Elastic Deformation Energy for Beams and Thin Filaments Electrostatic

MAE 545: Lecture 12 (10/27)

Electrostatic Elastic deformation energy for energy for beams and bending DNA thin filaments Poisson-Boltzmann equation Let’s assume some mean-ﬁeld electric potential ( ~r ) throughout the cell. Local density of mobile ions carrying charge z ↵ e 0 . 3 z↵e0(~r )/kB T z↵e0(~r )/kB T d ~r n e = N n↵(~r )=n↵e ↵ ↵ Z Charge density of mobile ions

z↵e0(~r )/kB T ⇢mobile ions(~r )= z↵e0n↵e ↵ X Poisson equation 4⇡ 2(~r )= ⇢(~r ) r ✏ Poisson-Boltzmann equation

2 4⇡ z e (~r )/k T (~r )= ⇢ (~r )+ z e n e ↵ 0 B r ✏ macroions ↵ 0 ↵ " ↵ # X For a given distribution of macroions Poisson-Boltzmann equation must be solved self-consistently for the electric potential ( ~r ) .

2 Dissociation of charge from a plate y + + + density of positive counterions + + + e0(y)/kB T + n(y)=ne + + Poisson-Botzmann equation

2 4⇡ e (y)/k T (y)= (y)+e ne 0 B r ✏ 0 h i < 0 charge density of the plate + + 2kBT y kBT ✏ + (y)= ln 1+| | y0 = e0 y0 ⇡e + + + +  0| | + + + y0 - Guoy-Chapman length thickness of diffusive boundary layer that shields a charged membrane density of positive counterions ⇡ 2 1 electric ﬁeld n(y)= | | 2 2✏kBT (1 + y /y0) @(y) y 2⇡ 1 | | E(y)= = | | @y y ✏ (1 + y /y ) dy n(y)= | | | | 0 Z 3 Debye-Hückel approximation Let’s assume that electrostatic energy due to the mean ﬁeld electric potential is small compared to kBT. Local density of mobile ions carrying charge z ↵ e 0 .

z↵e0(~r )/kB T 3 z↵e0(~r )/kB T n↵(~r )=n↵e d ~r n↵e = N↵ z e (~r ) Z n (~r ) n 1 ↵ 0 ↵ ⇡ ↵ k T n↵ N↵/V ✓ B ◆ ⇡ Charge neutrality

z↵n↵ =0 ↵ X Charge density of mobile ions

z↵e0(~r )/kB T ⇢mobile ions(~r )= z↵e0n↵e ↵ X e2(~r ) ⇢ (~r ) 0 z2 n = ` ✏(~r ) z2 n mobile ions ⇡ k T ↵ ↵ B ↵ ↵ B ↵ ↵ X X

4 Debye-Hückel approximation

Charge density of mobile ions e2(~r ) ⇢ (~r ) 0 z2 n = ` ✏(~r ) z2 n mobile ions ⇡ k T ↵ ↵ B ↵ ↵ B ↵ ↵ X X Poisson equation 4⇡ 2(~r )= [⇢ (~r )+⇢ (~r )] Debye screening length r ✏ macroions mobile ions

4⇡ (~r ) 2 2 2 =4⇡` z n (~r )= ⇢macroions(~r )+ 2 D B ↵ ↵ r ✏ D ↵ X

Electric potential for a Electrostatic interaction between macroions point charge ⇢ (~r )= z e (~r ~r ) macroions m 0 m ⇢macroions(~r )=ze0(~r ) m X zme0 ~r ~r / (~r)= e| m| D ✏ ~r ~r m m ze0 r/D | | (~r)= e X ✏r Einteractions 1 zne0(~rn) zmzn`B ~r ~r / = = e| m n| D k T 2 k T ~r ~r B n B m

Negative unit charges separated Electrostatic energy by distance b along the rod. 2 0 0 e d d + 0 0 0 mn mn dmn/ V (dmn)= 2 e 24✏R b m 1 n 0 Vtot = V (d ) 0 N mn dmn m

