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Calculating melting temperatures for oligonucleotide duplexes provides an instructive exercise in the application of thermodynamics to biochemistry. Hybridized complementary oligonucleotide sequences are important in a number of molecular biology applications, including polymerase chain reaction (PCR) primers, nucleotide probes, gene arrays, silencing RNA, and single nucleotide polymorphisms. Melting temperatures are important parameters used in designing experimental conditions for these applications

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644
Journal of Chemical Education
ã
Vol. 86 No. 5 May 200
9 ã
www.JCE.DivCHED.org
ã
© Division of Chemical Education
Research: Science and Education
Advanced Chemistry Classroom and Laboratory
edited by
Joseph J. BelBruno
Dartmouth CollegeHanover, NH 03755
Calculating melting temperatures or oligonucleotide duplexes provides an instructive exercise in the application o thermodynamics to biochemistry. Hybridized complementary oligonucleotide sequences are important in a number o molecu-lar biology applications, including polymerase chain reaction (PCR) primers, nucleotide probes, gene arrays, silencing RNA, and single nucleotide polymorphisms. Melting temperatures are important parameters used in designing experimental conditions or these applications. A number o empirical equations based on (G + C) base content, duplex length, and the salt concen-tration have been used to estimate the melting temperatures o oligonucleotide duplexes
(1, 2).
However, melting temperatures are more accurately calculated with a thermodynamic approach that takes into account the concentration o the oligonucleotide
(2–4).
Tis approach has been discussed brieﬂy in some text-books
(5, 6)
and has been incorporated into online calculators
(7, 8).
Since biochemistry has become a more integral part o the curriculum or chemistry majors, an in-depth treatment o this topic can also be used in physical chemistry classes. Te melting temperature,
m
, o an oligonucleotide duplex is deﬁned as the temperature at which 50% o the duplex is dis-sociated. A student laboratory exercise describing how
m
can be determined has been published
(9).
By assuming that there is a two-state equilibrium that is maintained throughout the melting process, the
m
can be related to the standard Gibbs energy or the dissociation as
eq
Δ
ln
G RT K
° = −
(1) where the variables have their standard deﬁnitions.
Te fraction of duplex melted is usually measured spec-troscopically, so that
K
eq
values have been determined as a function of temperature
(9, 10).
Tis provides the data for van’t Hoﬀ plots, that allows the standard enthalpy and entropy to be calculated
(9, 10).
From the
Δ
H
° and
Δ
S
° of melting for a large number of oligonucleotides, it has been possible to use nearest-neighbor analysis to derive
Δ
H
° and
Δ
S
° values for all possible base-stacking interactions
(3, 11–14).
Any oligonucle-otide duplex can then be treated as a combination of nearest neighbors to get
Δ
H
° and
Δ
S
° values, from which the
T
m
can be predicted.
From biochemistry courses and texts, most students develop an ingrained habit o thinking that the energetics and hence, the melting temperatures or oligonucleotides are determined solely by the diﬀerence in the number o hydrogen bonds between GC and A base pairs. Tis misconception arises rom the act that hydrogen bonding between bases determines the speciﬁcity o duplex ormation and rom the strong correlation between
m
and (G + C) base content
(15).
However, it is ofen not appreci-ated that melting o the duplex does not abolish hydrogen bonds to the bases, since afer disassociation the bases are hydrogen bonded to water. Further, students commonly think o single-stranded oligonucleotides as rigid linear structures, rather than as ﬂexible molecules with unstacked bases as is illustrated in Fig-ure 1. Tereore, this topic helps students to ocus on the losses o hydrophobic interactions and o
π
−
π
interactions that occur
Thermodynamics of Oligonucleotide Duplex Melting
Sherrie Schreiber-Gosche and Robert A. Edwards*
Department of Biological Sciences, University of Calgary, Calgary, AB T2N 1N4, Canada;
*redwards@ucalgary.ca
Figure 1. Diagrammatic representations of the melting of short oligonucleotide chains for the three cases described in the text: (A) a non-self-complementary duplex; (B) a self-complementary duplex; and (C) a probe or primer at a higher concentration than the concentration of its hybridization site.
A
T
A T G CT A C G
G
C
T
A
C
G
A
G
C
A G C TT C G A
T
A
G
T
C
BA
T
A
C
G
T
A
G
C
T
A
G C
A T G C
A
T
T
G
GCA
A
G
C
A
A
G T C............
C
T A C G
T
T
C
G
G
T
A
C
C
© Division of Chemical Education
ã
www.JCE.DivCHED.org
ã
Vol. 86 No. 5 May 200
9 ã
Journal of Chemical Education
645Research: Science and Education
when bases become unstacked, as an important determinant o melting temperature
(11, 16).
