DNA Complexes

May 13, 2009 - For the shorter and longer DNA, the estimated optimal wrapping length, lopt is ≈15.69 and ≈12.25 nm, respectively, for dendrimers t...
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Biomacromolecules 2009, 10, 1720–1726

Analytical Model Study of Dendrimer/DNA Complexes Khawla Qamhieh,*,† Tommy Nylander, and Marie-Louise Ainalem‡ Physical Chemistry, Center for Chemistry and Chemical Engineering, Lund University, P.O. Box 124, SE-221 00 Lund, Sweden Received January 16, 2009; Revised Manuscript Received April 14, 2009

The interaction between positively charged poly(amido amine) (PAMAM) dendrimers of generation 4 and DNA has been investigated for two DNA lengths; 2000 basepairs (bp; L ) 680 nm) and 4331 bp (L ) 1472.5 nm) using a theoretical model by Schiessel for a semiflexible polyelectrolyte and hard spheres. The model was modified to take into account that the dendrimers are to be regarded as soft spheres, that is, the radius is not constant when the DNA interact with the dendrimer. For the shorter and longer DNA, the estimated optimal wrapping length, lopt is ≈15.69 and ≈12.25 nm, respectively, for dendrimers that retain their original size (Ro ) 2.25 nm) upon DNA interaction. However, the values of lopt for the dendrimers that were considered to have a radius of (R ) 0.4Ro) 0.9 nm were 9.3 and 9.4 nm for the short and long DNA, respectively, and the effect due to the DNA length is no longer observed. For lopt ) 10.88 nm, which is the length needed to neutralize the 64 positive charges of the G4 dendrimer, the maximum number of dendrimers per DNA (Nmax) was ≈76 for the shorter DNA, which is larger than the corresponding experimental value of 35 for 2000 bp DNA. For the longer DNA, Nmax ≈ 160, which is close to the experimental value of 140 for the 4331 bp DNA. Charge inversion of the dendrimer is only observed when they retain their size or only slightly contract upon DNA interaction.

Introduction The general trend in the recent development in biology, pharmaceutics, and medicine is closely related to the utilization of nanotechnology. One of the examples is in gene therapy where DNA has to be delivered into a cell to, for example, correct the genetic defects of damaged sites. Here the challenge is to be able to transport DNA, which generally is a very large, stiff and therefore bulky polyelectrolyte (PE), through the cell walls. This can be achieved by condensing DNA with an oppositely charged specimen, such as positively charged surfactants or PEs. In recent years, DNA condensation has been studied extensively by using different surfactants like cetyl trimethylammonium bromide (CTAB) or dioctadecyl dimethylammonium bromide (DODAB), histones, as well as poly(amido amine) (PAMAM) dendrimers, polylysine, or poly(ethylene imine) (PEI) as a way of replacing viral vectors as gene carriers for in vivo transfer.1-9 Dendrimers are built up through chemical synthesis producing a uniform globular nanostructure with a low degree of polydispersity. PAMAM dendrimers contain amido amine groups emanating from a central ethylenediamine core, where the charge, size and internal conformation depend on the number of generations. They are highly charged, which makes them very efficient DNA condensing agents, with electrostatic forces stabilizing the formed complexes. The PAMAM dendrimer structure is also very sensitive to the ionic strength where a more extended conformation is formed at low ionic strength due to the intramolecular electrostatic repulsion. Dendrimers of fourth generation (G4) have been shown to be soft and flexible with a dense core.10-12 Significant advances in theoretical modeling13-23 and computer simulations24-32 have been made on the complexes of * To whom correspondence should be addressed. E-mail: khawlaq@ gmail.com. † Permanent address: Al-Quds University, College of Science and Technology, Physics Department, Jerusalem, Palestine. ‡ ¨ rberg. Ainalem nee´ O

hard spheres (macroions) and PEs. The complexation in systems containing more than two macroions has been addressed by Nguyen and Shklovskii,21-23 Kunze et al.16 and Schiessel et al.19 Wallin and Linse performed a series of Monte Carlo simulations including determination of the free energy of the formation of complexes between macroions and PEs.24-28 Computer simulations have recently been performed on dendrimers with charged terminal groups and oppositely charged linear chains by Welch and Muthukumar,33 Lyulin et al.,34,35 and Maiti et al.36 Analytical consideration of such systems is restricted due to their complexity. The objective of this work is to provide further understanding of the formation and the structure of dendrimer/DNA complexes by using a theoretical model of the complex formation. We will base our modeling on experimental data for the formation of aggregates between DNA of different lengths and PAMAM dendrimers of generation 4. Throughout this paper, the term complex will be used for the interaction between one dendrimer and the part of the DNA chain wrapping around it and the structure formed between the entire DNA chain and a multiple of dendrimers will be termed an aggregate. The model will allow us to determine the charge of the complex as well as how much of the DNA chain is wrapped around the dendrimers.

