Ion-dependent DNA Configuration in Bacteriophage Capsids
Pei Liu, Javier Arsuaga, M. Carme Calderer, Dmitry Golovaty, Mariel Vazquez, Shawn Walker
IIon-dependent DNA Configuration in Bacteriophage Capsids
Pei Liu , Javier Arsuaga , M. Carme Calderer , Dmitry Golovaty , MarielVazquez , and Shawn Walker School of Mathematics, University of Minnesota, Twin Cities, MN, 55455 Department of Molecular and Cellular Biology, and Department of Mathematics,University of California Davis, Davis, CA, 95616 Department of Mathematics, The University of Akron, Akron, OH, 44325 Department of Mathematics, and Department of Microbiology and MolecularGenetics, University of California Davis, Davis, CA, 95616 Department of Mathematics, Louisiana State University, Baton Rouge, LA, 70803November 26, 2020
Abstract
Bacteriophages densely pack their long dsDNA genome inside a protein capsid. The con-formation of the viral genome inside the capsid is consistent with a hexagonal liquid crystallinestructure. Experiments have confirmed that the details of the hexagonal packing depend onthe electrochemistry of the capsid and its environment. In this work, we propose a biophys-ical model that quantifies the relationship between DNA configurations inside bacteriophagecapsids and the types and concentrations of ions present in a biological system. We introducean expression for the free energy which combines the electrostatic energy with contributionsfrom bending of individual segments of DNA and Lennard–Jones-type interactions betweenthese segments. The equilibrium points of this energy solve a partial differential equation thatdefines the distributions of DNA and the ions inside the capsid. We develop a computationalapproach that allows us to simulate much larger systems than what is currently possible usingthe existing simulations, typically done at a molecular level. In particular, we are able to esti-mate bending and repulsion between DNA segments as well as the full electrochemistry of thesolution, both inside and outside of the capsid. The numerical results show good agreementwith existing experiments and molecular dynamics simulations for small capsids.
Bacteriophages are viruses that infect bacteria. Icosahedral double-stranded (ds)DNA bacte-riophages pack their genome in a roughly spherical protein capsid. The length of the genome ison the order of tens of microns, while the diameter of the capsid is in the 10 to 50 nm range. Thetightly packaged DNA molecule forms liquid crystal phases that have been observed and confirmedby both experimental and theoretical studies since the 1980’s [1]-[10]. The infectivity of bacte-riophages stimulated a number of potential applications of bacteriophages, ranging from phagetherapy [11] and drug discovery [12, 13] to the food industry [14].The structure and properties of the viral DNA inside protein capsids have been modeled usingboth continuum [15, 16] and molecular simulations, including Monte–Carlo [17, 18, 19], energyminimization [20], Brownian dynamics [21, 22, 23, 24, 25], Langevin dynamics [26] and moleculardynamics [27]. These molecular-level methods are capable of predicting the precise trajectory ofthe viral genome and elucidating liquid crystalline properties [23]. However, computational costs1 a r X i v : . [ c ond - m a t . s o f t ] N ov imit the size of a system that can be simulated to genomes that are only a few thousand basepairsin length.In [28, 29] we adopted a point of view that DNA packed inside the capsid is in a columnarhexagonal liquid crystalline state and that the equilibrium configuration of the system is determinedin competition between DNA bending, electrostatic, and entropic effects [30]. In [28] we used acontinuum mechanics model to study the liquid crystal structure of the encapsidated genome. Thismodel is motivated by the data obtained with cryo-Electron Microscopy (cryoEM). The data showsthat DNA is in a disordered state in the middle of the capsid, while it forms an ordered structurein a vicinity of the capsid wall [5, 31, 32, 33, 34]. The ordered region features locally parallel DNAsegments that form a triangular lattice on a perpendicular cross-section. In [28] the encapsidatedDNA is characterized by a director—a unit vector representing the preferred local orientation ofDNA segments—and a scalar order parameter reflecting the degree of order of the DNA packing.This parameter can be assumed to be equal to 1 in the ordered region, while it equals 0 in thedisordered region. The model incorporates three energy contributions for the entropic cost of thedisordered region, the bending of the DNA