Release of cationic polymer-DNA complexes from the endosome: Atheoretical investigation of the proton sponge hypothesisShuang Yang and Sylvio May Citation: J. Chem. Phys. 129, 185105 (2008); doi: 10.1063/1.3009263 View online: http://dx.doi.org/10.1063/1.3009263 View Table of Contents: http://jcp.aip.org/resource/1/JCPSA6/v129/i18 Published by the American Institute of Physics. Additional information on J. Chem. Phys.Journal Homepage: http://jcp.aip.org/ Journal Information: http://jcp.aip.org/about/about_the_journal Top downloads: http://jcp.aip.org/features/most_downloaded Information for Authors: http://jcp.aip.org/authors

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Release of cationic polymer-DNA complexes from the endosome:A theoretical investigation of the proton sponge hypothesis

Shuang Yang and Sylvio Maya�

Department of Physics, North Dakota State University, Fargo, North Dakota 58105-5566, USA

�Received 1 August 2008; accepted 7 October 2008; published online 14 November 2008�

Polyplexes are complexes composed of DNA and cationic polymers; they are promising transportvehicles for nonviral gene delivery. Cationic polymers that contain protonatable groups, such aspolyethylenimine, have been suggested to trigger endosomal escape of polyplexes according to the“proton sponge hypothesis.” Here, osmotic swelling is induced by a decrease in the endosomal pHvalue, leading to an accumulation of polymer charge accompanied by the influx of Cl− ions tomaintain overall electroneutrality. We study a theoretical model of the proton sponge mechanism.The model is based on the familiar Poisson–Boltzmann approach, modified so as to account for thepresence of ionizable polyelectrolytes within self-consistent field theory with assumed ground statedominance. We consider polyplexes, composed of fixed amounts of DNA and cationic polymer, tocoexist with uncomplexed cationic polymer in an enclosing vesicle of fixed volume. For such asystem, we calculate the increase in osmotic pressure upon moderately decreasing the pH value andrelate that pressure to the rupture tension of the enclosing membrane. Our model predicts membranerupture upon pH decrease only within a certain range of free polymer content in the vesicle. Thatrange narrows with increasing amount of DNA. Consequently, there exists a maximal amount ofDNA that can be incorporated into a vesicle while maintaining the ability of content release throughthe proton sponge mechanism. © 2008 American Institute of Physics. �DOI: 10.1063/1.3009263�

I. INTRODUCTION

Complexes formed from cationic polymers andDNA—so called “polyplexes”—constitute one of the mostattractive nonviral delivery systems of genetic material intoliving cells.1–3 Transfection rates of nonviral vectors gener-ally lag behind their viral counterparts. However, nonviralvectors, including polyplexes, offer a number of advantagessuch as nonimmunogenicity and large-scale manufacturabil-ity which have stimulated the development and optimizationof sophisticated vectors with improved transfection rates.

Nonviral vectors are typically internalized into cellsthrough endocytosis upon which they become enclosed in atransport vesicle derived from the plasma membrane—theearly endosome.4 Further along the endocytotic pathway thevector may be trafficked to late endosomes and, eventually,to lysosomes where an increasingly acidic environment acti-vates degradative enzymes. One of the bottlenecks in thedelivery pathway is the timely endosomal �or lysosomal� re-lease of the vector in order to escape enzymatic degradation.Viral vectors are equipped with the fusion machinery of theviral host whereas cationic lipid-based nonviral vectors �“li-poplexes”� often contain fusogenic lipids �such as phosphati-dylethanolamine� that trigger their endosomal release.5 Therelease of polyplexes can be enhanced by attaching exog-enous endosomolytic agents such as fusogenic peptides tothe cationic polymer.2 Yet, some cationic polymers have spe-cial properties that enable them to trigger their endosomalrelease through an independent putative mechanism, referred

to as the “proton sponge hypothesis.”6 Here, the cationicpolymer contains groups that become protonated at �orslightly below� physiological pH. Upon a decrease in theendosomal pH value from 7 to about 5 the charge of thecationic polymer increases, invoking a corresponding influxof negatively charged Cl− ions to maintain overall electro-neutrality. In addition, the cationic polymer’s newly acquiredcharges may cause swelling of the polyplexes. Both influxand swelling induce an increased osmotic pressure and thus adestabilization of the endosome. There is considerable ex-perimental support—both qualitative and quantitative—forthe mechanism underlying the proton sponge hypothesis.6–12

At the same time, a multitude of other factors,3 including theunpackaging efficiency of the complexes and transfer to thenucleus after endosomal release, also affect transfectionactivity.13–15

Examples of polyplexes that are candidates to escape theendosome according to the proton sponge hypothesis includecationic polymers such as polyethylenimine �PEI�,16,17 polya-midoamine �PAMAM�,7 various dendrimers �including PEIand PAMAM in their dendrimeric form�,18 partially acety-lated PEI,14,15,19 PEI grafted with hydrophobic chains,20 andpolymers containing imidazole groups.21 Especially PEI-based polyplexes, which were introduced in a seminal workby Boussif et al.,8 constitute a strong proton sponge becauseevery third atom along the polymer backbone is a �poten-tially charged� nitrogen. At physiological pH a rather smallfraction, 15%–20%, of all protonatable amine groups are ac-tually charged; endosomal pH 5 increases that fraction toabout 45%.22

The opposite charges of PEI and DNA generally drivea�Electronic mail: sylvio.may@ndsu.edu.