Experimental Section G4 PAMAM dendrimers containing 64 surface primary amine groups and with a diameter of 4.5 nm have been used as the vehicle for condensation of linear salmon sperm DNA (2000 base pairs (bp)) and linearized DNA plasmids (4331 bp).37-39 The formation of these complexes has been extensively studied using several experimental techniques like dynamic light scattering, steady-state fluorescence spectroscopy, cryogenic transmission electron microscopy, and gel ¨ rberg et electrophoresis. Results based on these experiments enabled O al. to propose a binding model in which the complex formation process was shown to be cooperative, with the coexistence of fully condensed DNA and free DNA in its native form found already at low dendrimer

10.1021/bm9000662 CCC: $40.75  2009 American Chemical Society Published on Web 05/13/2009

Analytical Model Study of Dendrimer/DNA Complexes

Biomacromolecules, Vol. 10, No. 7, 2009 Fcompl(l) =

{

e2Z2(l) for |Z(l)| < Zmax 2εR |Z(l)|kBTω(Z(l)) for |Z(l)| . Zmax

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(2)

kB is the Boltzmann constant, T is the absolute temperature and ω(Z(l)) ) 2 ln(|Z(l)|lBκ-1/R2). Zmax is the effective charge of the sphere and of the order of ωR/lB,40 and Z(l) is expressed as

Z(l) ) Z - l/b

(3)

The second term, Fchain (L - l), is the total entropic electrostatic free energy of the remaining chain (L - l), given by Figure 1. Proposed binding model between DNA of contour length L, radius r, and distance between negative charges b and G4 PAMAM dendrimers modeled as hard spheres of radius R. The DNA is shown to wrap around the dendrimer with the length of the wrapping part equal to l, and a distance between the centers of two neighboring dendrimers, D(N,l). The model is in accordance with the cooperative ¨ rberg et al.37 binding model proposed by O

concentrations.37 Also, it was concluded that it is possible to form discrete aggregates composed of one DNA molecule and several dendrimers in dilute solutions. Figure 1 shows the binding of dendrimers ¨ rberg et al.37 For the shorter to a DNA chain in accordance with O DNA molecule roughly 35 dendrimers, compared to 140 for the longer DNA, is binding to each DNA molecule. Longer DNA molecules therefore appear to bind a higher fraction of G4 dendrimers, which means that the correlation between the number of bound G4 dendrimers and the DNA length is not linear. Theoretical Basis. To analyze the experimental results of the dendrimer/DNA complex formation, we adopted a theoretical model developed by Schiessel, who considered complexes formed between positively charged hard sphere macroions and a persistent linear PE.19 Analytical Model of the System. The first approximation is to regard the dendrimer as a hard sphere of radius R ) 2.25 nm and charge Ze ) 64 e, and the DNA as a semiflexible rod of radius r ) 1 nm and length L . R. We will consider two lengths of DNA, for which experimental data previously has been obtained. The contour length L for the shorter DNA (2000 bp) is 680 nm and for the longer DNA (4331 bp), 1472.5 nm. The charge of DNA per unit length is -e/b ) -e/0.17 nm and the persistence length is lp ) 50 nm, which is large compared to R. The experimental data for DNA condensation by G4 dendrimers was recorded in monovalent salt solutions, which is taken into account through the Bjerrum length, lB ≡ e2/εkBT, and the Debye screening length, κ-1 ) (8cπlB)-1/2, where ε is the dielectric constant of the solvent and cs is the salt concentration. Experimental studies were performed in 10 mM NaBr resulting in κ-1 ) 3 nm, which is comparable to the size of the dendrimers but smaller than L. However, the derivation according to Schiessel assumes that κ-1 is large compared to the sphere radius (low ionic strength and small particles), that is, κR , 1, while in our case it is 0.75. This will be discussed further below where we will argue that this assumption does not affect the results to a large extent as the apparent radius of the dendrimer in the complex is smaller than that of the free dendrimer and the electrostatic term for the interaction within the aggregate is comparably small. As DNA is a highly charged PE, the distance b between charges is smaller than lB ) 0.7 nm using the dielectric constant of water at room temperature.19,37 Calculation of the Free Energy for the Dendrimer/DNA Complex. According to the model by Schiessel for electrostatic complexation,19 the total free energy of the system consisting of one dendrimer and one DNA chain can be expressed as