molecule in the ordered region, and the DNA-DNAinteractions. Numerical results show that the DNA segments wind around the axis of the capsidin the ordered region and predict the osmotic pressure inside the capsid, as well as the size of thedisordered region. The DNA-DNA interaction term is a macromolecular interaction model of theform introduced by de Gennes and Kleman [35, 36] and by Oswald and Pieransky [37].Because electrochemistry plays a significant role in the packing [38, 39], folding [40, 41] and ejec-tion of the viral genome [31, 42, 43], here we build upon the methods in [28] to further characterizehow ions affect DNA-DNA interactions in the ordered region of the genome, their distribution, andenergetics inside the capsid.The DNA chain is negatively-charged, with a linear charge density of about 6 e /nm, where e isthe elementary charge. The aqueous environment with a high ionic concentration plays an essentialrole in screening the electrostatic repulsion between the DNA segments and neutralizing the overallcharge distribution. Experiments and molecular simulations have shown that the encapsidatedDNA structure is sensitive to, and can be controlled by, the ionic conditions [43, 44, 40, 31,41]. For example, single molecule studies and molecular dynamics simulations show that highconcentrations of positive ions may induce DNA condensation, significantly increasing the shearstresses of the DNA molecule and reducing the pressure inside the capsid [38, 27]. With increasingsalt concentration, and the resulting increase in concentrations of positive counterions, the spacingbetween two DNA segments is reduced, given the relatively lower contribution from the DNAself-repulsion and bending energy [41].Ions affect the DNA conformation inside the viral capsid according to two possible mechanisms.The first one is through mean-field electrostatic interactions, which account for basic DNA-DNArepulsion and can be described by the Poisson–Boltzmann theory [45]. The second one is by chang-ing the DNA persistence length. This goes beyond the mean-field description of the electrostaticsand can be accounted for through the Debye–H¨uckel theory with charge renormalization [45, 46].Various approximations have been proposed to model the dependence of the persistence lengthon the environmental ionic conditions. These include the Odijk–Skolnick–Fixman (OSF) modelfor high-ionic conditions [47, 48], the OSF-Manning formula that offers a correction to the OSFmodel for low ion concentrations [49], and the Netz-Orland model agrees with a wide range ofexperimental data using two fitting parameters [50], and an interpolation formula with four fittingparameters that works for the whole ionic strength range [51]. Here we use the OSF model sincethe goal of this work is to consider ionic solutions comparable to those found in biological systems.In addition to considering the electrostatic contributions, we use the Oseen–Frank energy todescribe the bending energy of the hexagonal chromonic liquid crystal structure [29], and introducethe Lennard–Jones energy to account for the interaction between nearby DNA segments.This paper is organized as follows. Section 2 and the appendix present the details of theanalytical and computational model of the packaged DNA in the presence of ions. Section 3describes numerical results for a small “virtual” capsid and for the virus P4. Our observations are2n qualitative agreement with previously published experimental and simulation results [27, 41]. Inparticular, we show that (a) the center region of the capsid cannot be occupied by ordered DNAsegments due to prohibitively large bending energy; (b) the DNA has ordered packing near thecapsid wall while the repulsion between the DNA segments prevents the DNA density from beingtoo large; (c) the presence of ions decreases variation of the DNA density in the ordered state; (d)the distance between nearby DNA segments decreases with increasing ionic concentrations.The approach proposed here provides a fast alternative to molecular simulations to model therole of ionic conditions on the packing of DNA in bacteriophage capsids, and captures both thequalitative behavior and most of the quantitative aspects of the system. In this section, we focus on the local number density of the ordered DNA basepairs and theions. We propose the free energy of the system by combining the liquid crystal theory with theelectrolyte theory. Then the equilibrium distributions of the DNA and ions is given by minimizingthis total free energy, which is governed by a set of nonlinear partial differential equations.