THE JOURNAL OF CHEMICAL PHYSICS 129, 185105 �2008�

0021-9606/2008/129�18�/185105/9/$23.00 © 2008 American Institute of Physics129, 185105-1

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the formation of complexes between the two macroions. Yet,shape �globular versus toroidal aggregates�, microscopicstructure, and compactness of the polyplex depend on vari-ous factors such as molecular weight and degree of branch-ing of the cationic polymer, the amount of added salt, andhydrophobic interactions.16,23–26 An important experimen-tally controllable parameter is the molar ratio N / P of PEInitrogen atoms to DNA phosphate groups. At 20% proto-nated nitrogen N / P=5 leads to electroneutrality between PEIand DNA. Optimal transfection is commonly achieved forN / P�5. For example, Choosakoonkriang et al.27 reportedoptimal transfection at N / P=6–10, where relatively small��100 nm�, positively charged polyplexes are formed. AtN / P=6–10 it was found that 86% of the PEI moleculeswere in a free �uncomplexed� form. Each polyplex containedon average 3.5 plasmids and 30 molecules of PEI. The DNAdensity inside the polyplex was found significantly smallerthan that in toroidal condensates induced by multivalentions.28 That is, only 10% of the polyplex volume is occupiedby DNA �in contrast to 72% within toroidal aggregatesformed by multivalent cations29�.

In the present work we study the proton sponge hypoth-esis using a theoretical model. Our major goal is to identifyconditions at which acidification of the endosome leads torupture of the endosomal membrane. More specifically, weshall consider a vesicle with enclosed polyplex and free, un-complexed, cationic polymer. The structure of the polyplexwill be calculated based on a modified Poisson–Boltzmannmodel that was suggested in Refs. 30–34. We note that thereis a multitude of other approaches to model complex forma-tion between oppositely charged polymers, as reviewedrecently.35–37 The model used in the present work is simpleand enables us to calculate the free energy of a polyplex �andthus also the corresponding osmotic pressure�. We shall showthat rupture of the enveloping endosomal membrane upon amoderate decrease in the pH value is a feasible process, butonly if a sufficiently large amount of free, uncomplexed, cat-ionic polymer is also present in the endosome.

II. THEORY

We consider a polyplex formed from flexible cationicpolymers and DNA molecules. As DNA is comparativelystiff �with a persistence length of about 50 nm� we model itas a straight cylinder with radius r0 and uniform negativelinear charge density � �number of charges per unit lengthalong the DNA backbone�. We adopt the common Wigner–Seitz cell model where DNA rods are parallel and form ahexagonal array. The hexagonal Wigner–Seitz cell aroundeach rod is conveniently approximated by a cylindrical cellof the same volume and corresponding radius R, as shown inFig. 1. The cylindrical unit cell renders all physical quantitiesdependent on only the radial distance r from the long axis ofthe DNA. Hence, the volume of the cylindrical unit cell �ofunit length L along its long axis� is V=�R2L. The model weemploy in the present work was introduced by Borukhov etal.;32 it is based on the Poisson–Boltzmann theory supple-mented by a polymer contribution. The polymer contributionis described by the ground state dominance of self-consistent

field theory.38 Here, the cationic polymer is modeled by anorder parameter �=��r� which specifies the local volumedensity �2 of the polymer as function of the radial position r.The phenomenological free energy per unit cell, F, expressedin units of the thermal energy kBT, can be written as

F

kBT=

1

8�lB�

V

dv����2+ �V

dv�n+ lnn+

n0+ − n+ + n0

+�+ �

V

dv�n− lnn−

n0− − n− + n0

−�+ �

V

dv�a2

6����2 +

�

2�4�

+ �V

dv�2�� ln�

p+ �1 − ��ln

1 − �

1 − p�

+ �p − �V

dv�2� . �1�

The first line in Eq. �1� describes the electrostatic energy ofthe unit cell in terms of the dimensionless electrostatic po-tential �=��r�=e� / �kBT� and the Bjerrum length lB

=e2 / �4���0kBT�, where � denotes the electrostatic poten-tial, e is the elementary charge, � is the dielectric constant ofwater, and �0 is the permittivity of free space. The secondand third lines in Eq. �1� account for the ideal mixing entro-pies of the salt ions; n+=n+�r� and n−=n−�r� are the localconcentrations of positive and negative monovalent salt ions,respectively. Positive and negative salt ions are in thermalequilibrium with a reservoir of bulk concentrations n0

+ andn0

−, respectively. Note that if there is no charged polymerpresent in the bulk then n0

