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Article Cite This: ACS Omega 2019, 4, 7255−7264
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Effect of the Protonation Level and Ionic Strength on the Structure of Linear Polyethyleneimine Caleb E. Gallops, Chang Yu, Jesse D. Ziebarth, and Yongmei Wang* Department of Chemistry, The University of Memphis, Memphis, Tennessee 38152, United States
ACS Omega 2019.4:7255-7264. Downloaded from pubs.acs.org by 185.14.192.50 on 04/22/19. For personal use only.
S Supporting Information *
ABSTRACT: Polyethylenimine (PEI) is a highly studied vector for nonviral gene delivery, with a high transfection efficiency that has been linked with its pH responsiveness. Atomistic molecular dynamics simulations of a linear 40 mer PEI chain were performed for nine protonation states and various NaCl concentrations to examine how the structure of PEI depends on pH and salt concentration. PEI continuously expands as it transitions from being unprotonated to fully protonated; however, we observe that two different regimes underlie this expansion. Sparsely protonated chains behave as weakly charged polyelectrolytes whose expansion is associated with the reduction of intrachain hydrophobic interactions. In contrast, the expansion of densely protonated chains with increased protonation involves increasing chain stiffness and breaking intrachain hydrogen bonds. The weakly to highly charged transition occurred at ∼40% protonation, suggesting it may occur in endosomal conditions. These results provide a microscopic picture of changes in PEI structure during the gene delivery process.
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INTRODUCTION Polyethylenimine (PEI) is an effective first-generation polycationic transfection vector used in nonviral gene delivery.1,2 Cationic PEI chains bind to the anionic phosphate backbone of nucleic acids, forming polyplexes and facilitating the delivery of genetic materials into cells. The success of PEI as a nonviral gene delivery vector has long been linked with its ability to promote the escape of the polyplex from endosomes, preventing the degradation of the delivered genetic material before it can reach the nucleus. However, although the endosomal escape process remains a highly studied and debated area within the field nonviral gene delivery, several potential mechanisms for how PEI contributes to endosomal escape have been described.3,4 A key feature underlying these mechanisms is that PEI is not fully protonated at physiological pH (∼7.4) and therefore it becomes increasingly protonated in the acidic endosomal environments encountered by gene delivery polyplexes.5 The most highly studied endosomal escape mechanism, known as the “proton sponge effect”, is directly related to the pH-buffering capacity of PEI. According to the proton sponge hypothesis, the presence of PEI increases the osmotic pressure within the endosome, as PEI absorbs protons that are pumped into developing endosomes. The osmotic pressure continues to increase as more protons are pumped in and Cl− passively enters the endosome to neutralize the charge. Eventually, the endosome ruptures due to high osmotic pressure, allowing PEI/nucleic acid polyplexes to escape.6 Other potential mechanisms suggest that cationic PEI chains, either free in solution or within polyplexes, interact with and destabilize the phospholipid membrane of the endosome or that PEI swells as endosomal pH decreases, leading to disruption of the endosome.3,4,7−12 Changes in the protonation of PEI during endosomal acidification would also play a role in these © 2019 American Chemical Society
mechanisms, as the pH responsiveness of PEI could impact its interactions with membranes and nucleic acids within polyplexes. Despite the importance of the pH-responsive protonation and properties of PEI in its role in gene delivery, the structural understanding of these properties and how they impact interactions with nucleic acids and phospholipid membranes in aqueous solutions is limited. In response, computer simulations of these systems have been employed to investigate the atomistic behavior of PEI and PEI/nucleic acid polyplexes. In a previous study by this group, the pH-dependent protonation of PEI for several different salt concentrations was investigated through Monte Carlo simulations5 producing theoretical titration curves that were in accord with previous experiments;13 these results have also been compared with subsequent experiments.14,15 Atomistic molecular dynamic (MD) simulations have been used to show the spontaneous formation as well as quantify the physical properties of PEI/ nucleic acid complexes.16−19 Furthermore, several studies have used atomistic MD to examine the structure and behavior of linear and branched PEI when alone in solution at various protonation states.20−22 These studies were able to show that both forms of PEI elongate at higher protonation states (i.e., at lower pH) and that the solvation shell and ion structure surrounding the PEI depends on the protonation state. However, these studies examined PEI in salt-free conditions either at only three protonation states20,22 or only when the chain was at most half protonated.21 Recently, Basu et al. performed a series of atomistic molecular dynamic simulations of a 20 mer PEI chain at seven different protonation states and Received: January 8, 2019 Accepted: April 11, 2019 Published: April 22, 2019 7255
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(mol Å2) restraints on the chain for 200 ps followed by a restrained NPT equilibration at a pressure of 1 atm for 500 ps. Under the same conditions, an unrestrained 10 ns NPT equilibration was performed. The production simulations were performed for another 90 ns. The temperature was held constant at 300 K using Langevin dynamics with a collision frequency of 1.0 ps−1. A 10 Å cutoff was employed along with the particle mesh Ewald method to treat long-range electrostatic interactions, and the integration time step of the simulations was 2 fs. Simulation trajectories were visualized with VMD 1.9.2.29 Average quantities were determined by averaging over 45 000 snapshots taken during the 90 ns of the production simulations (a snapshot every 200 ps). The radius of gyration and the average end-to-end distance were calculated to quantify the extension of the PEI. The persistence length Lp was calculated to quantify the stiffness of the chain according to the following equation
two salt concentrations and found that there is a transition in the pH- or charge-dependent behavior of PEI when it becomes more than 50−60% protonated.23 Here, we continue these efforts to understand the pH-dependent behavior of PEI through atomistic molecular dynamic simulations. We simulate a 40 mer linear PEI chain with nine different protonation states ranging from unprotonated to fully protonated in multiple salt concentrations to add to an understanding of how the structure of PEI adjusts in response to changes in ionic strength and pH, such as the changing conditions that are encountered during the gene delivery process. Among various types of polyelectrolytes, PEI is unique in the sense that the charge is located along the chain backbone, not carried on the side groups, as in polystyrene sulfonate, polyacrylic acid, and polylysine. If all amines on a PEI chain are protonated, then the distance between two positively charged amines, N+−N+ distance, is shorter than the Bjerrum length, lB, where the Bjerrum length is the length at which the Coulombic potential between two unit charges is equal to the thermal energy. In an aqueous solution, the Bjerrum length is typically taken as 7.1 Å. Theoretical discussions of polyelectrolyte conformations use a dimensionless parameter, Γ = lB/A, where A is the charge-to-charge distance. A polyelectrolyte is considered weakly charged if Γ < 1 and highly charged if Γ > 1. As the protonation level of a PEI chain gradually increases, the chain transitions from being a weakly charged polyelectrolyte to a highly charged polyelectrolyte. Thus, studying the conformational change of a single PEI as a function of the protonation level and ionic strength provides an interesting test case for examining prevalent polyelectrolyte theories.
