Stability of Human telomeric G-Quadruplex DNA with
Actinomycin-D
Rushi Ghizal
Physics Department
Integral University,Kursi Road,
Lucknow, 226066.U.P,India.
rushi_ghizal@yahoo.co.in
Gazala Roohi Fatima
Seema Srivastava
Physics Department
Integral University,Kursi Road,
Lucknow, 226066.U.P,India.
gazala03@gmail.com
Physics Department
Integral University, Kursi Road,
Lucknow, 226066. U.P,India.
seemasr1@rediffmail.com
ABSTRACT
The theory of cooperative transitions has
been applied to explain the stability and melting
behavior of Telomeric G-Quadruplex DNA. The
transition profiles and heat capacity curves are
best explained in terms of two variable
parameters, namely nucleation and propagation
parameters. Greater stability is achieved in
presence of K+ ions, compared with Na+ ion. It is
ascribed to dipole and dipole-induced
transitions.
Keywords
G-Quadruplex, transition profiles, heat capacity.
Fig 1. Chemical structure of Actinomycin D
1. INTRODUCTION
In recent years the study of small molecules
that selectively target and stabilize the Gquadruplex structures have emerged as an area
of tropical interest. This DNA structure motif
(G-quadruplex) has emerged as a novel and
exciting target for the discovery and design of
new class of anticancer drugs [1-3].
Recently Jason S.H, et al. [4] have reported
thermodynamic and structural characterization
of human telomeric G-quadruplex.In their work
an interaction between actinomycin D (Fig. 1)
and
human
telomeric
sequence
d[AGGG(TTAGGG)3] have been presented.
The thermal melting profiles were obtained for
both free and bound Na+ and K+ forms of the Gquadruplex DNA structure using differential
scanning calorimetry. Their study shows that in
presence of K+ ions, guanine forms a more
stable tetramolecular complex, compared with
Na+ ions.
In the present study, we have used the theory
of co-operativity for a finite system to generate
the transition profiles and λ-point anomaly in
heat capacity at transition point. The heat
capacity measurements of Jason S.H, et al. [4],
have provided the input information. The
alteration in nucleation parameter, which is
inverse measure of binding strength, reflects the
effect of actinomycin D binding.
2. THEORY
The stability of G-Quadruplex is directly
related to the strength and number of hydrogen
bonds. They are also responsible for cooperativity enhancement. Thus, the melting of
such molecules, which involve both positional
and orientational disordering, can be treated as
two-phase problem. Thus we can modify Zimm
and Bragg theory [5]. In brief, the above theory
consist of writing an Ising matrix for two phase
system, the bonded state and unbounded state.
As discussed earlier [5-10], the Ising matrix 𝑀
can be written as-
1
𝑓𝑟2
𝑓𝑟
1
2
𝑓𝑘
1
2
𝑓𝑟
𝑓𝑟 𝑓𝑟
𝑀 = 𝑓𝑘
𝑓𝑟2 𝑓𝑘2
1
1
2
1
2
𝑓ℎ [𝑓 𝑓
ℎ 𝑟
1
2
0
𝑓𝑘2 𝑓ℎ2
0
1
2
2
[ 𝑓ℎ ]
1
(1)
1
2
𝑓ℎ 𝑓ℎ ]
Where, 𝑓𝑟 , 𝑓ℎ and 𝑓𝑘 are corresponding base
pair partition function contributions in the three
states (i.e ordered, disordered and boundary or
nucleation). G-G and T-T have been assumed to
have the same statistical weight because
according to crystallographic studies reported
recently [6], both thymine and guanine are
hydrogen-bonded in a tetrad complex in the
same manner. The hydrogen bond lengths are
approximately the same. The eigen values for M
are given by:
1
1
(3)
𝑓𝑘2
1
0
1
1
𝑉=
𝑓𝑟 𝑓𝑘
1
1
2
𝑓ℎ
The partition function for a 𝑁 -segment chain is
given by:
𝑍 = 𝑈 𝑀𝑁−1 𝑉
(4)
The matrix 𝑇 which diagonalizes M consists of
the column vectors given by:
𝑀𝑋 = 𝜆𝑋
(5)
Where
𝑋1
𝑋 = [ 𝑋2 ]
𝑋3
By substituting the value of 𝑀 from Eq. 1 in Eq.
