities and expressed as a temperature dependence of the single-stranded DNA persistent length.40 In accordance with this, the extent of DNA duplex formation ought to be significantly dependent on the single-stranded DNA chain length: The duplex formation in shorter oligonucleotides can safely be viewed as a two-state transition, but this is not the case for the longer oligonucleotides.39 With this in mind, we may think that the order of the phase transition represented by the duplex formation could be dependent on the DNA chain length, with the duplex formation in shorter DNA being a phase transition of the second order, whereas that in the longer ones is a phase transition of the first order. Such effects are well-known in the physics of finite clusters (a cross-over between the phase transitions of the first and the second-kind; see, e.g., refs 41, 42). Hence, we may
J. Phys. Chem. B, Vol. 113, No. 14, 2009 4703 conclude that eq 12b is a good approximation for long DNA duplexes, but not for the short ones. The present model is in accordance with that presented by eq 12a, since the former supports the conclusion of the latter that the phase transition at T0 cannot be visible on the ∆Cp(T) curves. Instead, since we have associated ∆Cp(T) with the defacto temperature change in entropy of the DNA duplex when considering the model of eq 12a, the zero value of ∆Cp(T) in the close proximity of T0 means that the latter may be considered a kind of Kauzmann zero-entropy/zero-enthalpy temperature.29,43,44 The presence of the Kauzmann temperature should indicate that the range of dynamical cooperativity in the whole DNAcounterion-water system becomes really infinitely long at this temperature point. That there are practically no observable special points on DNA ∆Cp(T) plots in the vicinity of T0 is supported by numerous systematic calorimetric experiments (see, e.g., refs 45-49 and references therein). 7. Generalized Theory of Thermochemical Stability and Equation 12b Is eq 12b just a handy and compact empirical formula, or it does have deeper thermodynamical roots ? We shall address this question by considering temperature effects on general thermodynamical equilibria. For this purpose, we assume that the heat capacity Cp is essentially dependent on temperature, but its particular functional form will remain unknown. Besides, we consider just two arbitrary temperatures, Tmin and Tmax, but define no standard temperature, unlike what is usually done. Then, for the change in Gibbs free energy when going from Tmin and Tmax, or vice versa, we have
G(Tmin) ) G(Tmax) +
∫TT
max
min
Cp(τ)dτ - Tmin
∫TT
max
min
Cp(τ) dτ τ
Cp(τ) ∂2G(τ) )τ ∂τ2 (this follows from the Gibbs-Helmholtz equation) Tmax ∂2G(τ) G(Tmin) ) G(Tmax) - T τ dτ + min ∂τ2
∫
Tmin
∫TT
max
min
(
)
( )
∂2G(τ) dτ ∂τ2
(18)
Equation 18 is an integral-differential equation that can be formally solved using the standard unilateral Laplace transform (F(s) ) limεf0 + ∫ -∞ εe-st f(t) dt, for real t g 0) with respect to the variable t ) Tmin (Such an approach would give no solution, if we chose t ) Tmax. Hence it is important to note that Tmax > Tmin holds from now on, but still without loss of generality):
G(s) )
2sG(Tmax) + (1 - sTmax)G'(Tmax) 2s2
(19a)
1 G(Tmin) ) G(Tmax) + (Tmin - Tmax)G'(Tmax) (19b) 2 where G(s) is the Laplace transform of G(Tmin), whereas G′(Tmax) stands for the derivative of G by temperature taken at the point Tmax. It can readily be seen that -G′(Tmax) is then just entropy at the temperature Tmax, according to the Gibbs-Helmholtz
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Starikov and Norde´n
equation. With this in mind, eq 19b can be recast in the following form:
1 G(Tmin) - G(Tmax) ) (Tmax - Tmin)S(Tmax) (19c) 2 with Eq 19c being, in fact, well-known from the standard thermodynamics of chemical reactions, if we take Tmax ) 298 K and S(Tmax) as twice the difference between the entropies of products and reactants. The only difference is that here we work only with the Tmax > Tmin temperature range, as mentioned above. But since we have assumed that our temperature Tmax is also arbitrary, we now try to solve eq 19b with respect to G(Tmax). If we take the G(Tmin) as a parameter weakly dependent on temperature, then eq 19b will be an elementary differential equation, so that we immediately arrive at the desired solution:
G(Tmax) ) G(Tmin) + C(Tmin - Tmax)2
(20)
which (by properly choosing the parameter G(Tmin) and the arbitrary integration constant C) can be transformed into the expression for the temperature-dependent Gibbs free energy change, when the heat capacity Cp is a linear function of temperature (see, e.g., ref 5). However, in the general case, both Tmin and Tmax are variables in the linear differential equation given by eq 19b. Thus, we may also look for the solutions of eq 19b in the form of G(Tmax) ) exp(λTmax), and, after solving the corresponding characteristic equation with respect to λ, we get the desired solution as follows:
G(Tmax) ) 2Tmax Tmin - Tmax 2Tmax 2Tmax exp LambertW G(Tmin) Tmin - Tmax Tmin - Tmax G(Tmin)
(
(
))
(21)