⟨cos θij⟩ = e−L0 |i − j| / Lp
(1)
where θij is the angle between a vector tangent to the polymer’s ith monomer and another vector tangent to the jth monomer, displaced L0|i − j| along the contour of the chain, where L0 is the average monomer-to-monomer distance, and |i − j| is the number of monomers between the two vectors. The vector is defined by the nitrogen atoms of two neighboring monomers. As will be discussed more fully later, since PEI nitrogen atoms are separated by two CH2 groups, L0 was found to have two values, corresponding to a trans configuration and a gauche+ or gauche− configuration. As the populations of trans and gauche states were found to vary, an average was determined separately for each simulation. An exponential decay function was fit to a plot of cos θij vs |i − j| to determine the constant L0 ,
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METHODS The atomistic simulations were performed in the CUDA implementation of AMBER 1424 using the generalized AMBER force field (GAFF) for the PEI chain and ionsjc_tip3p for monovalent ion parameters.25 GAFF was used for the Lennard-Jones parameters of PEI chain and a HF/6-31G* calculation with the RESP charge method as implemented in Gaussian0326 to determine the partial charges of the atoms in the PEI chain. A single 40 mer PEI chain with varying protonation, ranging from 0 to 40 protonated monomers, was studied. The protonation state of the chain is characterized by α, the fraction of nitrogen atoms that are protonated; chains with α = 0, 0.125, 0.25, 0.375, 0.5, 0.625, 0.75, 0.875, and 1 were studied here. Protonated sites were regularly spaced along the chain, so that, for example, every fourth nitrogen was protonated when α = 0.25 and every other nitrogen was protonated when α = 0.5. This arrangement of protonated sites was used in consideration of previous works, as it avoids having the energetically unfavorable situation where nearest-neighbor nitrogens were both protonated until it was necessary.5,27 The PEI chain was placed in a periodic box with approximate dimensions of 100 Å × 100 Å × 100 Å that was filled with TIP3P water and randomly located Na+ and Cl− ions using the LEAP module in AMBER.28 The systems were minimized with 10 000 steps of the steepest descent followed by 10 000 steps of conjugate gradient using a restraint of 10 kcal/(mol Å2) on the nonhydrogen atoms of the PEI. A subsequent unrestrained minimization process was performed with 5000 steps of the steepest descent and 5000 steps of conjugate gradient. Each system was heated by raising the temperature from 0 to 300 K at NVT conditions with 10 kcal/
Lp
and Lp was then calculated using the for that simulation. Figures S1−S5 show an example of the decay function obtained. Values for , , , and other parameters, as well as radial distribution functions, were determined using the cpptraj module of AMBER.24
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RESULTS AND DISCUSSION Changes in Intrachain Interactions with Increased Protonation of Linear PEI. The dependence of the protonation of a single PEI chain on pH has been studied earlier in our group using a Monte Carlo titration model.5 At any given pH, one may define the fraction of nitrogen atoms that are protonated, α, which theoretically speaking is an ensemble average over different protonation microstates. Variation in α with pH is salt dependent, but, in general, α changes from a low value (∼0.2) when pH > 7 to around 0.9 when pH drops below 3. This behavior is believed to contribute to endosomal escape and high transfection efficiency of PEI-based nonviral gene delivery.1,2 From our coarse-grained Monte Carlo titrations, we observed that the chain expands as α increases to reduce electrostatic repulsion between protonated sites. Here, we investigate, at atomic-level detail, how chain conformations change with protonation levels through atomistic molecular dynamics simulation of a single linear PEI chain with 40 repeating units of ethylene imine. The protonation level α in the simulations varies from 0 to 1. For intermediate values of α, we adopted a simple approach: the charges are arranged evenly along the chain 7256
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whenever possible. This is justified by the observation that in the Monte Carlo titration, doublets of two charged amines (i.e., nearest-neighbor amines that are both protonated) only rarely occur when the protonation level α is less than 0.5.5 The conformation of the PEI chain depends on several factors, including electrostatic repulsion between charges on the chain, hydrophobic interactions, and interactions with the solvent and salt ions, which all change with its protonation. When α = 0, there is no electrostatic repulsion between protonated nitrogens along the chain and the chain forms a very compact structure that allows for interactions between its hydrophobic methylene groups (Figure 1a). This result
Figure 2. Average number of times that monomers i and j of the PEI chain contain carbon atoms that are within 4.15 Å of each other as a function of |i − j|, the separation between the monomers along the chain sequence for simulations in salt-free conditions (only counter ions are included). The fraction of protonated nitrogens in the chain, α, is given in the figure legend. The α values of 0, 0.125, 0.25, 0.5, 0.375, and 0.5 are represented by the red, blue, green, purple, and turquoise lines, respectively.
