Potential of Mean Force of Polyethylenimine-Mediated DNA Attraction

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Potential of Mean Force of Polyethylenimine Mediated DNA Attraction Sampada Bagai, Chongbo Sun, and Tian Tang J. Phys. Chem. B, Just Accepted Manuscript • DOI: 10.1021/jp308132y • Publication Date (Web): 04 Dec 2012 Downloaded from http://pubs.acs.org on December 13, 2012

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The Journal of Physical Chemistry

POTENTIAL OF MEAN FORCE OF POLYETHYLENIMINE MEDIATED DNA ATTRACTION Sampada Bagai, Chongbo Sun and Tian Tang* Department of Mechanical Engineering, University of Alberta, Edmonton, Alberta, Canada, T6G 2G8. *Corresponding author: [email protected]

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Abstract

The aggregation of DNA molecules induced by cationic polymers is of importance to applications in gene delivery. In this work, we performed a series of umbrella sampling molecular dynamics simulations to calculate the potential of mean force (PMF) between two DNA molecules in the presence of polyethylenimine (PEI) molecules using the weighted histogram analysis method. The distance between the centers of mass of the two DNAs was chosen as the reaction coordinate, and the location and depth of the global minimum in the PMF curve were used to gauge the compactness and stability of the formed aggregate. The effects of PEI to DNA charge ratio (N/P charge ratio) and protonation state of the PEI were investigated. The DNA aggregation was found to be more favorable at higher PEI/DNA charge ratios and higher PEI protonation ratios. Compared with small multivalent ions, PEIs give rise to stronger DNA attraction even with an N/P charge ratio at which the DNAs are over neutralized. The effect of changing the protonation ratio on the compactness of the aggregate is more prominent than changing the N/P charge ratio, while the depth of the PMF well is more strongly influenced by the N/P charge ratio.

KEYWORDS: Umbrella sampling, WHAM, N/P charge ratio, Protonation ratio.

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1. Introduction The phenomenon of condensing DNA chains into compact particles, which can fit into spaces with equivalent radii of 1µm or less, is known as DNA condensation.1 DNA condensation has been of interest to researchers in various fields, as it not only explains how genetic information is packed but also has potential applications in non-viral gene delivery.1 The phenomenon that DNA condensation can occur in the presence of cationic ions has been observed in experimental settings.2,3 Gosule et al.2 demonstrated that adding spermidine in a test tube with low ionic strength (0.001 M) aqueous buffer could cause spontaneous DNA condensation. In an experiment by Wilson et al.,3 it was shown that DNA was condensed from its extended coil form upon the addition of trivalent or tetravalent cations. Not all types of cationic ions have the ability to condense DNA. It has been shown experimentally that multivalent cations are capable of condensing DNA while monovalent and divalent ions lack this ability.4,5 Cationic polymers have also proven to be successful in the condensation of DNA.6,7 One of the cationic polymers, polyethyleneimine (PEI) is able to form toroidal nanoscale aggregates with DNA molecules,8,9 which can subsequently facilitate in-vivo as well as in-vitro delivery of oligonucleotides and plasmid DNAs.10 Compared with other candidate synthetic gene carriers, PEI has a high cationic charge density due to its structure in which every third heavy atom is a protonable amino nitrogen.10 This gives PEI a high buffering capacity at different pH values. Also, PEI has the ability to transfect a wide variety of cells,11 making it a potential non-viral gene delivery carrier that shows great promise in clinical settings.12 Many experiments have been conducted to investigate DNA condensation and aggregation facilitated by PEI. A dynamic light scattering study showed that PEI was able to aggregate small DNAs (150 base pairs), giving it an advantage over smaller cations that form aggregates only 3



