RESEARCHARTICLE

Journal of Computational and Theoretical NANOSCIENCE Volume 4, Issue 3, Pages 521–528, 2007

DOI:10.1166/jctn.2007.014

Formation and Stability Conditions ofDNA-Dendrimer Nano-ClustersAli Nikakhtar1� † , Asadollah Nasehzadeh1, and G.Ali Mansoori2

1Shahid Bahonar University, Kerman 76175, Iran2University of Illinois at Chicago, (M/C 063) Chicago, IL 60607-7052, USA

In this report a mathematical model, which is based on the electrostatic potential and elastic/mechanical properties, is presented for formation and stability prediction of DNA-dendrimer nano-cluster. This model is built upon the assumption of the DNA inextensible semi-flexible polymer model (also known as worm-like chain model) and application of the non-linear Poisson-Boltzmann equation. The electrostatic free energies (free energies for assembling of fixed charges and mobile ions), and also the elastic free energies of DNA chain wrapped around the dendronized polymer are predicted. The effects of the mechanical properties of DNA such as bending, twisting, and bending-twisting interactions on the stability of nano-cluster with different conformations in terms of free energy are investigated. The effect of temperature on the free energy is also investigated and the enthalpy and entropy of the systems are calculated at different ionic strengths. The effects of ionic strength on the free energy of nano-cluster formation from DNA and dendronized polymer (free energy of nano-cluster formation) and thermodynamics stability of their conformations are studied. The reported properties will help us to understand the electrical and mechanical properties of DNA and the formation and stability conditions of DNA-Dendrimer nano-clusters.

Keywords: Dendrimer, Dendronized Polymer, DNA, DNA Delivery, DNA-DendrimerNano-Cluster, Drug Delivery, Gene Therapy, Nano-Cluster Stability, Nano-ClusterFormation, Poisson-Boltzmann Equation.

1. INTRODUCTION

Investigation into the nano-cluster formation between DNA molecules of flexible chains which possess sur-face negative charges and ionic-macromolecules possess-ing opposite (positive) charges is an important subject in many aspects of nanotechnology, especially in nano drug delivery and gene therapy.1–7 Dendrimers are one of the primary ionic-macromolecules which are proposed as the DNA carrier. 8� 9

Physical and chemical properties of dendrimers are an outgrowth of their shape as well as the presence of charged groups on their surface. Highly positive charge density with no i mmunogenetic or carcinogenetic10 effect, together with high solubility i n water has l ed to the use of dendrimers as an efficient carrier f or i n vivo DNA delivery.11 The dendronized polymers are new type of den-drimers which have a cylindrical shape and are capable to produce nano-clusters with DNA i n a l iquid media ( see Fig. 2 of Ref. [1]). The nano-cluster is formed by

†Present address: Chemistry Department, University ofBirjand, Birjand, Iran

wrapping the DNA around the dendronized polymer. Elec-tron microscopy photographs of dendronized polymer andnano-clusters have revealed a cylindrical shape for suchnano-clusters.12

Since shape, size, surface charge density, and chemi-cal structure of these nano-clusters play an important rolein their efficiency for DNA delivery and their activitiesfor transfiction, therefore electrostatic,1�13–15 elastic,16–23

and thermodynamic investigations of their behavior areinevitable.

We have already reported on the electrostatic propertyprediction and simulation of DNA-dendronized polymernano-clusters based on the nonlinear Poisson Boltzmann(PB) equation.1�24–31 In our previous work the densityof charges was applied and electrostatic potentials wereresulted by solving the nonlinear PB equation.