which can have either trans or gauche+/gauche− states (see Supporting Information Figure S6 for an illustration), with L0 being larger in trans state than in gauche+/gauche− states. Both and the distribution of the torsional angles depend on the protonation state of the chain (Figure 3). The measured average value changes with the protonation level, α, from 3.1 Å, when α = 0−3.9 Å, to when α = 1 (Figure 3a), with more significant increases in occurring for α > 0.5. Additionally, examination of the distributions of observed L0 values (Supporting Information Figure S7a) shows that L0 in a gauche state is fixed at 3.0 Å, whereas L0 for the trans state increases slightly, with the peak changing from 3.75 to 4.0 Å as the protonation level increases. Most of the increase in the trans L0 occurs for α > 0.5 protonation levels for which there will be appreciable amounts of charged nearest-neighbor amines that will electrostatically repel each other. We also see a transition in the behavior of the relative populations of gauche and trans states of the N−C−C−N torsional angle when the chain has high amounts of protonation (Figure 3b,c). When α = 0, the combined total of gauche states is ∼90% of the dihedral angle population, a result in agreement with quantum mechanical investigations of the conformation of 1,2-ethanediamine in the gas phase.30 The gauche state in 1,2-ethanediamine was 1.65 kcal/mol lower in energy than the trans state due to energetic stabilization from hydrogen bonding in the gauche state. Using a Maxwell− Boltzmann distribution, this energy difference can be approximately be translated to relative populations of 94% for gauche and 6% for trans, similar to the values found for α = 0. As the protonation level begins to increase, populations of gauche states begin to diminish slightly. When the chain has high charge, the gauche population begins to decrease more significantly and the chain adopts nearly all trans configurations (90%) when it is fully protonated. These results are relatively insensitive to the presence of salt, as the gauche state percentage is similar for all α values at salt concentrations of 0, 150, and 500 mM (Figure 3c). The impact of the protonation of PEI on the N−C−C−N torsion angle has been examined previously,20,23 with results that largely agree with those found here. Specifically, Basu et al. found that large reductions in gauche state populations only begin when α is larger than 66%, similar to our finding of a large drop in gauche state population between α = 50 and 75%, and that the N−C−C−N torsion angle populations do not depend highly on salt concentration.23 The relationship between protonation and the
Figure 1. Visualizations of PEI chains for the fraction of protonated amines, (a) α = 0, (b) α = 0.25, (c) α = 0.5, (d) α = 0.75, and (e) α = 1.0. To see a monomeric structure of the chain, the images are at different scales. Values of Rg averaged over 90 ns of simulation time are given above the respective images for each chain.
correlates with the observation that, in reality, linear PEI has very low solubility in basic solution when it would have a low amount of protonation. As α increases and nitrogens become protonated, the electrostatic repulsion between these charges increases and the chain expands, reducing the amount of hydrophobic interactions (Figure 1b−d). When α = 1, the chain adopts extended structures that minimize electrostatic interactions between charges (Figure 1e). To quantitatively examine changes in hydrophobic interactions in PEI as a function of α, we counted the number of times that monomer i of the chain contained a carbon atom that was within 4.15 Å of a carbon atom of monomer j as a function of |i − j|, the separation between monomers i and j in the chain sequence, for |i − j| ≥ 3 (Figure 2). A cutoff distance of 4.15 Å was selected after examination of C−C radial distribution functions. When α = 0, there are a significant number of hydrophobic interactions throughout the chain, correlating with the compact configuration shown in Figure 1a. As α increases and the chain expands, the number of hydrophobic interactions quickly decreases and there are only occasional C−C interactions when α > 0.25. To further examine changes in the chain structure with the protonation state of the PEI chain, we monitored , the average monomer−monomer distance, and a key torsional angle related to this distance. Here, the monomer−monomer distance refers to the distance between the nitrogen atoms of adjacent neighboring repeat units, −(NH2−CH2−CH2)−, regardless of the protonation of the amine. Since these two nitrogen atoms are separated by two methylene groups (CH2), L0 depends on the torsional angle defined by N−C−C−N, 7257
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that A = /α. We showed in the previous section that, although increases with α, it remains relatively constant when α is low. Here, we take ∼ 3.1 Å, suitable for PEI with low amounts of protonation, and set A = lB = 7.1 Å, to predict that linear PEI transitions from being weakly charged to highly charged when α ∼ 3.1/7.1 = 0.43. This protonation level occurs roughly at pH ∼ 7.5 Hence, one can expect that when pH drops below 7, the number of amines that are protonated are sufficient for the PEI chain to be considered a highly charged polyelectrolyte. We can examine this prediction by determining the charged amine−charged amine (N+−N+) distance from the simulations (Figure 4). The average N+−N+ distance shown in Figure 4a
Figure 3. (a) as a function of the protonation level, α, in saltfree simulations. (b) Distributions of torsional angle N−C−C−N for different values of α provided in the figure legend in salt-free simulations. The α values of 0, 0.25, 0.5, 0.75, and 1 are represented by the blue, red, green, purple, and turquoise lines, respectively. (c) Percent of populations of torsional angle gauche states at different α. The solid bars represent the gauche population at salt concentrations 0 mM (blue), 150 mM (red), and 500 mM (green). Calculated percent population using dihedral angles less than −120° and greater than 120° for trans and dihedral angles between −120 and 120° for gauche.
