Analytic theory of DNA condensation Y. Ishimoto Theoretical Physics Laboratory, RIKEN, Wako 351-0198, Japan N. Kikuchi Institut fu¨r Physik, Johannes Gutenberg-Universit¨at Mainz, Staudinger Weg 7, D-55099 Mainz, Germany (Dated: February 2, 2008) 7 0 WeintroduceanovelmodelforDNAcondensation(whip-toroidtransition)usingthepathintegral 0 method in the framework of the non-linear sigma model on a line segment. We show that some of 2 itsclassical configurationsexhibittoroidalforms,andthesystemhasphasetransitionsfrom awhip n totoroidal phases with aparameter c= W2l `2Lπ´2. Wealso discuss stability and finitesizeeffect on a thesestates. J PACSnumbers: 87.14.Gg, 87.10.+e,64.70.Nd,82.35.Lr 0 3 Since in living cells DNA is often packagedtightly, for interactions can be given by the path integral ] t instance, inside phage capsids, DNA condensation has f o drawn much attention [1, 2, 3, 4, 5, 6] as well as its ~r(L)=R~,u~(L)=u~f s biochemical/medicalimportance in the emerging field of G(~0,R~;~ui,~uf;L,W)= −1 [~r(s)]e−H[~r,u~,W] (1) t. gene therapy. In fact, when we put condensing agents N Z~r(0)=~0,u~D(0)=u~i a m as multivalent cations into DNA solution, it can cause with a constraint ~u2 = 1 [8, 9]. s is the proper time DNAtoundergothecondensationfromaworm-likechain | | along the stiff polymer chain of total contour length L. - d (whip) to toroidal states [1, 2, 3, 4]. ~r(s) denotes the pointing vector at the ‘time’ s in our n A double stranded DNA chain can be modelled, for three dimensional space while ~u(s) ∂~r(s) corresponds o ≡ ∂s c example, by a semiflexible homopolymer chain [7, 8, 9]. to the unit orientation vector at s. is the normali- N [ To increase our understanding of “whip-toroid transi- sation constant. Following Freed et al. [8, 26], the di- tion”, semiflexible homopolymers in a poor solvent con- mensionless Hamiltoniancan be written by [~r,~u,W]= 3 H v dition (i.e effective interactions between polymer seg- 0Lds [H(s)+VAT(s)]whereH(s)andVAT(s)arethelo- 7 ments are attractive) have been investigated as simple Rcalfree Hamiltonianandthe attractiveinteractionterm, 7 models [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, respectively: 4 22, 23, 24]. Simulations using Monte Carlo, Langevin 7 approaches or Gaussian variational method, calculated l ∂ 2 0 H(s) = ~u(s) , (2) phase diagram for the semiflexible chain in a poor sol- 2(cid:12)∂s (cid:12) 5 (cid:12) (cid:12) 0 vent [10, 11, 12, 13, 14, 15, 16]. In theoretical works, (cid:12) s (cid:12) / existing phenomenological models balance the bending VAT(s) = W(cid:12) ds(cid:12)′δ(~r(s) ~r(s′)). (3) at and surface free energies to estimate toroidal properties − Z0 − m [17, 18, 19, 20, 21, 22, 23, 24]. It becomes increasingly l is the persistence length and W is a positive coupling probable that toroid is the stable lowest energy state — - constant of the attractive interaction between polymer d the groundstate. We note,however,that the theoretical segments. Thermodynamic β = 1/(k T) is implicitly n aspectsoftheworksassumeaprioritoroidalgeometryas B included in l and W. This will be revived when we con- o the stable lowest energy state with no theoretical proof c sider the thermodynamic behaviours of the system. In [21]. Moreover, compared to the theory of coil-globule : what follows, we express ~r by the unit tangent vector ~u v transition of flexible chains [7, 25, 26, 