Electrostatic theory of the acidity of the solution in the lumina of viruses and virus-like particles

Recently, Maassen et al. measured an appreciable pH difference between the bulk solution and the solution in the lumen of virus-like particles, self-assembled in an aqueous buffer solution containing the coat proteins of a simple plant virus and polyanions. [Maassen, S. J.; et al. Small 2018, 14, 1802081] They attribute this to the Donnan effect, caused by an imbalance between the number of negative charges on the encapsulated polyelectrolyte molecules and the number of positive charges on the RNA binding domains of the coat proteins that make up the virus shell or capsid. By applying Poisson-Boltzmann theory, we confirm this conclusion and show that simple Donnan theory is accurate even for the smallest of viruses and virus-like particles. This, in part, is due to the additional screening caused by the presence of a large number of immobile charges in the cavity of the shell. The presence of a net charge on the outer surface of the capsid we find in practice to not have a large effect on the pH shift. Hence, Donnan theory can indeed be applied to connect the local pH and the amount of encapsulated material. The large shifts up to a full pH unit that we predict must have consequences for applications of virus capsids as nanocontainers in bionanotechnology and artificial cell organelles.


Introduction
The co-assembly of the coat proteins of many simple viruses with their single-stranded (ss) RNA seems to be primarily driven by electrostatic interactions between the negatively charged genome and the positively charged, disordered RNA binding domains on the coat proteins. [1][2][3][4] It is no surprise, then, that under appropriate solution conditions virus coat proteins spontaneously encapsulate not only homologous but also heterologous ssRNAs, synthetic polyanions, surface-functionalized nanoparticles and so on. [5][6][7][8][9][10][11] Interestingly, the co-assembly does not necessarily produce virus-like particles of the same size or T number or even the same shape as that of the native virus. Size and shape selection seems to be controlled on the one hand by mass action and hence stoichiometry, and on the other by a compromise between the interaction between the coat proteins and that between the coat proteins and cargo. 8,12,13 Virus coat proteins, such as that of Brome Mosaic Virus, Cowpea Chlorotic Mosaic Virus and Hepatitis B Virus, can actually be made to self-assemble into shells in the absence of a negatively charged cargo by changing the acidity and/or salinity of the solution, removing the RNA binding domain of the coat proteins or chemically modifying this domain. [14][15][16] The spontaneous assembly is in that case driven by hydrophobic and other types of attractive interaction, involving, e.g., ionic and hydrogen bonds. 17,18 Because virus coat proteins are able to encapsulate molecular cargo, either spontaneously from solution or by attaching it chemically or physically to, e.g., the RNA binding domain of the protein, there is a significant interest in utilising this property for application purposes in targeted drug delivery, tomography, controlled catalysis and metamaterials. [19][20][21][22] Recently, virus-based artificial organelles have been suggested as a viable route to be used in living cells for therapeutic purposes, restoring or even adding cellular activity. 23 These artificial organelles contain catalytically active particles such as enzymes. The reason that encapsulated catalytic particles can actively process substrates present in the solution, is that the capsid shell is permeable to these molecules but only if they are not too large in size. If too large, substrates cannot diffuse through the pores in the shell that include ones with a diameter as large as a few nanometers. 24 The semi-permeability of capsids, in particular that of plant viruses, has been known for some time. 25 Catalytic activity, in general, is strongly pH and ionic strength dependent, suggesting that the control of the physico-chemical conditions of the solution in the lumen of the virus capsid or any other type of proteinaceous shell used for the same purpose must be of paramount importance. 26 Interestingly, so far this issue seems to have met with relatively little attention in the artificial organelle community, 23 even though recent experiments by Maassen et al. demonstrate that the pH in the lumen of a virus-like particle can be much lower than that of the bulk solution. 27 It is interesting to mention that the lumina of carboxysomes, which are proteinaceous shells that act as organelles in bacteria, also seem to have a lower pH than the bulk solution both in vitro and in vivo. 28 The reason that the acidity and ionic composition of the lumen of a protein shell can be much different than that of the bulk solution is the presence of a net immobile charge in that lumen, that is, a net charge not associated with translationally mobile species that can freely diffuse in and out of the shell, and the result of what is commonly known as the Donnan effect. 