Therefore, eq 21 is in fact an essentially nonlinear difference equation, which can be solved by numerical iteration. A comparison of eq 21 with eq 12b shows that the latter is identical to the former, if we assume that (a) we have not two, but only one variable, namely, the temperature difference Tmin - Tmax; (b) the temperature dependence of the exponential in the Lambert W function expression can be neglected; and (c) it is enough to carry out only one iteration with the G(Tmin) as the proper initial condition. To this end, eq 12b can be considered an approximation to eq 21 and is hence possessed of the profound thermodynamical sense. The general nonlinear difference equation eq 21 may definitely have stable unique solutions (point attractors), periodical, or even chaotic solutions (limit cycles or strange attractors, respectively). However, these interesting topics are outside the scope of our study here. 8. Discussion: Thermodynamics of DNA Hybridization from the Standpoint of Room Temperature Metastability/ Premelting Concept Here we would like to demonstrate how the ideas presented above help in bringing many seemingly disparate experimental findings under one and the same conceptual roof.
Indeed, a more detailed insight into the physical-chemical mechanisms of DNA hybridization could be gained from the systematic experimental studies on DNA-PNA hybridization (PNA stands for peptide nucleic acid, where the negatively charged deoxyribose phosphodiester backbone is substituted with the electroneutral N-(2-aminoethyl)glycine having a significant dipole moment, see, e.g., refs 50-54 and references therein) which are important for parametrizing models of gene targeting.55 Specifically, a noticeable and peculiar sequence dependence of apparent association constants for the pentadecamer PNA-DNA duplex formation when the PNA strand is surface-immobilized52 could indicate that the hybridization is not a simple two-state process but involves at least one precursory or outer-sphere complex, in accordance with the findings for DNA-DNA duplexes (see, e.g., refs 39, 40 and references therein). As anticipated, this conclusion does not seem to apply to shorter (decamer) PNA-PNA, PNA-DNA and DNA-DNA duplexes.53 Moreover, a higher thermal stability of PNA-RNA compared to PNA-DNA duplexes52 correlates well with the higher conformational flexibility of DNA strands. On the other hand, there ought to be no interdependence between electrostatics and the DNA chain length, when speaking of the duplex stability, since a significant length-independent (10- to 15-mers) destabilization of PNA-DNA duplex compared to PNA-PNA duplex was observed at low to moderate NaCl concentrations (1 M), where the electrostatic effects are well-known to saturate, destabilization of PNA-PNA, PNADNA, and DNA-DNA duplexes follows an analogous trend irrespective of their backbone charges,50 as anticipated. Besides, the anionic destabilization effects at high salt concentrations were found to follow the Hofmeister series especially for PNA-PNA and PNA-DNA complexes,50 which indicates that hydrophobic interactions play a dominant role in the stability of both of these complexes. Finally, the additional stability of PNA-DNA duplexes over the analogous DNA-DNA duplexes is due to the more favorable entropic contribution, whereas PNA-PNA duplexes outweigh the analogous DNA-DNA duplexes in enthalpic contributions to their thermal stability.50 Concerning the physical nature of the presumable precursory or outer-sphere complexes (or the single-stranded self-transitions, according to the terminology of refs 39 and 40), which interfere in the two-state hybridization of longer DNA/PNA chains, ref 54 has investigated formation of two DNA-PNA duplexes with the same sequence of base pairs, but with the backbones interchanged. Interestingly, the binding free energies of these two duplexes are noticeably different from each other, so that both placing of guanine residues on the DNA strand and decreasing the fractional pyrimidine content of the PNA strand give the most favorable duplex free energy. Such a finding may be interpreted as a consequence of intrastrand base stacking propensities and/or some specific base-backbone coupling, which could effectively interfere the duplex formation and promote the intermediary conformational states. Qualitatively similar effects were also observed during the PNA-thymine strand invasion into the double-stranded poly(dA)-poly(dT) DNA, with both PNA and DNA being rather short (6- to 10-mers).51 Indeed, this invasion process was shown to require at least two PNA strands for its rate-limiting step to be promoted, with the DNA base pair opening being the substantial part of the activated state. Furthermore, the PNA-DNA complex formation was found to involve intermediate states and a definite nucleation step