27, 28], which andthereforetheHamiltonian (~u)intermsof~u. Hence, i X are well described by field theoretical formalism[25, 26], H theGreenfunctionG ~0,R~;~u ,~u ;L,W becomesapath thereisnosimplemicroscopictheory,whichcontainsthe i f r (cid:16) (cid:17) a salientphysicstodemonstratethewhip-toroidtransition integral over ~u with a positive coupling constant W, re- of the semiflexible polymer. Difficulties in formulating gardless of~r, theory results specifically from the local inextensibility constraint of the semiflexible chain [8, 9], which makes G= u~f [~u(s)]δ Lds~u(s) R~ e−H[u~,W], (4) the theory non-Gaussian, and also from the non-local Zu~i D (cid:16)R0 − (cid:17) nature of the attractive interaction along the polymer chain, which makes the theory analytically intractable. where we used ~r(L) = Lds~u(s) and the Jacobian is 0 Inthisletter,weinvestigatethewhip-toroidtransition absorbed by N which isRneglected here. The delta func- tion selects out the end-to-end vector. When W = 0, of a single semiflexible homopolymer chain with attrac- our free dimensionless Hamiltonian with the constraint tion,usingpathintegralmethodandthenonlinearsigma ~u(s)2 = 1 can be interpreted as the low energy limit model on the line segment. | | of a linear sigma model on a line segment, or quantum In the continuum limit, the Green function (end-to- equivalently a nonlinear sigma model on a line segment, end distribution) of a stiff polymer chain with attractive rather than some constrained Hamiltonian system. 2 We begin with O(3) nonlinear sigma model on a line Without any loss of generality, we assume a > 0 and segment which is nothing but a quantum mechanics of a n Z . Introducing N(s) [as/2π] by Gauss’ symbol + ∈ s ≡ s limited time s [0,L] with a constraint. The constraint [31], we obtain dt~u(t) = dt~u(t) = = ~u2 =1 restric∈ts the value of ~u on a unit sphere S2. Its s dt~u(Rts)−=2π/0a. ThereforeR,s−th4eπ/aattractive p·o·t·en- |po|lar coordinate decomposition in the free Hamiltonian s−2πN(s)/a tRial is given by V (s) = W N(s). Note that N(L) (~u)= l Lds ∂~u(s)2 gives the action: AT − · H 2 0 | | represents the winding number of the classical solution R (7)alongagreatcircleofS2. Finally,anintegrationover l L S[θ ,ϕ ] = ds (∂θ )2+sin2θ (∂ϕ )2 . (5) syieldsthedimensionlessHamiltonianwithourclassical u u u u u 2Z solutions: 0 (cid:2) (cid:3) This is calledthe nonlinearsigmamodelsince the action L L [~u,W]= dsH(s)+ dsV (s) is O(3) symmetric but some of its transformations are AT H Z Z 0 0 realised nonlinearly. Minimizing the action (5) in terms N(L)−1 of θu and ϕu yields the classical equations of motion: = Lla2 W 2π k+ 2π aL N(L) N(L) 2 − a a (cid:18)2π − (cid:19) ∂2+ sin2θu(∂ϕ )2 θ = 0,  Xk=1  (cid:20)− 2θ u (cid:21) u Ll π u = a2 WL N(L) 1 (N(L)+1) . (8) ∂2+2(∂θu)cotθu∂ ϕu = 0. (6) 2 − · n − aL o (cid:2) (cid:3) The first term denotes the bending energy, and the sec- Our aim is now to explore classical solutions of eq.(6) ond and the third terms are thought of as ‘topological’ andtostudythewhip-toroidphasetransitioninthepres- terms from the winding number. ence of attractive interactions. Consider classical solu- tions of eq.