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with long DNAs (at least 400 base pairs).13 Dunlap et al.9 compared the DNA condensates formed by PEI and lipospermine and found that the condensates formed by PEI were more compact than those formed by lipospermine. PEIs have a wide range of molecular weights, protonation ratios and degrees of branching, and these properties affect the transfection efficiency of PEIs.8,9,14 It has been shown that high molecular weight PEIs in general yield a higher transfection efficiency compared with low molecular weight PEIs, but they are also associated with high cytotoxicity that restricts their clinical usage.8,15 The role of protonation ratio of PEI was investigated by Utsuno and Uludag,14 who suggested that PEI bound with DNA more rapidly in a low pH value environment where the PEI possesses a higher protonation ratio. Dai et al.16 studied the effect of PEI structure on cellular uptake and transfection efficiency by examining polyplexes formed by DNA with linear as well as branched PEIs. They found that branched PEI was more effective in causing DNA condensation and resulted in better cellular uptake than its linear counterpart. The PEI to DNA charge ratio (N/P charge ratio) is another important factor affecting DNA condensation.17 It was observed that high N/P charge ratio created a net positive charge on the DNA-PEI polyplexes, improved cellular uptake and extended DNA retention in the nucleus.18 On the other hand, while increasing the N/P charge ratio increases the transfection efficiency, it also increases the cytotoxicity.19 In general, it has been found that the size and stability of the nanoparticles of DNA-PEI polyplexes significantly affect the delivery and transfection process, with more compact and stable particles showing better cellular penetration and uptake.20,21 There have been several theoretical studies on the mechanisms governing cation-induced DNA condensation and aggregation. Oosawa22 proved that the addition of counterions reduced the repulsion between like charged polyelectrolytes. Manning23 proposed that counterions condensed 4


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on a polyelectrolyte such that its linear charge density was lowered to a certain value. While these works demonstrated that electrostatic screening of DNA charges by counterions could reduce the repulsion between DNAs at close separation,24 they were not able to fully explain DNA condensation. In particular, electrostatic screening can only reduce the distance between the segments on the DNA, whereas DNA in a condensed form involves lateral contact of the segments.25 This led to the study of other mechanisms that are not based on electrostatic effects. In the simulation work by Sun et al.26 where the aggregation of two and four short DNA molecules in the presence of PEIs was investigated, it was found that in addition to electrostatic screening, polyion bridging (a polyion simultaneously binds to multiple DNA molecules) is an important mechanism responsible for polycation-medicated DNA aggregation.26 Other mechanisms proposed to explain DNA condensation by cations also exist, such as the formation of Wigner crystal by counterions,27 zipper motif model28 and the creation of long range attractive hydration forces because of the counterions.29 Recently, molecular dynamics (MD) simulation has become a useful tool to study cation, especially polycation medicated DNA aggregation and condensation. Use of such an approach is driven both by the need to represent the atomic details of DNA-cation interaction30,31 and by the fast growing computing capacity. Using MD simulations, Ziebarth et al.32 compared the complex formation of DNA with PEI and with poly-L-lysine (PLL), and concluded that PEI was more efficient in neutralizing the DNA. Sun et al.33 studied the effects of PEI’s protonation ratio and architecture on DNA/PEI complexation. They found that the PEIs primarily bind to the DNA backbone through the formation of hydrogen bonding with the backbone oxygens. PEIs of higher protonation ratio bind to the DNA mainly through direct hydrogen bonding, resulting in more stable complex with DNA than PEIs of lower protonation ratio for which indirect interaction 5



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medicated by water molecules plays an important role in binding. A series of MD simulations were also conducted to study the aggregation of multiple DNAs mediated by PEI molecules and structural analysis on the formed aggregate was performed.26 The analysis revealed several mechanisms contributing to the aggregation. However, it did not explain the kinetics or spontaneity of the aggregation process. The spontaneity of the aggregation process can be gauged by the binding free energy of the system. Although computationally demanding, calculation of binding free energies has been shown to be useful in various fields including medicine and biotechnology.34 Different methods have been proposed to perform free energy calculations, such as the molecular mechanics/ Poisson Boltzmann surface area (MM-PBSA) method35 and the linear interaction energy (LIE) method.36 Potential of mean force (PMF) calculation is another reliable and accurate method for determining the strength of interaction or binding.34 First introduced by Kirkwood, PMF is calculated along a predefined reaction coordinate function