0 dmn = b m n 1 nx | | f(x)= n(1 + nx)e L = Nb n=1 X 7 Bending of charged rod

Negative unit charges separated Electrostatic energy by distance b along the rod. 2 Le0 kBT L`B Vtot 2 f(b/)= 2 f(b/) b m ⇡ 24✏R 24 R 1 n 1 0 N nx dmn f(x)= n(1 + nx)e e0 e n=1 0 X Mathematical trick x R 1 nx e R g(x)= e = 1 e x n=1 X f(x)= g0(x)+xg00(x) ✓ x x x e (1 + x + xe e ) f(x)= x 3 (1 e ) d0 = b m n mn | | L = Nb 8 Bending of charged rod

Negative unit charges separated Electrostatic energy by distance b along the rod. k T L` 1  L V B B f(b/)= e tot ⇡ 24 R2 2 R2 b m Bending rigidity due to n 1 electrostatic energy 0 N e dmn 0 kBT e E `Bf(b/) 0 ⇡ 12 How this compares to measured R R bending rigidity for DNA?  = kBT `p

b 0.17nm 1nm ✓ ⇡ ⇡ ` 0.7nm `p 50nm B ⇡ ⇡  ` 0.7 E B f(b/) 104 0.12 0 d = b m n  ⇡ 12`p ⇡ 12 50 ⇥ ⇡ mn | | ⇥ L = Nb 9 L.R. Comolli et al. / Virology 371 (2008) 267–277 273

Phage Packaging and Ejection Forces 857

moved farther apart. For trivalent and tetravalent ions the An analogous formula holds for the repulsive-attractive physics is quite different. In this regime there is an attractive regime. Note that this expression reports the free energy of interaction with a preferred spacing d0 between the strands, the inverse spool configurations when a length L has been and the measurements are well fit by packaged by relating R to L via Eq. 7. We will now show that the spacing between the strands varies in a systematic way 2 d0 ds Gint L; ds p3F0L c 1 cds exp ÿ during the packing process, reflecting the competition be- ð Þ¼ ð Þ c tween bending and interaction terms. ffiffiffi 1 DNA packaging in bacteriophage virusesc2 1 cd d2 d2 ; (10) ÿð 0Þÿ2ð 0 ÿ s Þ DNA spacing in packed capsids where c 0.14 nm and d 2.8 nm, for example, in the case Our model makes a concrete prediction for the free energy typical DNA is packaged ¼ 0 ¼ of cobalt hexamine as the condensing agent. For ds , d0 the Gtot of packaged DNA in any phage and for a wide range of interaction is strongly repulsive and for ds . d0 it is solution conditions. To find Gtot for a particular phage, we bacteriophage by a motor The attractive.whole This expression DNA is a goodis representation of the free minimize Eq. 11 by varying ds, under the constraint that energy of interactions between the DNA molecules for L given by Eq. 7 is equal to the length of DNA already packagedspacings ds lessin than 2-5 or equal min. to the preferred value d0 (Rau packaged. The expressions for N(R9) will differ for different and Parsegian, 1992). In viral packaging we encounter capsid geometries. Most capsids are icosahedral and we interaxial spacings in exactly this range and hence we will idealize them as spheres. Some capsids, e.g., f29, have Velocityuse this of free energyDNA to study packing the effects of repulsive-attractive a waist. We idealize them as cylinders with hemispherical capsid interactions on encapsidated DNA. caps. Eventually, we will find that the geometry does not is ~50-200It is important to notenm/s. that in the experiments of Rau et al. affect the overall free energy of packing as long as the (1984) and Rau and Parsegian (1992) DNA was confined in internal volume of the idealization is the same for each the same way (except for the lack of bending) as it is within geometry, once again reflecting the importance of the Packaginga phage capsid. This motors means that the free energy Gint obtained parameter rpack, introduced earlier. Before we specialize to from their measurements accounts for multiple effects, in- particular geometries (see Fig. 3) we observe that tail cluding electrostatics, entropy of the DNA (Odijk, 1983) and N R9 z R9 =ds where z(R9) is the height of a column producecounterions, force and any hydration ~60 phenomena pN. (Strey et al., 1997). ofð hoopsÞ¼ð ofð DNAÞ situatedÞ at radius R9. Using this fact and Given that we have now examined the separate contribu- differentiating Eq. 7 with respect to ds, while holding L tions arising from DNA bending, entropy, and interaction constant, gives us dR=dds p3Lds=2pRz R . Mini- DNA is tightly packed ð Þ ¼ ÿð ð ÞÞ terms, we now write the free energy of the encapsidated mizing Gtot with respect to the interstrand spacing ds gives insideDNA in thethe repulsive capsid: regime, ffiffiffi Rout z R9 ð Þ jpkBT jpkBT R R9 dR9 p3F0 exp ds=c 2 2 2 R : (12) R out VDNA/Vcap > 0.5 out ðÿ Þ¼ R ds ÿ ds R9z R9 dR9 2pjpkBT N R9 RR Gtot L; ds ð ÞdR9 ffiffiffi ð Þ ð Þ¼ p3d R9 s ZR R This equation represents a balance between the bending ffiffiffi Bending energy terms and the interaction terms. Note that if the size L.R. Comolli et al. / Virology 371 (2008) 267–277 schematic273 of|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl 2 {zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} ds empty capsid capsid with DNA 1 p3F0 c 1 cds L exp : (11) of the capsid is fixed then longer lengths of packed DNA ð Þ ÿ c imply smaller radii of curvature for the hoops since the packaged DNAffiffiffi in Interaction strands want to be as far away as possible from each other for bacteriophage | fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl 29 {zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