(6) with a trial solution θ˙ = 0. The first The non-zero winding number of the classicalsolution u in the ~u space means that the polymer chain winds in equation of (6) leads to sin2θ (ϕ˙ )2 = 0. Thus the so- lution is either θu = 0,π2,π our ϕ˙uu = 0. The solutions tahroeu~rnsdpathceeassecwoenldl.cTlahsasticiasl,wsohluentioan>(72L)π,stcaorntfifgourmraitniognas θ =0,π or ϕ˙ =0 with θ˙ =0 are equivalent to having u u u toroidal shape since a constant ~u. Accordingly, classical solutions reduce to θu = π2 or ~u = const. Substitution of θu = π2 into the 1 sin(as+b) sin(b) secondequationofmotion(6)gives∂2ϕu =0. Therefore, ~r(s)= a1{cos(as+b)− cos(b}) , (9) we have two types of classical solutions: −a{ − } const.   ~u(s)=const. and stabilise itself by attracting neighbouring segments. or We call such classical solutions the “toroid states.” π θ = and ϕ =as+b, (7) Whenever a increases and passes through the point 2πn 2 u for n Z , another toroid state appears with the iLn- + ∈ creased winding number n. Note that the radius of the where a,b are constants. By symmetry argument, we toroid state is given by 1. When the chain of contour statethatthesolutions(7)representthegeneralsolutions a length L winds N(L) times we have the N(L) circles of [29]. That is either a constant ~u(s) (rod solution) or a each length 2π and the rest L 2πN(L) . The second rotation at a constant speed along a great circle (toroid a − a solution). and third terms in the secon(cid:0)d line of eq.((cid:1)8) result from theformerandthelatterrespectively. When0<a 2π, Now we consider the attractive interaction term (3). ≤ L thechaincannotwindlikethetoroidstates. Bothendsof It is difficult to interpret it in the context of quan- thechainarenotconnectedtoeachother,thuscanmove tum theory due to its non-local nature along the poly- freelyaswellasanyotherpartsofthechainfluctuate. As mer chain. However, we can solve them with our clas- long as the totalenergy of the chain does not exceed the sical solutions (7). Let us rewrite eq.(3) by ~r(s) ~r(s′) = sdt~u(t) s′dt~u(t) = sdt~u(t), that i−s, bending energy of 2πL2l at a = 2Lπ, they can whip with V (s) =R0 W sds−′δR0 sdt~u(t) . HRse′nce the problem zero winding number. We call such low-energy states AT − 0 s′ the “whip states.” Although the definition includes fluc- is now reducedRto the (cid:0)oRne in the(cid:1)~u space: finding non- tuations around the classical solutions, unless otherwise s zero values of δ dt~u(t) with the classical solutions s′ stated,weprimarilyrefertotheclassicalsolutionsofsuch (7). That is to (cid:0)fiRnd ~u(s′)(cid:1)for a given s, which satisfies states, which are rather bowstrings than whips. s dt~u = 0 . In the polar coordinates, this is expressed s′ We now explore the exact energy levels of the whip s s Rby dt sinθ cosϕ = 0, dt sinθ sinϕ = 0, and s′ u u s′ u u and toroid states, and discuss the phase transitions be- s s′dRt cosθu = 0. The firstRclassical solution does not tween these states. The dimensionless Hamiltonian of Rsatisfy these equations and thus derives no attractive the second classical solution (7) is a function of l,L,W interactions. If we substitute the second classical solu- and a: tion of eq.