F(l) ) Fcompl(l) + Fchain(L - l) + Fcompl-chain(l) + Felastic(l) (1) where l is the length of the part of the chain wrapped around the sphere (dendrimer). The first term, Fcompl(l), is the electrostatic charging free energy of a spherical complex of charge eZ(l)

Fchain(L - l) =

kBT Ω(r)(L - l) b

(4)

where Ω(r) ) 2 ln(4ξκ-1/r), and ξ ≡ lB/b is called the Manning parameter.41 The third term, Fcompl-chain(l), is the electrostatic free energy of the interaction between the complex and the remainder of the chain, given by

Fcompl-chain(l) = Z*(l)KBT ln(κR)

(5)

where Z*(l) is the effective charge of the complex. For complexes of low charge, Z(l) < Zmax, the effective charge obeys Z*(l) ) Z(l). For complexes of higher charge compared to the macroion, Z(l) > Zmax and Z*(l) ) Zmax. The final term in eq 1, Felastic(l), is the elastic (bending) free energy

Felastic(l) =

kBTlp R2

l

(6)

When analyzing eq 1, Schiessel considered the two cases, |Z(l)| < Zmax and |Z(l)| > Zmax, separately. For |Z(l)| > Zmax, eq 1 gives

F(l) = B-l/b + const kBT

(7)

with

B- )

lpb R2

-Ω-ω

(8)

For |Z(l)| < Zmax and by using eq 1, he obtained

lB F(l) l = ZkBT 2R b

(

)

2

+ Al/b + const

(9)

where all contributions, linearly dependent on the wrapping length (bending energy, electrostatic interaction between the complex, and the free chain and release of the chain counterions) are combined in the quantity A

A)

lpb R2

- ln(κR) - Ω

(10)

According to eq 9 and as described in further detail by Schiessel,19 the optimal wrapping chain length, lopt, obtained at the free energy minimum will be equal to

lopt ) liso -

AR ξ

(11)

where liso ) Zb is the isoelectric chain length needed to compensate the charge of the sphere. Calculation of the Free Energy for the Dendrimer/DNA Aggregate. For a system consisting of one DNA molecule and N number of dendrimers, the total free energy can be expressed as

F(N, l) ) NF(l) + Fint(N, l)

(12)

where F(l) is the total free energy of the dendrimer/DNA complex as expressed in eq 1. Fint is the interaction between the hard (nonpenetrable) spheres, which are decorating the polymer chain and is obtained from summing over the electrostatic repulsion between all complexes within

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one chain. As defined in Figure 1, D(N,l) is the center-to-center distance between two dendrimers next to each other and equal to (L - Nl + 2NR)/N for hard spheres. The repulsion is electrostatic and if ND(N,l) . κ-1, and in the limit that D(N,l) is small compared to κ-1 (very low ionic strength) but larger than 2R (no excluded volume effects), then we can write the interaction force with the following approximation of an (repulsive) electrostatic interaction

Fint(N, l) = ΛKBT

NlBZ2(l) D(N, l)

(13)

where the quantity Λ is a logarithmic factor of the order ln(κ-1/D). We here note that this expression for the interaction is approximate, and a more rigorous treatment of the electrostatic interaction force is not possible within our analytical framework. It is worth noting that this term will be small if the complex charge is close to neutral, when the neighboring spheres are sufficiently far apart or at high ionic strength. If the macroions (dendrimers) are equally spaced and well-separated along the PE chain and if L/N greatly exceeds liso and R, eq 12 can be used to give an approximate expression for the optimal wrapping length

lopt = liso -

AR 2ΛRN 1ξ L

(

)

(14)