Various aspects of the elastic/mechanical properties ofDNA under different range of forces have been analyzedby various investigators.32–36 The variety of proposed mod-els which describe the elastic behavior of linear polymersusually start with the random walk theory where the poly-mer path is modeled by a sequence of uncorrelated stepsin space.37 For polymers of simple chemical structure or

521

Email addresses: AN1: alinikakhtar@yahoo.com; AN2: asadnasehzadeh@yahoo.com: GAM: mansoori@uic.edu

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for very long chain polymers, the random walk model isshown to be sufficient. However, the random walk modelis not found sufficient enough for polymers which exhibita certain bending stiffness in their structure that preservesthe direction of a monomer in the polymer with respectto the two adjoining monomers over a persistence length.These polymers are semi flexible because their monomersegments can not freely bend with respect to one anotherand the bending of chain requires energy. DNA is semiflexible and therefore the wrapping of DNA around thedendronized polymer requires energy. Thus in this case,the classical inextensible semi flexible polymer model,or worm-like chain model, is well adapted to DNA.37�38

A proper mathematical model,20�39–41 which is based onthe electrostatic potential and elastic/mechanical proper-ties, may be applied to predict the behavior of DNA-dendronized polymer nano-cluster formation.

In the present report we introduce a formation andstability prediction model for DNA-dendronized polymernano-cluster considering the semi-flexible nature of DNA.The stability model presented here will provide an insightinto the formation and conformations of DNA-dendronizedpolymer and its stability during DNA delivery.

The specific feature of the present work which is a com-plementary to our previous reports1�24–31 is to investigationthe thermodynamics of DNA and dendronized polymersystems in order to determine the formation and stabil-ity of nano-cluster in terms of free energy, enthalpy, andentropy. It should be mentioned that dendronized poly-mers are a new class of synthesized dendrimers and theirnano-cluster formation with DNA has not yet been experi-mentally investigated. To our knowledge there is no exper-imental published data available to be compared with ourcomputational results. However, the theoretical computa-tions reported here, which are based on sound physicalprinciples, will be quite useful for formation and stabilityanalysis of DNA-dendronized polymer nano-clusters.

2. THEORY AND METHOD

The free energy of a nano-cluster, which is formed bywrapping a DNA (as a semi-flexible rod) around a den-dronized polymer (rigid cylinder) due to surfaces charges(see Fig. 2 in Ref. [1]) in an ionic solution with definedionic strength, can be formulated by considering the com-bination of the following three contributions:(1) Electrostatic contribution:1 DNA and dendronizedpolymer in solution are charged macromolecules thus thereare electrostatic interactions between them and also thereare electrostatic interactions between these charged macro-molecules and small mobile ions in the ionic solution.(2) Elastic contribution:23 DNA is a semi-flexible macro-molecule and under the electrostatic attraction could bewrapped around the dendronized polymer. However, wrap-ping a DNA around a dendronized polymer requires con-sideration of bending and twisting energies.

(3) The “chemical” (i.e., non-electrical) contribution: Theorigin of chemical free energy is the chemical preferenceof the ions forming the surface charge for the surface overthe bulk or the tendency to the electrolytic dissociation ofgroups in the surface. This contribution may be omittedin dealing with interactions in which the surface chargeremains constant.

Here this third contribution is omitted because the func-tional groups on the surface of DNA and dendronizedpolymer in nano-cluster in a biological pH range of about7 (pH ∼ 7.36–7.42) are completely dissociated. As a resultthe surface charges remain constant. The first two con-tributions are described in more detail in the followingsection.

2.1. Electrostatic Contribution

The electrostatic free energy is a result of the electrostaticpotential of charging process. It is the energy which isneeded to assemble the charges on the surface of a nano-cluster. It is also the driving force to move the mobileions towards the nano-cluster and to change the distribu-tion and direction of mobile ions and solvent around thenano-cluster.

The electrostatic free energy for this process is definedby the following equation:42

�Gelec =∫V

{kT

∑i

cbi

[1− exp

(−zie

kT

)]

+�f −[�(�

)2

8�

]}dV (1)

where [volt] is the electrostatic potential in the vol-ume element dV [m3], � is the dielectric constant, and ziis the charge number of ith mobile ion, cbi [particle/cm3]is the concentration of the ith, ion in the bulk of elec-trolyte solution, k is the Boltzmann constant, and T isthe absolute temperature. In order to calculate the elec-trostatic free energy it is necessary to know electrostaticpotential and charge density at every point in the system.The local concentration of mobile ions is given by theBoltzmann’s electrostatic expression (cbi e

−ezi /kT ). Theelectrostatic potential can be calculated by the Poisson-Boltzmann (PB) equation. The nonlinear PB equationfor DNA-dendronized polymer nano-cluster system, whichincludes positive and negative fixed charges, in an ionicsolution has been derived as:1�13