−=n0+=n0, where n0 denotes the

bulk salt concentration. Line four in Eq. �1� expresses thefree energy contribution of the cationic polymer; the twoterms in that line embody the connectivity of polymer chains�with a being the Kuhn length� and effective excluded-volume interactions �with interaction constant ��. The term inline five of Eq. �1� describes the entropic contribution to thefree energy of the dissociation equilibrium along the cationicpolymer chains. Here, �=��r� is the average charge fraction�with 0 � 1� per ionizable group of the cationic polymer.Dissociable groups along the polymer chains are describedaccording to a noninteracting lattice gas model with nominalprobability p to find a given site occupied when ��0. Simi-

R

r0

L

ξ

FIG. 1. Schematic representation of a cylindrical unit cell of radius R andunit length L. The cell is centered about a DNA segment of radius r0

=1 nm and �=6 negative charges per unit length L=1 nm. The aqueousregion of the cell contains pointlike mobile salt ions and cationic polymer.

185105-2 S. Yang and S. May J. Chem. Phys. 129, 185105 �2008�

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larly, ��r� is the probability to find an ionizable groupcharged when exposed to an external potential �. Note firstthat, for simplicity, we assume each monomeric unit of thecationic polymer to carry one single dissociable group�hence, the prefactor �2�. Second, p, the charge fraction ofthe cationic polymer in a bulk solution �where ��0�, isregulated by the pH value of the aqueous solution.22

Finally, the last term in Eq. �1� enables us to conservethe volume p occupied by cationic polymer in the unit cellof the polyplex. To this end, we need to choose the corre-sponding Lagrangian multiplier appropriately. We note thatall integrations in Eq. �1� extend over the volume V of theunit cell �with the understanding that the cationic polymer issterically excluded from the interior of the DNA, and withthe assumption that there is no electric field inside the DNA,the latter being justified because of the cylindrical symmetryof the unit cell�.

In thermal equilibrium the free energy adopts its globalminimum with respect to all unconstrained variables. Func-tional minimization of the unit cell’s free energy F=F�n+ ,n− ,�� yields the ion concentrations n+=n0 exp�−��,n−=n0 exp���, the local charge fraction of the cationic poly-mer

� =1

1 +1 − p

pe�

, �2�

and the following relation for the polymer order parameter:

a2

6�2� = ��3 + �� ln

�

p+ �1 − ��ln

1 − �

1 − p+ �� − .

�3�

Using these distributions �and inserting them into the expres-sion for the local charge density �=e�n+−n−+�2�� allows usto express Poisson’s equation �2�=−4�lB� /e as well as Eq.�3� as two coupled differential equations,

�2� = 4�lB�2n0 sinh � −�2

1 + e��1 − p�/p� ,

�4�a2

6�2� = ��3 − � ln�1 − p + pe−�� − � ,

that specify the potential � and polymer order parameter �.As mentioned above, the Lagrangian multiplier can bechosen so as to fix the total volume p= Vdv�2 occupied bythe cationic polymer in the unit cell.

Note that due to the cylindrically symmetric unit cellEqs. �4� constitute two ordinary differential equations �withthe Laplacian �2=d2 /dr2+ �1 /r�d /dr� for ��r� and ��r�.They must be solved subject to appropriate boundary condi-tions at r=R and at r=r0. In the former case, at r=R,symmetry dictates vanishing slope of both ��r� and ��r�,implying

� ��

�r�

R

= 0, � ��

�r�

R

= 0. �5�

In the latter case, at r=r0, the boundary conditions accountfor the fixed charges attached to and for the depletion ofpolymer from the surface of the DNA. Thus,

� ��

�r�

r0

= 2�lB

r0, ��r0� = 0. �6�

As discussed previously,30,33 the condition ��r0�=0 appearsas the nonadsorbing limit of the more general case��� /�r�r0

=��r0� /D, where the length D is related to the non-electrostatic adsorption strength of the polymer to the mac-roion’s surface.

Solving numerically the differential Eqs. �4�, subject tothe boundary conditions, Eqs. �5� and �6�, yields ��r� and��r� for any given volume V of the unit cell �or, equivalently,R� and volume p of polymer in the unit cell. Inserting ��r�into the equilibrium distributions for n� and � allows us—together with ��r�—to calculate the free energy F=F�p ,V� per unit cell according to Eq. �1�. Note that wealso need to specify the parameter p �the charge fraction ofthe cationic polymer in the bulk�. Modeling approaches forthe relation p�pH� based on the dissociation equilibrium of apolyelectrolyte chain exist,39,40 but in the present work weshall simply use the experimentally reported values p=0.15and p=0.45 for pH values of 7 and 5, respectively.22

The free energy F=F�p ,V� is subject to all limitationsinherent in the Poisson–Boltzmann theory. These include theassumption of pointlike ions and the neglect of ion-ion cor-relations. Moreover, the underlying assumption of groundstate dominance within self-consistent field theory of thepolymer is strictly valid only in the limit of infinitely longlinear chains. Finally, the representation of the polyplex by aunit cell ignores both packing defects and surface effects ofthe polyplex. Nevertheless, the present approach appealsthrough its simplicity and makes predictions that can betested experimentally.