5, we get
𝜆1 = 2 [(𝑓𝑟 + 𝑓ℎ ) + {(𝑓𝑟 − 𝑓ℎ )2 + 4𝑓𝑟 𝑓𝑘 }2 ]
1
1
𝜆2 = [(𝑓𝑟 + 𝑓ℎ ) − {(𝑓𝑟 − 𝑓ℎ )2 + 4𝑓𝑟 𝑓𝑘 }2 ]
2
𝜆3 = 0
1
1
(𝜆1 −𝑓𝑟 )
(𝜆2 −𝑓𝑟 )
1 1
Since we are dealing with finite system, hence
the effect of initial and final states becomes
important. The contribution of the first segment
to the partition function is given by a row vector
𝑈. It is assumed that the initial state is
disordered state.
1
2
𝑈 = [(𝑓𝑟 , 0, 0)]
(2)
Similarly, the column 𝑉 gives the state of last
segment,
1 1
(𝑓𝑟2 𝑓𝑘2 )
(𝑓𝑟2 𝑓𝑘2 )
1 1
(𝑓ℎ2 𝑓𝑟2 )
1 1
(𝑓ℎ2 𝑓𝑟2 )
[(𝜆1 −𝑓ℎ )
(𝜆2 −𝑓ℎ )
𝑇=
1
1
1
−𝑓𝑟2 𝑓𝑘2
(6)
1
1
−𝑓ℎ2 𝑓𝑟2 ]
Similarly, we get 𝑇 −1 from the matrix equation:
𝑌𝑀 = 𝜆𝑌
(7)
Where
𝑌 = [𝑌1
𝑌2
𝑌3 ]
Again, by substituting the values of 𝑀 from Eq.
1 in Eq. 7, we get:
𝐶1
𝐶1
𝑇 −1 = 𝐶
2
𝐶2
1 1
1 1
𝑓𝑟2 𝑓𝑘2
𝑓𝑘 𝑓𝑟2 𝑓ℎ2
𝐶1 𝜆
1 (𝜆1 −𝑓ℎ )
1 1
𝑓𝑘 𝑓𝑟2 𝑓ℎ2
𝜆1
1 1
𝑓𝑟2 𝑓𝑘2
2 (𝜆2 −𝑓ℎ )
1 1
𝑓𝑘 𝑓𝑟2 𝑓ℎ2
1 1
[𝐶3
𝐶3
(8)
𝐶2 𝜆
𝜆2
𝑓𝑟2 𝑓𝑘2
𝐶3 𝜆
]
3 (𝜆3 −𝑓ℎ )
𝜆3
The normalizing constants are:
𝐶1 =
(𝜆1 −𝑓ℎ )
(𝜆1 −𝜆2 )
, 𝐶2 =
(𝜆2 −𝑓ℎ )
(𝜆2 −𝜆1
, 𝐶3 = 0
)
(9)
If we let 𝛬 = 𝑇 −1 𝑀𝑇 be he diagonalized form
of 𝑀, the partition function can be written as:
𝑍 = 𝑈𝑇𝛬𝑁−1 𝑇 −1
(10)
The extension of this formalism is straight
forward. The specific heat is related to molar
enthalpy and entropy changes in the transition
from state I to state II. From well-known
thermodynamic relations, free energy and
internal energy are 𝐹 = −𝐾𝑇𝑙𝑛𝑍 and 𝑈 =
−𝑇 2 (𝛿/𝛿𝑇)(𝐹/𝑇) respectively. Differentiating
internal energy with respect to temperature we
get the specific heat:
𝐶𝑣 =
1
𝛿𝑙𝑛𝑍
𝑁
𝛿𝑙𝑛𝑓𝑟
𝑚
(11)
]
𝑄𝑟 = 2 +
(1−𝑆)(2𝐴−1)
2𝑃
+
(1+𝑆){(2𝐴−1)𝑃 −1 +𝑠}
2𝑃 2 𝑁
)
(14)
1
− ( 3 ) [{(2𝐴 − 1)𝑃 − 1
2𝑃 𝑁
+ 𝑠}2(𝑠 + 1)]
(12)
Where
𝑃=
𝛿𝑠
+ {(2𝐴 − 1)𝑃 − 1 + 𝑠}}]
Solving the above equation, we get
1
2 𝑠𝛿𝑄
𝑟
𝛿𝑄𝑟
1
𝛿𝐴
= ( 2 ) [2𝑃(1 − 𝑠)
− 𝑃(2𝐴 − 1)
𝛿𝑠
2𝑃
𝛿𝑠
𝛿𝑃
− (1 − 𝑠)(2𝐴 − 1)
]
𝛿𝑠
1
+ ( 3 ) [𝑃 {(𝑠
2𝑃 𝑁
𝛿𝑃
𝛿𝐴
+ 1) {(2𝐴 − 1)
+ 2𝑃
+ 1}
𝛿𝑠
𝛿𝑠
The fraction of the segments in the disordered
form is given by
𝑄𝑟 = [ ] [
𝛿𝑇
∆𝐻
= 𝑁𝑘 (𝑅𝑇 ) (
Where ∆𝐻 is the molar change in enthalpy
about the transition point, 𝑇𝑚 is the transition
temperature, and
On substituting the values from Eqs. 1, 2, 3, 6, 8
and 9 in Eq. 10, the partition function becomes:
𝑍 = 𝐶1 𝜆1𝑁 + 𝐶2 𝜆𝑁
2
𝛿𝑈
With
(𝜆1 −𝜆2 )
𝑓𝑟
,𝑠=
𝑓ℎ
𝑓𝑟
𝑓𝑘
, 𝜎=
−
𝐴 = [(𝑓𝑟 − 𝑓ℎ )2 + 4𝑓𝑘 𝑓𝑟 ]
𝑓𝑟
,
𝛿𝐴
𝛿𝑠
1
2
Here 𝑠 is propagation parameter, which for
simplicity is assumed to be 1. In fact, in most of
the systems, it is found to be close to unity. If
𝐴𝑟 and 𝐴ℎ are the absorbance in disordered and
ordered states, respectively, the total absorption
can be written as:
𝐴 = 𝑄𝑟 𝐴𝑟 + (1 − 𝑄𝑟 )𝐴ℎ
(13)
𝛿𝑃
𝛿𝑠
= {
=
(𝑠−𝜎)𝑁
𝑍 2
𝑓𝑁
𝑟
(𝑠−1)
𝑃
𝜎
} × (𝑃3 ) × [−2 + 𝑁
, 𝜎=
(𝑠−2𝜎−1)
(𝑠−𝜎)
]
𝑓𝑘
𝑓𝑟
Where 𝜎 is nucleation parameter and is a
measure of the energy expanded/released in the
