J. Phys. Chem. B 2010, 114, 9905–9911
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Molecular Thermodynamic Model for DNA Melting in Ionic and Crowded Solutions Y. Liu, F. Kermanpour, H. L. Liu,* Y. Hu, Y. Z. Shang, S. I. Sandler, and J. W. Jiang* Department of Chemical and Biomolecular Engineering, National UniVersity of Singapore, 117576, Singapore, State Key Laboratory of Chemical Engineering and Department of Chemistry, East China UniVersity of Science and Technology, Shanghai 200237, China, and Department of Chemical Engineering, UniVersity of Delaware, Newark, Delaware 19716 ReceiVed: May 6, 2010; ReVised Manuscript ReceiVed: June 22, 2010
A molecular thermodynamic model is developed to predict DNA melting in ionic and crowded solutions. Each pair of nucleotides in the double-stranded DNA and each nucleotide in the single-stranded DNA are respectively represented by two types of charged Lennard-Jones spheres. The predicted melting curves and melting temperatures Tm of the model capture the general feature of DNA melting and match fairly well with the available simulation and experimental results. It is found that the melting curve is steeper and Tm is higher for DNA with a longer chain. With increasing the fraction of the complementary cytosine-guanine (CG) base pairs, Tm increases almost linearly as a consequence of the stronger hydrogen bonding of the CG base pair than that of adenine-thymine (AT) base pair. At a greater ionic concentration, Tm is higher due to the shielding effect of counterions on DNA strands. It is observed that Tm increases in the presence of crowder because the crowder molecules occupy a substantial amount of system volume and suppress the entropy increase for DNA melting. At a given concentration, a larger crowder exhibits a greater suppression for DNA melting and hence a higher Tm. At the same packing fraction, however, a smaller crowder leads to a higher Tm. 1. Introduction DNA melting is an important biological process in genetics and life science. During its melting, the double-stranded DNA (dsDNA) is dissociated into the single-stranded DNA (ssDNA) and the hydrogen bonds of dsDNA are broken, which maintain the double helix structure of DNA. The melting is usually regarded as a bioreaction dsDNA h 2 ssDNA, and occurs when DNA solution is heated or subjected to denaturant. Because of the different conformations, dsDNA and ssDNA exhibit significant differences in viscosity, optical activity, and other properties. The melting process is characterized by its melting curve, which is the degree of reaction as a function of temperature, and melting temperature (Tm) when half of the dsDNA is melted. Numerous experimental studies have investigated DNA melting. For example, the Tm of 92 different DNAs were measured over a wide ionic concentration range and a semiempirical formulation for the estimation of Tm was proposed.1 The heat capacities for the formation of DNA duplex structure were reported.2 The initial step of DNA hybridization and the reverse reaction of DNA melting were studied, and it was found that protein functionality affects DNA stability.3 Nevertheless, DNA melting and other important in vivo cellular processes usually occur in crowded environments, because all living cells consist of a large number of crowders, such as polymers, protein tubulins, and actin fibers, which could occupy as much as 40% of the total cell volume.4,5 This physical crowding can significantly alter the DNA melting process, and few experimental studies have examined crowding effects on DNA melting. From isothermal titration calorimetry and UV melting measurements, the Tm of DNA was found to increase by 2 to 5 °C in the * Corresponding author. E-mail:
[email protected] (J.J.);
[email protected] (H.L.).