� ·���r�����r� = −e

kT

{∑j

qf−j ���r−�rj�+

∑i

qf+i ���r−�ri�

}

+��2sinh���r� (2)

where �= ekT

is the dimensionless electrostatic potential.At any point of the system, q

f+i and q

f−j are positive

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A. Nikakhtar, A. Nasehzadeh, G.A. MansooriFormation and Stability Conditions of DNA-Dendrimer Nano-Clusters

J. Compu’l & Theor’l NANOSCIENCE 4(3): 521–528, 2007

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and negative fixed charges, respectively, � is the modifiedDebye-Hückel parameter, �2 = 1

�2 = 2e2I�kT

, � is the Debyelength, I is the ionic strength of the bulk solution, andr̄[m]

is the position vector of a charge. The analyticalsolution of Eq. [2] is only possible for charged sphericalmacromolecule systems which have simple and symmetriccharge density on their surface. Since the nano-cluster con-sidered here is a rather complicated system the analyticalsolution of the nonlinear PB equation is presently not pos-sible. Therefore the finite difference method has been usedto obtain electrostatic potential of the system. For a moredetailed explanation of this computational finite differencemethod please see Ref. [1].

2.2. The Elastic Contribution

The elastic free energy associated with the rigidity of aflexible rod does not favor the helical conformation. Inthis work DNA is considered to behave as an incompress-ible rod at constant temperature and pressure so that theapproximation dG = dF = dU is valid for it. Also DNAis assumed to behave as a semi flexible rod which resiststo bending and twisting. The linear conformation of DNAis the most stable conformation while the other conforma-tions (bending and twisting) require higher energies thanthe linear conformation. The shape and size of a bendedand twisted DNA can be parameterized by an arc lengthdenoted by s and defined as the length of the arc from areference point, or starting point, as shown in Figure 1.

We describe the rod by relating its local coordinateframe L to its frame L0 rigidly embedded in the curve(wrapped part of DNA) in its relaxed configuration at eachpoint of arc length. The helical space curve conformationis parameterized20 as:

�r�s�= R cos(

2�s%&

)�i+R sin

(2�s%&

)�j+ s

%�k (3)

k

Fig. 1. The length of space curve defined as the arc length and unitvectors which determine the coordinates of the nano-cluster and thelocal frame of each point of the DNA wrapped around the dendronizedpolymer.

by setting e1 = �u, e2 = �n, and e3 = �t, with +ei/+s = ,×ei;here , = (

,1�,2�,3

)is angular velocity of the frame

{ei}. These three unit vectors can be expressed by thefollowing equations:[, = (

,1�,2�,3

)]�t = d�r�s�

ds=−

(2�R%&

)sin

(2�s%&

)�i

+(

2�R%&

)cos

(2�s%&

)�j+ 1

%�k (4)

�u= d�t/ds� d�t/ds � = − cos

(2�s%&

)�i− sin

(2�s%&

)�j (5)

�n= �t× �u= 1%

sin(

2�s%&

)�i− 1

%cos

(2�s%&

)�j+

(2�s%&

)�k(6)

Then by using these three unit vectors it is possible todrive expression for the angular velocity as:

,1 = 0� ,2 =2�%&

(2�R%&

)and ,3 =

1%

(2�%&

)(7)

The general power series expansion of the elasticenergy with respect to deformations and truncated afterthe quadratic order in the deformations has the followingform32

�Gelas

kT=

∫ l

0ds

[A′

2,2

1 +A

2,2

2 +C

2,2

3 +B,1,3

](8)

The first two terms of the above equation are related tobending deformations. Parameters A and A′ are bendingpersistence lengths along the directions e1 and e2, respec-tively. The third term in the above equation represents thetwisting energy where C is the twisting-persistence length

In the above equation �i� �j� � are unit vectors in Cartesian coordinates that nano-cluster is defined in, R stands for radius of wrapping, % is the length of DNA per unit length of dendronized polymer, and & is the height per DNA turn (pitch) (see Fig. 1). The vectors �u�s�� �n�s�� �t�s) in Figure 1 are local unit vectors which are placed on the various points of DNA molecules and define the state of DNA rod, i.e., the direction of bending and twisting in the space.