29 This effect has long been known to cause pH gradients across lipid membranes in the context of vesicles. 30 In the experiments of Maassen et al., 27 the coat protein of cowpea chlorotic mottle virus (CCMV) and a 7.5 kDa random copolymer of styrene sulfonate and pH-sensitive fluorescent fluorescein methacrylate monomers spontaneously assemble in virus-like particles under a wide range of solutions pHs. The authors use the ratio of two excitation peaks to obtain the relation between the pH in free solution and that in the lumen. For pHs spanning values between 6.0 and 8.0, the pH inside the virus-like particle turns out to be about 0.4 pH units lower than that outside of it, that is, in the bulk solution. A simple Donnan theory put forward by the authors, presuming a uniform distribution of immobile charges in the lumen and ignoring both the presence of the protein shell holding the polyanionic cargo and the presence of a net charge on the outer surface of it, actually explains this observation. In the present context, Donnan theory presumes (i) local charge neutrality, meaning in this case that both the capsid lumen and the bulk solution outside the virus are electroneutral, and (ii) equal chemical potentials of the mobile ionic species in-and outside of the shells. 31,32 Typically, the well-known expressions for the electrochemical potentials of the charged mobile ionic species valid in dilute solution are used for the latter. Standard chemical potentials and the effects of non-ideality of the solution, insofar that these can be described by activity coefficients, can be absorbed in the Donnan potential, which actually acts as a Lagrange multiplier enforcing local charge neutrality. This implies that the theory should be valid even if the ionic solution does not behave ideally. 27 According to the theory, the dimensionless Donnan potential φ in the capsid lumen, defined as the Donnan potential energy scaled to the thermal energy, obeys the simple where the quantity γ ≡ ∆ρ/2ρ s is defined in terms of the difference ∆ρ = ρ i + − ρ i − of the number densities of positive and negative immobile charges ρ i ± , averaged over the volume of the lumen, and 2ρ s the overall density of mobile ionic species in the bulk solution (tacitly presumed to be monovalent). This quantity is dominated by the concentrations of added salt and buffer of the assembly mixture, and explains why in practise the Donnan potential and also the pH differential across the protein shell is virtually independent of the acidity of the solution, 27,28 unless some form of charge regulation takes place involving weakly ionic moieties on the cargo or the coat proteins themselves. 27,35 The pH shift, defined as the difference ∆pH = pH in − pH out between the solution pH in inside and pH out outside of the particle, is proportional to the Donnan potential, Conversely, we can use the fact that a large number of T = 3 viruses with a genome smaller than about 6000 nt seem to have an average degree of overcharging of +60%, 41 and convert this number into a pH shift. According to Donnan theory, this degree of overcharging translates to a pH shift of −1, if we take the average of 10 or so positive charges per RNA binding domain of those viruses. 41 Overall, relative degrees of overcharging ranging from −100% for empty capsid shells due to the absence of any charge compensation by immobile cargo because in that case ρ i − = 0, to +800% for encapsulated high-molecular weight poly(styrene sulfonate) have been reported. 18 This then suggests that, if we take eqs.
(1) and (2) at face value, there must be a large spread in pH shift in virus-like particles ranging in value from about −1 and +1 depending on the amount of encapsulated cargo, ionic strength and so on.
Of course, these estimates hold only if Donnan theory, which presumes a spatially uniform electrochemical potential in the lumen of the protein shell, actually applies on the scale of T = 1 and T = 3 viruses. Such small viruses measure between twenty to thirty nanometers in outer diameter, whilst their lumina are obviously even smaller than that, and, say, ten to twenty nanometers wide. 42 Typical electrostatic screening lengths are around one nanometer and any encapsulated polyanion should perhaps be expected to be concentrated primarily in the region where the RNA binding domains are located, which for CCMV for instance is estimated to be about three or four nanometers wide. 4,13,38 Theoretically, the effect seems to be smaller for linear polyanions than for ssRNA-like randomly branched ones, and for T = 1 particles this layer should extend almost to the centre of the cavity of the capsid. 43,44 One might attempt to construct a non-uniform Donnan theory, in a similar vein as was done by Odijk and Slok for densely packed double-stranded DNA in bacteriophages, 45 and by Philipse in the somewhat different context of charged colloids in a gravitational field. 46 In the present context of over-or undercharged virus-like particles such an enterprise would defeat our purpose, which is to obtain estimates from a simple theory that requires as few input parameters as possible, and that does not require taking recourse to numerical methods.