formation (uncoiling) of first turn of the
ordered/disordered state. It is related to the
uninterrupted sequence lengths [5]. The volume
heat capacity 𝐶𝑣 has been converted into
constant pressure heat capacity 𝐶𝑝 by using the
Nernest-Lindermann approximation [11-12].
𝐶𝑝2 𝑇
𝐶𝑝 − 𝐶𝑣 = 3𝑅𝐴0 (𝐶
𝑣 𝑇𝑚
)
(15)
Where 𝐴0 is a constant often of a universal
value [3.9 x 10-3 (K mol/J)] and 𝑇𝑚 is the
melting temperature.
3. RESULTS
3.1. Transition Profiles
When Actinomycin D binds to GQuadruplex DNA, the structure of DNA still
remains highly co-operative and hence two state
theory of order/disorder transitions is applicable.
The Zimm-Bragg theory is amended so as to
consider ordered and disordered states
(bounded/unbounded), as the two states coexist
at the transition point. The transition is
characterized mainly by the nucleation
parameter
and
overall
change
of
entropy/enthalpy, which are also the main
thermodynamic force driving the transitions.
The change in enthalpy obtained from
differential scanning calorimetric (DSC)
measurements takes all this into account. This is
evident from the enthalpy changes and the
changes in other transition parameter, such as
nucleation parameter (𝜎) and melting point
(Table 1 and Table 2). The result obtained from
the theoretical study suggested that the binding
of Actinomycin D increases the melting
temperature of G-Quadruplex DNA (for both K+
and Na+ forms). At saturation the melting point
of K+ G-Quadruplex DNA was increased with
9.375 K in comparison with free G-Quadruplex.
Similarly, the melting point of Na+ GQuadruplex was increased with 18.25 K The
sharpness of the transition can be looked at in
terms of half width and sensitivity parameter
defined as (∆𝐻/𝜎). The deviations in half-width
and sensitive parameters scientifically revealed
that the transition is sharp in case of unbounded
state and goes blunt with Actinomycin D
saturation. In case of 𝜆-transition, the same
trend in the sharpness of transition is seen
between the Actinomycin D boned as well as
unbounded curves. As, anticipated, the
sharpness is better in unbound state as compared
to bound state. A range of parameters, which
give transition profiles in best agreement with
the experimental measurements for binding of
Actinomycin D to G-Quadruplex DNA (K+ and
Na+ forms) are given in Table 1 and Table 2.
Table 1. Transition Parameters for Actinomycin D binding
to G-Quadruplex (K+ form).
Parameters
Unbonded
GQuadruplex
DNA
341.75
𝑇𝑚 (K)
4
1.6X10
∆𝐻(Kcal/Mbp)
0.0046
𝜎
56
𝑁
Half Width
12
(Exp)
Half Width
12
(Theo)
Sensitivity
3.478X106
parameter
(∆𝐻/𝜎)
G-Quadruplex
DNA bonded
with
Actinomycin D
351.125
1.1X104
0.00061
56
7.5
7.5
1.083X107
Table 2. Transition Parameters for Actinomycin D binding
to G-Quadruplex (Na+ form).