Figure 4. (a) Average distance between charged amines, A, as a function of the protonation level, α, for simulations in salt-free conditions. (b) Histogram plot of distances between charged amines, A, for values of α given in the figure legend. The α values of 0.125, 0.25, 0.375, 0.5, 0.75, and 1 are represented by the red, blue, green, purple, orange, and pink lines, respectively. The black dashed lines represent the transition from a weakly to a highly charged polyelectrolyte, where = lB.
relative fractions of gauche and trans states is likely related to changes in hydrogen bonding between nearest-neighbor amines, as charged doublets found when α > 0.5 lack lone pairs on nitrogen atoms that are required for hydrogen bonding and thus the extent of hydrogen bonding between nearest-neighbor amines decreases significantly for α > 0.5 (Supporting Information Figure S7b). Transition from Weakly Charged to Highly Charged Polyelectrolyte. Recall that the dimensionless parameter Γ quantifies the polyelectrolyte as highly (Γ > 1) or weakly (Γ < 1) charged and Γ = lB/A, where the Bjerrum length, lB, is the length at which electrostatic repulsion between two unit charges equals the thermal energy and A is the distance between linear charges along the polyelectrolyte. Thus, the protonation fraction, α, at which PEI transitions from being a weakly charged polyelectrolyte to a highly charged one occurs when lB = A. We may approximate A by assuming that the potentially charged sites in linear PEI are regularly arranged so
smoothly decays with α (the dependence is close to 1/α). One may draw a horizontal line when A = lB = 7.1 Å in Figure 4a; this line intercepts the data approximately at α = 0.38, a value slightly lower than the estimation made using . An examination of histogram plots of N+−N+ distances for different values of α also reveals an important connection between lB and α values near 0.38 (Figure 4b). For low values of α (i.e., α = 0.125 and 0.25), the large majority of N+−N+ distances are longer than lB whereas the large majority of these distances are shorter than lB for large values of α (i.e., α ≥ 0.5). When α = 0.375, there are significant numbers of N+−N+ distances both larger than and smaller than lB. Thus, we will adopt α = 0.38 as marking the transition from a single PEI chain being regarded as a weakly charged polyelectrolyte (Γ < 1) to a highly charged polyelectrolyte (Γ > 1). The observed value α = 0.38 is not too far from the earlier prediction, α = 0.43, whereas we acknowledge that both approaches have limitations, such as the simulations not taking into account the 7258
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bound to the PEI chain when protonation level α is greater than 0.25, consistent with the expectation from Manning’s prediction. Conformational Changes at Zero-Salt Limit. Figure 6a presents the change in chain extension of a single PEI chain in
different microstates that could coexist for a given α as the charges are arranged uniformly along the chain in our simulations. To further examine PEI’s transition from being weakly to highly charged, we examined the extent of counter ion condensation around the PEI as a function of its protonation state. Counter ion condensation is a known phenomenon discussed by Manning.31−33 Manning predicted that for a highly charged polyelectrolyte, characterized by Γ >1, a fraction of counter ions will bind to the polyelectrolyte chain to reduce the charge−charge interaction along the chain.31,34 Counter ion condensation around DNA, which is a highly charged polyelectrolyte with Γ estimated to be as high as 4, has been confirmed by theories, experiments, and simulations.35−38 It is of interest to examine if counter ion condensation occurred for PEI as the protonation level increases and changes from a weakly charged to a highly charged polyelectrolyte. Figure 5a presents the radial distribution functions of N−Cl−
Figure 6. Change in (a) radius of gyration Rg (squares) and end-toend distance R (triangles) and (b) persistence length Lp for a single PEI chain with 40 repeating units as a function of protonation level, α, for simulations in salt-free conditions.
terms of the radius of gyration, Rg, and end-to-end distance, R, as the protonation level increases from 0 to 1. The end-to-end distance, R, increases by more than a factor of 4 over the entire range of protonation, whereas Rg increases by a factor of ∼3. Values of Rg and R for linear PEI chains with 50 repeating units determined using atomistic molecular dynamics simulations have been previously reported for a few values of α.20,22 In most cases, the values reported here agree with these previous reports after accounting for the fact that the PEI chains in this study are only 40 mers. For example, Rg and R for a PEI chain with α = 0.5 has values of 22 ± 2 and 63 ± 20 Å, respectively, compared to previously reported values of ∼25 and ∼70 Å for 50 mer chains.20,22 An exception to this trend is that the unprotonated chain here adopted a more compact configuration than what has been reported previously. We attribute this difference to two factors. First, the smaller size of the chain in comparison with the simulations by Beu et al. was likely due to that study using force field parameters that resulted in higher torsional rigidity21 in the PEI chain, as the value of Rg (9 Å) found here was similar to that found by Choudhury and Roy20 (12 ± 1 Å) after accounting for the difference in chain length. Second, once formed, the compact configuration formed by the unprotonated chain remained relatively stable throughout the simulation, indicating that the end-to-end distance would be highly dependent on the specific configuration that was formed. Finally, we note that PEI is difficult to dissolve in basic solution; therefore, the behavior of unprotonated PEI in solution may have little applicable importance. Figure 6b presents the persistence length Lp determined from the simulation. (The method used to determine Lp is described in the Methods section.) The transition from a
Figure 5. (a) N−Cl− radial distributions for α = 0, 0.25, 0.5, and 1.0, as shown in the figure legend with a ∼1 M NaCl concentration. The α values of 0, 0.25, 0.5, and 1 are represented by the red, blue, green, and purple lines, respectively. (b) Excess g(r) is defined as the integral Å of the radial distribution above the bulk concentration: ∫ 20 2.85 Åg(r) − 1. The integration ranges from 2.85 Å, the distance of closest approach between N and Cl−, to 20 Å, where g(r) approaches 1, the bulk ion density, for all protonation states.