Even if the distribution of the immobile charges in the lumen is more or less uniform, as we expect it to be for T = 1-sized shells, we nevertheless think it is prudent to investigate under what conditions simple Donnan theory applies, and how accurate it actually is given that the mobile charges tend to not be uniformly distributed. 34 We note also that within By applying Poisson-Boltzmann theory to model virus-like particles, we are able confirm that predictions for pH shifts obtained from Donnan theory are quite accurate also for the smallest, that is, the T = 1-sized particles, even for ionic strengths much below physiological.
The reason is that in practice the overall concentration of immobile ionic species in the capsid lumen is very much larger than the ionic strength of the buffer solution, and because of this also that of the mobile ionic species. This gives rise to a smaller effective screening length in the capsid lumen, which may in fact also be inferred from earlier work on the electrostatics of soft particles. 34,47,48 The impact of a surface charge is relatively minor, in part caused by the presence of the capsid shell. Interestingly, a net charge on the shell that has the same sign as the net immobile charge in the lumen potentially increases the accuracy of Donnan theory.
The remainder of this paper is organised as follows. First, in the section Theory, we

Theory
To set up the Poisson-Boltzmann theory, we need to construct a model. In our model, we presume that (i) the geometry of the problem obeys spherical symmetry and (ii) all charges associated with the RNA binding domains of the coat proteins and cargo (the "immobile" charges) are uniformly distributed in a spherical volume of radius R 1 > 0. See also Figure 1.
So, the number densities ρ i ± (r) = ρ i ± H(R 1 − r) of positive and negative immobile charges are step functions of the radial co-ordinate r ∈ [0, ∞), with ρ i ± (as before) their mean densities in the cavity and H(R 1 − r) = 1 for r < R 1 and H(R 1 − r) = 0 for r ≥ R 1 the usual Heaviside step function. The number density associated with the net immobile space charge  51 We also note that any transient exposure of the RNA binding domains and/or cargo to the outside of the capsid shell, 52 e.g., via the pores, only renormalizes the charge densities of the lumen and the surface.
Using the known pK a s of basic and acidic residues of surface-exposed amino acids, for the (dimensionless) potential ψ I = ψ in region I, so for radial distances 0 ≤ r < R 1 , for the potential ψ II in region II where R 1 ≤ r < R 2 , and for the potential ψ III region III where r ≥ R 2 , where we refer again to Figure 1. These equations have to be supplemented with boundary conditions that enforce overall charge neutrality, radial symmetry and continuity of the potential for all radial positions r: i) We have not been able to solve the above set of differential equations exactly. Hence, we follow the prescription of Lifson, 34 who approximately solved the Poisson-Boltzmann equations for the related problem of a model polyelectrolyte chain in which region II is absent as is the surface charge. (A similar approach in a slightly different geometry was used, e.g., by, Ohshima. 48 ) Hence, we write ψ I (r) = φ + ∆ψ I (r) for 0 ≤ r < R 1 , where φ is a constant background potential and ∆ψ I (r) a position-dependent correction to that constant background potential. If we presume that |∆ψ I | |φ|, we can Taylor expand sinh ψ I = sinh(φ + ∆ψ I ) = sinh φ + cosh φ × ∆ψ I (r) + · · · . If we insert this in eq. (3), and realise that the function ∆ψ I is by construction a varying function of r, we obtain the identity sinh φ = γ for the constant background potential, and for the spatially varying part, where It transpires that φ must indeed be the Donnan potential as we find it to obey eq. Since the potential decays significantly in the intermediate region II, we apply the Debye-Hückel approximation in region III and linearise eq. (5), to obtain By comparing eq. (6) with eq. (8), we are able to conclude that the Debye length in region I must be a factor Γ = √ cosh φ ≥ 1 smaller than that in the outer region III. 34 This enhanced electrostatic screening is stronger the larger the magnitude of the Donnan potential. Indeed, for Donnan potentials stronger than the thermal energy, so for |φ| 1, we have Γ ∼ |γ|, implying that in that case the effective Debye length in region I, λ D /Γ, scales as 1/ |∆ρ|.
The magnitude of this effective Debye length is then determined by the mismatch between the number of immobile positive and negative charges in the lumen, and must then be virtually independent of the concentration of salt in the bulk solution.
The remaining set of equations, eqs. (4), (6) and (8) can now straightforwardly be solved in polar co-ordinates and using the boundary conditions quoted above in order to fix all the integration constants. For the dimensionless potential in region I, we find with a parameter that depends on the inner and outer radii of the protein shell R 1 and R 2 , the degree of overcharging via the Donnan potential φ and the net surface charge density ∆σ. Clearly, if |u sinh(R 1 Γ)/R 1 Γ| |φ|, then ψ I ∼ φ for r < R 1 , and the Donnan potential provides a good representation of the electrical potential in region I, the lumen of the protein shell.