For Z(l) < Zmax, that is, for large wrapping lengths, the total free energy of the dendrimer/DNA complex is quadratic in l and eq 12 can be expressed as

lBN 2 ΛN2lBZ2(l) F(N, l) l = Z (l) + A N - NωZ + + ln(κR)NZ kBT 2R b D(N, l) (15) We will analyze the experimental results for the complex formed between dendrimers and DNA by using the model by Schiessel. According to Manning, (1 - ξ-1)L/b, counterions condense on such a chain reducing the effective charge density to 1/lB.41 Schiessel ignored ξ-1 in eq 4 because of the assumption that ξ , 1. For our system, ξ ) 4.117, and as a result, ξ-1 is taken explicitly into account in the second term in eq 1, which states the total entropic electrostatic free energy of the remaining chain (L - l). Equation 4 therefore has to be expressed as

Fchain(L - l) =

kBT Ω(L - l)(1 - ξ-1) b

The constant A in eqs 14 and 15 then becomes

(16)

A)

lpb R2

- ln(κR) - Ω(1 - ξ-1)

(17)

Results and Discussion By applying eq 14, which is an approximation of lopt and derived from eq 12, the optimal wrapping length around one G4 dendrimer for 2000 and 4331 bp DNA, respectively, can be estimated. For the shorter DNA lopt ≈ 13.32 nm if N ) 35, as found in the experimental study using salmon sperm DNA. If the estimated linker length, D′, is 5.43 nm, which is the value found for the wrapping of DNA around histone octamers inside cell nuclei,42 the total resulting length of the chain, Lres ) N(lopt + D′), is 665.4 nm and less than the original contour length for salmon sperm DNA (680 nm). Lres ) 680 nm for an optimal length of ≈13.38 nm and a D′ of 6.05 nm. For the longer DNA where the experimental value of N is 140 for the linearized plasmid DNA, the optimal wrapping chain length is ≈13.57 nm and Lres ) 2660.8 nm for D′ ) 5.43 nm. This value of Lres is much larger than the original contour length (1472.5 nm). We therefore conclude that eq 14 does not give a physically realistic estimation of the optimal wrapping length in the dendrimer/DNA complex for both sizes of DNA. As an alternative for estimating the optimal wrapping length of DNA around a dendrimer, the original free energy equation, eq 15, can be used by taking the first derivative of the free energy with respect to the wrapping length l and equal it to zero. As was described in the section for the free energy calculation of dendrimer/DNA aggregates, the dendrimerdendrimer spacing D(N,l) is set to (L - Nl + 2NR)/N for hard (nonpenetrable) spheres and Figure 2 and Table 1 show the results. The optimal wrapping length of the shorter DNA is ≈15.69 nm, which gives a ratio of ≈1.11 relative to the circumference of the dendrimer. This optimal length value is ≈4.24 nm larger than liso, which is equal to Zb ) 10.88 nm. The effective charge of the dendrimer/DNA complex is -28.29e, which means that the charge of the dendrimer is reversed. Charge inversion can be observed if the total linear charge is larger than the dendrimer charge. The linker length between the complexes, D′ ) D - 2R, is ≈3.73 nm, and Lres ) N(lopt + D′) is 680 nm. The optimal wrapping length of the longer DNA is ≈12.25 nm, which gives a ratio of ≈ 0.86 relative to the circumference of the dendrimer and is 1.37 nm longer than liso. If we add the estimated wrapping length of all

Figure 2. Total free energy as a function of the wrapping length for a system of (a) 35 positively charged macroions modeling G4 dendrimers and an oppositely charged PE representing DNA of 2000 bp (L ) 680 nm) and (b) 140 dendrimers and DNA of 4331 bp (L ) 1472.5 nm). The macroions are considered to be nonpenetrable hard spheres (R ) 2.25 nm) with the dendrimer-dendrimer spacing equal to D(N,l) ) (L - Nl + 2NR)/N.

