Influence of salt and viral protein charge distribution on encapsidation of single-stranded viral RNA molecules
aa r X i v : . [ phy s i c s . b i o - ph ] J un Influence of salt and viral protein charge distribution on encapsidation ofsingle-stranded viral RNA molecules
Antonio ˇSiber
1, 2 and Rudolf Podgornik
1, 3 Department of Theoretical Physics, Joˇzef Stefan Institute, SI-1000 Ljubljana, Slovenia Institute of Physics, P.O. Box 304, 10001 Zagreb, Croatia Department of Physics, University of Ljubljana, SI-1000 Ljubljana, Slovenia
We examine the limits on viral composition that are set by the electrostatic interactions effectedby the charge on the viral proteins, the single-stranded viral RNA molecule and monovalent saltions in the solution. Within the mean-field model of viral energetics we demonstrate the primeimportance of the salt concentration for the assembly of a virus. We find that the encapsidationof the viral RNA molecule is thermodynamically suppressed in solutions with high concentrationsof monovalent salt. This effect is significantly less important in viruses with proteins whose chargedistribution protrudes into the interior of the capsid, leading to an increase in the stability of suchviruses in solutions with high salt concentrations. The delocalization of positive charge on the capsidprotein arms thus profoundly increases reliability of viral assembly in high-salt solutions.
PACS numbers: 87.15.Nn,41.20.Cv,82.35.Rs
Viruses are a prime example of precise spontaneous as-sembly. It was more than fifty years ago since Fraenkel-Conrat and Williams demonstrated that fully infectioustobacco mosaic viruses (TMV) could be created simplyby mixing the viral RNA molecules together with the vi-ral proteins . Under the right conditions (pH and salin-ity), the viruses formed spontaneously, i.e. without anyspecial external constraints.There is no unanimous view concerning the physicalinteractions that guide the viral self-assembly. It is oftensupposed that specific interactions acting between the vi-ral genome and its proteins guarantee the precise assem-bly. However, this cannot be the entire story since it hasbeen demonstrated that (i) empty viral protein coatings(capsids) assemble, at least when the amount of salt inthe bathing solution is large enough , and (ii) the filledvirus-like particles form even when the viral genome isreplaced by noncognate RNA molecules .The role of salt in the viral assembly is of essential im-portance. Already in the early studies of TMV assembly it was found that the viral proteins can be assembledin capsid-like structures at sufficiently high salt concen-trations even if the pH of the solution is high enoughto prohibit the assembly in low ionic concentrations (orcauses an alkaline degradation of the assembled viruses).It is experimentally documented that upon lowering thepH of the high-salt solution ( > empty viral capsids form. All these experi-mental findings clearly indicate that there is a nonspecificinteraction of electrostatic origin acting between the viralproteins and RNA molecules. This interaction depends crucially on the concentration of salt ions in the bathingsolution.The aim of this letter is to decipher the role of saltin the assembly of viruses that contain single-strandedhighly negatively charged (one elementary charge per nu-cleotide) RNA (ssRNA) molecule. The proteins of suchviruses typically carry positive net charge at physiolog-ical pH. For many of these relatively simple viruses ithas been experimentally demonstrated that they can bespontaneously assembled in vitro . Spontaneous assem-bly takes place only if the energy of the capsid/genomecomplex is favorable - as proposed already by Caspar andKlug . In the context of electrostatic interactions, thismeans that the number of charges on the ssRNA and thecapsid must be related and this relation should dependalso on the amount of added salt.We approach the problem by representing the viral ss-RNA as a generic flexible polyelectrolyte with effectivemonomer size a , and pe charge per monomer, where e isthe electron charge and 0 < p <