Nonlinear Enthalpy-Entropy Compensation represented by a PNA2-DNA invasion triplex of about five bases,51 which physically renders the PNA-DNA invasion a kind of phase transition of the 1st order, even irrespective of extreme shortness of the DNA duplexes under study. Besides, salts and groove-binding DNA-duplex ligands decrease the rate of PNA invasion, whereas intercalating agents increase it.51 To this end, it is important to note that such groove-binding ligands as haipin polyamides, distamycin, and Hoechst 33258 ought to stabilize the overall DNA duplex structure and/or efficiently compete against PNA for binding groove-exposed functional groups of DNA bases with high affinity (see, e.g., ref 56 and references therein), thus hampering the PNA invasion. Meanwhile, intercalators, being inserted into DNA duplexes, may significantly disturb and destabilize DNA structure in their neigborhood, thus facilitating PNA invasion. Most probably, the idea of duplex room-temperature metastability (in the form of length-dependent crossover between the first- and second-order phase transitions discussed above) can also be relevant to DNA strand-exchange reactions catalyzed by RecA/Rad51 type of enzymes (there is strong and specific protein-DNA coupling that significantly restructures one of the DNA strands, thus speeding up the search for the corresponding complementary strand to form the proper DNA duplex57,58). Interestingly, this same concept ought to be crucial for understanding the modalities of self-assembly in DNA nanostructures addressable in terms of their DNA-base code (see, e.g., refs 59 and 60 and references therein) as well. But we will not dwell on these two topics here. Another finding, which is extremely interesting and important in the context of the present communication, reveals that thermodynamical parameters of DNA/PNA melting/hybridization may follow not only the nonlinear EEC given by eq 1, but also the valid conventional linear EEC (see, e.g., refs 3 and 54 and references therein). To observe the latter trend, one has to plot the corresponding enthalpic versus entropic changes for various DNA/PNA/RNA duplexes as a whole, but not just those for particular nucleotide doublets. It is therefore tempting to assume that the nucleic acid duplex metastability at room temperature ought to be driven by local, short-range, largely temperature-independent effects that most probably are dictated by DNA base sequences. On the other hand, there should be a significant degree of cooperativity among these local metastable effects: such a cooperativity is obviously long-range (alongside the conventional polyanion electrostatics, or even contrary to it, if we take into account collective “plasmon-like” modes in DNA and its surrounding,12-16 which are capable of dynamical screening of electrostatics), temperature-dependent, and thus capable of promoting and regulating stability of the whole duplex. Hence, a nucleic acid duplex can in principle be considered as a result of nonequilibrium spatio-temporal pattern formation, where nonlinear and dissipative molecular dynamics ought to pay a mandatory contribution.61 The above discussion reveals that there ought to be some intimate inter-relationship (very specific, but still not well understood) between nucleic acids molecular dynamics and thermodynamics. It is therefore very encouraging that, with the advent of diverse sophisticated experimental approaches to study single molecules, dynamical aspects of nucleic acid hybridization/dehybridization may become increasingly clearer. For example, nanomechanical detection of DNA melting dynamics has been successfully carried out on microcantilever surfaces with high precision.62-64 The principle of this method is based upon the changes in microcantilevel surface stress that occur when one cDNA strand melts and diffuses away from the other,
J. Phys. Chem. B, Vol. 113, No. 14, 2009 4705 resulting in alterations of the electrostatic, counterionic, and hydration-mediated coupling forces among the remaining neighboring surface-grafted DNA molecules, so that the mechanical responses are directly translated into the pertinent Gibbs free energy changes due to macromolecular interactions. References 62-64 demonstrate that, although melting of the DNA duplexes under study (15-, 20-, and 25-mers have been studied) in general follows the conventional two-state model, the longer the duplexes, the more complex are time dependences of the surface stress during their melting. This is in full accordance with the above-mentioned ideas about the involvement of some intermediary “premelting” states into the melting of longer DNA duplexes, which could most probably be of purely dynamical nature and thus prove the intrinsic metastability of the DNA duplexes already at room temperature. Another group of modern experimental data on DNA melting is connected with the use of magnetic tweezers to probe the shearing force of matched sequence (see, e.g., refs 65 and 66 and references therein). Remarkably, although the results of such experiments can be explained using a simple model by P.-G. de Gennes (where both intra- and interstrand dynamics of DNA duplexes are represented by rigid elastic springs),66 there is definitely much more to the story. Indeed, a theoretical study on DNA force-induced slippage dynamics65 reveals that the response of DNA duplex to the shear force is drastically dependent on DNA sequence: DNAs with regular periodic sequences can be opened by sliding, which requires much lower thermodynamic critical shear force than unraveling, which is the only possible physical mechanism by which nonperiodic DNAs open. With this in mind, we can readily rationalize the recent experimental results on kinetic recognition of AT-rich DNA duplexes by threading intercalation of various binuclear ruthenium complexes.67,68 Specifically, the rate of the latter strongly depends on the nucleobase composition of DNA, so that they rather quickly intercalate into a periodic duplex poly(dAdT)2, but extremely slowly into mixed-sequence DNAs. Along with this, two specific