Analytical Model Study of Dendrimer/DNA Complexes

Biomacromolecules, Vol. 10, No. 7, 2009

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Table 1. Analytical Model Results for the Interaction between G4 Dendrimers and DNA for Two Different Contour Lengths of DNAa L (nm)

N

680 35 1472.5 140

lopt (nm) lopt - liso (nm) Z*compl (e) D(N,l) (nm) D′ ) D - 2R (nm) Zlinker (e) Lres ) N(lopt + D′) (nm) ratio lopt/2Rπ Fmin (kBT) 15.69 12.25

4.81 1.37

-28.29 -8.05

8.24 2.76

-21.9

3.73 -1.73

680 1472.5

1.11 0.86

-26111 -99507

a The dendrimer is considered to be a nonpenetrable sphere with the dendrimer-dendrimer spacing equal to D(N,l) )(L - Nl + 2NR)/N. L is the contour length of the DNA, N is the number of dendrimers obtained experimentally, and lopt is the wrapping chain length per one dendrimer. The difference between the optimal wrapped chain length and the length needed to neutralize the dendrimer charges is lopt - liso, Z* is the charge of the complex and D is the center-to-center dendrimer spacing. D′ is the length of the DNA linking two neighboring dendrimers, Zlinker is the charge of the linker, Lres is the contour length obtained using the calculations, and the ratio expresses the relation between the wrapped part of the chain and the dendrimer circumference. Fmin is the minimal free energy of the system.

Table 2. Analytical Model Results for the Interaction between G4 Dendrimers and the Shorter DNA (2000 bp)a x (R ) xRo)

lopt (nm)

lopt - liso (nm)

Z*compl (e)

D(N,l) (nm)

D′ ) D - 2R (nm)

ratio lopt/2πR

Fint (kBT)

1.00 0.90 0.80 0.70 0.60 0.55 0.50 0.40

15.69 14.28 13.04 12.00 11.09 10.60 10.21 9.30

4.81 3.40 2.16 1.12 0.21 -0.28 -0.67 -1.58

-28.29 -20.2 -12.71 -6.588 -1.235 1.647 3.941 9.294

8.23 9.19 9.98 10.56 11.03 11.29 11.46 11.92

3.73 5.14 6.38 7.41 8.33 8.82 9.21 10.1

1.10 1.12 1.15 1.20 1.30 1.37 1.44 1.64

-2402 -1192 -475 -126 -1.43 -7.79 -44 -244

Fmin (kBT)

Fint/Fmin

-26111 -25409 -24465 -23659 -22064 -20862 -19215 -13583

0.09 0.04 0.01 0.005 0.00006 0.0003 0.002 0.018

a The dendrimer is considered to be a hard sphere with the dendrimer-dendrimer spacing equal to D(N,l) ) (L - Nl + 2NR)/N, where N ) 35 and L ) 680 nm. R is the radius of the sphere where x defines the fraction of the noncompacted sphere of radius R ) Ro. Fint is the electrostatic repulsion between complexes and Fmin is the minimal total free energy of the system. The remaining parameters included in the model are defined in the footnote of Table 1.

Table 3. Analytical Model Results for the Interaction between G4 Dendrimers and the Longer DNA (4331 bp)a x (R ) xRo)

lopt (nm)

lopt - liso (nm)

Z*compl (e)

D(N,l) (nm)

D′(N,l) (nm)

ratio lopt/2πR

Fint (kBT)

Fmin(kBT)

Fint/Fmin

1.00 0.90 0.80 0.70 0.60 0.5 0.45 0.40

12.25 11.95 11.65 11.34 10.98 10.45 10.00 9.40

1.37 1.07 0.77 0.46 0.10 -0.43 -0.88 -1.48

-8.05 -6.29 -4.53 -2.71 -0.588 2.529 5.176 8.705

2.77 2.62 2.47 2.31 2.23 2.32 2.53 2.91

-1.73 -1.43 -1.13 -0.82 -0.47 0.067 0.51 1.11

0.86 0.93 1.03 1.14 1.29 1.47 1.57 1.66

187 203 19 78 4.4 70 171 73

-99507 -98780 -97238 -94178 -88246 -76659 -67082 -53381

0.009 0.002 0.0002 0.0008 0.00005 0.00091 0.0025 0.0013

a The dendrimer is considered to be a hard sphere with the dendrimer-dendrimer spacing equal to D(N,l) ) (L - Nl + 2NR)/N, where N ) 140 and L ) 1472.5 nm. The parameters included in the model are defined in the footnote of Tables 1 and 2.