1. The polyelectrolyteconcentration, Ψ( r ) and electrostatic potential, Φ( r ),are treated as continuous real-valued fields that minimizethe mean-field ground state dominance free energy of thepolyelectrolyte/capsid/salt system , F , in the subspaceof fixed total number of polyelectrolyte monomers, N , sothat F = Z f ( r ) d r − µ (cid:18)Z d r Ψ( r ) − N (cid:19) , (1)where µ is the Lagrange multiplier enforcing the condi-tion of fixed number of monomers, and f ( r ) = k B T (cid:20) a ∇ Ψ( r )) + v r ) (cid:21) + (cid:2) ec + ( r ) − ec − ( r ) − pe Ψ( r ) + ρ p ( r ) (cid:3) Φ( r ) − ǫ ǫ ∇ Φ( r )) + X i = ± (cid:8) k B T (cid:2) c i ( r ) ln c i ( r ) − c i ( r ) − (cid:0) c i ln c i − c i (cid:1)(cid:3) − µ i (cid:2) c i ( r ) − c i (cid:3)(cid:9) . (2)Here T is the temperature, k B is the Boltzmann con-stant, c ± are the concentrations of + and − monovalentsalt ions, with c ± being their bulk concentrations, and µ ± their chemical potentials, ǫǫ is the permittivity ofwater, and v is the (non-electrostatic) excluded volumeof the polyelectrolyte chain. The density of charge lo-cated on the capsid proteins is denoted by ρ p ( r ). Weshall consider this charge density to be fixed, i.e. weshall investigate the stability of the assembled capsids atfixed (physiological) pH. The variation of the free energyfunctional with respect to fields Ψ, Φ and c ± yields twocoupled non-linear equations - the generalized polyelec-trolyte Poisson-Boltzmann equation , and the Edwardsequation . These equations are solved numerically withthe requirement that the polyelectolyte density ampli-tude field vanishes at the interior capsid radius, R . It isknown that the asphericity of ”spherical” (icosahedral)viruses increases with the mean radius of the virus .However, deviations from the perfectly spherical shapeare generally small even for quite large viruses , thus ourapproximation of spherical symmetry is not expected tobe a serious limitation. We do not explicitly account forthe mechanical elasticity of the polyelectrolyte, i.e. ourapproach is taylored to flexible polyelectrolyte molecules,the ssRNA in particular. Our approach cannot accountfor the details of the RNA conformation, such as itsbranched and locally double stranded structure, or itspossible dodecahedral ordering in the vicinity the cap-sid, a feature that has been investigated recently .Since the size of empty capsids is usually the same as inthe fully functional viruses (at least in a range of pH andsalinity values ), it is reasonable to fix the capsid ra-dius at a prescribed value, corresponding to the preferredmean curvature of the empty capsid and to examine theenergetics of the filled capsid depending on the amountand type of enclosed polyelectrolyte and the concentra-tion of salt ions in the bathing solution. Figure 1a) dis-plays the free energies of the polyelectrolyte/capsid/saltsystem as functions of the number of monomers ( N ) inthe polyelectrolyte for three different salt concentrationsin the bulk bathing solution. In this calculation, we haverepresented the protein charges as a uniformly chargedinfinitely thin spherical shell of surface charge density σ = 0 . e /nm and inner radius R =12 nm. This shouldbe representative for typical ssRNA viruses . In Fig1b) we represent the polyelectrolyte concentration profilefor several polyelectrolyte lengths and for ”physiological”salt concentration of c =100 mM. There are several im-portant messages that can be read directly off this figure.First of all, the energetics of viral capsids is profoundlyinfluenced by the concentration of salt. Second, there isa critical number of monomers that can be thermody-namically packed within the capsid. This happens at the FIG. 1: Panel a): The free energies of the polyelec-trolyte/capsid/salt system as functions of the number ofmonomers. The parameters used are p = 1, a = 0 . v = 0 .