As explained above t he bending and t wisting f ree ener-gies of a r od with t he l ength * are r elated t o i ts helical conformation. The configuration of an i nextensible poly-mer such as helical conformation are specified by t hree orthonormal unit vectors �u�s�� �n�s�� �t�s� along t he chain, where �t�s� i s t he axial direction vector of t he DNA dou-ble helix and u�s� i s a unit vector perpendicular t o �t�s� and pointing from one backbone chain to the other, �n�s� = �u�s� × �t�s�. It proves to be convenient to use Eüler angles

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A.. Nikakhtar, A. Nasehzadeh, G.A. MansooriFormation and Stability Conditions of DNA-Dendrimer Nano-Clusters

J.. Compu’l & Theor’l NANOSCIENCE 4(3): 521–528, 20

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and the last term is the contribution by bending-twistingcoupling where B is the coupling constant. When ,1 = 0indicates that the bending deformation is only along thedirection e1 and therefore, it is clear that bending andtwisting deformation is independent and bending-twistingcoupling must be eliminated to calculate elastic freeenergy. The expressions for bending and twisting free ener-gies resulting from ,2 and ,3 are

�Gbend = A

2*R2

(2�%&

)4

(9)

and

�Gtwis = C

2

(2�%2&

)2

* (10)

The total free energy of nano-cluster is then given by:

�Gtotal = �Gelec +�Gelas = �Gelec +�Gbend +�Gtwis

(11)The enthalpy and entropy of nano-cluster can be also

calculated by using the Gibbs-Helmholtz equation:(+��G/T �

+�1/T �

)p

= �H (12)

2.3. Nano-Cluster Formation

Since the free energy is a state function then the ther-modynamic path depicted in Figure 2 can be chosen tocalculate the free energy of nano-cluster formation know-ing the other free energies described above for which weknow their equations and which can be directly estimated.

In order to calculate the total free energy of nano-cluster formation, the reagents, i.e., DNA and dendronizedpolymer are assumed to be initially in separate solu-tions of 1:1 salt (such as NaCl) with the same ionicstrengths. The surface charge densities of DNA and den-dronized polymer, each, produce an electrostatic filed inthe solution which causes the mobile ions and solvent(water) molecules to rearrange around the polyelectrolyte.Consequently the system will have a new free energy.The DNA in its initial separate solution is assumed tobe neither bended nor twisted. Since DNA has surfacenegative charges and is also flexible, then on mixing theDNA solution with dendronized polymer solution, DNA

Surface chargedDNA (aq)

Surface chargeddendronized polymer

(aq)

Unchargeddendronizedpolymer (aq)

UnchargedDNA (aq) +

Surface chargednano-cluster (aq)+

∆Gelec3

∆Gelas

–∆Gelec1 –∆Gelec2

Unchargednano-cluster (aq)

∆GNC

will be adsorbed by dendronized polymer and will wraparound it. Consequently a nano-cluster with the lowest freeenergy is formed. The free energy of nano-cluster forma-tion can be calculated by using the thermodynamic pathshown in Figure 2 which is represented by the followingequation.

�GNC =−�Gelec1 −�Gelec2 +�Gelas +�Gelec3 (13)

The new arrangements of ions and water moleculesaround nano-cluster, the initial conformation of DNAand dendronized polymer and also the resistance elas-ticity of DNA are included in this equation. The resis-tance elasticity is responsible for bending and twistingfree energies and is a characteristic property of the DNAstructure.

As an example we use the same system which wasstudied in Ref. [1]. It consists of a dendronized polymerwith the length of 50 Å and 80 positive charges and aDNA with the length of 435 Å and 255 negative charges.The electrostatic potential of nano-cluster which is formedby these two polyelectrolytes is calculated by using thenonlinear Poisson-Boltzmann equation as discussed inour previous work.1 The calculations are repeated at dif-ferent ionic strengths and at different temperatures forseveral nano-clusters of different conformations in orderto predict the formation and stability conditions of thenano-cluster.