dendrimers on one DNA chain (140 × 12.25 nm), we end with a value that is larger than the DNA contour length and the calculated linker length between the complexes is therefore negative (-1.73 nm), which is not a reasonable value. The effective charge of the dendrimer/DNA complex is -8.05 e, which means that the DNA interaction leads to charge reversal of the dendrimer. The resulting chain length is 1472.5 nm. The aggregate formed by one DNA chain and N numbers of dendrimers have a net negative charge of -1760 e and +298 e for the short and long DNA, respectively. Dendrimers as Soft Spheres. Dendrimers are known to be soft structures of high flexibility,10-12 and because their internal structure and size is controlled by the intramolecular electrostatic repulsion, they are likely to contract into smaller globules when they interact with the oppositely charged DNA. To express this effect, we use eq 15 for a hard sphere (dendrimer), but we take into account a reduction of its size, Ro. This will affect the distance between two dendrimers, the wrapping length as well as the charge of the complex. The results from the calculation of DNA wrapping around spheres of varying size are shown in Tables 2 and 3 for the shorter DNA of 2000 bp (L ) 680 nm) and for the longer DNA of 4331 bp (L ) 1472.5 nm), respectively. Table 2 shows that lopt decreases from 15.69 to 9.30 nm when x ) R/Ro decreases from 1 to 0.4, where Ro ) 2.25 nm and is equal to the original radius of the dendrimer. The values of lopt - liso decrease and turn negative for lower x-values, that is, smaller spheres, which indicate that the complexes will only be negatively charged if the dendrimers retain or only slightly

contract when they interact with the DNA. This is apparent when comparing the effective charge of the complexes, Z*compl, which changes from -28.29e to 9.29e when the dendrimers contract, that is, when x is changed from 1 to 0.4. As N is equal to 35, as obtained experimentally, the linker length, D′, is increased for smaller dendrimers. In addition, the ratio of lopt relative to the sphere circumference is increased, which means that DNA is wrapping with more than one turn around the model dendrimer, for example, 1.64 for x ) 0.4. Table 3 shows that the optimal wrapping length for the longer DNA decreases from 12.25 to 9.40 nm when x decreases from 1 to 0.4, that is, for smaller spheres. For x-values between 1 and 0.6, the wrapping length is longer than liso and the value of lopt - liso turns negative around the same value of x as for the shorter DNA. This indicates that the complexes will have a net positive charge for small enough dendrimers for the case of N ) 140. The value of Z*compl also changes from -28.29e for x ) 1 to 9.294e for x ) 0.4. For N ) 140, the linker length, D′, increases with the size of the dendrimers and turns from being a negative physically unrealistic value to being positive for x e 0.5. The ratio of lopt relative to the sphere circumference is also increased, and the wrapping length is just as for the shorter DNA molecules, longer than 1 turn for smaller spheres, that is, 1.66 for x ) 0.4. The results suggest that for dendrimers modeled as hard spheres, a more reasonable optimal wrapping length is obtained if the dendrimer contracts into a sphere of smaller radius. This is particularly apparent for the longer DNA, where the sum of the estimated wrapping length over all dendrimers per DNA

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Figure 3. Optimal wrapping length, lopt, as a function of x ) R/Ro for the longer (black circles) and shorter (white circles) DNA of 4331 and 2000 bp, respectively. N ) 35 for the shorter and 140 for the longer DNA.

Figure 5. Minimum free energy at different optimal wrapping lengths as a function of the maximum number of spheres N per one PE, that is, G4 dendrimers per one DNA chain, at different sphere radii for DNA of 2000 bp (L ) 680 nm) in 10 mM NaBr solution with a dendrimer-dendrimer spacing of D(N,l) ) (L - Nl + 2NR)/N.

molecule exceeds the contour length if we assume that the diameter of the dendrimer do not change due to interaction. The minimum in free energy, Fmin, becomes shallower as the efficient radius, that is, Rox, decreases. This is a consequence of inversed square dependence of the elastic (bending) free energy, Felastic, on the radius of the sphere (eq 6). The elastic (bending) free energy becomes, as expected, more important for smaller than for larger spheres. We also note from Tables 2 and 3 that the Fint term, that is, the (electrostatic) interaction between the complexes, in particular, for realistic values of R, is relatively small or negligible in comparison with the terms of that account for the interaction between the DNA and the dendrimer in the complex. Thus, our assumption regarding this interaction is not likely to introduce any errors of significance in our calculations. In Figure 3, lopt is presented for both the long and the short DNA as a function of the sphere radius, where x ) 0.4 equals the smallest model of the dendrimer. For both lengths of DNA, lopt decreases with the radius of the dendrimer (at constant dendrimer charge), which is likely to be a consequence of the cost of bending of the DNA around a smaller sphere. This is noticeable from the fact that for smaller spheres the lopt is not dependent on the length of the DNA. The effect of reversing the charge of DNA when complexed with a cationic compacting agent has attracted considerable attention as a factor to consider when designing systems for gene delivery and was observed by Lyulin et al. for simulated complexes consisting of a dendrimer and a linear PE.34 This