presence of poly(ethylene glycol) (PEG) with different molecular weights and dextran T-70.6 In contrast to the general excludedvolume argument, crowding-mediated effects were found to include an enthalpy contribution. It was further proposed that crowding agents act as a metabolic buffer and exert “genuine” buffering activity.6 The Tm of 8-, 17-, and 30-mer DNA duplexes were determined in solutions containing 20 wt % ethylene glycol (EG), PEG 200, PEG 1000, and PEG 8000.7 The study revealed that both the length of DNA and the size of cosolute are crucial to DNA stability under crowding conditions. Experimental polymerase chain reaction (PCR) confirmed that the Tm of cDNA-DNA and DNA-RNA hybrids increased by up to 8 °C in the presence of macromolecular crowders such as Ficolls and polyvinyl pyrrolidones.8 There are a number of theoretical studies for DNA melting. Based on the Peyrard-Bishop (PB) model, a theoretical model was proposed.9 However, the model parameters were not determined nor was the theory compared with experiment. By using a modified self-consistent phonon approximation (MSPA), the effect of hydrogen bonds on DNA melting was examined.10 A coarse-grained model, in which each nucleotide was modeled as a base site and a backbone site, was used in simulation to predict the melting curve and dynamic properties of DNA melting.11,12 Similarly, a bead-pin model was proposed by representing sugar group as a bead and base as a pin.13 Also, a more sophisticated model was reported,14 in which each base pair consisted of a phosphate site, a sugar site, and a base site. The model precisely predicted the Tm of DNA in different ionic concentrations. However, theoretical studies of DNA melting in crowded environments are rare. In this work, a molecular thermodynamic model is developed for DNA melting in both ionic and crowded solutions. The numerical calculations with the model are much simpler and less time-consuming than computer simulations, and thus
10.1021/jp104121q 2010 American Chemical Society Published on Web 07/12/2010
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Figure 1. Schematic illustration of DNA melting from double-strand DNA to single-strand DNA: dsDNA h sDNA. Each pair of nucleotides in the dsDNA (a) and each nucleotide in the ssDNA (b), respectively, are represented by two different types of coarse-grained particles. A nucleotide consists of a base and a backbone; the backbone possesses phosphate and deoxyribose groups.
provide valuable insight into the process of DNA melting. The model is the extension of a recent work by Jiang and co-workers for the crowding effects on biochemical reactions such as protein folding, coagulation, and polymerization.15 This model predictions were consistent with experimentally observed crowding effects based on the preferential binding or preferential exclusion of the crowders. It was found that the reactivity in a crowded medium could be enhanced by several orders of magnitude and could vary with the relative sizes of the reactant, product, and crowder as well as with the crowder concentration. For example, in Figure 3a of ref 15, with the product (folded protein) size of σP ) 2.4 nm, a small crowder (σc ) 0.3 nm) stabilizes the unfolded protein and inhibits the protein folding; a relatively large crowder (σc ) 0.4 and 0.5 nm) promotes the