The bacteriophage capsid is a protein enclosure that typically has icosahedral shape. Forsimplicity, we describe it as a rigid sphere of radius r , B = { ( r, θ, z ) | − r ≤ z ≤ r , ≤ θ < π, ≤ r ≤ (cid:113) r − z } , (1)where ( r, θ, z ) are cylindrical coordinates. We suppose that the unit vectors (cid:126)e r , (cid:126)e θ and (cid:126)e z point inthe directions of increasing r , θ and z , respectively. As shown in Figure 1(a), the DNA chain windsin the direction of (cid:126)e θ , around the z -axis which is assumed to be perpendicular to the plane of thefigure. It is worth mentioning that any shape of the capsid with axial symmetry can be treated ina similar way as described in the later sections.To reduce the problem further, we postulate that the director field (cid:126)n is equal to the unit vector (cid:126)e θ , which is shown to be a good approximation in [28]. This allows us to assume that the entiresystem is rotationally symmetric so that all equations are independent of θ . The intersection ofthe DNA with the r − z plane is a hexagonal lattice, as illustrated in Figure 1(b). The cross-sectional density is a function in space, denoted as m ( r, z ). Thus, the concentration of DNA is c ( r, z ) = ηm ( r, z ), where η is a constant representing the line number density along the DNAchain. Here we choose η = 3 nm − , corresponding to the rise of one basepair of DNA along thechain (i.e. about 0 .
34 nm) [52].There are N ionic species in the system, whose valences and concentrations are denoted by z i and c i ( r, z ) for 1 ≤ i ≤ N , respectively. The exterior of the capsid containing ions is described asa large cylinder Ω = { ( r, θ, z ) | − L ≤ z ≤ L, ≤ θ < π, ≤ r ≤ L } (2)with the height and the radius L , where L > r so that the cylinder contains the capsid B in itsinterior. We define the total free energy of the system as follows: E cap [ c i ( r, z )] = (cid:90) B k | (cid:126)n × ∇ × (cid:126)n | dx + 12 (cid:90) Ω N (cid:88) i =0 z i ec i φdx + k B T (cid:90) Ω [ γc log c + N (cid:88) i =1 c i log c i ] dx + (cid:90) B f ( c ) dx. (3)3 (a) -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5-2-1.5-1-0.500.511.52 (b) d Figure 1: Configuration of packed DNA inside a bacteriophage capsid. (a) Side view parallel to theaxial direction. The black circle represents the protein capsid, the blue curve describes the orderedstructure of the DNA chain. The red dots in the center describe the disordered core region. (b):Side view perpendicular to the axial direction, each dot represents the intersection of the DNAstrand with the cross-section. The dots are locally arranged in a hexagonal lattice structure. Notethat the distance between neighboring DNA segments may change in space as a function of thedistance to the center of the capsid.The first term comes from the Oseen–Frank free energy, describing the bending energy of theDNA segments. Note that the energy from splay and twist vanishes because (cid:126)n = (cid:126)e θ . The bendingcoefficient k is proportional to the number density of DNA segments and to the persistence length (cid:96) p of DNA [25, 15]: k = k B T (cid:96) p m . (4)The dependence of the DNA persistence length (cid:96) p on the ionic condition is modeled using the OSFtheory, (cid:96) p = (cid:96) + q π (cid:80) Ni =1 z i e c i . (5)Here e is the elementary charge, and qe is the DNA line charge density. (cid:96) is a constant representingthe persistence length of the DNA when the ionic concentrations approach infinity. It can beobtained by fitting the experimental data [51].The second term in Eq. (3) describes the electrostatic energy, where φ is the mean electricalpotential, given by Poisson’s equation: − (cid:15) ∇ φ = N (cid:88) i =0 z i ec i . (6)Here (cid:15) is the dielectric coefficient. z = q/ ( ηe ) describes the valence of one DNA base-pair. Theboundary condition is Dirichlet φ = 0 on ∂ Ω, which describes the overall charge neutrality in thelarge box Ω.The third term in Eq. (3) captures the contribution from the entropy of DNA and all ionicspecies. Here the entropic density of the two-dimensional hexagonal structure of DNA is propor-tional to k B T m ( r, z ) log m ( r, z ), multiplied by the factor of 2 πr , where r is the radius of theDNA segment. This expression is equivalent to the DNA entropy in (3). It also contains a con-stant weight γ that accounts for the fact that DNA is a polymer, unlike the mobile ions. This is atypical assumption in Flory–Huggins theory for polymers [53, 54, 55].4he fourth term in Eq. (3) describes the interaction between the DNA molecules inside