pairs for α = 0, 0.25, 0.5, and 1.0. There are three chloride shells around the N atoms, the first peak at ∼3.5 Å corresponds to the Cl− ions that are adsorbed directly to the nitrogen. The second and third peaks correspond to the Cl− ions that are coordinated around neighboring N atoms or are located nearby in the solvent. The radial distribution functions, g(r), presented in Figure 5a have been normalized to the bulk ion density. Thus, if Cl− ions condense onto the chain, the density of bound ions should be greater than the density of ions in the bulk solution and g(r) will be >1. Figure 5b presents the integral of g(r) − 1 from 2.85 Å, the distance of the closest approach between N and Cl−, and 20 Å, where the Cl− concentration approaches the bulk value for all PEI protonation states. It is clear that there are excess Cl− ions 7259
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weakly charged polyelectrolyte to a highly charged one has a marked difference in the variation of the persistence length Lp. In regime I, where the chain is a weakly charged polyelectrolyte, Lp remains relatively constant; however, Rg and R both increase. This implies that the chain begins to swell in this regime although the chain stiffness reflected in terms of Lp did not change. In regime II, the highly charged polyelectrolyte regime, Lp, Rg, and R, all increase. The differences in the changes in Lp, Rg, and R with protonation may be associated with changes in the hydrophobic interactions discussed earlier. In regime I, the protonation α increases the intrachain electrostatic repulsion, breaking hydrophobic interactions between methylene units that are separated by several repeating units along the chain sequence. Hence, this results in an increase for Rg and R. However, Lp remains relatively constant, reflecting that the chain maintains local flexibility and many N−C−C−N torsional angles remain in the gauche configuration. In regime II, the N+−N+ distance becomes shorter than the Bjeruum length lB; there is significant electrostatic repulsion between the charged amines, giving rise to a large increase in persistence length Lp, and a simultaneous increase in Rg and R. Regime II also includes the protonation states for which charged doublets begin to form (i.e., α > 0.5). Thus, the increased stiffness in this regime is likely related to the N−C−C−N torsional angle increasingly being in the trans configuration. Basu et al. recently determined the persistence length of PEI at several protonation states using atomistic molecular dynamics simulations.23 Their results roughly agree with the data shown in Figure 6b, as they found that Lp ≈ 5 Å for uncharged PEI and increases to ∼40 Å for fully charged chains, with the majority of this increase occurring when the chain was highly charged. We next examine if the conformational statistics of the PEI chain can be described by the Worm-Like-Chain (WLC) model. The WLC model is often used for stiff chains like DNA and has been widely used to analyze DNA conformations and dynamics.39 The WLC model predicts a relationship between R2, Lp, and Lc defined in eq 2 and another relationship between R2g, Lp, and Lc, as shown in eq 340 R2 = 2Lp(Lc + Lp(e−Lc / Lp − 1)) R g2 =
Lp3 ij Lp yz 1 (1 − e−Lc / Lp)zzz LpLc − Lp2 + 2 jjj1 − j z 3 Lc k Lc {
Figure 7. Comparison of values of (a) R (squares) and (b) Rg (triangles) from simulations in salt-free conditions and predicted according to WLC model (dashed lines) as a function of α. The predictions of the WLC model were based on values of Lp (circles) and Lc that were determined using separate data from the simulations.
values observed in the simulations, they do reproduce the trend of an increase with α. This is consistent to what was observed by Brunet et al. when using eqs 2 and 3 for predicting Lp.39 In that study, the end-to-end distance R of DNA was experimentally determined and used to calculate Lp with eq 2. The values of Lp obtained using eq 2 were consistently higher than what is commonly accepted. To further examine when PEI can be characterized using the WLC model, we examined the ratio / as a function of α and again found that the WLC model reasonably describes the conformations of PEI only after the protonational level increased and the chain becomes a highly charged polyelectrolyte chain (see Figure S8 and additional discussion in the Supporting Information). Dependence of Persistence Length on the Ionic Strength. The dependence of persistence length on the ionic strength, especially for DNA, has been a subject of debate in the literature.39,41,42 Initially, Odijk43 and Skolnick and Fixman (OSF)44 proposed that the persistence length of a rodlike highly charged polyelectrolyte such as DNA should depend on Debye screening length κ−1, where lB is the Bjerrum length and A is the distance between the two unit charges along the chain.