stability of folded protein at low concentrations and enhances the protein folding. If the product size is increased to σP ) 4.8 nm, both small and large crowders (σc from 0.3 to 3.0 nm) inhibit the protein folding. The latter case is consistent with what was discussed by Zhou et al.5 In section 2, the molecular model and parameters are introduced to examine DNA melting in ionic and crowded solutions. In section 3, the melting curves and melting temperatures are predicted for DNA with various base pairs in different ionic and crowded solutions; the predictions are compared with available experiment and simulation results. The concluding remarks are given in section 4. 2. Theoretical Framework 2.1. Molecular Model. DNA melting dsDNA h 2 ssDNA is considered to occur in an ionic and/or crowded aqueous solution. Water as the solvent is treated as a background continuum with dielectric constant εr ) 78. At equilibrium, the system consists of dsDNA, ssDNA, ions, and/or crowders. As illustrated in Figure 1, the dsDNA and ssDNA are modeled as freely jointed polyion chains. For simplicity, all chains are regarded as homopolymers with chain density Fi and segment
density Fs,i ) FiLi, where Li is the chain length. Each pair of nucleotides in the dsDNA and each nucleotide in the ssDNA are respectively represented by two different types of coarsegrained charged particles. The interactions between the particles are modeled by using the Lennard-Jones (LJ) plus Coulombic potentials
uij(r) ) 4εij
(
σij12 r12
-
σij6 r6
)
ZiZje2 + 4πε0εrr
(1)
Here εi is the well depth, σi is the collision diameter, Zi is the magnitude of charge on particle i, e ) 1.6022 × 10-19 C is the elementary charge, and ε0 ) 8.8542 × 10-12 C2 N-1 m-2 is the permittivity of vacuum. The cross parameters εij and σij are calculated from the combining rules εij ) (εiεj)1/2 and σij ) (σi + σi)/2. We assume that the initial number density of dsDNA is Ft and the degree of melting is R, and then the densities of dsDNA (Fds) and ssDNA (Fss) at equilibrium are
{
Fds ) Ft(1 - R) Fss ) 2FtR
(2)
Because the ions and inert crowders do not take part in DNA melting, their densities are regarded as constants. The free energy of the system can be expressed as
A ) AQ + Aid + Aex
(3)
where AQ is the standard free energy
AQ ) V
∑ FiAiQ
(4)
Model for DNA Melting
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where AQi is the standard free energy of species i, V is the system volume. Aid in eq 3 is the free energy of ideal-gas contribution
Γ ) πlB 2
)(
)
(13)
Fs,iσh,iZi π / 1+ 1 + Γσh,i 2∆
Fs,iσh,i3 1 + Γσh,i
)
(14)
∑ Fs,i i
Aid ) kBTV
∑ Fi[ln(FiΛi) - 1]
(5)
where kB is the Boltzmann constant, T is the absolute temperature, and Λi is the de Broglie thermal wavelength. Aex in eq 3 is the excess free energy which includes the contributions from the LJ interaction, Coulombic interaction, and chain connectivity
Aex ) ALJ + ACoul + Achain
∑
(∑
aiF*i + i i)1 8
Fs,iεx
i
6
∑ biGi i)1
∑ i
2
πPnσh,i2 Zi 2∆
(
∑ i
2
The excess free energy due to the chain connectivity of polyion can be calculated from the two-particle cavity correlation functions (CCF)20–22
(6)
The contribution of the LJ interaction to the excess free energy can be expressed16
ALJ ) V
Pn )
(
1 1 + Γσh,i
)
βAchain ) Fi(1 - Li)V ln yiiLJ-Coul(σii)
(15)
The CCF of the charged hard-sphere fluid can be derived as22