by

and is related to the average distribution

.37 Calculating PMF from a direct MD simulation is not

practical due to the presence of large energy barriers that prevent sufficient sampling over the range of the reaction coordinate. 38 One way to overcome this limitation is to perform umbrella sampling simulations39 where an additional biasing potential (e.g. a harmonic potential) is applied to

in order to cross the energy barriers. A series of MD simulations are performed with

the biasing potential applied at different windows of ; the results obtained from all the windows are then unbiased and used to calculate the PMF for the system.38 An effective method for calculating PMF using the results from umbrella sampling simulations is known as the weighted histogram analysis method (WHAM).40 The main principle of WHAM is based on the maximum overlap method proposed by Bennett.41 It involves assigning a weighting factor to minimize the 6


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statistical error when estimating the unbiased distribution function from the simulation data. The advantage of WHAM over other methods is that it utilizes all the information from umbrella sampling simulation, explicitly calculates and minimizes the statistical errors. It allows multiple overlaps in the probability distribution to obtain better sampling and more accurate result, and can be applied to multidimensional cases as well.40 WHAM has been widely used to calculate PMF for various processes such as protein-ligand interactions,42,43 base flipping and base opening of DNA,44,45 DNA and lipid assembly.46 In the area of DNA condensation and aggregation, Dai et al.5 studied the interaction of two DNA molecules in the presence of multivalent counterions and calculated the interaction potential using umbrella sampling MD simulations. Despite the great interest in polycation mediated DNA condensation and aggregation, the quantification of DNA-DNA attraction in presence of polycations, to the best of our knowledge, is still absent. Calculating PMF of polycation, such as PEI, mediated DNA attraction allows us to examine the size and stability of the formed DNA-polycation aggregates and address how they can be affected by various material/structure parameters involved in the system. These issues are of great importance in understanding the role of polycations in DNA aggregation and ultimately in polycation-based gene delivery. In this work, we use WHAM to study the interaction between two DNA duplexes in the presence of PEI molecules. A series of MD simulations incorporating umbrella sampling are performed to obtain the PMF along the center to center distance between the two DNAs. PMF for systems involving different N/P charge ratios and different PEI protonation states are used to compare the stability of the DNA aggregates and the strength of the DNA-DNA attraction in these systems. The remainder of the paper is organized as follows. In Section 2, we describe the simulated systems, the method used to perform the simulation as well as the PMF calculations 7



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using WHAM. Simulation results and analysis are presented in Section 3. Conclusions are given in Section 4.

2. Method 2.1 DNA-PEI systems simulated The Dickerson-Drew B-DNA dodecamer 47 d(CGCGAATTCGCG)2 is used in the simulation, which consists of 24 nucleotides with a total charge of -22 in its fully deprotonated state. The PEI (Figure 1) simulated in this work has thirteen amine groups with a molecular weight of 586 Da.33 Two protonation ratios are used: 46% and 23%. The former corresponds to 6 out of the 13 amine groups being protonated while the latter corresponds to 3 out of the 13 amine groups being protonated (Figure 1). These protonation ratios are close to the results obtained by Utsuno et al.,14 where 600 Da PEI was found to have 47% protonated amines at pH 6 and 21% protonated amines at pH 8. A total of five systems were simulated, with four of them containing 46% protonated PEIs and the last containing 23% protonated PEIs. Each system contains two DNAs but may have a different number of PEIs. The four systems with 46% protonated PEIs contain 8, 6, 4 and 2 PEIs respectively, and are in turn referred to as systems 2D-8P, 2D-6P, 2D-4P and 2D-2P throughout the paper. These four systems are simulated and compared in order to address the effect of N/P charge ratio on the PMF. The fifth simulated system consists of two DNAs and eight 23% protonated PEIs, and is referred to as system 2D-8P(23%). Since system 2D-8P(23%) has the same N/P charge ratio as system 2D-4P, their comparison allows us to study the effect of PEI’s protonation state on the PMF.