Parameters
𝑇𝑚 (K)
∆𝐻(Kcal/Mbp)
𝜎
𝑁
Half Width
(Exp)
Half Width
(Theo)
Sensitivity
parameter
(∆𝐻/𝜎)
Unbonded
GQuadruplex
DNA
333
2.4X104
0.015
56
8
G-Quadruplex
DNA bonded
with
Actinomycin
D
351.25
1.5X104
0.00099
56
7.5
8
7.5
1.6X106
1.515X107
Experimental Cp
1
Absorbance
Specific Heat (Kcal/mol K)
1
0.75
Theoretical Cp
Predicted Value
0.5
0
290
0.5
320
350
Temperature (K)
0.25
0
290
300
310
320
330
340
350
360
Temperature (K)
Fig. 2 Heat Capacity and transition profile of free K+
form of the G-quadruplex DNA structure.
370
Experimental Cp
Theoretical Cp
1
1
Predicted Value
Absorbance
Specific Heat ( (Kcal/mol K)
3.2. Heat Capacity
Heat capacity, second derivative of the free
energy, has been calculated by using Eq. 14 and
15, which is used to characterize conformational
and dynamical states of a macromolecular
system. These heat capacities with 𝜆-point
anomaly along with their transition profiles for
unbounded G-Quadruplex DNA (K+ form) is
shown in Fig. 2 and for Actinomycin D bonded
to G-Quadruplex DNA (K+ form) is shown in
Fig. 3. The results revealed that the theoretically
obtained heat capacity profiles agreed with the
experimentally reported one and could be
brought almost into concurrence with the use of
scaling factors. Minor insignificant deviations at
the tail end is primarily due to the presence of
various disordered states and presence of short
helical segments found in random coil states.
The predicted values of absorbance for both
unbounded and bonded state are also given in
Fig. 2 and Fig. 3.
0.75
0.5
0
0.5
290
320
350
Temperature (K)
0.25
0
290
300
310
320
330
340
350
360
370
Temperature (K)
Fig. 3 Heat Capacity and transition profile of K + form of the
G-quadruplex DNA structure bonded with Actinomycin D
Similarly the heat capacities and their
transition profiles for unbounded G-Quadruplex
DNA (Na+ form) is shown in Fig. 4 and for the
Actinomycin D bonded to G-Quadruplex DNA
(Na+ form) is shown in Fig.5. Again one
observes that the theoretically obtained heat
capacity profiles agreed with the experimentally
reported ones. The sharpness of the transition
can be characterized by half width of the heat
capacity curves that are in good agreement in
both experimental and theoretical graphs. Again
the predicted absorbance is shown in Fig. 4 and
Fig. 5 for unbounded and bonded states. It must
also be noted that the heat capacities and the
transition profiles of G-Quadruplex (K+ form
and Na+ form) revealed that greater stabilization
is achieved in presence of K+ ions, compared to
Na+ ions.
Experimental Cp
Absorbance
Specific Heat (Kcal/mol K)
Predicted Value
1
1
Theoretical Cp
0.5
0
0.75
290
320
350
Temperature(K)
0.5
0.25
0
290
300
310
320
330
340
350
360
370
Temperature (K)
Fig. 4 Heat Capacity and transition profile of free Na+ form
of the G-quadruplex DNA structure.
Experimental Cp
Theoretical Cp
REFERENCES
1
Predicted Value
Specific Heat (Kcal/mol K)
Absorbance
1
0.5
0.75
0
290
0.5
320
350
Temperature (K)
0.25
0
290
300
310
320
330
340
350
360
370
Temperature (K)
Fig. 5 Heat Capacity and transition profile of Na+ form of
the G-quadruplex DNA structure bonded with Actinomycin
D
4. CONCLUSION
The present study concluded that the DNA
molecule is an extremely co-operative structure
and when Actinomycin D binds to it, the cooperativity is not so much disturbed. Therefore
the amended Zimm Bragg theory (phase
transitions theory) can be effectively applied to
it. It generates the experimental transition
profiles and 𝜆-point heat capacity anomaly
successfully. These results will allow us to
assess the thermodynamic profile of binding
process. Our theoretical data also demonstrates
that the binding of Actinomycin D to GQuadruplex DNA is an endothermic process and
that the binding increases the melting
temperature of the quadruplex. Therefore the
theoretical analysis presented in this study can
be implicated to understand bimolecular
interactions and may also be applied in
biomedical industries for drug design and
development.
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