(2)
(3)
Thus, eqs 2 and 3 can be used to predict R and Rg for the PEI chain in each simulation based on values of Lc and Lp for the chain in that simulation. The contour length Lc is calculated from the simulations by multiplying the average monomer-tomonomer distance L0 by N, the number of monomers on the chain, Lc = NL0 where L0 was determined in simulations and presented in Figure 4a, whereas values of Lp were determined as discussed in the Methods and shown in Figure 6. Figure 7 presents the comparison between the predicted Rg and R according to the WLC model (dashed lines) and the actual values determined in simulations. Two things can be concluded from this comparison. First, in regime I, the weakly charged polyelectrolyte regime, the WLC model is not applicable; the predicted R and Rg remain essentially flat since Lp and Lc are essentially constant. There are other mechanisms that contribute to the increase in Rg and R. Second, in regime II, the highly charged regime, although these equations consistently underestimate the actual R and Rg
L POSF = L0 +
lB 4κ 2A2
(4)
LOSF p
This implies that is linearly dependent on ionic strength as 1/I. OSF theory assumes that the polyelectrolyte is rodlike at zero-salt limit. As the ionic strength increases, the electrostatic repulsion between charged monomers becomes screened and the electrostatic contribution to the persistent length decreases. A plot of tthe persistent length as a function of ionic strength should exhibit 1/I dependence. This dependence was confirmed with dsDNA in some experimental studies,45 but more recent studies over a broader range of ionic strength and divalent cations showed discrepancies.39 Various modifications to OSF theory have been proposed in the literature,42,46,47 most of them trying to take ion condensation phenomena discussed by Manning into account, leading to a 7260
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dependence of Lp with 1/I began to emerge especially when α = 1. (Figure 8c). However, we found that the interpolation formula proposed by Brunet et al. (eq 5) fits the data better than the linear dependence on 1/I, especially for α = 0.5. Additionally, we find the somewhat surprising result that, at high salt concentration, the persistence length of the half protonated chain is less than that of the chain with α = 0.25. This behavior may be explained by considering that the α = 0.5 chain is in the highly charged regime and is thus predicted to attract and condense significantly more counter ions than the α = 0.25 chain, reducing intrachain electrostatic repulsion and increasing flexibility. The fully protonated chain is much stiffer than the other chains for all ionic strengths studied. Conformational Changes of PEI at Different Salt Concentrations. Figure 9 presents the observed conforma-
more complicated dependence on ionic strength. Recently, Brunet et al. examined many of these modifications by employing high-throughput tethered particle motion and found that they are inconsistent with empirical data. In the end, they proposed a nontheoretical interpolation formula for large ranges of ionic strengths (eq 5) and showed that it fits agreeably to experimentally measured persistence lengths of dsDNA,39 where L0p is the persistence length with no salt, L∞ p is the persistence length at extremely high salt concentration, I0 is the critical ionic strength, and δ is a scaling constant. Lp =
Lp∞
+
Lp0 − Lp∞ 1+
δ
() I I0
(5)
The dependence of Lp on 1/I for linear PEI with α = 0.25, 0.5, and 1 is shown in Figure 8. When α = 0.25, PEI is weakly charged and Lp fluctuates around a constant value for all I studied, showing no ionic strength dependence (Figure 8a). The average value is represented by the solid line. This shows that when PEI is weakly charged, OSF theory is not applicable. In the highly charged polyelectrolyte regime, the linear
Figure 9. Change of Rg with protonation level, α, at different salt concentrations given in the figure legend. The fits for the NaCl concentrations of 0, 150, and 500 mM are represented by the blue, red, and green dashed lines, respectively. The NaCl concentrations of 0, 150, and 500 mM are represented by the blue diamonds, red squares, and green triangles, respectively.
tional change of the PEI chain at three different salt concentrations: zero salt and 150 and 500 mM salt. The Rg value of the chain in 500 mM salt is consistently lower than that in the zero-salt limit. The transition from a weakly charged polyelectrolyte to a highly charged one occurred at the same protonation level in different salt conditions since the average charge−charge distance (shown in Figure 4a for the zero-salt condition) remained the same at different salt concentrations. Rg for various values of α has also been recently calculated using molecular dynamics simulations by Basu et al. for 20 mer PEI chains in 50 and 150 mM salt solutions.23 Despite differences in the chain lengths and force field parameters, the change in Rg as a function of α for PEI chains at relatively high salt (150 mM in Basu et al., 150 and 500 mM in this study) has many consistent features in the two simulation studies. At very low protonation, there is little change in Rg with increasing α but, as α continues to increase, there is an increase and then a plateau in Rg near α = 0.5. Finally, Rg increases with α for α > 0.5. To examine how changes in Rg depend on pH, we combined data showing the variation in Rg with α from the atomistic simulations described here (Figure 9) with data showing the variation in α with pH from our previous Monte Carlo titrations5 (Figure 10a). Combining the results of the studies gives a plot of Rg versus pH for three different salt concentrations, as shown in Figure 10b. At pH ≥ 8, the single PEI chain has low amounts of protonation and the chain is collapsed, which is consistent with the fact that PEI is insoluble in basic solution. As the pH is lowered, the chain expands, resulting in a larger Rg. Notably, Figure 10b shows that linear PEI undergoes a significant expansion for pH changes that are
Figure 8. Variation of persistence length of a single PEI chain with ionic strength, I, at three different protonation levels (a) α = 0.25; (b) α = 0.5; (c) α = 1.0. The solid line in (a) is the average Lp since there is no apparent 1/I dependence. The solid lines in (b) and (c) are fits to eq 5. 7261
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with increasing acidity starting at a pH of slightly above 6 before plateauing near pH 3. In comparison, our simulations predict that PEI size plateaus at a similar pH but that it begins to expand at a more basic pH than that found in experiments. However, direct comparisons with this experiment are complicated by the factors including interchain interactions in the experiments and solubility effects (i.e., the increase in PEI size from pH 8 to 6 in the simulations may be mostly due to PEI conformations and protonation states that would not be soluble).