The dielectric constant of the aqueous solution (themedia for the nano-cluster) is dependent on temperature.Therefore the dielectric constant for every temperature42 isused to calculate the electrostatic potential of nano-clustersystem. The electrostatic free energy in terms of differentcontributions of fixed and mobile charges and the entropicfree energy of mixing of mobile species and solvent aredetermined by using the electrostatic potential of systemand finite difference method using the same procedure asbefore.1

3. RESULTS AND DISCUSSION

3.1. Nano-Cluster Stability Conditions

The bending and twisting free energies of nano-cluster ofdifferent conformations calculated using Eqs. (9) and (10)are presented in Figures 3 and 4, respectively. Accordingto these two figures the bending free energy is about oneorder of magnitude larger than the twisting free energyfor a given pitch. Also the bending free energy decreases,while the twisting free energy increases as the size of thepitch increases.

According to Eq. (9) the free energy of bending isdirectly proportional to, both, the length of the wrappedpart of DNA (*�, and the fourth power of the inverselength of DNA per unit turn (%&). These two terms willdecrease by increasing the size of the DNA pitch (&).The size of the pitch, which is the same as the length

Fig. 2. Thermodynamic path for the energy estimation of the nano-cluster.

A.. Nikakhtar, A. Nasehzadeh, G.A. MansooriFormation and Stability Conditions of DNA-Dendrimer Nano-Clusters

J.. Compu’l & Theor’l NANOSCIENCE 4(3): 521–528, 20

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0

20

40

60

80

100

120

140

160

10 20 30 40 50 60 70

Pitch (Å)

∆Gbe

nd/k

T

Fig. 3. Bending free energy of the wrapped DNA versus the DNA pitchfor different conformation of nano-clusters at the biological solution ionicstrength of 0.1 M. The size of the DNA pitch is equivalent to the lengthof dendronized polymer per one turn of wrapped DNA.

of dendronized polymer per one turn of wrapped DNA isdenoted by & . The length of wrapped DNA per unit lengthof dendronized polymer is denoted by %. Therefore thelength of DNA per one turn is %& .

According to Eq. (10) the free energy of twisting isa function of three variables *, %& , and %. It increaseswith increasing * (The length of wrapped part of DNA). Itdecreases with increasing % (length of wrapped DNA perunit length of the cylindrical dendronized polymer). It alsodecreases with increasing %& (DNA length per turn).

The effect of temperature on the electrostatic freeenergy, Eq. (1), and total free energy, Eq. (11), of nano-cluster with different conformations is shown in Figure 5.According to Figure 5 variation of temperature has sim-ilar effects on the electrostatic free energy and the totalfree energy. In this figure the minimum free energy foreach temperature corresponds to the most stable conforma-tion. For example, according to Figure 5 at T = 283315 Kthe most stable conformation is at a DNA pitch of & =30 Å length when only electrostatic free energy is consid-ered. By considering the total free energy (which includes

0

1

2

3

4

5

6

7

8

9

10 20 30 40 50 60 70

∆Gtw

is/k

T

400

450

500

550

600

650

700

750

800

850

900

0 10 20 30 40 50 60 70

Pitch (Å)

∆G/k

T

283.15 elec283.15 total303.15 elec303.15 total323.15 elec323.15 total

Fig. 5. The effect of temperature on the total free energy and total freeenergy of nano-cluster of different conformation at the biological solutionionic strength of 0.1 M.

electrostatic and elastic free energies) the most stableconformation shifts to a DNA pitch of & = 35 Å length.Another interpretation of the data reported on Figure 5 isthat with consideration of the elasticity of DNA the DNApitch size increases and its elasticity actually destabilizesits electrostatic conformation.

To calculate the effect of temperature on the totalfree energy we have used the Gibbs-Helmholtz equation,Eq. [12]. The result of this calculation for & = 35 Å isshown in Figure 6. According to Figure 6 the total freeenergy at & = 35 Å changes linearly with inverse temper-ature for the temperature range which is examined. Theslope of this plot gives us the total enthalpy of nano-cluster,�Htotal =−568 kJ ·mol−1 which indicates that the chargingof nano-cluster is an exothermic process.