The potential in the protein shell (region II) obeys which is not uniform, unless u = 0. This happens exactly if φ = 4π∆σR 2 /(1 + R 2 ), in which case the uniform Donnan potential extends all the way to the boundary with region III, the outer surface of the shell. The potential in region III decays with increasing radial distance r under all circumstances, and reads The somewhat unwieldy expressions eq. (9) -(12) agree with those of Lifson for the corresponding case R 1 = R 2 and ∆σ = 0. 34 Lifson found excellent agreement with a numerical solution of the equations for all cases investigated, that is, for fixed γ = 5 and varying R 1 > 1, and for fixed R 1 = 5 and varying γ ≥ 0.5.
To calculate the local pH and relate that to the value in the bulk solution for r → ∞, in terms of the scaled variables r, R 1 and ψ I . Note that eq. (13) applies to any positively charged mobile ionic species, whilst for negatively charged ionic species we only need to replace the minus sign in the exponential by a plus sign. Finally, the pH differential between the interior of the capsid and the bulk solution is given by which we can calculate once the potential in cavity is known, ψ I . Notice that for |u| → 0, we have J = 1 and we retrieve eq. (2) noting that log 10 e = 1/ ln 10 with e Euler's number.
To calculate the pH shift, ∆pH, we insert eq. (9) into eq. (13), and by applying a suitable change of variables we find from eq. (14) We have not been able to exactly solve this integral. We expect that under most experimentally relevant conditions |u sinh(R 1 Γ)/R 1 Γ| 1, allowing us to Taylor expand the exponential and obtain to second order in the parameter u, In the limit R 1 Γ 1, the corrections in powers of u are small only if |φ − 4π∆σR 2 | 1.
For R 1 Γ 1, they are of the order (φ − 4π∆σ)/(R 1 Γ) 2 implying that as long as the radius of the lumen is much larger than the effective screening length in it, the pH shift should be dominated by the Donnan potential. In the opposite limit, |u sinh(R 1 Γ)/R 1 Γ| 1, the perturbation approach that we invoke to solve the Poisson-Boltzmann equation in region I should break down. The reason is that in that case |∆ψ I | is not necessarily small compared to |φ| near the edge of the lumen. Hence, we do not discuss this limit any further.

Results
To investigate in more detail the limits of applicability of Donnan theory, we compare its predictions with our perturbation theory. From equation (9) we read off that provided |u sinh(R 1 Γ)/R 1 Γ| |φ| we have ψ I (r) φ for all 0 ≤ r ≤ R 1 , implying that in that case the Donnan potential accurately describes the electrical potential in the lumen of the capsid shell. This happens if at least one of two conditions is met: i) The presence of a net charge on the outer surface of the shell compensates for the drop in the potential in the shell, so if φ = sinh −1 γ 4π∆σR 2 /(1 + R 2 ). In that case, we have ψ I ψ II φ, and most of the drop of the potential happens outside of the shell, that is, in region III. This shows that a net surface charge, if not too large and of the same sign as the net immobile space charge present in the lumen, makes the prediction of Donnan theory more accurate than without it; ii) If the concentration of salt or the magnitude of the Donnan potential is sufficiently large so that the effective screening length of the solution in the cavity is much smaller than the radius of the lumen, that is, if For weak degrees of over-or undercharging |γ| 1, this implies that R 1 1, whilst for large degrees of overcharging |γ| 1, the dimensionless radius R 1 1/ √ γ may actually be substantially smaller than unity.
From this we conclude that in the present context Donnan theory has a wider range of applicability than is sometimes thought. It is neither restricted to low Donnan potentials 58 nor to low ionic strengths, 59 as in fact is already clear from the early work of Lifson. 34 It seems that modelling a continuous immobile space charge distribution by a series of fixed, localised charges, as was done by Grodzinsky et al. 58 and Huster et al., 59 would lead us to underestimate the range of applicability of Donnan theory.