thecapsid. Considering the fact that the DNA chain is tightly packed, we use the standard 6-12Lennard–Jones potential to represent interactions between DNA segments. Ignoring attractionterm we model repulsion between neighoring segments by f ( c ) ∝ c d . Given the hexagonal latticestructure of DNA, the distance d satisfies d ∝ c . Thus, we set f ( c ) = αk B T c , (7)where α is a coefficient controlling the strength of the repulsion.Now the total energy simplifies to E cap [ c i ( r, z )] = (cid:90) B k r dx + 12 (cid:90) Ω N (cid:88) i =0 z i ec i φdx + k B T (cid:90) Ω [ γc log c + N (cid:88) i =1 c i log c i ] dx + k B T (cid:90) B αc dx. (8) In this section we derive the set of partial differential equations governing the equilibriumdistributions of DNA inside the capsid as well as the distributions of ions inside and outside ofthe capsid. The chemical potential of each species can be obtained by computing a variation ofthe total energy (8) with respect to the concentrations of that species. Then the concentrations ofDNA and the ions are given implicitly by a modified Boltzmann’s distribution. The concentrationscan now be determined with the help of the Poisson’s equation, taking the form of a modifiedPoisson–Boltzmann equation.We first compute the chemical potential of DNA, µ = δE cap δc ( r, z ) = z eφ + γk B T (log c + 1) + χ (cid:18) (cid:96) p ηr + 7 αc (cid:19) , in Ω , (9)where χ = 1 in B , and χ = 0 in Ω / B . In equilibrium, the chemical potential of DNA must beconstant in Ω and Ω / B , respectively, i.e., µ = (cid:40) µ b,in , in Ω; µ b,out , in Ω / B . (10)The equation (10) describes the fact that the capsid B is not permeable to the DNA. Indeed, DNAcan only be packaged inside or ejected from the capsid through the connector region and modellingthe packaging/ejection process is beyond the scope of this article. Here µ b,in and µ b,out are twoconstants to be determined from the mass conservation of DNA, (cid:90) Ω c dx = N ; (cid:90) B c dx = N p . (11) N is a number representing the total basepairs of DNA in the system, and N p represents thenumber of DNA basepairs that is packaged in the capsid B . Equation (9), (10) and (11) together,implicitly determine the equilibrium distribution of the DNA, which can be viewed as a modifiedBoltzmann’s distribution. Since equation (9) is highly nonlinear, an explicit form of the distributionis not expected.Likewise, the chemical potentials of ions are, µ i = δE cap δc i ( r, z ) = z i eφ + k B T (log c i + 1) − χ k B T c q z i πηr ( (cid:80) Ni =1 z i ec i ) , in Ω . (12)5t should be noticed that the capsid B is permeable to the ions. At equilibrium, the chemicalpotential of each ionic species should be a constant in Ω, µ i = µ bi , i = 1 , , · · · , N, (13)where µ bi is a constant to be determined from the mass conservation of the i th species, (cid:90) Ω c i dx = N i . (14)Here N i describes the number of the i th ion in the system. Equations (12), (13) and (14) give theimplicit distribution of ions.Combining these distributions with Poisson’s equation (6), we obtain a closed PDE system,which is in the form of a modified Poisson–Boltzmann’s equation. Directly solving this PDEsystem is complicated due to the highly nonlinearity. Instead, we consider the following modifiedPoisson–Nernst–Planck equations based on the gradient flow approach, − (cid:15) ∇ φ = N (cid:88) i =0 z i ec i ,∂∂t c i = ∇ · J i = ∇ · (cid:32) k B T ∇ c i + c i ∇ ( z i eφ − χk B T c q z i πηr ( (cid:80) Ni =1 z i ec i ) ) (cid:33) , i = 1 , , · · · , N,∂∂t c = ∇ · J = ∇ · (cid:18) k B T γ ∇ c + c ∇ ( z eφ + χ k B T (cid:96) p ηr (cid:19) . (15)The boundary conditions on ∂ Ω are, (cid:40) φ = 0; J i · (cid:126)e n = 0 , i = 0 , , · · · , N. (16)The interface conditions on ∂ B are, [ φ ] = 0;[ (cid:15) ∇ φ · (cid:126)e n ] = 0;[ µ i ] = 0 , i = 1 , , · · · , N ;[ J i · (cid:126)e n ] = 0 , i = 1 , , · · · , N ; J · (cid:126)e n = 0 . (17)Here [ · ] represents the jump of a given quantity across the capsid wall and (cid:126)e n is the unit normalvector of the interface ∂ B . It is straightforward to verify that this system satisfies the followingequation, implying that the energy is decreasing in time: ddt E cap = − (cid:90) Ω N (cid:88) i =0 J i c i dx. (18)When t → ∞ , the solution of equation (15) approaches the equilibrium described by equations(9)–(13). We developed an efficient numerical method for solving the convection-diffusion system(15). The details of the numerical algorithm are summarized in the Appendix. Estimation of the radial probability distribution of DNA and ions.