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The initial configurations of the systems described above are shown in Figure 2, with systems 2D-8P and 2D-8P(23%) having the same initial configuration. For each system, the two DNAs at the beginning of the simulation had their principle axes aligned parallel to each other as well as to the axes of the PEIs. Two PEIs were placed in between the two DNAs such that their centers of mass (COMs) were equidistant from the COMs of the two DNAs. For systems 2D-8P, 2D-6P, 2D-4P and 2D-8P(23%), the remaining PEIs were placed around the DNAs such that their COMs were at a distance of 20 Å from the COM of the nearest DNA molecule. In specifying the initial configuration, we have considered the two important mechanisms governing PEI mediated DNA aggregation that was identified in our previous work26, which are polyion bridging and charge neutralization. We placed two PEIs in between the two DNAs so that they can form polyion bridges. The rest of the PEIs were placed around the DNA in order to capture the effect of charge neutralization. Each system was placed in a water box large enough to make sure that each solute molecule was at a distance of at least 16 Å from the edge of the water box, i.e. the distance between the outermost PEI and the closest PEI in the periodic boxes is 32 Å. The distance between the DNA molecule and the closest DNA in the periodic boxes is therefore 72 Å. To maintain neutralization, certain amounts of sodium or chlorine ions were added to the water box by replacing equal number of water molecules. The details of the five simulated systems are tabulated in Table 1. 2.2 Simulation details The simulations were run using NAMD48 with CHARMM general force field49 for the PEIs and CHARMM 27 force field50 for the rest of the molecules. TIP3P model51 for water, Particle Mesh Ewald method52 for calculating electrostatic interaction, periodic boundary condition and a time step of 2 fs were used for all the simulations. SHAKE algorithm53 was used to constrain the 9



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hydrogen bonds and the non-bonded interactions were cut off at 12 Å. VMD54 was used to visualize the systems and analyze the trajectories obtained from the MD simulations. For each simulation, the system was first minimized for 5000 steps in order to eliminate bad contact. This includes 2000 steps of minimization with constrained solute molecules followed by 2000 steps of minimization with harmonically restrained solutes and 1000 steps of unrestrained minimization. Afterwards, the system was gradually heated from 0 to 300 K in 20 ps. A harmonic restraint of 10 kcal/mol/Å2 was then applied on the non-hydrogen atoms of the solutes at 300 K and 1 bar pressure, and dynamics was run for 4 ns to allow the ions and water molecules around the DNAs and PEIs to relax. After removal of the restraint, the system was then simulated in an isothermalisobaric ensemble condition at 300 K temperature and 1 bar pressure for 20 ns. Such simulation length has been shown to generate dynamic equilibrium. Such simulation length has been shown to generate dynamic equilibrium. Specifically, we did a convergence test where we chose different 10 ns time slots from our simulation and calculated the PMF based on each of them. The depth of the PMF well was shown to be converging, confirming the attainment of equilibrium. The trajectory of each simulation was saved every 1000 steps and the data from the last 10 ns of the 20 ns simulation were used for analysis. 2.3 Umbrella sampling and PMF calculation In the following, we describe how the umbrella sampling and PMF calculation were performed for each of the five systems described above. In order to perform umbrella sampling, we chose the distance between the COMs of the two DNAs as the reaction coordinate x and umbrella sampling windows were created at 1 Å interval along the reaction coordinate. A total of 29 windows were created with the value of the reaction coordinate varying from 22 Å to 50 Å, which provided sufficient sampling for the entire range of the reaction coordinate. For each 10