(7) into the equations, we have cosθ (s) = 0, u RHRsssse′′nddcttecsiownse((aahtta++vebb))s=o=lua1ti(a1osni(nsc:(oasss(a+−s′bs+)′−b=)s−i2nnc(πao/ss(′aa+s>b+)0)b,)=n)0=∈anZ0d.. H≡clL(2ala,l2,+L,πWaW)N(L)(N(L)+1)−WL·N(L). (10) 3 This matches with the first classical solution when the whip phase to whip-toroid co-existence phase transi- N(L) = 0 for a = 0 is defined. Accordingly, the above tion may occur. On the other hand, when c > 1, there a 2 expression is valid for all classical solutions. always exists at least one (meta-) stable state for some Consider first a case with L, W, and l fixed. By defi- non-zero value of a with positive winding number N(L). nition, (a) (a,l,L,W) is continuous in the entire The number of minima are roughly given by the width cl H ≡H regionofa 0andisasmoothfunctionineachsegment: of the region for N, i.e., N (c) N (c). For example, U L ≥ − a ∈ h2πLN,2π(NL+1)i for N ∈ Z≥0. However, it is not warheeantclea≥st4t,hrNeeUm(ci)n−imNaLw(ict)h>pos3i,tivaendwitnhdeirnegfonruemthbeerres smoothateachjointofthesegments: aL Z . Weplot, 2π ∈ + greater than 1. When 0 < c < 4, the condition of hav- in Fig.1, the energy level as a function of a for different ing three minima is c > 9. To summarise, when c > 9 values of c, showing qualitative agreement with Conwell 4 4 there exist at least three minima with different winding etal. [5]. GivenN(L)=N isfixed,theHamiltonian(10) numbers. This occurence of the multiple local minima is because the energy is given by the balance between the Energy Level bending energy and the attractive potential energy. The 6 former is a monotonically increasing function of a while the latter is monotonically decreasing but not smooth 4 c=1/2 function of a. This non-smoothness is a source of mul- c=27/16 2 c=9/4 tiple stable states. Note that our precise analysis shows c=4 that the number of minima could be reduced in some 0 cases, for example, one can confirm it shortly by plot- H(a)Wl-2 ting the Hamiltonian(12) with some finite size effecton. The existence of multiple minima indicates the first or- -4 derphasetransitionsbetweenthesestablestates. Infact, -6 whentheenergyoftheN =1stabletoroidstatebecomes zero,thewhipdominanttotoroiddominantphasetransi- -8 tionoccurs. Suchavalueofcis27/16. Whenc>27/16, -10 the toroid states will dominate the action. Further dis- 0 1 2 3 4 5 x cussionsonthephasetransitionscanbeseenbychanging the value of c [30]. FIG. 1: The dependence of the energy (a) on x = aL/2π and c. (a) is scaled by the factor of √HWl for convenience. Onecanplotthecriticalvalueofcwheretheminimum The defiHnition of c will appear shortly. of the N-th segment emerges and vanishes. The lower boundoftheN-thsegmentisc(N)= N2 ,whiletheupper L N+1 takes a minimum at a = a (N) πWN(N +1) 1/3. bound is c(N)=(N+1)2. So, when c(N)<c<c(N), the N- c ≡ Ll U N L U Accordingly, each segment falls into o(cid:0)ne of three cas(cid:1)es: th segment has a minimal and (meta-) stable point. For example, when 1 < c < 4, the first segment a [2π,4π] (i) When a (N) 2πN, (a) is a monotonic function 2 ∈ L L c ≤ L H (i.e. N =1) has a minimal point at a=ac(1). inthe segmentandtakes its minimum ata= 2πN. L Sofarwehavedealtwiththeclassicalsolutions,which (ii) When 2πN < a (N) < 2π(N+1), (a) behaves are derived from the first derivative of the action. Thus, quadraticLinaandctakesitsminLimumHata=a (N). they may correspond to the global/local minima of the c actionintheconfigurationspace. However,thesolutions (iii) When 2π(N+1) < a (N), (a) is monotonic in the arenotnecessarilystableunlesswetakeintoaccountthe L c H segment and takes its minimum at a= 2π(N+1). attraction, since the second derivative of the action is L zero or even negative in many cases. When we take the Thefirstandthirdcasesarephysicallylessrelevantsince