The radial proba-bility distribution is defined as, P i ( r ) = (cid:82) π (cid:82) π c i ( r, θ, φ ) r sin φdθdφ (cid:82) Ω c i ( r, θ, φ ) r sin φdrdθdφ . (19)6ere ( r, θ, φ ) correspond to the spherical coordinates, and r sin φ is the Jacobian used to convertfrom Cartesian to spherical coordinates. The radial probability distribution describes the proba-bility of finding one DNA segment on a sphere of radius r centered at the origin. It is related withthe structure factor obtained in experiments such as the X-ray or neutron scattering techniques. Estimation of the distance between DNA segments.
The distance d between parallel DNAsegments is estimated using the hexagonal lattice structure with known density c : √ d · c η = 1 . (20)Here √ d represents the area of a hexagon of diameter d , and c η = m is the cross-sectionaldensity. So, d = (cid:115) η √ c . (21) In this section, we present the numerical results obtained by solving equation (15) for particularbacteriophages. The parameters are chosen based on the real biological systems. Here η = 3 nm − ,describing the fact that one basepair of DNA corresponds to about 0 .
34 nm of length along itsstrand [52]. The line charge density is approximately q = 6 e/ nm [56], so that z = qηe = 2. Thedielectric coefficient is set to be the dielectric constant of water at room temperature, i.e. (cid:15) = 78.To determine the two parameters α and γ in the model, we require the contribution from eachterm in the total energy to be comparable with each other. In the following, γ = 0 . r = 12 . N = 4500 bp. To consider the equilibriumstate, we assume that all the DNA is inside the capsid thus N p = N . The average concentrationof DNA is c a = 3 N / (4 πr ) ≈ .
55 nm − . We set α = 0 . − nm . There are two ionic speciesNa + (or Mg ) and Cl − in the system. The overall charge neutrality requires, z N + z N + z N = 0 . (22) Figure 2, shows a quadrant of the DNA density (left pane) and the ionic densities for Na + (middle) and Cl − (right) in the r − z plane. The DNA density outside of the capsid is identicallyzero, since we assume that the DNA is completely packaged. Additionally, there is a core regionclose to the central axis of the capsid (the z -axis in the figure) where the DNA density is negligible.The radius of this region is about 2 nm and indicates that the DNA molecule cannot be perfectlyordered at the center core of the capsid due to its bending rigidity. Further away from the center,the DNA density increases, showing that DNA tends to stay close to the capsid, mainly dueto the effects of the DNA bending energy. The contribution from the Lennard–Jones repulsionprevents the DNA from condensing at the capsid, and instead forces the DNA density to be nearlyhomogeneous in the region outside the inner core. Interestingly the figure also shows a sharptransition zone between the inner core and the outer region.The distribution of Na + is shown in Figure 2(b). Na + follows the distribution of DNA becausethe negatively charged DNA attracts (or absorbs) positive charges. Since the capsid is permeableto ions, outside of the capsid the Na + density is very small (though not zero). As expected, thedistribution of Cl − behaves in an opposite manner (Figure 2 (c)). Cl − ions are repelled from theregion where the DNA is located, and are displaced to the inner core region and to the regionoutside of the capsid. 7igure 2: Number density in the r − z plane. (a): DNA; (b): Na + ; (c): Cl − In conclusion, our results show that inside the capsid, NaCl dissociates into its positive andnegative ions. Positive ions mostly associate with the DNA molecule, while negative ions areexpelled to the center of the capsid (where DNA is mostly absent) and to the region outside thecapsid.