(7) yijhs-Coul(σh,ij) ) gijhs(σh,ij) exp[gijMSA(σh,ij) gijhs(σh,ij)] exp[βuijCoul(σh,ij)] (16)
where ai and bi are functions of the reduced temperature T* ) kBT/εx. F* ) ΣFs,iσx3 is the reduced density with with K
σx3 )
K
∑ ∑ XiXjσij3
(8)
i)1 j)1
gijhs(σh,ij) ) K
εx )
K
∑ ∑ XiXjεijσij3
1 σx3 i)1
1 + ∆
πσh,iσh,j
i
2
4∆ σh,ij
where Xi is the mole fraction of component i. Gi is related to density by
{
(1 - F)/(2γ) i)1 Gi ) -[FF*2(i-1) - 2(i - 1)Gi-1]/(2γ) 1 < i e 6 (10) with F ) exp(-γσ*2) and γ is an adjustable parameter and usually chosen to be 3.16 For the Coulombic interaction, the charged LJ fluid is approximated as a charged hard-sphere fluid with the hardsphere diameter σh,i being calculated from the Baker-Henderson (BH) theory17
1 + 0.2977/(βεi)
βuijCoul(σh,ij) )
ωi )
[
F Z
(
∑ 1 +s,iΓσi h,i ΓZi + i
πPnσh,i 2∆
)]
+
Γ V 3π (12)
where β ) 1/(kBT), lB ) βe2/(4πε0εr) is the Bjerrum length, ∆ ) 1 -(π/6)∑iFs,iσh,i3, and Γ is the scaling parameter which can be estimated iteratively from eqs 13 and 14
lBZiZj σh,ij
(18)
(19)
(20)
From eq 16, the CCF due to the Coulombic interaction is
yijhs-Coul(σh,ij) yijhs(σh,ij)
) exp[gijMSA(σh,ij) gijhs(σh,ij)] exp[βuijCoul(σh,ij)] (21)
and the CCF of LJ fluid is23 5
3
4π2σh,ijlB
2 2πlB[Zi - πPnσh,i /(2∆)] Γ(1 + Γσh,i)
(11) Then, the excess free energy due to the Coulombic interaction is derived analytically from the mean-spherical approximation (MSA)18,19
Γ2ωiωj
where
yijCoul(σij) )
σi 1 + 0.33163/(βεi) + 0.0010471/(βεi)2
βACoul ) -lBV
(17)
(9)
j)1
gijMSA(σh,ij) ) gijhs(σh,ij) -
σh,i )
∑ Fs,iσh,i2
yLJ(σ) ) gLJ(σ) ) 1 +
5
∑ ∑ aij(F*)i(T*)1-j
(22)
i)1 j)1
where aij are constants. With eqs 21 and 22, the CCF of the charged LJ fluid can be approximated as
yijLJ-Coul(σij) ) yijLJ(σij)yijCoul(σij)
(23)
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TABLE 1: Lennard-Jones Potential Parameters and Charges of DNA Sites and Ions diameter σ (nm) ssDNA dsDNA Na+ Cl-
0.43 0.542 0.235 0.445
TABLE 2: Base-Dependent Enthalpy and Entropy Changes for the Dissociation of Base Pairs31
well depth ε/kB (K)
charge (e)
558.71 1117.4 65.422 50.325
-1 -2 +1 -1
∆H
At a given temperature T, the free energy A is a function of density from eqs 3-23 and the system-specific molecular parameters. Combining the equations above with eq 2, the free energy A is then only a function of the degree of melting R. Using that the free energy A reaches a minimum at thermodynamic equilibrium under constant temperature and volume, the degree of melting R can be calculated by minimizing the free energy. 2.2. Molecular Parameters. Each nucleotide in the ssDNA is represented by a charged LJ sphere, as shown in Figure 1. The sphere has a charge -e and a diameter σss ) 0.43 nm, which is the distance between two nucleotides in the ssDNA.24 Drukker et al. proposed a simplified backbone-base model to study the dynamics of DNA denaturation.11 They represented each nucleotide by two sites, one was the backbone sugar plus phosphate group and the other was the base. The LJ parameters of the backbone and base sites were εbackbone/kB ) 150 K, σbackbone ) 0.24 nm; εbase/kB ) 225 K, σA ) σG ) 0.3 nm and σT ) σC ) 0.24 nm.11 The subscripts A, G, T, and C refer to adenine, guanine, thymine, cytosine. In our model, a nucleotide in the ssDNA is considered as a single site. Based on geometrical analysis, the LJ well depth of the single