Because 25 °C is ofen called a “standard temperature” in introductory chemistry courses, it is necessary to explain to stu-dents that standard Gibbs energies can be calculated and related to
K
eq
at any temperature o interest. emperatures o interest include
m
and
m
− 5 °C, because a temperature about 5 °C less than
m
is ofen used to attain conditions in which most o the oligonucleotide primer is annealed to the template. Te thermodynamic approach also helps students appreciate that the melting temperature or a primer depends upon the concentra-tion o oligonucleotide used. Melting temperatures are ofen rapidly approximated at the bench rom the number o A and bases,
N
A
, and G and C bases,
N
CG
, in one o the strands according to
(1, 2)
m
T
=
22 4
N N
AT GC
+
(2)Although this equation is easy to use, it is not accurate
(2).
Te thermodynamic approach equips students to calculate more accurate
m
values by explicitly taking into account the depen-dence o
m
on duplex concentration, as is required or modern primer design
(4).
Tis approach also provides a more rigorous ramework or understanding the inﬂuence o mismatches on melting temperature, which is needed or site-directed mutagen-esis or dealing with the possibility o incorrect annealing.
Melting Equilibrium
Te thermodynamic approach involves two steps. First, data rom a nearest-neighbor analysis are used to obtain the standard entropy and standard enthalpy o the stacking interactions be-tween base pairs in the duplex. Ten, the melting temperature is obtained as the midpoint where the duplex, composed o strands X and Y, is 50% dissociated in the equilibrium
(9, 10)
X
+
Y X
⋅
Y
(3)Te
K
eq
or dissociation is related to the total oligonucle-otide strand concentration (i.e.,
C
tot
)
(6, 12).
Te concept o
C
tot
is easily understood i it is thought o as the sum o the single strand concentrations at a high temperature where the duplex is completely dissociated:
C
tottot tot
X Y
=
[ ]
+
[ ]
(4)However, it is important to remember that the thermodynamic data have been obtained under initial conditions consisting o a duplex that has been heated to separate, with no extra single stands (neither X nor Y) being present. Under these condi-tions, the total strand concentration is twice the initial duplex concentration. Te relationship between
K
eq
and
C
tot
is conceptually diﬀer-ent in three common cases, which are diagrammed in Figure 1: (i) melting o non-sel-complementary duplexes, (ii) melting o sel-complementary duplexes (i.e., palindromes), and (iii) melt-ing o a probe or primer that is at much higher concentration than the concentration o its complementary hybridization site. Tese three cases are important because some o the thermody-namic measurements have utilized a sel-complementary strand, in which the strand base pairs with itsel; however, others have utilized non-sel-complementary duplexes. In addition, many applications or melting temperatures utilize the melting tem- perature o a probe or primer that is at much higher concentra-tion than the concentration o its complementary hybridization site on large DNA molecules.Te relationships between concentrations, between
K
eq
and
C
tot
, and between the melting temperature and
C
tot
are tabulated or each o these three cases in able 1. Students can be expected to rationalize the srcin o any o the concentration relationships and be able to obtain rom the concentration relationships the relationship between
K
eq
and
C
tot
or each o the cases. Te equations or
m
in terms o
Δ
H
°,
Δ
S
°, and
K
eq
in able 1 can be obtained by substituting
Δ
H
° −
Δ
S
° or standard Gibbs energy into eq 1 and isolating the temperature to obtain
T H S R K
eq
= °° −ΔΔ
ln
(5)Substituting one hal or the raction dissociated (i.e.,
f
= ½) and the relevant equations or
K
eq
in terms o
C
tot
into eq 5 leads to the tabulated equations or
m
.
Base Stacking
Te notation that is used or stacking interactions consists o speciying the bases that are stacked as nearest neighbors next to each other, or example, an AG stacking interaction with an adenine base stacked on a guanine base as shown on the upper lef o Figure 2. For a single oligonucleotide strand, taking into account direction, there are 16 possible nearest neighbors (e.g., AA, A, AC, AG, and so orth; where the ﬁrst base is closer to the 5
′
-end o the oligonucleotide). Te disruption o each o these base-stacking interactions contributes a unique quantity to the standard enthalpy and entropy o melting. Note that the energetics are similar, but not identical, or nearest-neighbor couplets with a diﬀerent order (e.g., the energetics o AG stack-ing is not identical to GA energetics). Although there is some inconsistency in the notation used or the direction o the second strand
(3),
most authors use a notation in which the strands are shown antiparallel (e.g., AG/C means 5
′
-AG-3
′
base paired with 3
′
-C-5
′
to produce an A stacked on a G and a C stacked on a ).