III. RESULTS AND DISCUSSION

Under experimental conditions polyplexes are producedby mixing DNA with an excess solution of cationic polymer.After endosomal uptake of the polyplex and, usually, someamount of free cationic polymer, the total amount of cationicpolymer remains fixed during the decrease in the endosomalpH. If polyplex is present together with free cationic poly-mer in the endosome, thermodynamic equilibrium is estab-lished between these two phases.

We develop our thermodynamic model accordingly:First, in Sec. III A we calculate the equilibrium structure of apolyplex upon mixing DNA with excess cationic polymer.This will give rise to polyplexes with a certain equilibriumratio �N / P�p of PEI nitrogen atoms to DNA phosphategroups. We then investigate the structure of the polyplex atfixed �N / P�p as function of decreasing pH value. Second, inSec. III B we analyze the rupture mechanism of the endo-some. To this end, we consider the coexistence of polyplexand free cationic polymer in the endosome. Below we shall

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argue that it is appropriate to fix the amount of cationic poly-mer in both phases which leaves the spatial extension of eachphase as the only thermodynamic degree of freedom. Weshall study whether a pH decrease will increase the lateraltension of the lipid membrane enclosing the endosome be-yond the rupture tension.

In all our calculations we use the following fixed set ofparameters: r0=1 nm �radius of the rodlike macroion�, �=6 �number of charges per unit length L=1 nm along therod�, n0=0.09 /nm3 �corresponding to a 0.15M concentrationof monovalent salt�, a=0.5 nm �monomer size of the poly-mer�, and �=0.4a3 �excluded-volume parameter�. Thesechoices are motivated by typical experiments using PEI �or asimilar cationic polymer� and DNA.

A. Polyplex energy and structure

As specified above, we first consider the equilibriumstructure of a polyplex formed upon mixing DNA with ex-cess cationic polymer. To this end, let us assume we dissolvea certain amount of DNA in an excess aqueous solution ofcationic polymer with volume fraction �b=0.006a3 /nm3

=0.000 75 of monomer segments �corresponding to a con-centration of 10 mM�.8 To investigate polyplex formation inthis situation we need to consider an appropriate thermody-

namic potential F̃. Recall that in F�p ,V� �defined in Eq. �1��the total amount p of polymer in the unit cell appears asthermodynamic variable. Yet, for thermal equilibrium withan excess polymer solution the chemical potential of thepolymer is fixed. We thus define the new thermodynamic

potential F̃� ,V�=F�p ,V�−p, where —introduced as aLagrangian multiplier in Eq. �1�—now appears to be thepolymer’s chemical potential. Note also that the presence ofcationic polymer in the bulk demands n0=n0

+=n0−− p�0

2 inorder to ensure overall electroneutrality.

The relevant excess free energy of forming a complex,33

starting with a dilute DNA solution �where the DNA mol-

ecules are separated by large distances� is �F̃� ,V�= F̃� ,V�− F̃0, where F̃0=V���b

4 /2−�b2� is the free energy

of a homogeneous polymer solution. In Fig. 2 we plot

�F̃� ,V=�R2L� as a function of the cell radius R. As is well

known from previous studies32,41 there is a minimum in �F̃at intermediate R. The minimum results from the connectiv-ity of the polymer and thus reflects nonelectrostatic interac-tions between the polyelectrolyte charges. In the presentcase, the minimum is located at R=1.8 nm, and the corre-sponding amount of polymer located in the unit cell is p

=1.29 nm3; see the inset of Fig. 2. Hence, within the poly-plex �N / P�p=p / ��a3�=1.73.

Let us now analyze the energetics and structure of thepolyplex for fixed amount of polymer p=1.29 nm3 in theunit cell. Here, the relevant thermodynamic potential isF�p ,V�; see Eq. �1� and previous work by Podgornik.42 Fig-ure 3 shows F�p=1.29 nm3, V=�LR2� as a function ofcell radius R for different choices of p. Compare the curvefor p=0.15 to the corresponding curve in Fig. 2 �which isalso derived for p=0.15�: The minimum in the free energyfor fixed chemical potential of the polymer shifts from R

=1.8 nm to the somewhat larger value of R=2.3 nm whenconserving the amount of polymer in the unit cell. Indeed,fixing p removes the pressure from the polymer outside theunit cell and thus the cell must expand. We note that R=2.3 nm leads to a volume density of slightly less than 20%for DNA in the polyplex. This value is somewhat larger thanthe �10% volume density estimated by Clamme et al.28

based on spectroscopic measurements.Before discussing the free energy F�p ,V� for varying p

it is useful to present a number of structural properties of thepolyplex. A decrease in the pH value of the ambient solutionleads to an increasing protonation of the cationic polymerand thus to larger p. We expect that an initially weaklycharged polymer �p=0.15� will exhibit a more favorableelectrostatic interaction with the DNA upon increasing itspropensity to adopt a higher charge fraction. This is corrobo-rated in Fig. 4 which displays ��r�2 for various p at R=5 nm. �Qualitatively similar results are found for differentchoices of R.� Upon increasing p the polymer condensesmore tightly onto the DNA rod. We also remark on the be-havior of the reduced electrostatic potential ��r�, which isplotted in the inset of Fig. 4, again for different p �and fixed