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window i (i = 1, 2 …29), the COM of each DNA was subjected to a harmonic restraint with a , where the force constant k = 2.0 kcal/mol/Å2

biasing potential in the form of and

is the equilibrium separation associated with the ith biasing potential. The choice of the k

value is based on previous studies involving binding of molecules with similar size to our simulated molecules.5,46 Following WHAM, the trajectories collected from all the 29 simulations were used to calculate the PMF. Specifically, the entire range of the reaction coordinate, i.e., from 22 Å to 50 Å, was divided into bins of size 0.25 Å. Such a size was determined by a convergence test on the dependence of the PMF on the bin size. A histogram was then constructed for the number of counts in each bin vs. the reaction coordinate. Figure 3 shows such a histogram for the 2D-8P system. Once the histogram was created, the unbiased probability P as a function of the reaction coordinate and the free energy shift Fi for each umbrella sampling simulation were calculated using the WHAM equations below.40

(1)

(2) (3)

In these equations,

is the total number of umbrella sampling simulations (= 29 in our case),

gives the total number of counts in simulation i and simulation i that fall into the bin centered at x. temperature. The biasing potential

is the number of counts in

is the Boltzmann constant and

is the

as given in Eq. (3) is twice the harmonic potential

applied on each DNA since both DNAs are subjected to the harmonic restraint.


and

were 11


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solved iteratively from the above equations to a satisfactory convergence, after which the PMF was calculated using (4)

The datum of the PMF was adjusted such that at sufficiently large value of

the PMF became

zero. 3. Results and Discussion Following the simulation and analysis methods described above, the PMF calculations were carried out for all the five systems. In the following, we first demonstrate the basic characteristic of the PMF curve using the example of the 2D-8P system. A comparison is then made among the systems with different N/P charge ratios and PEI protonation states, which allows us to address the stability and compactness of the aggregate under different conditions. 3.1 PMF for system 2D-8P The PMF for system 2D-8P is plotted along the reaction coordinate as shown in Figure 4. As the COM distance between the two DNAs (x) increases, the curve shows an initial decreasing branch which is characteristic of repulsive interaction between the DNA molecules. The curve reaches its global minimum at 23.6 Å, beyond which there is an overall increasing trend in the PMF till 35 Å; this branch represents attraction between the two DNA molecules. After 35 Å, the curve fluctuates without overall increase in the PMF, indicating negligible interaction beyond this separation. The global minimum of the PMF is -6.7 kcal/mol located at x = 23.6 Å. This is in accordance to the structure obtained from the simulation study of Sun et al.,26 where the shortest distance between two DNAs aggregated by eight 600 Da 46% protonated PEIs was found to be 23.2 Å.

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The size and stability of the DNA aggregates are crucial for gene delivery and transfection20,21 and are important criteria for determining the extent of the DNA aggregation. The location of the global minimum in the PMF curve is the COM distance at which the interaction between the two DNAs is neither repulsive nor attractive. That is, it is the equilibrium separation between the two DNAs and is an indicator of the compactness of the aggregate. In addition, the depth of the PMF well, i.e., the absolute value of the PMF at its global minimum, is an indicator of the stability of the aggregate. A larger depth of the PMF well corresponds to larger amount of energy needed to dissociate the aggregate and hence higher stability. In a previous study, Dai et al.5 performed MD simulations to investigate the attraction between DNA molecules in the presence of multivalent ions and calculated the interaction potential between two parallel 10-base pair DNAs due to these multivalent ions. Three types of ions were used in their simulations: putrescine (divalent), spermidine (trivalent) and spermine (tetravalent). Several observations can be made on the characteristics of multivalent ion and polyion medicated DNA aggregations. First, the minimum of the interaction potential for putrescine, spermidine and spermine occurs respectively at x = 23.8 Å, 22.8 Å and 23.7 Å, with the depth of the potential well valued at kcal/mol),