effect is in qualitative agreement with the predictions of the correlation theory for a hard-sphere model.23 To estimate the maximum number of dendrimers per discrete aggregate with one DNA chain we used eq 15 to calculate the total free energy and to find the conditions for the minimum energy of the system for D(N,l) ) (L - Nl + 2NR)/N. Figure 4 displays the free energy of the system as a function of the number of spheres (dendrimers) N per PE (DNA) for (a) the shorter and (b) the longer DNA when R ) Ro ) 2.25 nm. For the free energy minima, N ) 105 and 228 for DNA of 2000 and 4331 bp, respectively. The estimated optimal wrapping length is equal to the value of liso, 10.88 nm. Both DNA lengths indicate that the value of N is higher compared to the experimentally obtained data where N ) 35 and 140 for 2000 and 4331 bp, respectively. To further investigate this, the calculation was repeated for the dendrimers when considered to be soft spheres, that is, when the radius of the dendrimer decreases due to the interaction with DNA. Figures 5 and 6 display the free energy of the system as a function of the number of dendrimer spheres N per PE (DNA) for the shorter and the longer DNA, respectively. Figure 5 and the corresponding data in Table 4 display that the optimal number of dendrimers per DNA decreases for smaller dendrimer radii. This means that the more the dendrimer contracts upon DNA interaction, less dendrimers will bind to the DNA molecule. Figure 6 and the corresponding data in Table 5 display the same trend as that previously observed for the shorter DNA.

Figure 4. Minimum free energy at different optimal wrapping lengths as a function of the number of dendrimers per one DNA chain for a system of positively charged macroions resembling G4 dendrimers and an oppositely charged PE resembling DNA of (a) 2000 bp (L ) 680 nm) and (b) 4331 bp (L ) 1472.5 nm) with a dendrimer-dendrimer spacing of D(N,l) ) (L - Nl + 2NR)/N and lopt ) 10.88 nm.

Analytical Model Study of Dendrimer/DNA Complexes

Figure 6. Minimum free energy at different optimal wrapping lengths as a function of the maximum number of spheres N per one PE, that is, G4 dendrimers per one DNA chain, at different sphere radii for DNA of 4331bp (L ) 1472.5 nm) in 10 mM NaBr solution with a dendrimer-dendrimer spacing of D(N,l) ) (L - Nl + 2NR)/N. Table 4. Maximum Number of Dendrimers per One DNA Molecule of 2000 bp (L ) 680 nm) for Varying Dendrimer Radiia x (R ) xRo)

Nmax

Fmin kBT

1.00 0.90 0.80 0.70 0.60 0.55 0.50 0.40

105 99 92 87 83 80 78 74

-72889 -68472 -63449 -58405 -52066 -47765 -42694 -26704

a The dendrimer-dendrimer spacing is D(N,l) ) (L - Nl + 2NR)/N and lopt ) 10.88 nm. Nmax is the maximum number of dendrimers and Fmin is the minimum total free energy of the system. The remaining parameter x is defined in the footnote of Table 2.

Table 5. Maximum Number of Dendrimers per One DNA Molecule of 4331 bp (L ) 1472.5 nm) for Varying Dendrimer Radiia x (R ) xRo)

Nmax

Fmin kBT

1.00 0.90 0.80 0.70 0.60 0.5 0.45 0.40

228 213 201 189 178 169 165 160

-157857 -147873 -138104 -126153 -112448 -92212 -77834 -57682

a The dendrimer-dendrimer spacing is D(N,l) ) (L - Nl + 2NR)/N and lopt ) 10.88 nm. Nmax is the maximum number of dendrimers and Fmin is the minimum total free energy of the system. The remaining parameter x is defined in the footnote of Table 2.