FIG. 2. Excess free energy per unit cell, �F̃, as function of cell radius R.The polyplex coexists with cationic polymer of fixed volume fraction �b

=0.000 75 in the bulk. The charge fraction of the cationic polymer in thebulk is p=0.15. The inset shows the corresponding amount �volume per unitcell� of cationic polymer, p. The dashed lines in the inset mark the equi-librium values for R and p.

FIG. 3. The free energy per unit cell F�p=1.29 nm3,V=�LR2� of a poly-plex as function of cell radius R, plotted for different values of p. Note that�in contrast to Fig. 2� the amount of cationic polymer p=1.29 nm3 is fixed.

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p=1.29 nm3 and R=5 nm�. The most interesting feature isthat for sufficiently high p the potential ��r� becomes posi-tive, indicating an overcompensation of the DNA charges bythose of the polymer and salt ions. This phenomenon is oftenreferred to as “overcharging”43 and can occur as a result ofion-ion correlations in a simple electrolyte,44 but also as aresult on nonelectrostatic interactions in more complexelectrolytes.45 For example, it has been shown that the pres-ence of rodlike ions �two like charges connected by a rod offixed distance� can lead to overcharging, even within thePoisson–Boltzmann theory where ion-ion correlations areignored.46 Similarly in the present system, the connectivityof the cationic polymer introduces nonelectrostatic interac-tions that lead to the observed overcharging.

Another structural property of the polymer solution isthe local charge fraction ��r� which is plotted in Fig. 5, againfor p=1.29 nm3 and R=5 nm �with qualitatively similarresults for different cell sizes�. Clearly, close to the highlycharged DNA rod the protonation of the cationic polymer isincreased. The extent of charge regulation is most pro-nounced for a weakly charged cationic polymer wherecharge regulation drastically affects the polymer’s chargedensity in the vicinity of DNA. For example, �= p=0.15 in ahomogeneous bulk phase increases to �=0.8 close to theDNA.

Once �2 and � are known we can compute the totalnumber of charges

Qt =1

a3�V

dv�2� �7�

carried by the polymer in the unit cell. Note that Qt

=��N / P�p��� reflects the average charge fraction ���= Vdv�2� / Vdv�2 of the cationic polymer in the unit cell.In Fig. 6 we display Qt�R� for different p. In the limit of asmall unit cell, R→r0, all excess salt ions are released andthe polymer alone �through regulation of its local chargefraction �� ensures overall electroneutrality, implying�N / P�p=1 / ��� and thus Qt=�. In the other limit, R�r0, thetotal charge Qt carried by the polymer is somewhat largerthan it would be in the absence of DNA, where Qt= pp /a3,because polymer close to the DNA has �� p. Interestingly,the behavior of Qt�R� can be nonmonotonic—see, for ex-ample, the case p=0.55 in Fig. 6. Here, starting at R=5 nm the numbers of charges carried by the polymer andDNA are nearly identical. Yet, because of the large cell sizeonly a fraction of the polymer takes part in neutralizing theDNA charges. The remaining part of the neutralization is stillcarried out by positively charged salt ions �counterions� thatreside close to the DNA. Upon decreasing the cell size Rsome polymer is brought closer to the DNA where it partici-pates in neutralizing the DNA, with concurrent release ofcounterions and some decrease in Qt.

After discussing the structure of the polyplex we resumeour discussion of the free energy shown in Fig. 3. Increasingthe polymer’s propensity p to adopt a higher charge fractionhas significant implications for polyplex stability. A moderateincrease from p=0.15 to p=0.45 leads to a deeper minimumat a smaller cell size R. Upon further increasing p beyondp=0.45 the minimum becomes more shallow and a secondmetastable minimum F=0 at R→� �that is, for a disinte-grated polyplex� appears. At about p=0.55 the second mini-mum turns stable, implying that the condensed polyplex ismetastable. Recalling Fig. 6 we note that this occurs roughlywhen the total charge carried by the polymer equals that ofthe DNA. For even larger p �starting at p=0.7� the free en-

FIG. 4. Local volume fraction of cationic polymer, ��r�2, for variouschoices of p, ranging from p=0.15 to p=0.7. In all cases, p=1.29 nm3 andR=5 nm. The inset shows the corresponding dimensionless potentials ��r�.

FIG. 5. Local fraction of charge, ��r�, for various choices of p, rangingfrom p=0.15 to p=0.7. In all cases, p=1.29 nm3 and R=5 nm.