By using the Gibbs-Helmholtz equation the totalenthalpy of various DNA pitch sizes is calculated and theresults of calculations are reported in Figure 7 at threedifferent solution ionic strengths. According to Figure 7the total enthalpy decreases with increasing the size of theDNA pitch, i.e., as the size of the pitch increases moreheat is released. This reveals that by increasing the size

y = –567.95x + 7.3822

R2 = 0.9924

5.3

5.35

5.4

5.45

5.5

5.55

5.6

5.65

5.7

0.003 0.0031 0.0032 0.0033 0.0034 0.0035 0.0036 0.0037

T–1 (K–1)

∆Gto

tal /T

(kJ

/K m

ol)

Fig. 6. The effect of temperature on the total free energy of nano-clusterwith conformation of contest size of the pitch (35 Å) at the biologicalsolution ionic strength of 0.1 M.

Pitch (Å)

Fig. 4. Twisting free energy of wrapped DNA versus DNA pitch for different conformation of nano-clusters at the biological solution ionic strength of 0.1 M.

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–2500

–2000

–1500

–1000

–500

0

10 20 30 40 50 60 70

Pitch (Å)

∆H (

kJ/m

ol)

0.1 M

0.05 M

0.005 M

Fig. 7. The effect of ionic strength on the enthalpies of different con-formations of nano-clusters.

of the DNA pitch, the adsorption of mobile ions on thenano-cluster is increased.

Moreover Figure 7 indicates that when the ionic strengthdecreases the enthalpy of the system also decreases (moreheat is released). This can be interpreted as the follow-ing: At low concentrations of mobile ions, the repulsionbetween the mobile ions of the same charges is less thanat high concentrations. Also, the permittivity of electro-static field into the solution is increased by decreasing theconcentration of the mobile ions. Therefore the releasedheat from adsorption of mobile ions on the nano-cluster isincreased.

From the data for total Gibbs free energy and totalenthalpy and using equation � = ��H −�G�/T we havecalculated the entropy of nano-clusters of different con-formations. The results of this calculation are reported inFigure 8. According to this figure the calculated entropyof nano-cluster at various conformations is negative anddecrease by increasing the size of the DNA pitch. It shouldbe pointed out that the entropy data of the nano-clusterreported in Figure 8 represents the degree of the orderin the system. Figure 8 indicates that the nano-cluster

–18

–16

–14

–12

–10

–8

–6

10 20 30 40 50 60 70

∆S (

kJ/K

mol

)

0.1 M

0.05 M

0.005 M

system becomes more ordered as the size of the DNA pitchincreases. As the size of the DNA pitch increases morespace on the surface of the nano-cluster will be availableto opposite sign mobile ions to be attracted by the nano-cluster. At lower ionic strengths than the biological ionicstrength of 0.1 the entropy of the system becomes morenegative. Therefore the permittivity of electrostatic fieldwill be larger and the order in the solution will be morethan the solution with higher ionic strengths.

3.2. Formation of DNA and Dendronized PolymerNano-Cluster

The electrostatic free energies of a 435 Å DNA with 255negative charges and 50 Å dendronized polymer with 80positive charges, which have been calculated at differentsolution ionic strengths, are shown in Figure 9. Accordingto Figure 9 the electrostatic free energy of dendronizedpolymer is smaller than the electrostatic free energy ofDNA at the same solution ionic concentration. This ismainly due to the difference in their number of surfacecharges. Also according to Figure 9, when the ionicstrength of the solution increases, the electrostatic freeenergies of, both, DNA and dendronized polymer decrease.This is because when the ionic strength is increased theconcentration of opposite charge ions absorbed around thepolyelectrolyte is increased. Thus the electrostatic field ofpolyelectrolyte is decreased in the system.

By using Eq. [13] the formation free energy of thenano-cluster is calculated. The results of the calcula-tions for three different solution ionic strengths versusthe DNA pitch size are reported in Figure 10. Accord-ing to Figure 10, the formation free energies of mostof the conformations at various ionic strengths are nega-tive. This indicates that for most of the conformations thenano-cluster formation is spontaneous. The formation freeenergy of nano-cluster at various solution ionic strengths

0

100

200

300

400

500

600

700

800

900

1000

0 0.05 0.1 0.15 0.2 0.25

Ionic strength (M)

∆Gel

ec/K

T

DNADendrimer

Fig. 9. Electrostatic free energy of reactants DNA ��� and dendronizedpolymer (�) at various ionic strengths and at constant temperature of298.15 K.