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CONCLUSIONS We used atomistic molecular dynamics simulations to study the structural properties of 40 mer linear PEI at 9 protonation states ranging from an unprotonated to a fully protonated chain at various salt concentrations, a wider range of conditions than have been examined in previous studies.20,21,23 Our simulations show that the PEI continues to gradually expand as the fraction of protonated sites increases from 0 to 1; we do not observe a sharp change in the PEI conformation, such as the one that has been experimentally observed for P2VP.49,50 However, we do observe that there are two general regimes of behavior as the protonation of PEI increases that results from two related factors. First, when a fraction of ∼0.4 of the PEI amine groups are protonated, PEI transitions from being a weakly charged polyelectrolyte to a highly charged polyelectrolyte. Second, increasing the fraction of protonated amines above 0.5 necessitates the formation of doublets of charged nearest-neighbor amines. Because PEI charges are along its backbone, these doublets must be close to each other (within ∼4 Å), reinforcing that the chain should behave similar to a highly charged electrolyte. When the chain has low protonation (i.e., in the weakly charged polyelectrolyte regime when the formation of doublets is rare), the chain expands as the protonation fraction increases, as electrostatic repulsion between charged groups begins to overcome hydrophobic interactions that hold the chain in compact conformations at very low protonation states. However, we find that the average monomer−monomer distance along the chain as well as the persistence length is relatively constant in this regime. In contrast, when the chain has high protonation (i.e., in the highly charged polyelectrolyte regime when doublets form), increases in protonation state are accompanied by increases in the stiffness/persistence length of the chain and in the average distance between monomers as intrachain hydrogen bonding diminishes and N−C−C−N dihedral angles increasingly adopt trans configurations. Several of these results agree with recent simulations by Basu et al., who also found that there are two regimes of behavior of PEI corresponding to weakly and highly charged states and that changes in dihedral angle populations were associated with the transition between these two regimes. Additionally, both studies found that analytical theories for polyelectrolytes were not able to adequately describe the behavior of PEI across the full range of possible protonation states. One important feature of the simulations in this study is that they can be used to understand changes in the structure of linear PEI during the biological environments encountered in its role as a gene delivery vector. This process is believed to involve the PEI gene delivery complex being injected into blood (pH ∼7.4), entering cells through endocytosis, and, eventually, the PEI aiding in the release of the complex from acidic endosomes that may have pH as low as 4.5.3 On the
Figure 10. (a) Titration data obtained using our house-developed Monte Carlo code for a PEI chain of length 40 in three salt concentrations. (b) The predicted dependence of Rg on pH by combining the molecular dynamic simulation results in Figure 9 and the titration data in panel (a). The NaCl concentrations of 0, 150, and 500 mM are represented by the blue, red, and green solid lines, respectively.
likely to be encountered during the gene delivery process. Whereas blood pH is regulated around 7.4, endosomes and lysosomes are more acidic with pH values that range from 6.5 to 4.5.3 Thus, the Rg of PEI may increase by approximately a factor of two as it transitions through the endosome pathway during gene delivery. The concentration of Cl− within endosomes is also known to increase from approximately 40 mM to as high as 100 mM during their development,48 likely leading to further expansion of the PEI chain.10 However, we note that the computational titrations used to determine the protonation of PEI as a function of pH likely overestimate the impact of ionic strength;5,15 therefore, increasing salt concentration may not have as large of an impact on PEI conformation, as indicated by Figure 10. The results shown in Figure 10b can be compared with previous experimental investigations of the conformational change of a different polyelectrolyte, poly-2-vinylpyridine (P2VP), as a function of pH.49,50 As the pH was lowered, P2VP become more protonated and the chain expanded, resulting in a decreased diffusivity that was measured by singlechain fluorescence correlation spectroscopy. The experiments revealed that the conformation transition was very sharp for P2VP, allowing for the identification of a critical pH for the transition. The critical pH was found to decrease when the salt concentration decreased. The transition in PEI is less sharp compared to that reported for P2VP, partly due to the fact that PEI has buffering capacity. The chain picks up the protons gradually as the pH is lowered, unlike P2VP, in which the transition occurs more abruptly. However, the conformation transition in PEI also occurs at lower pH in lower salt, consistent with behavior reported for P2VP. Additionally, one recent experiment has provided a plot of the hydrodynamic radius of linear PEI as a function of the observed pH.23 In this experiment, the size of PEI in 10 and 150 mM salt increases 7262