Next, we computed the radial probability distribution of DNA and ions, as explained in themethods section. Figure 3 shows the probability distribution for DNA (black), Na + (red) and Cl − (blue). Distance to Origin P r obab ili t y DNANaCl (a) 0 10 20 30
Distance to Origin P r obab ili t y DNANaCl (b)
Distance to Origin P r obab ili t y DNAMgCl (c)
Figure 3: Probability distribution for DNA (black), Na + (red) and Cl − (blue) under different ionicconditions. (a): 100 mM NaCl; (b): 1 M NaCl; Right: 100 mM MgCl .The probability distribution of DNA inside the capsid increases monotonically with the distancefrom the center core, mirroring the increase of the DNA density. Our results are in agreement withthose reported in [27] with a small difference in the DNA distribution. Figure 3 shows only onelocal maximum of the DNA probability distribution next to the protein capsid, while the work of[27] identified two discrete layers of DNA. This difference is due to the fact that in our model, theDNA double helix is not explicitly described, instead its discrete structure is implicitly given bythe averaged DNA concentration. The distribution probabilities for ions are consistent with thosepresented in Figure 2(a) and clearly show the co-localization of positive ions with the DNA, andof positive and negative ions.Next, we performed a simulation experiment in which we over saturated the sample with 1.6MNaCl (Figure 3 (b)). We observe the Na + and Cl − curves apparently overlap unlike the other twopanels. To understand this behavior, we focus on the renormalized density profile as a function of8 . We define the average of the density c i ( r ) on a sphere of radius r as: ρ i ( r ) = (cid:82) π (cid:82) π c i ( r, θ, φ ) sin φdθdφ (cid:82) π (cid:82) π sin φdθdφ , (23)Figure 4 shows the results for three different concentrations of N aCl . With increasing salt con-centration, we observe a decrease on the maximum of the DNA curve (near the protein capsid).As expected the ion concentration increases both inside and outside the viral capsid. We note thatas the overall concentration of
N aCl increases, the concentration of Cl − in the region enclosedby the capsid increases faster than outside, while the average concentration of N a + in the regionenclosed by the capsid increases slower than outside. The combination of these two events makesthe two curves similar.Finally, we simulated in the presence of 100 mM of MgCl. Results are very similar to thosepresented for NaCl (Figure 3(c)) and are consistent with the molecular simulation results in [27].The slope of the DNA curve decreases with increasing salt concentration due to the inverse rela-tionship between persistence length l p and ionic strength. Weakening the energy contribution ofDNA bending allows for a higher DNA probability distribution near the center core and therefore amore homogeneous distribution of the DNA inside the capsid. The distribution of the ions (Mg and Cl − ) mirrors that of the experiment with NaCl. Distance to Origin C on c en t r a t i on DNANaCl (a)
Distance to Origin C on c en t r a t i on DNANaCl (b) 0 5 10 15 20 25
Distance to Origin C on c en t r a t i on DNANaCl (b)
Figure 4: Average radial distribution ρ i ( r ) under different ionic conditions. (a): 100 mM NaCl;(b): 166 mM NaCl; (c): 1M NaCl. We investigated the average inter-strand separation of DNA as a function of the salt concentra-tion. It has been observed that as the concentration of positive ions increases the average distancebetween DNA segments decreases, in both molecular simulation [27] and experiment [41].Figure 5 describes the inter-strand distance of DNA under different ionic conditions. Sincethe value of c , and therefore d , is a function of the spatial location we computed their values attwo different cross-sectional locations. In Figure 5(a), the curve labeled (11 ,
0) corresponds to theDNA spacing near the protein capsid, and the curve labeled (7 ,
0) corresponds to DNA spacing halfway between the center and the protein capsid. Both curves show that the inter-strand distancedecreases with higher salt concentrations. The main reason is because the persistence length ismonotonically decreasing with ionic strength, facilitating the packing of DNA into an orderedhexagonal structure. This result is in agreement with the experimental observations reported in[27, 41]. 9
500 1000 1500
NaCl Concentratioin (mM) DN A S pa c i ng ( n m ) (7,0)(11,0) Figure 5: Inter-strand distance of DNA as a function of NaCl concentration, at cross-sectionallocations (11 ,
0) and (7 ,
0) in a capsid with radius r = 12 . As discussed earlier, ionic concentrations affect both the shielding of negative charges alongthe DNA molecule and the persistence length of DNA. To better understand the origin of theobserved difference in strand-separation as a function of ionic concentration, we investigated thecontribution of the different components of the energy to the total energy.