FIG. 6. The total amount of charge per unit cell carried by the cationicpolymer according to Eq. �7�. Different curves correspond to different p asindicated. In all cases, p=1.29 nm3.

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ergy becomes monotonically decreasing with R. Beyond that,a condensed polyplex can no longer exist without applicationof an external pressure.

B. Rupture mechanism of endosome

Enclosing polyplex by a lipid bilayer �as is the case inthe endosome� fixes the amount of cationic polymer. In thiscase, we find the pressure exerted by the polyplex from thefree energy in Eq. �1� through

� = − � �F

�V�

p

. �8�

We refer the � as osmotic pressure because mobile salt ionsand water �but not cationic polymer and DNA� can still beexchanged for a polyplex enclosed by a lipid vesicle.

Mechanical rupture of a lipid vesicle will occur if theosmotic pressure � inside the vesicle creates a sufficientlylarge lateral tension in the enclosing membrane. The lateraltension T required for mechanical rupture of a lipid bilayerhas been determined experimentally.47 It is on the order ofT=Tc�1�10−3 N /m. The relation between the lateral ten-sion and the �osmotic� pressure for a spherical vesicle ofradius Rv is given by Laplace’s law

T = 12�Rv. �9�

Representing the endosome by a spherical vesicle of radiusRv=100 nm leads to the critical osmotic pressure

�c =2T

Rv= 2 � 104 Pa, �10�

beyond which rupture will occur. We shall use this value for�c in our discussion below.

In Fig. 7 we plot the osmotic pressure � as a function ofthe unit cell’s volume V for p=0.15 to p=0.45. Note thatboth curves are calculated from the corresponding ones inFig. 3 using Eq. �8�. Clearly, the pressure exerted by a poly-plex at fixed volume V becomes smaller for increasing pfrom p=0.15 to p=0.45. We remark that � increases at large

V �with V�20 nm3� but would be negative then. Only adrastically larger increase in p beyond p=0.7 �see Fig. 3�would lead to an increase in the osmotic pressure � for all V.Yet, we are not aware of any evidence that would indicatesuch high values of p to be biologically relevant. In sum-mary, we conclude from Fig. 7 that polyplexes usually tendto shrink when exposed from physiological pH to a moder-ately more acidic environment. Differently expressed, poly-plexes in the endosome are not expected to osmoticallyswell, and thus do not contribute directly to the functioningof the proton sponge. This prediction is consistent with arecent experimental finding48 of increased polyplex compact-ness for decreasing pH value.

Calculating the osmotic pressure � on the basis of Fig. 3using Eq. �8� assumes that the volume of the entire endo-some is filled with condensed polyplex. This, of course, isnot necessarily the case. In fact, the role of polyplexes beingpositively charged due to the presence of excess cationicpolymer at the surface has been pointed out.49 For example,Clamme et al.28 reported that in the mixture of PEI and DNAthey used for transfection, the vast majority �86%� of PEImolecules was in a free uncomplexed form. Thus, it is appro-priate to assume a certain amount of excess polymer in theendosome. In the following we shall argue that the presenceof excess cationic polymer can provide a large contributionto the osmotic pressure. This contribution increases with p toan extent that may lead to the rupture of an enclosing lipidbilayer.

Consider free �uncomplexed� cationic polymer enclosedin a lipid vesicle. The change in osmotic pressure upon de-creasing the pH value can be computed using the free energyin Eq. �1� for a homogeneous system of cationic polymer. Inthis case, both � and � are spatially constant, and thePoisson–Boltzmann equation

0 = 2n0 sinh � − �2 1

1 +1 − p

pe�

�11�

becomes an algebraic relation for ����. The correspondingfree energy is

F

kBT= V� �

2�4 − 2n0�cosh � − 1� − �2 ln�1 − p + pe−��� ,

�12�

from which the osmotic pressure can be obtained throughEq. �8�. In Fig. 8 we plot the osmotic pressure � as a func-tion of the polymer’s volume fraction for two different val-ues of p. Clearly, changing p from p=0.15 to p=0.45 in-creases the osmotic pressure by almost an order ofmagnitude. Correspondingly, there is a rather wide range ofpolymer densities �starting in our case with �2=0.0022� forwhich our model predicts membrane rupture upon decreasingthe pH value from 7 to 5.

We summarize our findings obtained so far. If polyplexfills the entire volume of an enclosing vesicle then a moder-ate decrease in the pH value is expected to decrease theosmotic pressure. Yet, if polyplex is replaced by free poly-mer we expect a large increase in osmotic pressure. Conse-

FIG. 7. The osmotic pressure � exerted by a polyplex as function of cellvolume V for p=0.15 and p=0.45. The osmotic pressure is calculated ac-cording to Eq. �8� using the data given in Fig. 3. The critical osmotic pres-sure �c=2�104 Pa and the corresponding cell volumes for p=0.15 andp=0.45 are indicated. In all cases, p=1.29 nm3.