Pitch (Å)

Fig. 8. The effect of ionic strength on the entropy of different

confor-mations of nano-clusters.

A.. Nikakhtar, A. Nasehzadeh, G.A. MansooriFormation and Stability Conditions of DNA-Dendrimer Nano-Clusters

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–350

–300

–250

–200

–150

–100

–50

0

50

100

150

200

0 10 20 30 40 50 60 70

∆GN

C/k

T

C = 0.005 M

C = 0.05 M

C = 0.1 M

computations performed using the mathematical model theconditions under which DNA and dendronized polymerspontaneously form nano-clusters are well described anddefined.

Symbols and Abbreviation

A�A′ Elastic constants of bendingB Coupling constant of bending and

twistingci Local Salt concentration ith ioncbi Salt concentration in bulk solutionC Elastic constants of twistinge Charge of electronF Helmholtz free energyG Gibbs free energyH Enthalpy�i� �j� �k Three orthonormal unit vectors

in the Cartesian coordinateI Ionic strength of the bulk solutionk Boltzmann constant* Length of DNAL Local coordinate frameL0 Local coordinate frame of referencep Pressureqf+n Positive fixed chargeqf−n Negative fixed charge�r Position vectorR Radius of wrappings Arc lengthT Absolute temperatureU Internal energyV Volumezi Number of charge{�u�s�� �n�s�� �t�s�} Three orthonormal unit vectors on

the local frame

Greek Letter

� Dielectric constant Electrostatic potential [volt]�f Charge density of fixed charges�= e

kTDimensionless electrostatic potential

� Modified Debye-Hückel parameter���r− �ri� Kronecker Delta� Debye length�Gelec Electrostatic free energy�Gelas Elastic free energy�Gtwis Twisting free energy�Gbend Bending free energy�Gtotal Total free energy, sum of electrostatic

and elastic free energies% Length of DNA per unit length of

dendronized polymer& Height per turn (size of pitch),= (

,1�,2�,3

)Angular velocity of the frame

Pitch (Å)

Fig. 10. The effect of ionic strength on the free energy of nano-cluster formation for different conformation of nano-clusters.

passes thought a minimum at the DNA pitch size of 32 Å and increases when the ionic strength increases.

A comparison between different free energies contribut-ing to the formation free energy shows that the electrostatic free energy of DNA is the major one followed by the den-dronized polymer and nano-cluster free energies.The free energies of bending and twisting have minor contributors to the formation free energy.

4. CONCLUSION

We previously reported1 the electrostatic potential which is generated in an electrolyte solution surrounding the DNA-dendronized polymer nano-clusters. Such nano-clusters have found applications in nanotechnology, in general, and in nano drug delivery and gene therapy, in particular. I n this work, we reported a new method for electrical and mechanical stability prediction as well as formation of DNA-dendronized polymer nano-clusters.

The mathematical model presented here will pro-vide us an insight into the formation conditions of DNA-dendronized polymer and its stability during DNA delivery.

In this report the DNA worm-like chain model (which is a realistic model for DNA) is applied to calculate the elastic free energies. The plot of free energy per temper-ature versus inverse temperature (in the range of 278.15–323.15 K) was found to be linear and was used to determine the enthalpy and entropy of nano-cluster for various confor-mations in solution medium with different ionic strengths. It is concluded that the entropy and enthalpy of system are decreased by decreasing the ionic strength and increasing the size of the DNA pitch.

The stability of nano-cluster can be represented by free energy of nano-cluster formation since the most preferable reaction is accompanied with the most nega-tive free energy. A thermodynamic path is proposed to calculate the total free energy of nano-cluster formation between DNA and dendronized polymer. Based on the

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A.. Nikakhtar, A. Nasehzadeh, G.A. MansooriFormation and Stability Conditions of DNA-Dendrimer Nano-Clusters

J.. Compu’l & Theor’l NANOSCIENCE 4(3): 521–528, 20

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