05 nm , T = 300 K, σ =0.4 e /nm , and R =12 nm. Theresults are shown for three different bulk concentrations of saltions, c = 10 , N ≈ c =100mM. The curves displayed correspond to N =100, 300, 500,700, 900, 1100, 1300, and 1500. The lines are styled so thatthe length of their dashes is proportional to N . point when the total energy of the system becomes largerthan the energy of the empty capsid ( N = 0); in Fig. 1these are the points at which the full lines intersect withthe dotted horizontal lines for given bulk concentrationsof the salt. For polyelectrolytes larger than this criticalsize, formation of empty capsids is thermodynamicallypreferable. While it is generally easier to form capsids atelevated salt concentration (this is seen from the smallervalues of the free energy at high salt, irrespective of thenumber of monomers), it is more difficult to form filled capsids. For low salt concentrations, the critical numberof monomers is such that the total polyelectrolyte chargeis about two times larger in magnitude from the chargeon the capsid. However, as the salt concentration in-creases, the critical number of monomers decreases. For c = 700 mM, it is only about 100, twelve times smallerthan the critical number in low salt ( c = 10 mM). Wedefine the optimal number of monomers as the one thatminimizes the free energy for a given salt concentration(in agreement with Ref. 16). In low-salt solutions, thishappens when the total polyelectrolyte charge approxi-mately equals the capsid charge, but in elevated salt, theoptimal number of monomers decreases. In contrast toRef. 16, we find that the optimal number of monomers issuch that the magnitude of charge on the polyelectrolyteis always smaller than the protein charge.These findings can be better understood by examiningthe polyelectrolyte density in the capsid [Fig. 1b)]. Dueto attraction between the polyelectrolyte and the capsid,there is always a maximum in the polyelectrolyte concen-tration at a distance ∼ = a from the capsid [distributionssimilar to that shown in Fig. 1b) have been experimen-tally observed - see e.g. Ref. 7]. When the numberof monomers is larger than the optimal one, the poly-electrolyte density becomes finite throughout the capsid,filling the capsid core, although the maximum in the den-sity close to the capsid is still distinguishable even for c as large as 700 mM (not shown). At high salt, the elec-trostatic interactions are screened and the polyelectrolyteentropy becomes important in the total balance of freeenergy. In this regime, it becomes energetically favor-able even for quite short (depending on exact amountof salt) polyelectrolyte to delocalize over the whole cap-sid as there is enough salt to efficiently screen the capsidcharge even in the absence of the polyelectrolyte. We findthat sub-optimal polyelectrolyte conformations are al-ways such that the polyelectrolyte is located only withina shell close to the capsid, while the super-optimal con-formations are extended throughout the whole capsid,irrespectively of the salt concentration. A similar resultwas found by authors of Ref. 13, who used a discretizedversion of a model akin to ours that does not include theeffects of salt, however.Not all ssRNA viruses can be represented by an in-finitely thin shell of positive protein charge density.While this may be a reasonable approximation for virusesas dengue or yellow fever, it is certainly a poor approx-imation for e.g. cucumber mosaic virus (see Fig. 2),tomato aspermy virus and the much investigated cow-pea chlorotic mottle virus . These viruses are known tohave specifically shaped capsid proteins, so that their N-terminal tails are highly positively charged and stretched.When virus of this type is fully assembled, the capsidprotein ”arms” protrude into its interior. It has recentlybeen suggested that the existence of highly basic capsidpeptide arms can explain the proportionality between thenet charge on the capsid proteins and the total length ofthe ssRNA viral genome . Here we are more interested ininvestigating whether such delocalization of the proteincharge influences the dependence of stability of a virus onthe salt concentration. We represent the capsid chargedensity as ρ p ( r ) = Q c / (4 πr ξ ) , R − ξ < r < R, (3)and ρ p ( r ) = 0 otherwise, i.e. we treat the capsid pep- FIG. 2: One quarter of the cucumber mosaic virus capsid(strain FNY). The image was constructed by applying thegroup of icosahedral transformations to the RCSB ProteinDatabank entry 1F15 and all atoms in the resulting structurewere represented as spheres of radius 3.4 ˚A (which is theexperimental resolution ). They were colored in accordancewith their distance from the geometrical center of the capsid. tide arms as strongly stretched polyelectrolytes, but notnecessarily of brush type, of length ξ , carrying in totala charge Q c per capsid. The actual charge distributionin real viruses depends on the amino acid content of thecapsid peptide arms, but it is worth mentioning that re-sults practically indistinguishable from those shown inFig. 