NaCl Concentration (mM) E nee r g y ( k B T ) BendingElectrostaticLenard-Jones
Figure 6: The energy from bending, electrostatic and Lenard–Jones for increasing NaCl concen-trations.Figure 6 shows the contribution of the electrostatic, bending and Lennard–Jones energies tothe total energy of the system. Although the bending and Lenard–Jones energy are larger invalue than the electrostatic energy, they are weakly dependent on the ionic concentrations, whileelectrostatic is more sensitive.As the salt concentration increases, the overall bending energy andelectrostatic energy decrease and the Lennard–Jones potential increases slightly. The decrease inelectrostatic energy is because of the screening in the electrical potential; the decrease in bendingenergy is explained due to the decrease in persistence length, and therefore the Lenard–Jonesenergy increases because of the smaller strand-separation.10 .5 Simulations for P4 phage
The genome of bacteriophage P4 is 11 . r = 22 . N = 11 . N p = N , i.e., the entire DNA is inside the capsid. The average concentrationof DNA is c a = 3 N / (4 πr ) = 0 .
24 nm − . We set α = 0 . − nm .The behavior of the density distributions of DNA and ions are similar to Fig. 2. The core-region at the center of the capsid has no (ordered) DNA, which has a radius of about 2 nm. Fig.7 describes the inter-strand distance of DNA under different ionic conditions for P4 phages. Sincethe average concentration of DNA is smaller than the virtual bacteriophage discussed before, thedistance between nearby DNA segments in P4 phage is larger. Numerical results again showsthe increasing ionic concentration causes the decreasing in the DNA spacing, which suggests thisobservation is true for a variety of bacteriophages unless additional phenomena are taken intoaccount.Figure 7: Inter-strand distance of DNA as a function of NaCl concentration, at cross-sectionallocations (22 ,
0) and (11 ,
0) in a capsid with radius r = 22 . Ions are essential in multiple biological processes. In bacteriophages ionic concentrations havebeen shown to play a key role in the packaging [38, 39], folding [41, 58] and delivery [40] of theviral genome. In this work we take a continuum mechanics approach to quantitatively describethe role played by ions in the folding of DNA inside the phage capsid.Our approach expands on previous work where we introduced a novel chromonic liquid crystalmodel for DNA inside a bacteriophage capsid [29, 28]. This model built on pioneer continuummechanics work by Tzlil and colleagues [25], and on the concept of director introduced by Klugand Ortiz [15]. However, the mechanics model in [25] implemented a phenomenological formula-tion of DNA-DNA interactions inspired by the works of de Gennes and Kleman [35, 36] and byOswald and Pieransky [37], that is not amenable to a detailed description of ionic interactions inthe environment. To address this issue we propose a model that introduces explicitly the ionicconcentrations, their diffusion, and their interaction with the DNA molecule.The proposed ionic model describes the distribution of ions both inside and outside the bacterio-phage capsid, and the average structure of the DNA packaged inside capsids. The model combinesthe Oseen–Frank energy from liquid crystal theory with salt-dependent persistence length, the elec-trostatic potential energy between charges, and the Lennard–Jones interaction potential betweenDNA segments. A key aspect of our model is that it incorporates effects of the ionic conditions11n the DNA-DNA interaction and their role in modulating the persistence length of the DNAmolecule.Our results are in agreement with those presented by Cordoba and colleagues [27]. The modelpredicts the distribution of ions relative to the DNA molecule, and the distribution of DNA insidethe capsid. We find that positive ions colocalize with the DNA molecule at relatively low ionicconcentrations (but not diluted as found