effects can be observed, namely, (a) more than one helix turn of AT base pairs is required for efficient threading intercalation, and (b) the rates of the latter are highly sensitive to small geometric variation of the threading complexes. The first effect is suggestive of the collective property, such as the duplex breathing dynamics, that ought to be the rate-limiting step of the threading, whereas the second effect is indicative of an intimate involvement of the intercalating molecule into the base-pair opening process.67 Finally, the dissociation rates for the above-mentioned DNA-drug complexes determined by their sequestration using the conventional SDS assay reveal that there should be some intermediate involving both SDS micelles and DNA-threaded complex.68 Thus, based upon the theoretical results,65 we may assume that both association and dissociation of the DNA-threaded complexes intimately correlate with the dynamics of DNA duplex slippage due to shear forces, which is greatly facilitated for periodic sequences compared to the nonperiodic ones. To this end, DNA duplexes with base pair slippage are in principle observable under some special experimental conditions.69 9. Conclusions We have demonstrated that the phenomenon of nonlinear EEC embodied in eq 1 does not solely deliver a handy empirical formula, but is also firmly based upon the conventional equilibrium thermodynamics. This conclusion opens the door to systematically studying qualitative physical-chemical principles of nucleic acid structure formation. Further approximative
4706 J. Phys. Chem. B, Vol. 113, No. 14, 2009 analysis of eq 1 has shown that there are two equally plausible standpoints that can describe the actual phenomena behind the nonlinear EEC involved. First of all, the behavior in the temperature range around the parameter T0 of eq 1 can be viewed from the standpoint of thermodynamical metastability theory based upon eq 12a, if we assume that the heat capacity change ∆Cp is finite but effectively independent of temperature. It can be shown that the latter assumption is in fair agreement with eq 1. Specifically, we may consider this temperature T0 a point of some reversible nonequilibrium phase transition between low-temperature and hightemperature metastable states of DNA duplex. The physical nature of the low-temperature and high-temperature metastability of DNA duplexes ought to be quite different from each other. It should however be noted that the metastability theory, as it stands, turns out to be incapable of anyhow describing the melting of the high-temperature metastable structure of DNA duplex. The problem of the joint description of both the transition at T0 and the high-temperature melting of DNA duplexes can be solved by using the model of eq 12b, that is, assuming some non-negligible temperature dependency of the ∆Cp. In this approximation, the T0 can be viewed as a kind of Kauzmann zero-entropy/zero-enthalpy temperature. We have also shown in passing that, for systems obeying linear EEC only, it is in no way possible to assume that the ∆Cp is perfectly temperature independent. To sum up, we have been able to demonstrate that there are three (with no further information available, equally plausible) principal levels of physical description of DNA melting/ hybridization. First, DNA structure assembly/disassembly can be viewed from the standpoint of the conventional equilibrium thermodynamics without taking special care of the ∆Cp value (by simply setting it equal to zero) This approximation level could be enough for some “quick-and-dirty” (e.g., comparative) analysis of experimental findings in any temperature range, but is definitely inapplicable for the systematical studies, because of the essential temperature dependence of ∆Cp. Second, it is also possible to assume that the ∆Cp is finite and independent of temperature. At this approximation level it is not possible to describe the high-temperature DNA melting, but only some transition between two different metastable states of DNA duplexes in water solutions, in the vicinity of the ice melting point. The latter transition can, for example, be of potential interest in cryobiology, so that the “finite and temperature-independent ∆Cp” approach can be reliably used to interpret the corresponding experimental findings. Third, both the above-metioned “ice melting point” transition and the high-temperature DNA melting can be described using one and the same approach, if the ∆Cp is assumed to have some arbitrary, significantly nonlinear temperature dependence. We can suggest using this level of approximation for systematical studies on DNA structural stability in a very broad temperature range. However, such an approach is applicable to rather long DNA duplexes only, where duplex formation and/or melting cannot be considered just a phase transition of the second order owing to a considerable number of the relevant intermediary states. The equal plausibility of these three approximation levels is guaranteed by the validity of eq 1, describing nonlinear entropy-enthalpy compensation for the standard thermodynamical parameters of the high-temperature melting of doublet units in DNA duplexes. Besides, the applicability of all approximations invoked is also ensured by the generalized
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