(3.5 kcal/mol) and

(1.18

(5.3 kcal/mol), respectively.5 Compared with the

depth of the PMF well for PEI (valence = +6) mediated DNA attraction, 6.7 kcal/mol, there is a general trend of having strong DNA-DNA attraction with cations of higher valence, which can imply that more stable DNA aggregates are formed by cations of higher valence. It should be pointed out that in the work mentioned before,5 the DNA charges are just neutralized by the multivalent ions (except for the case of spermidine where there is a net charge of +2 neutralized by Cl- ions), while in the current simulation, the two DNAs and eight PEIs carry a net charge of +4, i.e., the DNAs are over neutralized by the PEIs. Despite this, stronger DNA-DNA attraction 13



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is found for PEI mediated DNA aggregation. This is due to the PEI’s ability to interact locally with the DNA and bring the DNAs together via polyion bridging.26,33 The polyion bridges formed by PEIs are much stronger than the bridges formed by small multivalent ions, contributing to larger attraction between the DNAs. Second, from the data in the current work and in the work by Dai et.al.,5 a clear relation between the valence of the ions and the equilibrium separation between the two DNAs cannot be identified. In particular, spermidine, despite having a smaller valence than spermine, gives rise to a smaller equilibrium separation between the two DNAs,5 and the equilibrium separations associated with spermine and PEI are approximately the same. It is interesting to note that experimentally different conclusions also exist as to whether polycations can form more compact DNA aggregates compared with small multivalent ions. In the experiments by Kasyanenko and Afanasieva,55 the DNA aggregation in the presence of different trivalent ions (Fe3+, La3+, [Co(NH3)6]3+, spermidine ions) and cationic PLL molecules was compared. They observed that the increase in valence of the cations led to more reduction in the DNA volume.55 On the other hand, a different experiment reported that for the same cationic charge, smaller ions formed more compact and stable DNA condensates.5 A possible explanation for this is that while polycations have higher valence and can aggregate the DNAs through stronger polyion bridging,26 their larger size and the steric effect resulting from it can prevent them from forming more compact DNA aggregate compared with small cations. Finally, comparing Figure 4 with the curve relating the interaction potential to the interduplex distance5, Figure 4 is much less smoother and shows many local minima along the curve. We do not think this is a numerical issue because we have run a large number of simulations (29 for each system as compared to 4 simulations for each system5) with umbrella sampling windows densely distributed along the reaction coordinate and have obtained well overlapped histogram 14


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(see Figure 3). Instead, we attribute this to the fact that compared with small multivalent ions, PEI is a more flexible molecule and can adopt more configurations in the MD simulation. Throughout this paper, only the global minimum in the PMF and the associated DNA separation value will be used to gauge the strength of the DNA attraction. 3.2 Effect of N/P charge ratio Various studies in the literature have shown that the N/P charge ratio plays an important role in polycation mediated DNA aggregation.17-19 The effect of N/P charge ratio is studied in this work by comparing the PMF for systems 2D-8P, 2D-6P, 2D-4P and 2D-2P with N/P charge ratio varying from ~1 to ~0.25. The PMF curves for systems 2D-6P, 2D-4P and 2D-2P are respectively shown in Figure 5 (a)-(c). The location of the global minimum in each curve and the depth of the PMF well are given in Figure 6. For the 2D-6P system, the PMF curve has a global minimum at 25.6 Å at which the PMF value is -4.04 kcal/mol. For the 2D-4P system, the minimum is obtained at 26.1 Å with a PMF value of -2.92 kcal/mol. For the 2D-2P system, the minimum is obtained at 28.9 Å which corresponds to a PMF value of -1.96 kcal/mol. It is clear from Figure 6 that as the N/P charge ratio increases, the equilibrium separation of the DNAs decreases and the depth of the PMF well increases, both varying nearly linearly with the N/P charge ratio. This suggests that systems with higher N/P charge ratio are able to form not only more compact but also stronger DNA aggregate. The enhanced stability with increasing N/P charge ratio is also reflected through the smoothness of the PMF curve. PMF curves given in Figure 4 and Figure 5 shows an increase in fluctuation, i.e., an increase in the number of local minima along the reaction coordinate, as the number of PEIs reduces. This implies that at a lower N/P charge ratio, there is an increase in the probability of finding the DNAs at these local minima, which are shallow and associated with lower stability. In fact, in an earlier work 15