in common standard phage buffers), while negative ionsare displaced either to the center core of the capsid, where DNA is mostly absent due to the highbending required to fill this volume, or outside of the capsid. The DNA-DNA interactions at thisionic concentration are mostly repulsive [59]. Therefore a higher concentration of DNA is foundin very close proximity to the capsid. At saturated ionic concentrations our model predicts adisplacement of the DNA away from the capsid, and an overall ionic saturation of the capsid andsurrounding environment.The model also captures the contribution of the different energy terms to the total energy ofthe system and how the ionic conditions affect this contribution. In [60] it was reported that bothelectrostatic and entropic effects account for most of the energy of the system. Here we find thatthe bending energy and the Lennard–Hones potential also plays an important role. Our resultsare in agreement with experimental results obtained by Qui and colleagues [41].Based on the agreement of our results with known DNA and ionic distribution inside the viralcapsid, intra-strand DNA distance, and the energies, we conclude that the model presented inthis paper can capture the structure of the packaged phage DNA under physiological conditions.Also, as a continuum model, solving the equations numerically is much faster compared with theapproaches based on molecular simulations, and can be applied to large bacteriophages.Several improvements of the model can be considered. In our model, the structure of theDNA segments is implicitly given by the averaged concentrations. In order to capture the discretelayer structure, as was done in [27, 59], we need to add in the pairwise correlation energy fromthe hard-sphere repulsion. Furthermore, although the use of Odijk–Skolnick–Fixman theory tomodel the ion dependent persistence length is shown to be sufficient and successful with high ionicconcentrations, extensions to lower ionic conditions can be made by employing different theories.Our approach can also be extended to study the packing process, by varying the length of DNA, N p , (in bp) packed inside of the capsid.Experiments [9] and molecular dynamics simulations [60] have shown that the addition ofpolyvalent cations introduce attractive effects between DNA segments and promote the formationof toroidal structures [61]. This suggests promising avenues for future research. Appendix: Numerical Method
To solve the pesudo-time-dependent problem, we first simplify the equation using non-dimensionalvariables, ˜ φ = βeφ , ˜ x = x/L c , ˜ c = c L c . Here L c = 1 nm . For simplicity we drop all the tildes inthe non-dimensional equations: −∇ φ = 4 π(cid:96) B N (cid:88) i =0 z i c i ,∂∂t c i = ∇ · J i = ∇ · ( ∇ c i + c i ∇ µ i,ex ) ,∂∂t c = ∇ · J = ∇ · ( γ ∇ c + c ∇ µ ,ex ) . (24)Here (cid:96) B = e π(cid:15)k B T is the Bjerrum length. The excess chemical potentials are given by µ ni,ex = z i φ n − χ c n q z i πηr ( (cid:80) Ni =1 z i c ni ) , µ n ,ex = z φ n + χ (cid:96) p ηr . Using these non-dimensional quantities, the total12nergy is reformulated as follows: βE cap [ c i ( r, z )] = (cid:90) B c ηr (cid:32) (cid:96) + z η π (cid:80) Ni =1 z i c i (cid:33) dx + 12 (cid:90) Ω N (cid:88) i =0 z i c i φdx + (cid:90) Ω [ γc log c + N (cid:88) i =1 c i log c i ] dx + (cid:90) B αc dx. (25)We then use an Implicit-Explicit scheme for the time discretization, c n +1 i − c ni τ = ∇ · (cid:16) ∇ c n +1 / i + c n +1 / i ∇ µ n +1 / i,ex ) (cid:17) ,c n +10 − c n τ = ∇ · (cid:16) ∇ c n +1 / + c n +1 / ∇ µ n +1 / ,ex ) (cid:17) . (26)Here the concentrations at half time grids are interpolated using c n +1 / i = ( c ni + c n +1 i ) /
2, whichcorrepsonds to the Crank–Nicolson algorithm for diffusion equations. The excess chemical poten-tials at half time grids are extrapolated using µ n +1 / i,ex = (3 µ ni,ex − µ n − i,ex ) /
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