FIGURE 3 Idealized geometries of viral capsids.

Fig. 5. Images of partially packaged genomes. Shown here are representative images of particles projected at different orientations, with (top to bottom) 0%, 32%, 50nm 51%, 78%, and 100% of the genome packaged. For all partially packaged particles, we see an isotropic grey density distribution filling the interior of the capsids, ⇠ indicating that the DNA occupies all the available10 volume. One quite faint, outermost layer starts to appear in top views of some of the particles with 78% of the genome packaged (leftmost image in the 78% row).

Biophysical Journal 88(2) 851–866 constrained to adopt a close-packed local order, most probably (2000) used ATP-γS to block active DNA packaging motors. with some degree of local hexagonal packing. Cryo-electron microscopy was then used to obtain three- In a previous work aimed at elucidating the structure of the dimensional reconstructions of the particles with a partial φ29 motor in actively packaging particles, Simpson et al. genome contained in the capsids. The interior of the capsids had

Fig. 5. Images of partially packaged genomes. Shown here are representative images of particles projected at different orientations, with (top to bottom) 0%, 32%, 51%, 78%, and 100% of the genome packaged. For all partially packaged particles, we see an isotropic grey density distribution filling the interior of the capsids, indicating that the DNA occupies all the available volume. One quite faint, outermost layer starts to appear in top views of some of the particles with 78% of the genome packaged (leftmost image in the 78% row). constrained to adopt a close-packed local order, most probably (2000) used ATP-γS to block active DNA packaging motors. with some degree of local hexagonal packing. Cryo-electron microscopy was then used to obtain three- In a previous work aimed at elucidating the structure of the dimensional reconstructions of the particles with a partial φ29 motor in actively packaging particles, Simpson et al. genome contained in the capsids. The interior of the capsids had Phage Packaging and Ejection Forces 857