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involving the simulation of two DNAs aggregated by two PEIs,26 after 200 ns simulation, the two DNAs were found to be separated by a distance of 55 Å, and there were no PEIs simultaneously connecting the two DNAs. The shallow well in the PMF as shown in Figure 5 (c) together with the existence of multiple local energy minima beyond 35 Å have contributed to the inability of forming sufficiently stable DNA aggregate that can sustain much longer simulations. The results here are consistent with experimental observation that higher N/P charge ratio facilitates the DNA condensation process and improves the uptake and retention of DNA in the cell and nucleus.19,56 For example, Guo et al.56 studied the effect of N/P charge ratio on the transfection

efficiency

of

polyplexes

formed

by

DNA

with

PLL

and

dioeoylphosphatidylethanolamine/cholesteryl hemisuccinate liposomes. They found that at a constant lipid/DNA ratio, an increase in the N/P charge ratio rapidly increased the transfection efficiency, and suggested that this result could be applied to PEI as well.56 3.3 Effect of PEI protonation ratio Protonation ratio of the PEI is a crucial factor in the process of DNA aggregation.8,14,33 In general, it has been observed experimentally that increasing protonation ratio promotes the stability of the aggregation.14,33 To study the effect of protonation ratio, we calculated the PMF for system 2D-8P(23%), where 3 out of the 13 nitrogens on the PEI are protonated. The obtained PMF curve is shown in Figure 7. Compared with the PMF (Figure 4) for the 2D-8P system that contains the same number of PEIs but at a higher protonation ratio (46%), Figure 7 shows a much shallower and wider PMF well with a more or less flat region that ranges from 30 to 35 Å along the reaction coordinate. The global minimum is located at 34.2Å, 48.7% more than the equilibrium separation obtained for system 2D-8P. The depth of the PMF well is 2.5 kcal/mol,

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more than 62% smaller than that of the 2D-8P system. These data demonstrate the formation of significantly smaller and stronger DNA aggregate by PEIs with higher protonation ratio. As mentioned before, because polycations can aggregate DNAs not only via charge neutralization but also via polyion bridging, the total charge that the polycations carry (or the N/P charge ratio) may not be the only factor that affects the properties of the DNA aggregate. To further explore this, we compare in Table 2 the PMF calculated for the three systems: 2D-8P, 2D-8P(23%) and 2D-4P. 2D-8P and 2D-8P(23%) contain the same number of PEI molecules, but have different N/P charge ratio due to their different protonation ratios. 2D-8P(23%) and 2D4P differ in both PEI number and protonation ratio, but have the same overall PEI charges and hence the same N/P charge ratio. Clearly among the three systems, 2D-8P, the one with the highest protonation ratio and N/P charge ratio, forms the most compact and stable DNA aggregate. The more interesting comparison is between 2D-8P(23%) and 2D-4P. The location of the global minimum in the PMF curve is at 26.1 Å for 2D-4P which is 8.10 Å less than the 34.2 Å obtained for 2D-8P(23%). In addition, the depth of the PMF well for 2D-4P is slightly larger than that for 2D-8P(23%). Therefore, it is clear that at the same N/P charge ratio, 46% protonated PEIs form more stable and much more compact DNA aggregate than 23% protonated PEIs. We can also observe from Table 2 that the effect of changing the protonation ratio on the compactness of the aggregate is more prominent than changing the N/P charge ratio. Specifically, changing the N/P charge ratio from 0.54 in 2D-4P to 1.09 in 2D-8P reduces the equilibrium separation by only 11.8%, whereas increasing the protonation ratio, at the same N/P charge ratio, from 23% in 2D-8P(23%) to 46% in 2D-4P reduces it by 23.6%. On the other hand, the depth of the PMF well is more strongly influenced by the N/P charge ratio. In particular, it is increase by 16% on doubling the protonation ratio (from 2D-8P(23%) to 2D-4P) 17