attractionintoaccount,weobservethefollowings. When theymeanno(meta-)stablepointinthesegment. So,we c 4,thetoroidstatesbecomestableunderthequantum ≥ focus on the second case. Introducing a new parameter fluctuations,sincedeviationsfromsuchtoroidstateswill c L 2W,theconditionforN oratobeinthesecond costalargepenaltyinenergy. Whencissmall,anumber ≡ 2π 2l of the whip states become equally or more probable to case(cid:0)tur(cid:1)ns out to be any toroid state. N (c)< N <N (c) for c 4, L U ≥ For large c, the ground state — the dominant toroid 1 N <N (c) for 0 c<4, (11) U state of the winding number N canbe estimated by the ≤ ≤ c inequalityrelationofc: c(N) <c<c(N). ItreadsN c. where N = [aL/2π], NL(c) ≡ 2c (cid:16)1− 2c +q1− 4c(cid:17), and Utosrionigdtbheihs,avweescran=estLimLat=e t4hπalt. tOhueUrriaddeiaulstoofrooiudrscihd≃aevael NU(c)≡ 2c (cid:16)1+q1+ 4c(cid:17). zerothickness,bcut th2eπNficniteWsizLe effectoftheir crosssec- When N (c) 1 (i.e. c 1), the above second con- tions can be approximated by the hexagonally arranged U ≤ ≤ 2 dition vanishes and thus the whip states only survive at DNA chains with a van der Waals type interaction, i.e., low energy. Therefore, at the critical value of c = 1, with the effective nearest neighbour interactions. In the 2 4 case of N(L) 4, it leads to the modified Hamiltonian: 28.5 nm for B 1.15. The same argument for Sperm ≥ ∼ DNA packaged by protamines (L = 20.4µm) gives the (a) = Lla2 2πW (N(L)) analyticvaluerc =23.69B−25 25.48B−52 [nm],whichis H 2 − a VD comparabletoanexperimenta∼lresultr 26.25nm. Note c 2πW aL that the latter has a larger diameter of≃the segment and N(L) Gap(N(L)), (12) − a (cid:18)2π − (cid:19) is expected to have the weaker interaction with smaller B. where (N) 3N 2√3 N 1/4 and Gap(N) Vc(LDN()N<+VcD1<)−cU(VN≡D),(Non)e.−cFaonllofiwnpidngNtc−he≃sam2√e3pcro25cefdourrlearfgo≡er athreeTyhνaere=exipn15ocnoinennsitsmsteνonspttrwecdiatishcetstehd[e1in8e,xtph2ee2r,liimt2ee3rn,atta2ul4rly]e.fwoerHlrlockwn∼eovwLernν, c. By r L , we now estimate th(cid:0)e mea(cid:1)n radius of observation that the radius is independent of the chain c ≡ 2πNc the toroid(i.e. the averageof inner and outer radii) in a length[2,3, 4]. This mightsuggestthat the realinterac- physical system in more detail. A coupling constant can tion is not van der Waals like, or at least is not a single van der Waals type interaction. It should be noted here be given by W= 1 kǫ where k is the number of the lm(cid:16)kBT(cid:17) that combinations of our ideal toroid and its finite size electric dipoles in a monomer segment, which create van effectcangivearangeofν = 1 1 insomeregion. An- derWaalsinteractionsofthemagnitudeǫ. l denotesthe − ∼ 5 m other important and interesting remark is that, in fact, lengthofthe monomeralongthe chaincontour,takento when we apply Coulomb like interactions to our approx- be l 5bp=1.66nm in the end. Substituting N and m c imation,weobservesomeasymptoticbehaviourthatthe ≃ the above, we obtain radius remains constant as L changes. The precise anal- ysis is to be studied and given in the near future. 2 −2 −1 1 l 5 −1 1 2 kǫ 5 rc (6π) 5L5 =(6π) 5L5(lml)5 . ≃ (cid:18)W(cid:19) (cid:18)k T(cid:19) B Acknowledgments 1 This L5 dependence agrees with the proposed exponent Y.I.isgratefultoK.Nagayamaforhisdiscussionsand in the asymptotic limit [18, 22, 23, 24]. encouragement. 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