moved farther apart. For trivalent and tetravalent ions the An analogous formula holds for the repulsive-attractive physics is quite different. In this regime there is an attractive regime. Note that this expression reports the free energy of interaction with a preferred spacing d0 between the strands, the inverse spool configurations when a length L has been and the measurements are well fit by packaged by relating R to L via Eq. 7. We will now show that the spacing between the strands varies in a systematic way 2 d0 ds Gint L; ds p3F0L c 1 cds exp ÿ during the packing process, reflecting the competition be- ð Þ¼ ð Þ c tween bending and interaction terms. ffiffiffi 1 c2 1 cd d2 d2 ; (10) ÿð 0Þÿ2ð 0 ÿ s Þ DNA spacing in packed capsids where c 0.14 nm and d 2.8 nm, for example, in the case Our model makes a concrete prediction for the free energy ¼ 0 ¼ of cobalt hexamine as the condensing agent. For ds , d0 the Gtot of packaged DNA in any phage and for a wide range of interaction is strongly repulsive and for ds . d0 it is solution conditions. To find Gtot for a particular phage, we attractive. This expression is a good representation of the free minimize Eq. 11 by varying ds, under the constraint that energy of interactions between the DNA molecules for L given by Eq. 7 is equal to the length of DNA already spacings ds less than or equal to the preferred value d0 (Rau packaged. The expressions for N(R9) will differ for different and Parsegian, 1992). In viral packaging we encounter capsid geometries. Most capsids are icosahedral and we interaxial spacings in exactly this range and hence we will idealize them as spheres. Some capsids, e.g., f29, have use this free energy to study the effects of repulsive-attractive a waist. We idealize them as cylinders with hemispherical interactions on encapsidated DNA. caps. Eventually, we will find that the geometry does not It is important to note that in the experiments of Rau et al. affect the overall free energy of packing as long as the (1984) and Rau and Parsegian (1992) DNA was confined in internal volume of the idealization is the same for each the same way (except for the lack of bending) as it is within geometry, once again reflecting the importance of the a phage capsid. This means that the free energy Gint obtained parameter rpack, introduced earlier. Before we specialize to from their measurements accounts for multiple effects, in- particular geometries (see Fig. 3) we observe that cluding electrostatics, entropy of the DNA (Odijk, 1983) and N R9 z R9 =ds where z(R9) is the height of a column counterions, and any hydration phenomena (Strey et al., 1997). ofð hoopsÞ¼ð ofð DNAÞ situatedÞ at radius R9. Using this fact and Given that we have now examined the separate contribu- differentiating Eq. 7 with respect to ds, while holding L tions arising from DNA bending, entropy, and interaction constant, gives us dR=dds p3Lds=2pRz R . Mini- ð Þ ¼ ÿð ð ÞÞ terms, we now write the free energy of the encapsidated mizing Gtot with respect to the interstrand spacing ds gives DNA in the repulsive regime, ffiffiffi Rout z R9 j k T j k T ð ÞdR9 p p B p B R R9 3F0 exp ds=c 2 2 2 Rout : (12) Rout ðÿ Þ¼ R d ÿ d R9z R9 dR9 2pjpkBT N R9 s s RR Gtot L; ds ð ÞdR9 ffiffiffi ð Þ ð Þ¼ p3d R9 s ZR R This equation represents a balance between the bending ffiffiffi Bending energy terms and the interaction terms. Note that if the size |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl2 {zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} ds 1 p3F0 c 1 cds L exp : (11) of the capsid is fixed then longer lengths of packed DNA ð Þ ÿ c DNA packaging in bacteriophageimply smaller radii ofviruses curvature for the hoops since the ffiffiffi Interaction strands want to be as far away as possible from each other for 2r = 42nm |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} Packaging of DNA in bacteriophage 29 requires ~105 kBT.

h Bending energy 47nm =

 kBT `P 2 E L L FIGURE4 310 Idealizedk T geometries of viral capsids. b ⇠ 2r2 ⇠ 2r2 ⇠ ⇥ B

Loss of entropy

DNA length T S = kBT ln ⌦ L =6.8µm Estimate the entropy outside capsid with Biophysical Journal 88(2) 851–866 distance between ideal chain made of Nk Kuhn segments neighboring chains N ⌦ g k Nk = L/2`p 70 d 2.3nm ⇠ ⇡ ⇡ DNA persistence 2 T S NkkBT ln g 10 kBT length ⇠ ⇠ ` 50nm p ⇡ 11 Phage Packaging and Ejection Forces 857