site εss is approximately by εbackbone + εbase + (εbackbone εbase)1/2. A pair of nucleotides in the dsDNA is also modeled as a charged LJ sphere, but with different potential parameters from that in the ssDNA. By assuming a pair of nucleotides has the same packing fraction as the two constituent nucleotides, the diameter of the dsDNA sphere is σds3 ) 2σss3. The well depth εds is the summation of the ssDNA, εds ) 2εss. Sodium chloride (NaCl) is the electrolyte considered in this study for DNA melting; the parameters of Na+ and Cl- were adopted from David et al.’s work.25 Table 1 lists the molecular parameters for the ssDNA, dsDNA, Na+, and Cl-. For DNA melting dsDNA h 2 ssDNA, the change of the Q Q standard free energy is ∆AQ ) 2Ass - Ads, which is related to Q the equilibrium constant of melting K by
∆AQ ) -RTln KQ
M/M′
(24)
zˇ MN/M′N′
(kJ/mol)
zˇ ∆SMN/M′N′ (J/mol/K)
N/N′
A/T
T/A
G/C
C/G
A/T T/A G/C C/G A/T T/A G/C C/G
33.7 31.8 37.0 33.7 102.7 102.7 102.7 102.7
33.9 33.7 38.1 34.1 102.7 102.7 102.7 102.7
34.1 33.7 36.9 35.5 102.7 102.7 102.7 102.7
38.1 37.0 42.1 36.9 102.7 102.7 102.7 102.7
where nAT and nCG are the number of AT and CG base pairs in Q Q Q Q the dsDNA, respectively; ∆HAT, ∆HCG, ∆SAT, and ∆SCG are the changes of enthalpy and entropy for the dissociation of AT and Q Q CG base pairs. In addition, ∆SCG and ∆SAT can be calculated from Q Q Q Q Q ∆HAT ) (2∆HAA/TT + 2∆HTT/AA + 2∆HAT/TA + 2∆HTA/AT Q Q Q Q + ∆HAC/TG + ∆HAG/TC + ∆HCA/GT + ∆HGA/CT Q Q Q Q + ∆HTC/AG + ∆HTG/AC + ∆HCT/GA + ∆HGT/CA )/16
(27) Q Q Q Q ∆SAT ) (2∆SAA/TT + 2∆STT/AA + 2∆SAT/TA + Q Q Q Q Q + ∆SAC/TG + ∆SAG/TC + ∆SCA/GT + ∆SGA/CT + 2∆STA/AT Q Q Q Q + ∆STG/AC + ∆SCT/GA + ∆SGT/CA )/16 (28) ∆STC/AG Q
Q
In a similar manner, ∆HCG and ∆SCG can be calculated, where Q Q ∆ H MN/M′N′ and ∆ S MN/M′N′ are the base-dependent enthalpy and entropy changes as listed in Table 2.31 More specifically, Q Q ∆HAA/TT and ∆SAA/TT are the changes of enthalpy and entropy upon the dissociation of a nearest-neighbor doublet with base sequence 3′-AA-5′ and 5′-TT-3′ on the two strands, respectively. 3. Results and Discussion To illustrate the applicability of the model developed above, the melting curves and melting temperatures are predicted for DNA with various base pairs in ionic and crowded solutions, respectively. In all the calculations, the initial concentration of the dsDNA is set at 1 µM as in most experimental measurements. 3.1. Ionic Solutions. Figure 2 shows the melting curves for 5′-GCGTCATACAGTGC-3′ in NaCl solution (20, 50, and 120 mM). The fraction of CG bases in this DNA is about fCG )
where R is the ideal gas constant. In a liquid state, ∆AQ ≈ ∆GQ ) ∆HQ - T∆SQ. At a given temperature T, KQ can be evaluated by the changes of standard enthalpy ∆HQ and entropy ∆SQ
ln KQ ) -(∆HQ - T∆SQ)/RT
(25)
A sophisticated calculation of ∆HQ and ∆SQ requires basedependent parameters (A, T, C, and G).26–32 In this work, we assume that ∆HQ and ∆SQ only depend on the number of AT and CG pairs
{
Q Q ∆HQ ) nAT∆HAT + nCG∆HCG Q Q ∆SQ ) nAT∆SAT + nCG∆SCG
(26)
Figure 2. Melting curves for DNA 5′-GCGTCATACAGTGC-3′ (fCG ) 0.5714) in NaCl solution (20, 50, and 120 mM). Solid lines: prediction of this work; dashed lines: simulation;14 symbols: experiment.33
Model for DNA Melting
Figure 3. Melting curves for DNA with different base pairs (fCG ) 0.8) in 0.621 M NaCl.