185105-6 S. Yang and S. May J. Chem. Phys. 129, 185105 �2008�

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quently, if polyplex and free polymer coexist inside a vesicle,there may still be an increase in the osmotic pressure, depen-dent on the amounts of polyplex and free polymer. In thefollowing we apply our model to this situation. There are twodifferent limiting thermodynamic cases of coexisting poly-plex and free cationic polymer. The first allows for exchangeof the cationic polymer and the second suppresses it.

Concerning the first case, we may start with the freeenergy of the polyplex F� ,V�, where V is the volume of theunit cell and is the amount of cationic polymer present inthe unit cell. Thermal equilibrium of two unit cells indexed Aand B �corresponding to a condensed and a dilute phase�requires the coexistence equations

� �F

��

A

= � �F

��

B

, � �F

�V�

A

= � �F

�V�

B

,

�13�

FA − FB = �A − B�� �F

��

A

+ �VA − VB�� �F

�V�

A

to be fulfilled. These equations correspond to the familiar“common tangential plane construction.” The three equationsreflect three degrees of freedom, the size of a given phase aswell as its amounts of DNA and cationic polymer. Again, thesystem is able to exchange cationic polymer between thecondensed and dilute phases.

In the second case, one may consider coexistence be-tween a polyplex and free cationic polymer where exchangeof cationic polymer is suppressed and no DNA is present inthe free polymer phase. Here, only one equation, namely, theequality of the osmotic pressures in both phases �A=�B,determines thermal equilibrium. In the present work we shallconsider only this second case of completely suppressedpolymer exchange. We note that free polymer is likely to beattached to polyplex particles—individual polymer mol-ecules may actually partially penetrate into the polyplex andpartially participate in forming a polymer cushion on thepolyplex surface. Hence, endocytosis of polyplex is accom-panied by the uptake of some amount of free �uncomplexed�polymer. There are two reasons for why suppressed ex-

change of cationic polymer is the more relevant case forendocytosed polyplexes. First, the electrostatic interactionbetween an individual phosphate group of DNA and a singleprotonated amine group of PEI �or similar groups for othercationic polymers� may be low but the sum of all these in-teractions will be very large �compared to the thermal energykBT� for an entire polymer. Hence, the probability for desorp-tion of the polymer will be small. Second, long polymerswill bridge many cylinders,50 forming a network with theDNA in the polyplex. Hence, there are significant topologicalconstraints that render polymer diffusion slow. Indeed, as deGennes51 pointed out, if the polymer is long and adsorptionis strong, the adsorbed layer may form a glassy liquid. Forsuch a system, adsorption becomes an irreversible process,implying that the adsorbed amount of polyelectrolyte re-mains almost fixed.52

As discussed, we consider the case of suppressed poly-mer exchange between a polyplex and free cationic polymer.The two phases, polyplex and free polymer, occupy volumesVA and VB, respectively. The total volume Vtot=VA+VB

=4�Rv3 /3 corresponds to that of the enclosing vesicle �with

fixed Rv=100 nm�. If DNA of total length LDNA is present,the number of unit cells is NDNA=LDNA /L. Consider the casethat the osmotic pressure � is equal to the critical pressure�c=2�104 Pa. In this case we obtain the volume V of thepolyplex occupied per unit cell from Fig. 7. Hence, the vol-ume of free polymer contained in the vesicle is

f = �02�Vtot − VLDNA/L� , �14�

where �02, the polymer density that leads to �c, can be ob-

tained from Fig. 8. The linear relationship f�LDNA� is plot-ted in Fig. 9 for both p=0.15 and p=0.45 �corresponding tothe pH values of 7 and 5, respectively�. The two lines sepa-rate regions of mechanical instability of the enclosing vesicle

FIG. 8. The pressure exerted by free �uncomplexed� cationic polymer as afunction of monomer concentration �2 displayed for p=0.15 and p=0.45.The horizontal dashed line marks the critical osmotic pressure �c, beyondwhich a lipid vesicle of radius Rv=100 nm is expected to rupture.

FIG. 9. The relation f�LDNA� according to Eq. �14� at which the osmoticpressure �=�c is equal to the critical pressure needed for rupture of avesicle of radius Rv=100 nm. Here, f is the total volume of free �uncom-plexed� cationic polymer present in the vesicle and LDNA is the total lengthof DNA. The two straight lines correspond to p=0.15 and p=0.45. Withinthe hatched region vesicle instability is induced by increasing p fromp=0.15 to p=0.45. The inset replots the relation f�LDNA� in terms of theratio of free polymer to adsorbed polymer, �= f / �NDNAp�, as function ofthe volume fraction of DNA �DNA=3LDNAr0

2 / �4Rv3� in the vesicle.