moved farther apart. For trivalent and tetravalent ions the An analogous formula holds for the repulsive-attractive physics is quite different. In this regime there is an attractive regime. Note that this expression reports the free energy of interaction with a preferred spacing d0 between the strands, the inverse spool configurations when a length L has been and the measurements are well fit by packaged by relating R to L via Eq. 7. We will now show that the spacing between the strands varies in a systematic way 2 d0 ds Gint L; ds p3F0L c 1 cds exp ÿ during the packing process, reflecting the competition be- ð Þ¼ ð Þ c tween bending and interaction terms. ffiffiffi 1 c2 1 cd d2 d2 ; (10) ÿð 0Þÿ2ð 0 ÿ s Þ DNA spacing in packed capsids where c 0.14 nm and d 2.8 nm, for example, in the case Our model makes a concrete prediction for the free energy ¼ 0 ¼ of cobalt hexamine as the condensing agent. For ds , d0 the Gtot of packaged DNA in any phage and for a wide range of interaction is strongly repulsive and for ds . d0 it is solution conditions. To find Gtot for a particular phage, we attractive. This expression is a good representation of the free minimize Eq. 11 by varying ds, under the constraint that energy of interactions between the DNA molecules for L given by Eq. 7 is equal to the length of DNA already spacings ds less than or equal to the preferred value d0 (Rau packaged. The expressions for N(R9) will differ for different and Parsegian, 1992). In viral packaging we encounter capsid geometries. Most capsids are icosahedral and we interaxial spacings in exactly this range and hence we will idealize them as spheres. Some capsids, e.g., f29, have use this free energy to study the effects of repulsive-attractive a waist. We idealize them as cylinders with hemispherical interactions on encapsidated DNA. caps. Eventually, we will find that the geometry does not It is important to note that in the experiments of Rau et al. affect the overall free energy of packing as long as the (1984) and Rau and Parsegian (1992) DNA was confined in internal volume of the idealization is the same for each the same way (except for the lack of bending) as it is within geometry, once again reflecting the importance of the a phage capsid. This means that the free energy Gint obtained parameter rpack, introduced earlier. Before we specialize to from their measurements accounts for multiple effects, in- particular geometries (see Fig. 3) we observe that cluding electrostatics, entropy of the DNA (Odijk, 1983) and N R9 z R9 =ds where z(R9) is the height of a column counterions, and any hydration phenomena (Strey et al., 1997). ofð hoopsÞ¼ð ofð DNAÞ situatedÞ at radius R9. Using this fact and Given that we have now examined the separate contribu- differentiating Eq. 7 with respect to ds, while holding L tions arising from DNA bending, entropy, and interaction constant, gives us dR=dds p3Lds=2pRz R . Mini- ð Þ ¼ ÿð ð ÞÞ terms, we now write the free energy of the encapsidated mizing Gtot with respect to the interstrand spacing ds gives DNA in the repulsive regime, ffiffiffi Rout z R9 j k T j k T ð ÞdR9 p p B p B R R9 3F0 exp ds=c 2 2 2 Rout : (12) Rout ðÿ Þ¼ R d ÿ d R9z R9 dR9 2pjpkBT N R9 s s RR Gtot L; ds ð ÞdR9 ffiffiffi ð Þ ð Þ¼ p3d R9 s ZR R This equation represents a balance between the bending ffiffiffi Bending energy terms and the interaction terms. Note that if the size |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl2 {zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} ds 1 p3F0 c 1 cds L exp : (11) of the capsid is fixed then longer lengths of packed DNA DNA packagingð Þ inÿ c bacteriophage viruses imply smaller radii of curvature for the hoops since the ffiffiffi Interaction strands want to be as far away as possible from each other for 2r = 42nm Debye-Hückel electrostatic energy |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflbetween} charges on DNA

k T ` `B 0.7nm B B s/ ⇡ V (s)= e 1nm s ⇡ h

47nm = Electrostatic energy between two

neighboring chargedFIGURE loops 3 Idealized geometries of viral capsids.

number of charges per loop d 2⇡r N = DNA length b b 0.17nm L =6.8µm ⇡ ⇡ Biophysical Journal 88(2) 851–866 distance between 2⇡r rd✓ Vr = V (s(✓)) neighboring chains b ⇡ b Z d 2.3nm r 2 ⇡ pd2+(2r sin(✓/2))2/ ⇡ ✓ 2⇡r kBT `B e Vr = d✓ DNA persistence s 2 2 2 b ⇡ d +(2r sin(✓/2)) length Z 2 2 ` 50nm s(✓)= d +(2r sin(✓/2)) p p ⇡ p 12 Phage Packaging and Ejection Forces 857