Simpliﬁed Stacking Method
For double-stranded duplexes, there are only 10 unique stacking interactions or Watson–Crick base pairing, because the order on one o the strands dictates the order on the complemen-tary strand so that 6 stacking interactions are redundant. (e.g., AG/C = C/GA). Superﬁcially it might seem as though each double-stranded stacking interaction would have a redundant partner, however, this is not the case, and it is a useul exercise or students to consider the 16 possibilities and ﬁnd those that do have partners. For teaching purposes, it is possible to simpliy this to consider only three general sets o stacking interactions: (i) A or stacked on an A or , (ii) C or G stacked on a C or G, and (iii) all other interactions. Tis three-set simpliﬁcation ig-nores direction and is amenable to hand calculations (see below).
646
Journal of Chemical Education
ã
Vol. 86 No. 5 May 200
9 ã
www.JCE.DivCHED.org
ã
© Division of Chemical Education
Research: Science and Education
Table 1. Relationships Between Concentrations and Equilibrium Constants for Each of the Three Cases of Duplex Melting
Description of Case and EquilibriumRelationships Between Concentrations at Any Temperature
4,5
Relationships at the Melting Temperature (
T
m
).
6–8
Case 1Non-self-complementary duplex oligonucleotide.
1
X
+
YX
⋅
Y
X
⋅
YY
initial
f
=
[ ]
X
⋅
Y
initial
[ ]
=
[ ]
[ ]
X
=
[ ]
+
[ ]
+
[ ]
tot
X Y X
⋅
Y
C
2
=
[ ]
totinitial
XY
C
2
⋅
=−
eqtot
K f C
2 1
2
f f
( )
fraction dissociated:
f
= 1/2
= =
m
X
[ ]
m
[ ]
m
]
Y X
⋅
Y
[
eqtot
K C
=
4
4
mtot
T H S R C
=°° −
ln
ΔΔ
Case 2Self-complementary duplex oligonucleotide.
2
2XX
⋅
X
XX X
inital
f
=
[ ][ ]
2
⋅
X
tot
C
=
[ ]
+
2 X
⋅
X
[ ]
X
⋅
X
tot
C
=
[ ]
2
iinitial
eqtot
K f C f
=−
21
2
fraction dissociated:
f
= 1/2
m
=
]
m
X
[ ]
X
⋅
X
[
2
eq
K C
=
ttot
mtot
T H S R C
=°° −ΔΔ
ln
Case 3Oligonucleotide at high concentration hybridized to DNA at low concentration.
3
X
+
YX
⋅
Y
tot
YY
f
=
[ ][ ]
tot
X
[ ]
>>>
[ ]
>
Y
tot
X
⋅
Y
[ ]
≈ ≈
[ ]
X
tot t
C
to
X
[ ]
= +
[ ]
Y
[ ]
Y
tot
X
⋅
Y
[ ]
=−
eqtot
K f C f
1
fraction dissociated:
f
= 1/2
Y
m
[ ]
=
m
]
X
⋅
Y
[
eq tot
=
K C
mtot
=°° −
T H S R C
ΔΔ
ln
1
For case 1 the initial conditions consist of a duplex with no extra X or Y present.
2
For case 2 the strand is base paired with itself to produce a palindrome.
3
For case 3 a primer strand (X) is paired with DNA (Y) on the ﬁrst few rounds of PCR. Unbound single-stranded primer is also present.
4
The fraction of the duplex that is melted (dissociated) is
f
.
5
For case 3 the relevant DNA concentration (i.e., [Y]
tot
) is the concentration of DNA strands that contain the sequence complementary to the primer (i.e., the concentration of hybridization sites).
6
The subscript “m” is used to refer to the melting temperature (where half of the duplex is dissociated into single-stranded form(s)) and concentrations at that temperature.
7
Δ
H
º and
Δ
S
º in this article are for melting (i.e., the forward direct is dissociation of strands). Standard enthalpy and entropy are often tabulated for the reverse of melting (i.e., the formation of duplex).
mrrtot
T H S R C
=°° +ΔΔ
ln4
or
mrr tot
T H S R C
=°° +ΔΔ
ln
8
The derivations of the equations for
T
m
follow from
ΔΔ Δ Δ
G T RT K H T S
°
( )
= − = ° − °
ln
eq
where
Δ
H
º and
Δ
S
º can be treated as independent of temperature. Both
Δ
H
º and
Δ
S
º depend on temperature through
Δ
C
p
, but the values of
Δ
C
p
for oligonucleotide melting are not large
(4, 14).