Figure 4. Melting temperature versus NaCl concentration for DNA with different base pairs (fCG ) 0.8). Lines: prediction of this work; symbols: experiment.1
8/14 ) 0.5714. At low and high temperatures, the degree of melting R varies slightly with temperature. In the vicinity of the melting temperature Tm, however, R changes sharply. This is because the hydrogen bonds between ssDNA begin breaking as temperature increases closely to Tm; additionally, the repulsive interactions of the negatively charged ssDNA strands facilitate dissociation. As a consequence, the melting occurs in a narrow temperature range. The predicted melting curves from our model capture the general feature of DNA melting, though the predicted Tm are lower compared to experiment33 and simulation.14 The deviations may attribute to the approximations introduced in the model for such a complicated system (section 2.2). A better agreement could be achieved by more sophisticated description of the model or by adjusting the model parameters. Figure 3 shows the melting curves for DNA with different base pairs (10, 20, and 30 bp) but the same fraction of CG bases (fCG ) 0.8). As pointed out in section 2, the sequence of bases is not taken into account in our model; therefore, the theoretical predictions only depend on the number of base pairs (i.e., the length of DNA chain) and the fraction fCG. With increasing number of base pairs, Tm increases and the melting curve becomes steeper. Specifically, the temperature range of melting narrows from 35 K for 10 bp to 10 K for 30 bp. Consequently, it would appear that for a realistic DNA with thousands of base pairs, melting would occur in a quite narrow temperature range. The observed increase of Tm for DNA with a longer chain is a result of the larger number of hydrogen bonds that need be broken during melting. On the other hand, the longer negatively charged chains lead to stronger repulsive interactions, which promote the separation of the two strands upon melting, and hence the narrower temperature range of melting. Figure 4 shows the Tm for DNA with different base pairs (10, 15, 20, 25, and 30 bp) as a function of NaCl concentration.
J. Phys. Chem. B, Vol. 114, No. 30, 2010 9909
Figure 5. Melting curves for DNA with 30 base pairs and different CG fractions in 0.621 M NaCl.
Figure 6. Melting temperature versus fCG for DNA with 30 base pairs in 0.621 M NaCl. Lines: prediction of this work; symbols: experiment.1
Figure 7. Melting curves for DNA A20 in the absence and presence of crowder (1 M NaCl, σc ) 5 nm and εc/kB ) 20 K). Cc is the molar concentration of the crowder.
It is found from both theory and experiment that the Tm increases with increasing ionic concentration, in particular, sharply at low ionic concentrations and much less so at high ionic concentrations. The increase of Tm with ionic concentration is attributed to the increasingly important shielding effect of the counterions around DNA strands. Consequently, the electrostatic repulsions between DNA strands become weaker, and the melting occurs at a higher temperature. Furthermore, the sharp increase of Tm at low NaCl concentrations is because the significantly increased number of counterions near the DNA as a result of electrostatic attractions. While at high NaCl concentrations, most DNA charges are shielded and there is a limited space, the number of counterions near each DNA molecule increases only slightly. This leads to the observed marginal increase of Tm at high ionic concentrations. At a given ionic concentration, Tm tends to be a constant for sufficiently long DNA. Compared to experiment,1 the theory can predict the general trend of Tm. However, it
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Figure 8. Melting temperature for DNA A20 versus the (a) concentration Cc and (b) packing fraction ηc of crowder at different crowder size σc (1 M NaCl, εc/kB ) 20 K).