moved farther apart. For trivalent and tetravalent ions the An analogous formula holds for the repulsive-attractive physics is quite different. In this regime there is an attractive regime. Note that this expression reports the free energy of interaction with a preferred spacing d0 between the strands, the inverse spool configurations when a length L has been and the measurements are well fit by packaged by relating R to L via Eq. 7. We will now show that the spacing between the strands varies in a systematic way 2 d0 ds Gint L; ds p3F0L c 1 cds exp ÿ during the packing process, reflecting the competition be- ð Þ¼ ð Þ c tween bending and interaction terms. ffiffiffi 1 c2 1 cd d2 d2 ; (10) ÿð 0Þÿ2ð 0 ÿ s Þ DNA spacing in packed capsids where c 0.14 nm and d 2.8 nm, for example, in the case Our model makes a concrete prediction for the free energy ¼ 0 ¼ of cobalt hexamine as the condensing agent. For ds , d0 the Gtot of packaged DNA in any phage and for a wide range of interaction is strongly repulsive and for ds . d0 it is solution conditions. To find Gtot for a particular phage, we attractive. This expression is a good representation of the free minimize Eq. 11 by varying ds, under the constraint that energy of interactions between the DNA molecules for L given by Eq. 7 is equal to the length of DNA already spacings ds less than or equal to the preferred value d0 (Rau packaged. The expressions for N(R9) will differ for different and Parsegian, 1992). In viral packaging we encounter capsid geometries. Most capsids are icosahedral and we interaxial spacings in exactly this range and hence we will idealize them as spheres. Some capsids, e.g., f29, have use this free energy to study the effects of repulsive-attractive a waist. We idealize them as cylinders with hemispherical interactions on encapsidated DNA. caps. Eventually, we will find that the geometry does not It is important to note that in the experiments of Rau et al. affect the overall free energy of packing as long as the (1984) and Rau and Parsegian (1992) DNA was confined in internal volume of the idealization is the same for each the same way (except for the lack of bending) as it is within geometry, once again reflecting the importance of the a phage capsid. This means that the free energy Gint obtained parameter rpack, introduced earlier. Before we specialize to from their measurements accounts for multiple effects, in- particular geometries (see Fig. 3) we observe that cluding electrostatics, entropy of the DNA (Odijk, 1983) and N R9 z R9 =ds where z(R9) is the height of a column counterions, and any hydration phenomena (Strey et al., 1997). ofð hoopsÞ¼ð ofð DNAÞ situatedÞ at radius R9. Using this fact and Given that we have now examined the separate contribu- differentiating Eq. 7 with respect to ds, while holding L tions arising from DNA bending, entropy, and interaction constant, gives us dR=dds p3Lds=2pRz R . Mini- ð Þ ¼ ÿð ð ÞÞ terms, we now write the free energy of the encapsidated mizing Gtot with respect to the interstrand spacing ds gives DNA in the repulsive regime, ffiffiffi Rout z R9 j k T j k T ð ÞdR9 p p B p B R R9 3F0 exp ds=c 2 2 2 Rout : (12) Rout ðÿ Þ¼ R d ÿ d R9z R9 dR9 2pjpkBT N R9 s s RR Gtot L; ds ð ÞdR9 ffiffiffi ð Þ ð Þ¼ p3d R9 s ZR R This equation represents a balance between the bending ffiffiffi Bending energy terms and the interaction terms. Note that if the size |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl2 {zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} ds 1 p3F0 c 1 cds L exp : (11) of the capsid is fixed then longer lengths of packed DNA DNA packagingð Þ inÿ c bacteriophage viruses imply smaller radii of curvature for the hoops since the ffiffiffi Interaction Electrostaticstrands energy want to be as between far away as possible two from each other for 2r = 42nm |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} neighboring charged loops

number of charges per loop `B 0.7nm d ⇡ h 2⇡r 1nm

47nm = N = ⇡ b b 0.17nm FIGURE 3 Idealized⇡ geometries of viral capsids. 2 ⇡ pd2+(2r sin(✓/2))2/ 2⇡r kBT `B e Vr = d✓ 2 2 2 b ⇡ d +(2r sin(✓/2)) Z Electrostatic energy is exponentially p DNA length small for charges that are far apart. L =6.8µm Consider only charges in the range ✓ < d/r, such that s ( ✓ ) Biophysical d . Journal 88(2) 851–866 distance between | | ⇠ 2⇡r2k T ` 2d e d/ neighboring chains V B B r ⇠ b2 ⇥ r ⇥ d d 2.3nm r ⇡ ✓ 4⇡rkBT `B d/ DNA persistence Vr e s ⇠ b2 length 2 2 ` 50nm s(✓)= d +(2r sin(✓/2)) p ⇡ p 13 Phage Packaging and Ejection Forces 857