3 are obtained by assuming that ρ p = const . for R − ξ < r < R and zero otherwise, so that the totalprotein charge is still Q c (the robustness of the results ismostly due to the fact that ξ ≪ R ). Note that we donot account for steric repulsion acting between the cap-sid peptide arms and the viral ssRNA i.e. for the lossof interior capsid volume available to ssRNA resultingfrom the protrusion of pieces of capsid proteins into thecapsid interior. Figure 3 is analogous to Fig. 1 but forcapsid charge density represented by Eq. (3). The totalcharge on the capsid ( Q c = 724 e ) is the same in thesetwo cases studied. The arm length chosen ( ξ = 2 . (i) when the polyelectrolyte length is optimal, the viruseswith charge delocalized on the capsid arms are bound sig-nificantly stronger - this can be seen from smaller valuesof F achieved at the minimum when compared with thecorresponding values in Fig. 1 (for c =100 mM, the dif-ference in binding energies is about 450 k B T per capsid),and (ii) the optimal length of the polyelectrolyte is sig-nificantly less influenced by the salt concentration (for c = 10 mM, 100 mM, and 700 mM, the optimal monomer FIG. 3: The same as in Fig. 1 except for the protein chargedensity that is represented as in Eq. (3) with Q c = 724 e and ξ = 2.5 nm. numbers are 700, 650, and 550, respectively). Our sec-ond result is in rough agreement with the findings byBelyi and Muthukumar who estimate that the total ss-RNA charge and the total capsid charge are equal up toa quantity of the order of 10 elementary charges for saltconcentrations below about c = 100 mM . Our resultsthus bridge the two apparently contradictory previousattempts to describe the viral energetics and show thatthe spatial distribution of protein charge determines theimportant features of the energetics of viruses with regardto salt concentration. Note from Fig. 3b) that the thick-ness of the ssRNA ”shell” is determined by the length of the protein arms, unlike in the case of infinitely thinshell of viral protein charge where it is determined by a and v parameters of the polyelectrolyte. For superopti-mal polyelectrolyte lengths there appears a characteristic”two-humped” profile of the polyelectrolyte concentra-tion which results from an interplay of the four lengthscales involved in this case - the a and v parameters ofthe polyelectrolyte, the length of the capsid arms, ξ andthe Debye-H¨uckel screening length that depends on thesalt concentration .In summary, our results show that the influence of salton the energetics of viruses is quite important, especiallyfor viruses whose positive charge on the capsid interiormay be well represented as an infinitely thin shell. How-ever, the delocalization of positive charge on the capsidprotein arms profoundly increases the reliability of viralassembly in high-salt solutions. This effect also increasesresistance of the assembled viruses towards disassemblyin the solutions containing high concentration of (mono-valent) salt. Our results strongly suggest that the highlycharged delocalized capsid protein arms may offer an evo-lutionary advantage to viruses that have them. This doesnot conflict with the fact that the viruses we examinedare very simple ones, since, as Caspar and Klug alreadynoted ”viruses could not possibly exist before cells [and]the minimal viruses could be considered highly evolvedforms”. The delocalized capsid peptide arms may alsohave a role in the kinetics of the assembly, possibly speed-ing it up, and it is in this respect intriguing that weclearly see their effects also in the energetics of the as-sembled viruses.This work has been supported by the Agency for Re-search and Development of Slovenia under grants P1-0055(C) and L2-7080, the Ministry of Science, Educa-tion, and Sports of Republic of Croatia through ProjectNo. 035-0352828-2837, and by the National Foundationfor Science, Higher Education, and Technological Devel-opment of the Republic of Croatia through Project No.02.03./25. 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