185105-7 Proton sponge hypothesis J. Chem. Phys. 129, 185105 �2008�

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for p=0.15 and p=0.45. That is, both for increasing f atfixed LDNA or increasing LDNA at fixed f the osmotic pres-sure � in the vesicle will ultimately grow beyond �c. Thehatched region in Fig. 9 indicates loss of vesicle stabilityupon increasing p from p=0.15 to p=0.45. It suggests theamount of free polymer that needs to be present �in additionto the amount of polymer pNDNA forming the polyplex� inorder to induce membrane rupture by the proton spongemechanism. We note that our result is subject to our choicesof the vesicle radius Rv=100 nm, the initial 10 mM bulkpolymer concentration for polyplex formation �see Fig. 2�,the 0.15M concentration of monovalent salt, and the polymerparameters a=0.5 nm and �=0.4a3 �see Eq. �1��. Differentchoices would affect our results quantitatively but not quali-tatively.

Figure 9 recovers the results for a single phase system,either only polyplex and only free polymer, in the limits f

=0 and LDNA=0, respectively. In the former case, for f =0,increasing p from p=0.15 to p=0.45 decreases the osmoticpressure �. Hence, the proton sponge mechanism cannotwork in that limit. In the latter case, for LDNA=0, the pres-ence of a sufficiently large amount of �free� cationic polymeris indeed expected to cause rupture of the enclosing vesicleas predicted by the proton sponge hypothesis. An interestingpoint is the intersection of the two straight lines in Fig. 9.Here, the osmotic pressure �=�c remains constant upon in-creasing p from p=0.15 to p=0.45. To explain the reason forthis behavior we note that the polyplex condenses, leading toan expansion of the free polymer phase. At the same time thefree polymer acquires more charge. At the intersection of thetwo straight lines in Fig. 9 the pressure changes induced bythese two processes compensate exactly.

The relation f�LDNA� is replotted in the inset of Fig. 9 interms of the ratio �= f / �NDNAp� of free polymer to ad-sorbed polymer and volume fraction �DNA=3LDNAr0

2 / �4Rv3�

occupied by DNA in the vesicle. We observe that polyplexrelease according to the proton sponge mechanism is onlyfeasible for a DNA volume fraction �DNA of less than 20%.Decreasing the amount of DNA increases the ratio � of freeto complexed polymer in the vesicle. Note that this ratio isdirectly related to the total molar ratio N / P of PEI nitrogenatoms to DNA phosphate groups in the vesicle according to

N

P= �N

P�

p�1 + �� , �15�

where we recall �N / P�p to be the molar ratio of PEI nitrogenatoms to DNA phosphate groups in the polyplex �thus ex-cluding the free polymer�. Also recall that in the presentwork we have considered conditions leading to �N / P�p

=1.73; see the discussion following Fig. 2. Hence, the choiceof � in the inset of Fig. 9 is equivalent to specifying N / P=1.73�1+��.

Let us apply our results to the situation of small DNAcontent in the vesicle. A single plasmid of 20 kbp �kilobasepair� would correspond to the length of about 7 �m. In thiscase, the two limiting amounts of f =1�104 nm3 and f

=4�104 nm3 roughly specify the range at which our presentmodel predicts the proton sponge mechanism to work. Thesetwo values correspond to N / P= �N / P�p�1+ f / �NDNAp��

=3.6 and N / P=9.4, respectively, which is consistent with therange of molar ratios between PEI nitrogen atoms and DNAphosphate groups that is reported to be optimal fortransfection.28

IV. CONCLUSION

In this work we have investigated a theoretical model forthe proton sponge hypothesis. According to this hypothesisthe cationic polymer used to complex the DNA is also able tomediate endosomal escape through the ability to regulate itscharge. That is, when subject to a decrease in pH value from7 to 5 the increased polymer charge leads to the influx of saltions and thus to an increase in the osmotic pressure insidethe endosome. Our theoretical model confirms the feasibilityof this mechanism for typical system parameters. More spe-cifically, it predicts a relation between the amount of polymerand DNA that results in membrane rupture upon the decreasein pH from 7 to 5 �see Fig. 9�. When increasing the amountof vesicle-enclosed DNA the total ratio N / P �PEI nitrogenatoms to DNA phosphate groups in the vesicle� becomessmaller until, for a maximal amount of DNA, the protonsponge mechanism ceases to work. At this point the osmoticpressure does no longer increase upon a decrease in the pHvalue.

Our calculations also predict structure and energetics ofthe cationic polymer-DNA complexes �polyplexes�. Interest-ingly, we find increased condensation of the polyplex upon amoderate decrease in the pH value. Hence, polyplex alone isunable to induce vesicle rupture. Additional free �uncom-plexed� cationic polymer is needed for the osmotic pressureto increase. The ratio between free and complexed cationicpolymer needs to be well balanced �see Fig. 9� in order forthe vesicle to still be stable at pH 7 but unstable at pH 5.

In the present work we have only considered the case ofrestricted exchange of cationic polymer between polyplexand free polymer when enclosed in a lipid vesicle. That is,we assume the relation of free and complexed cationic poly-mer is fixed after endosomal uptake. While this might nottruly be the case under experimental conditions we still be-lieve it captures the underlying mechanism of the protonsponge correctly.

ACKNOWLEDGMENTS

This work was supported by the NSF through Grant No.DMR-0605883.

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