overestimates experiment at lower ionic concentrations and underestimates at high ionic concentrations. This implies that the shielding effect of counterions is not fully taken into account in the model. Experimental1 and simulation studies11 have demonstrated that the fraction fCG of CG base pair also affects DNA melting. Figure 5 shows the melting curves for DNA with 30 base pairs and different CG fractions (0.2, 0.5, and 0.8). The three melting curves have similar shape, and Tm increases with increasing fCG (decreasing fAT accordingly). The reason is the hydrogen bonding of the CG base pair is stronger than that of the AT base pair, as reflected in Table 2. Ideally, the CG base pair forms three hydrogen bonds in contrast to two for the AT base pair. Figure 6 shows the predicted and experimental Tm as a function of fCG. Both theory and experiment reveal that the Tm of DNA increases almost linearly with fCG. Although the sequence of base pairs is not incorporated in the theory, the predictions agree well with experimental data, particularly at high fCG. Furthermore, the agreement between theory and experiment in Figure 6 is better than those in Figure 2 and Figure 4. This is because the theory cannot fully incorporate the shielding effect, whereas it can relatively better describe the changes of enthalpy and entropy as a function of fCG (eq 26-28) with the experimentally measured data in Table 2. 3.2. Crowded Solutions. To examine DNA melting in crowded solutions by the theory, the crowder is modeled as LJ spheres (with well depth εc/kB and collision diameter σc) and the interactions between the crowder particles and DNA bases are modeled using the LJ potential. Figure 7 shows the melting curves for DNA A20 in the absence and presence of crowder. The crowder has a diameter of 5 nm and a concentration of 5 mM, representing 15 wt % PEG with a molecular weight of approximately 3000 g/mol.34 The two melting curves are similar, but the one in the presence of crowder shifts to a higher temperature and the Tm increases accordingly. The conformational entropy of DNA increases during melting process. However, as the crowder occupies part of the system volume, the extent of the entropy increase of DNA is inhibited. As a consequence, the melting is retarded and Tm increases. It is instructive to note that crowding effects was also observed in the binding of DNA with its polymerase35 and bioconjugate assembly between DNA and Au nanoparticles.36 Figure 8 shows the Tm for DNA A20 versus the concentration and packing fraction of the crowder at different crowder size σc. At a given crowder concentration, a larger crowder has a significantly greater effect on DNA melting. As seen in Figure 8a, the Tm remains nearly constant for small crowders (σc ) 0.4, 1, 2 nm) with increasing crowder concentration. However, a sharp rise in the Tm is observed for large crowders (σc ) 5, 7, 9 nm). It reveals that on the basis of crowder concentration, large crowders can enhance the stability of dsDNA and retard
DNA melting. This is consistent with the experimental observation that Tm increases with PEG concentration and molecular weight.6–8 The reason is large crowder molecules occupy a substantial amount of volume in the system and suppress the entropy increase during DNA melting. Interestingly, an opposite trend is observed in Figure 8b, i.e., at a given packing fraction, a smaller crowder exhibits a greater effect on DNA melting. With increasing packing fraction, the Tm rises sharply for a small crowder but slowly for a large one. At the same packing fraction, the interstitial gaps between the smaller crowder molecules are narrower from a geometric argument. Therefore, DNA molecules cannot easily enter the gaps of the smaller crowders compared to the larger ones. This reveals that based on packing fraction, a smaller crowder results in a greater inhibition of the entropy increase, and leads to a higher Tm for DNA melting. 4. Conclusions We have developed a molecular thermodynamic model to examine DNA melting. The aqueous medium (water) is considered to be a continuum, and the dsDNA and ssDNA are represented by homopolymeric chains with the nucleotides as charged Lennard-Jones spheres. The equilibrium constants of melting are estimated based on the experimentally reported changes of enthalpy and entropy. The model predicts the melting curve and Tm for DNA in ionic and/or crowded solutions, which are in good agreement with the experimental data and simulation results available in the literature. The melting curve varies sharply near Tm but slowly at low and high temperatures. For DNA with a larger number of base pairs (a longer chain), Tm is higher and the melting curve becomes steeper. For sufficiently long DNA, Tm approaches a constant. Also Tm is found to increase in an approximately linear manner with the fraction of CG base pairs because the hydrogen bonding of CG base pair is stronger than that of the AT base pair. Further, Tm increases with ionic concentration due to the increased shielding effect of the counterions on the electrostatic repulsions between DNA strands. In the presence of crowder, the melting is retarded and consequently Tm increases. This is attributed to the volume occupied by crowder molecules, which inhibits the entropy increase necessary for DNA melting. At a given concentration, a larger crowder is found to enhance the stability of DNA more than a smaller one, though the opposite is observed at a given packing fraction. Acknowledgment. This work was supported by the National University of Singapore (R279-000-238-112), the National Natural Science Foundation of China (Nos. 20736002 and 20706013), the Program for Changjiang Scholars and Innovative Research Team in University of China (No. IRT0721), and the 111 Project of China (No. B08021).
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