We now illustrate our findings by taking as model parameters those that we estimate for the T = 1 virus-like particles investigated by Maassen and collaborators. 27 We recall that these are formed by the spontaneous co-assembly of coat proteins of CCMV and poly(styrene sulfonate) copolymers. We set the dimensionless shell radii equal to R 1 = 6.5 and R 2 = 12,    The question arises what happens to the usefulness of Donnan theory if the ionic strength were a factor of, say, ten lower, so 15 mM instead of 150 mM. We would then have R 1 2.1 and R 2 3.8, which could indicate that Donnan theory might in that case be not quite as accurate. However, if we assume the overcharging to remain more or less constant, then the degree of overcharging becomes much more negative with γ = −11. In that case, we find φ = −3.1 and Γ = 3.3. This means that for a ten times smaller ionic strength we have R 1 Γ = 6.9, implying that Donnan theory should remain to be reasonable accurate. Indeed, the pH shift predicted by Donnan theory amounts to −1.3 and that from our Poisson-Boltzmann theory is also −1.3, presuming a dimensionless surface charge density of −0.055.
We conclude that an increase in the bulk screening length, which in principle would make the Donnan theory less accurate, is more than compensated for by an increase in the magnitude of the Donnan potential that decreases the effective Debye length in the lumen of the viruslike particles.
Let us now apply our theory to the experiments of Ren and collaborators, who encapsulated poly(styrene sulfonic acid) PSA of varying molecular weights from 13 to 990 kDa using the coat proteins of horseradish chlorotic ringspot virus or HCRSV, and predict the

Discussion and Conclusions
An imbalance in the number of localised positive and negative space charges in the lumina of viruses, virus-like particles and other types of protein shell, semi-permeable to mobile ionic species, gives rise to a Donnan potential difference between the bulk solution and the inside of the particles. This potential difference induces a pH differential that can potentially be large.  60 Donnan theory can be used to estimate the pH in the lumen of the virus-like particle.
As discussed in the Introduction, the predicted pH shifts can be as large as a full pH unit, in both positive and negative directions. If virus-like particles are used for catalysis, e.g., by way of encapsulated enzymes, then the activity of these enzymes may be influenced not only by the compartmentalisation itself but also by the local pH and ionic strength. 63,64 Since the charged state of enzymes depends on the pH, they would modify the Donnan potential self-consistently if their number is large enough. In fact, the coat protein itself might be involved in buffering activity. 27 This kind of charge regulation can be incorporated in both Donnan and Poisson-Boltzmann theory relatively straightforwardly. 35,50,65 For instance, if the cargo is a weak polyacid, such as polyacrylic acid, 60 we only need to modify the parameter γ = (ρ i + − ρ i − )/2ρ s , which now becomes a function of the local potential ψ I in the lumen, for .
This, of course, is the familiar Henderson-Hasselbalch equation, where in the second equality we have expressed the local pH in the lumen, pH in , in terms of the pH in the bulk solution, pH out . Further, pK a is the dissociation constant of the weak polyacid, and ρ i −,0 the concentration of chargeable groups on the encapsulated polyacid that we again presume to be uniformly distributed in the lumen. At the level of Donnan theory, ψ I = φ, whilst within our Poisson-Boltzmann theory, we would write again ψ I = φ + ∆ψ I , and expand sinh ψ I as well as γ to first order in ∆ψ I . The Debye length renormalisation factor Γ now obeys the equality Γ 2 = cosh φ − ∂γ/∂ψ I | ψ I =φ . As long as R 1 Γ remains sufficiently large, Donnan theory should again be reasonably accurate.
Eq. (18) tells us that the pH shift ∆pH now depends on the pH of the bulk solution pH out , except if |pH in − pK a | 1. For |pH in − pK a | 1, the pH shift can actually compensate for changes in the outside pH, leading to a constant pH inside the particle. This happens if ∂∆pH/∂pH out = −1, which within Donnan theory translates to ∂φ/∂pH out = − ln 10. It shows that charge regulation can indeed lead to a buffering effect, explaining the findings of Maassen and collaborators, who find a constant (negative) pH shift for solution pHs from 6 to 8 but a constant inside pH for solution pHs between 5 and 6. 27 In their particular case the buffering is arguably not caused by the encapsulated strong polyanion but by moieties on the coat protein itself.
We shall not dwell on this issue any further, and end by mentioning that the calculation involving charge regulation would allow us to establish the mean charge on any encapsulated weak polyacid and compare that with the mean charge of the same polyacid in free solution.
This, with the aforementioned experiments of Lim and collaborators in mind. 60 Note that charged state of the weak polyacids in free solution and that in the capsids are in principle not the same due to the impact of what essentially is the Donnan potential. 13 We leave this for future work.