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Sponge-Based Rupture of Endosomal Vesicles. ACS Nano 2018, 12, 2332−2345. (7) Wagner, E. Polymers for SiRNA Delivery: Inspired by Viruses to Be Targeted, Dynamic, and Precise. Acc. Chem. Res. 2012, 45, 1005− 1013. (8) Miyata, K.; Nishiyama, N.; Kataoka, K. Rational Design of Smart Supramolecular Assemblies for Gene Delivery: Chemical Challenges in the Creation of Artificial Viruses. Chem. Soc. Rev. 2012, 41, 2562− 2574. (9) Benjaminsen, R. V.; Mattebjerg, M. A.; Henriksen, J. R.; Moghimi, S. M.; Andresen, T. L. The Possible proton Sponge Effect of Polyethylenimine (PEI) Does Not Include Change in Lysosomal PH. Mol. Ther. 2013, 21, 149−157. (10) Funhoff, A. M.; van Nostrum, C. F.; Koning, G. A.; Schuurmans-Nieuwenbroek, N. M. E.; Crommelin, D. J. A.; Hennink, W. E. Endosomal Escape of Polymeric Gene Delivery Complexes Is Not Always Enhanced by Polymers Buffering at Low PH. Biomacromolecules 2004, 5, 32−39. (11) Walker, G. F.; Fella, C.; Pelisek, J.; Fahrmeir, J.; Boeckle, S.; Ogris, M.; Wagner, E. Toward Synthetic Viruses: Endosomal PHTriggered Deshielding of Targeted Polyplexes Greatly Enhances Gene Transfer in Vitro and in Vivo. Mol. Ther. 2005, 11, 418−425. (12) Fella, C.; Walker, G. F.; Ogris, M.; Wagner, E. Amine-Reactive Pyridylhydrazone-Based PEG Reagents for PH-Reversible PEI Polyplex Shielding. Eur. J. Pharm. Sci. 2008, 34, 309−320. (13) Smits, R. G.; Koper, G. J. M.; Mandel, M. The Influence of Nearest- and Next Nearest-Neighbor Interactions on the Potentiometric Titration of Linear Poly(Ethylenimine). J. Phys. Chem. 1993, 97, 5745−5751. (14) Lee, H.; Son, S. H.; Sharma, R.; Won, Y. A Discussion of the PH-Dependent Protonation Behaviors of Poly(2-(Dimethylamino)Ethyl Methacrylate) (PDMAEMA) and Poly(Ethylenimine-Ran-2Ethyl-2-Oxazoline) (P(EI-r-EOz)). J. Phys. Chem. B 2011, 115, 844− 860. (15) Curtis, K. A.; Miller, D.; Millard, P.; Basu, S.; Horkay, F.; Chandran, P. L. Unusual Salt and PH Induced Changes in Polyethylenimine Solutions. PLoS One 2016, 11, No. e0158147. (16) Ziebarth, J.; Wang, Y. Molecular Dynamics Simulations of DNA-Polycation Complex Formation. Biophys. J. 2009, 97, 1971− 1983. (17) Ziebarth, J. D.; Kennetz, D. R.; Walker, N. J.; Wang, Y. Structural Comparisons of PEI/DNA and PEI/SiRNA Complexes Revealed with Molecular Dynamics Simulations. J. Phys. Chem. B 2017, 121, 1941−1952. (18) Shen, J. W.; Li, J.; Zhao, Z.; Zhang, L.; Peng, G.; Liang, L. Molecular Dynamics Study on the Mechanism of Polynucleotide Encapsulation by Chitosan. Sci. Rep. 2017, 7, No. 5050. (19) Meneksedag-Erol, D.; Tang, T.; Uludaǧ , H. Molecular Modeling of Polynucleotide Complexes. Biomaterials 2014, 35, 7068−7076. (20) Choudhury, C. K.; Roy, S. Structural and Dynamical Properties of Polyethylenimine in Explicit Water at Different Protonation States: A Molecular Dynamics Study. Soft Matter 2013, 9, 2269−2281. (21) Beu, T. A.; Farcaş, A. CHARMM Force Field and Molecular Dynamics Simulations of Protonated Polyethylenimine. J. Comput. Chem. 2017, 38, 2335−2348. (22) Kim, I.; Pascal, T. A.; Park, S. J.; Diallo, M.; Goddard, W. A.; Jung, Y. PH-Dependent Conformations for Hyperbranched Poly(Ethylenimine) from All-Atom Molecular Dynamics. Macromolecules 2018, 51, 2187−2194. (23) Basu, S.; Venable, R. M.; Rice, B.; Ogharandukun, E.; Klauda, J. B.; Pastor, R. W.; Chandran, P. L. Mannobiose-Grafting Shifts PEI Charge and Biphasic Dependence on PH. Macromol. Chem. Phys. 2019, 220, No. 1800423. (24) Case, D. A.; Babin, V.; Berryman, J. T.; Betz, R. M.; Cai, Q.; Cerutti, D. S.; Cheatham, T. E., III; Darden, T. A.; Duke, R. E.; Gohlke, H.;et al.. AMBER 14; University of California: San Francisco, 2014.
basis of Monte Carlo titrations, this change in pH (7.4−4.5) should result in a significant increase in the fraction of protonated PEI amines (from α = ∼0.14 to ∼0.6) and a corresponding expansion of the PEI chain, with the Rg of the chain approximately doubling in this pH range Figure 10). Notably, these results indicate that the changing PEI protonation states encountered during gene delivery include PEI transitioning from behaving like a weakly charged polyelectrolyte to a highly charged one and that charged doublets may begin to form. Thus, PEI may begin to stiffen and the length between monomers may begin to increase during gene delivery, factors that have the potential to impact interactions between PEI and nucleic acids or membranes, accounting for the success of PEI as a gene delivery vector.
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ASSOCIATED CONTENT
* Supporting Information S
The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acsomega.9b00066. Plots of exponential decay functions fitted to cos θij vs |i − j| for protonation fractions of α = 0, 0.25, 0.5, 0.75, and 1 (Figures S1−S5); structural illustrations of L0 for trans and gauche torsional angles (Figure S6); probability distributions of monomer−monomer distances L0 and the number of hydrogen bonds between neighboring monomers for different α values (Figure S7); Lp/Lc and R2/Rg2 as a function of α and discussion of the values of these ratios in the simulations and analytical theories (Figure S8) (PDF)
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AUTHOR INFORMATION
Corresponding Author
*E-mail:
[email protected]. ORCID
Yongmei Wang: 0000-0002-7418-9489 Funding
We acknowledge the partial financial support for this work from NIH/NIGMS, 1R15GM106326-01A. Notes
The authors declare no competing financial interest.
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REFERENCES
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