The optimal number of dendrimers per DNA decreases for smaller dendrimer radii, which means that the more compressed the dendrimer becomes upon DNA interaction, less dendrimers will bind the DNA molecule. Again this can be explained as a consequence of the increased cost in the elastic (bending) free energy, Felastic, which is inversely dependent on the square of the radius of the sphere. The minimum in energy, Fmin, also becomes shallower as the efficient radius, that is, Rox, decreases. When the dendrimers are considered as soft spheres, the maximum number of dendrimers in the aggregate with one DNA chain is closer to the experimental values for both DNA lengths compared to if the dendrimers retain its radius (Ro ) 2.25 nm) upon DNA binding. For the shorter DNA, Nmax is ≈74 dendrimers, which is greater than the experimental value of 35 for salmon sperm DNA, while for the longer DNA, the Nmax

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value of 160 is close to the experimental value of 140 dendrimers for the linearized DNA plasmid. The results show that the apparent size of the dendrimer affects the results of the analytical model calculations and that the dendrimer should not be modeled as a hard sphere with a fixed radius. This can be explained by the observations from Brownian dynamics computer simulations performed by Lyulin et al.34,35 They found that the formation of complexes between a dendrimer and a PE leads to a large reduction of the dendrimer size as well as to a decrease of the PE dimension. It was concluded that the value of the mean square radius of gyration for the dendrimer in a complex with long chains, l > liso is closer to that for a neutral dendrimer and differ significantly from the mean square radius of gyration of an individual charged dendrimer under conditions where the Debye length is the same. On the other hand, the size of a dendrimer in a complex is not changed once the length of the oppositely charged linear PE chain is increased above liso. Here, the density of charged beads, which represents the charging sites on the PE is rather high (as expected for DNA), and their presence in the simulated complexes leads to the strong screening of electrostatic interactions. The monomers of the linear chain polymer also seemed to penetrate into the dendrimer due to the relatively open structure of the dendrimers.43,44 Lyulin et al. observed a rather high degree of chain penetration and the maximum in the distribution of the chain monomers was found to be located inside the corresponding dendrimer. In addition, Maiti et al. have found evidence for significant penetration of DNA inside the dendrimer using atomistic molecular dynamics simulations.36 The dendrimer penetration also increased with the size of the dendrimer or the dendrimer/DNA charge ratio. This effect of penetration is important as in particular, the strong penetration of a chain into a dendrimer may complicate the chain release, which would make the use of the dendrimers as nanocarriers difficult.34 However, they pointed out that it is possible to control the release by changing pH.36 Important to note is also that ¨ rberg et al. have showed the possibility of using studies by O heparin as an efficient route of releasing DNA when complexed with dendrimers of varying generation.38

Conclusions The value of the wrapping length was found to be larger than the isoelectric length necessary to compensate the total charge of the dendrimer when the dendrimer size was retained or the dendrimer was only slightly compressed. This is in agreement with the findings from Brownian dynamics computer simulations performed by Lyulin et al. for complexes formed by dendrimers.34 For the complexes containing dendrimers of Ro ) R and containing the longer DNA, the obtained wrapping length is close to the isoelectric value. These observations are in agreement with reports on the effect of the chain length on the adsorption of PEs on oppositely charged dendrimers, where the maximum adsorption is observed at some critical value of the polyion length and a further increase of the chain length leads to the polyion tail release and a consequent decrease of adsorption.34 However, for the dendrimers of the smallest size, that is, when the dendrimer is most contracted, the value of the wrapping length was found to be smaller than the isoelectric length. We can conclude that our results show that the radius of the dendrimer in the dendrimer/DNA complex has to be smaller than for the free dendrimer in order to accommodate the number of dendrimers to sufficiently reduce the DNA charge. This has also been verified experimentally. We also note that

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the cost in elastic (bending) free energy of the DNA chain becomes, as expected, more important for smaller than for larger spheres. To get more precise results when estimating the maximum number of dendrimers in the complex, it would be useful to analytically consider a system in which the impenetrable sphere is replaced by a charged spherical layer. The charge should then be distributed evenly through the whole volume and the linear chain monomers should have the possibility to be localized near the opposite charges inside this layer. Acknowledgment. We thank Helmut Schiessel, Per Linse, Rita Dias, Anna Carnerup, and Karin Schille´n for helpful discussions. The sixth EU framework program is acknowledged for work as being a part of a EU-STREP project with NEST program (NEONUCLEI, Contract 12967). The Linneus center of excellence on Organizing Molecular Matter through the Swedish Research Council is also thanked for financial support.

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