moved farther apart. For trivalent and tetravalent ions the An analogous formula holds for the repulsive-attractive physics is quite different. In this regime there is an attractive regime. Note that this expression reports the free energy of interaction with a preferred spacing d0 between the strands, the inverse spool configurations when a length L has been and the measurements are well fit by packaged by relating R to L via Eq. 7. We will now show that the spacing between the strands varies in a systematic way 2 d0 ds Gint L; ds p3F0L c 1 cds exp ÿ during the packing process, reflecting the competition be- ð Þ¼ ð Þ c tween bending and interaction terms. ffiffiffi 1 c2 1 cd d2 d2 ; (10) ÿð 0Þÿ2ð 0 ÿ s Þ DNA spacing in packed capsids where c 0.14 nm and d 2.8 nm, for example, in the case Our model makes a concrete prediction for the free energy ¼ 0 ¼ of cobalt hexamine as the condensing agent. For ds , d0 the Gtot of packaged DNA in any phage and for a wide range of interaction is strongly repulsive and for ds . d0 it is solution conditions. To find Gtot for a particular phage, we attractive. This expression is a good representation of the free minimize Eq. 11 by varying ds, under the constraint that energy of interactions between the DNA molecules for L given by Eq. 7 is equal to the length of DNA already spacings ds less than or equal to the preferred value d0 (Rau packaged. The expressions for N(R9) will differ for different and Parsegian, 1992). In viral packaging we encounter capsid geometries. Most capsids are icosahedral and we interaxial spacings in exactly this range and hence we will idealize them as spheres. Some capsids, e.g., f29, have use this free energy to study the effects of repulsive-attractive a waist. We idealize them as cylinders with hemispherical interactions on encapsidated DNA. caps. Eventually, we will find that the geometry does not It is important to note that in the experiments of Rau et al. affect the overall free energy of packing as long as the (1984) and Rau and Parsegian (1992) DNA was confined in internal volume of the idealization is the same for each the same way (except for the lack of bending) as it is within geometry, once again reflecting the importance of the a phage capsid. This means that the free energy Gint obtained parameter rpack, introduced earlier. Before we specialize to from their measurements accounts for multiple effects, in- particular geometries (see Fig. 3) we observe that cluding electrostatics, entropy of the DNA (Odijk, 1983) and N R9 z R9 =ds where z(R9) is the height of a column counterions, and any hydration phenomena (Strey et al., 1997). ofð hoopsÞ¼ð ofð DNAÞ situatedÞ at radius R9. Using this fact and Given that we have now examined the separate contribu- differentiating Eq. 7 with respect to ds, while holding L tions arising from DNA bending, entropy, and interaction constant, gives us dR=dds p3Lds=2pRz R . Mini- ð Þ ¼ ÿð ð ÞÞ terms, we now write the free energy of the encapsidated mizing Gtot with respect to the interstrand spacing ds gives DNA in the repulsive regime, ffiffiffi Rout z R9 j k T j k T ð ÞdR9 p p B p B R R9 3F0 exp ds=c 2 2 2 Rout : (12) Rout ðÿ Þ¼ R d ÿ d R9z R9 dR9 2pjpkBT N R9 s s RR Gtot L; ds ð ÞdR9 ffiffiffi ð Þ ð Þ¼ p3d R9 s ZR R This equation represents a balance between the bending ffiffiffi Bending energy terms and the interaction terms. Note that if the size |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl2 {zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} ds 1 p3F0 c 1 cds L exp : (11) of the capsid is fixed then longer lengths of packed DNA DNA packagingð Þ inÿ c bacteriophage viruses imply smaller radii of curvature for the hoops since the ffiffiffi Interaction Electrostaticstrands energy want to be as between far away as possible two from each other for 2r = 42nm |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} neighboring charged loops

number of charges per loop `B 0.7nm d ⇡ h 2⇡r 1nm

47nm = N = ⇡ b b 0.17nm FIGURE 3 Idealized⇡ geometries of viral capsids. 4⇡r`B d/ V k T e r ⇠ B b2 Electrostatic energy between all loops DNA length L =6.8µm L assuming only one V Vr Biophysical Journal 88(2) 851–866 distance between ⇠ 2⇡r ⇥ level of loops neighboring chains 2L`B d/ 4 d 2.3nm V kBT e 3 10 kBT ⇡ ⇠ b2 ⇠ ⇥ DNA persistence length (more accurate calculation would 5 get even closer to 10 kBT) ` 50nm p ⇡ 14 Deformations of macroscopic beams

beam beam made of undeformed beam cross-section material with Young’s modulus t L E0 stretching bending twisting