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and by 131% on doubling the N/P charge ratio (from 2D-4P to 2D-8P). It is known that increasing the N/P charge ratio increases cytotoxicity.19 The observations here suggest that by properly selecting the PEI protonation ratio and the N/P charge ratio, a good compromise may be found among the compactness of the DNA aggregate, its stability and cytotoxicity.

4. Conclusion We performed umbrella sampling MD simulations to study the interaction of two DNAs in the presence of PEI molecules. The PMF between the two DNAs was calculated using WHAM with the reaction coordinate being the COM distance between the two DNAs. The calculations allow us to quantitatively assess PEI mediated DNA attraction and address the effect of N/P charge ratio and PEI’s protonation state. Compared with small multivalent ions, PEIs give rise to stronger DNA attraction even with an N/P charge ratio at which the DNAs are over neutralized. As the N/P charge ratio and/or the protonation ratio of the PEI increases the DNA aggregate becomes more compact and stable. The effect of changing the protonation ratio on the compactness of the aggregate is more prominent than changing the N/P charge ratio, while the depth of the PMF well is more strongly influenced by the N/P charge ratio. Although in this study we have used PEI as a representative polycation, the results obtained here may be applicable to other polycation mediated DNA aggregation. References (1)

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TABLES Table 1. Details of the simulated systems Name of the system

2D-8P

2D-6P

2D-4P

2D-2P

2D-8P(23%)

No. of PEI

8

6

4

2

8

Protonation ratio of PEI

46%

46%

46%

46%

23%

N/P charge ratio

48/44=1.09

36/44=0.81

24/44=0.54

12/44=0.27 24/44=0.54

Type and No. of ions

4 Cl-

8 Na+

20 Na+

32 Na+

20 Na+

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Table 2.Comparison of PMF in systems 2D-8P, 2D-8P(23%) and 2D-4P.

System

No. of PEI

Protonation ratio of PEI

N/P charge ratio

2D-8P 2D-4P 2D-8P (23%)

8 4 8

46% 46% 23%

1.09 0.54 0.54

Location of global minimum in PMF (Å) 23.0 26.1 34.2

Depth of PMF well (kcal/mol) 6.7 2.9 2.5

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FIGURE CAPTIONS Figure 1. Molecular structure of the PEI simulated. The nitrogen numbers are given in red. For 46% protonated PEI nitrogens 2, 4, 6, 8, 11 and 13 are protonated whereas for 23% protonated PEI nitrogens 2, 6 and 11 are protonated. Figure 2. Initial configurations (at 0 ns) of the simulated systems each containing 2 DNAs and a different number of PEIs: (a) 2D-8P and 2D-8P(23%), (b) 2D-6P, (c) 2D-4P and (d) 2D-2P. Figure 3. Histogram for the 2D-8P system showing the number of counts for the entire range of the reaction coordinates. A total of 29 simulations were performed to generate the histogram. Different curves in the figure corresponds to simulations with different biasing potential (i = 1, 2, …, 29). Figure 4. PMF vs. DNA COM distance for system 2D-8P. Figure 5. PMF vs. DNA COM distance for systems (a) 2D-6P, (b) 2D-4P and (c) 2D-2P. Figure 6. Equilibrium separation between the DNAs and depth of the PMF well plotted against the N/P charge ratio for 46% protonated PEI mediated DNA attraction. Figure 7. PMF vs. DNA COM distance for 2D-8P(23%) system.

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FIGURES

Figure 1

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Figure 2

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Figure 3

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Figure 4

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Figure 5

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Figure 6

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Figure 7

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Table of Contents Image

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Potential of Mean Force of Polyethylenimine-Mediated DNA Attraction

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