Further, the small
Δ
C
p
dependent temperature contributions of enthalpy to
Δ
G
º and of entropy to
Δ
G
º are about equal in magnitude and opposite in sign, and therefore cancel each other out.
© Division of Chemical Education
ã
www.JCE.DivCHED.org
ã
Vol. 86 No. 5 May 200
9 ã
Journal of Chemical Education
647Research: Science and Education
Counting such interactions along an oligonucleotide sequence helps students to more clearly delineate between base pairing and base stacking, because they observe stacking interactions on couplets that do not base pair (e.g., AA, AG, etc.).
Calculating the Standard Enthalpies, Entropies, and Melting Temperatures
Empirical values or the
Δ
H
° and
Δ
S
° o each nearest-neighbor base-stacking interaction have been derived by ﬁtting the experimental standard enthalpies and entropies or many oligonucleotides as the sum o individual
Δ
H
° and
Δ
S
° nearest-neighbor components
(3, 11–14).
SantaLucia has presented, in a consistent ormat, the empirical
Δ
G
° values derived rom diverse sets o oligonucleotides by several investigators or the 10 diﬀerent stacking interactions
(14).
Tese parameters are relatively consistent with each other and the widely used uniﬁed set o
Δ
H
° and
Δ
S
° parameters adequately describes oligomer and polymer thermodynamics
(14).
Usually these are tabulated or base-stacking ormation (i.e., the reverse o melting as two strands come together). An entropy o “initiation” is also pres-
Figure 2. Diagram highlighting the difference between base-pairing and base-stacking interactions. The structure has been expanded slightly to show the regions where base-stacking interactions will occur. The hydrogen bonds come from base pairing, whereas the stacking interactions arise from the bases being on top of one another. Note that slight variations in the energetics of the stacking interactions occur between various base-pair combinations (not shown).Figure 3. Illustration of the stacking interactions on an oligonucleotide:
N
AA
is the number of occurrences in the single strand of AA, AT, TA, and TT stacking interactions;
N
CC
is the number of occurrences in the single strand of CC, CG, GC, and GG stacking interactions; and
N
other
is the number of occurrences in the single strand of AC, AG, TC, TG, CA, CT, GA, and CT stacking interactions. The complementary stand would have the same number of stacking interactions of each type (not shown).
5
′
-CTTTCATGTCCGCAT-3
∙
oligonucleotide:four AA, AT, TA, and TT base stackings
⇒
N
AA
= 4three CC, CG, GC, and GG base stackings
⇒
N
CC
= 3
ent, which accounts or the order arising rom the association o two strands to orm one complex. Empirical corrections have also been derived or end eﬀects, which essentially account or base pairs at the ends that have only one nearest neighbor. In the uniﬁed set o parameters compiled by SantaLucia the end eﬀects are included in the initiation parameters
(14).
Finally, or palin-drome duplexes a symmetry correction is required to account or the decrease in order when a duplex with end-to-end symmetry is changed into two single strands that do not individually have end-to-end symmetry. Tere are many tables o published values or the experi-mental thermodynamic parameters or oligonucleotide duplexes
(3, 4, 11–13, 17),
so the predicted values o examples chosen or students can be compared to experimental values. Examples may even be drawn rom an oligonucleotide primer or probe that is used or a particular experiment within the student’s range o experience. A sample is shown in Figure 3 and able 2 that includes the averaged parameters or the generalized equations. Tus, or example, an A base stacked on top o another A or a base will contribute 7.6 kcal/mol to the standard enthalpy. In the simpliﬁed calculations o able 2, there are enthalpy and entropy corrections or the ends. Te negative enthalpy correction is present because the bases at the ends do not have neighbors on both sides and thus are not held as ﬁrmly in place as the other bases. As a result, the end base may be transiently unstacked more ofen than an internal base, so there is slightly less avorable interaction between an end base and its neighbor. Te standard entropy correction is also dominated by entropy rom the transient unstacking o end bases. Since the end bases are already transiently unstacked in the double-stranded orm, the negative entropy correction adjusts the positive entropy o dissociation to be smaller or the end bases than or internal bases. Te positive contribution to entropy rom the separation o strands is buried in the entropy correction or the ends.A comparison between the accuracy o calculating
m
by the various methods is shown in able 3 or a set o duplexes that has the length range, NaCl concentration, and oligonucleotide concentrations that actually might be used or PCR. Because the empirical parameters or
Δ
H
° and
Δ
S
° apply to 1.0 M NaCl, it is necessary to correct or the experimental salt concentrations.

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