L(1 + ✏) R R strain ✏ radius of angle of twist curvature = ⌦L 2 2 E k✏ Eb A E C⌦ s = = t = L 2 L 2R2 L 2 2 4 4 k E t A E0t C E0t / 0 / / Bending and twisting is much easier than stretching for long and narrow beams! 15 Bending and twisting represented as rotations of material frame yˆ xˆ undeformed beam deformed beam ~e2 ~e2 ~e zˆ 1 ~e1 ~e3 s ~e3 rotation rate of Energy cost of deformations material frame d~ei = ⌦~ ~ei ds 2 2 2 ds ⇥ E = A1⌦1 + A2⌦2 + C⌦3 ~ 2 ⌦ = ⌦1~e1 + ⌦2~e2 + ⌦3~e3 Z bending around e1 bending around⇥e2 twisting around e⇤3

1 1 R1 = ⌦1 R2 = ⌦2 1 p =2⇡⌦3

16 Deformations of microscopic ﬁlaments

actin carbon nanotubes

DNA microtubule

Deformations of microscopic filaments can still be described with stretching, bending and twisting. Elastic constants (k, A, C) can be extracted from deformation energies of bonds and are in general not related to the microscopic thickness of filaments! Couplings between stretching, bending and twisting deformations may also be allowed by symmetries of filament shapes.

17 Elastic energy of deformations in the general form

Energy density for a deformed filament can be Taylor expanded around the minimum energy ground state twist-bend coupling Lds E = A ⌦2 + A ⌦2 + C⌦2 +2A ⌦ ⌦ +2A ⌦ ⌦ +2A ⌦ ⌦ 2 11 1 22 2 3 12 1 2 13 1 3 23 2 3 Z0  2 +k✏ +2D1✏⌦1 +2D2✏⌦2 +2D3✏⌦3 bend-stretch twist-stretch coupling coupling Energy density is positive definitive functional! In principle 10 elastic constants, but symmetries of filament shape A11,A22,A33,k >0 2 determine how many independent Aij

~e2 ~e1 beam cross-section ~e3

Beam has mirror symmetry through a plane perpendicular to ~e 3 .

Two beam deformations that are mirror images of each other must have the same energy cost!

19 How mirroring around ~e 3 affects bending and twisting?

~e2 beam ~e1 cross-section

~e3

bending around ~e1 bending around ~e2 twisting around ~e3

⌦1 ⌦2 ⌦3 mirror mirror image mirror image image

⌦1 ⌦ ⌦3 2 Note: mirroring doesn’t affect stretching 20 Beams with uniform cross- section along the long axis

Lds E = A ⌦2 + A ⌦2 + C⌦2 2 11 1 22 2 3 Z0  +2A12⌦1⌦2 +2A13⌦1⌦3 +2A23⌦2⌦3

2 +k✏ +2D1✏⌦1 +2D2✏⌦2 +2D3✏⌦3 mirror ⌦ ⌦ image 3 ! 3 Lds E = A ⌦2 + A ⌦2 + C⌦2 2 11 1 22 2 3 Z0  +2A ⌦ ⌦ 2A ⌦ ⌦ 2A ⌦ ⌦ 12 1 2 13 1 3 23 2 3 +k✏2 +2D ✏⌦ +2D ✏⌦ 2D ✏⌦ 1 1 2 2 3 3

Two mirror conﬁgurations A = A = D =0 have the same energy cost: 13 23 3

21 Beams with uniform cross- section along the long axis

~e2 ~e1 beam cross-section ~e3

Beam has mirror symmetry through a plane perpendicular to ~e 3 .

L ds 2 2 2 E = A11⌦1 + A22⌦2 + C⌦3+2A12⌦1⌦2 0 2 Z  2 +k✏ +2D1✏⌦1 +2D2✏⌦2

Twist is decoupled from bending and stretching!

22 Twist-bend coupling in propellers and turbines

wind turbine airplane propeller

ship propeller

Blades of propellers and turbines are chiral, therefore there is coupling between twist and bend deformations!