How adsorption influences DNA denaturation
A.E. Allahverdyan, Zh.S. Gevorkian, Chin-Kun Hu, Th.M. Nieuwenhuizen
aa r X i v : . [ c ond - m a t . s o f t ] M a y How adsorption influences DNA denaturation
A.E. Allahverdyan , Zh.S. Gevorkian , , , Chin-Kun Hu , and Th.M. Nieuwenhuizen Yerevan Physics Institute, Alikhanian Brothers St. 2, Yerevan 375036, Armenia, Institute of Radiophysics and Electronics, Ashtarak-2, 378410, Armenia, Institute of Physics, Academia of Sinica, Nankang, Taipei 11529, Taiwan Center for Nonlinear and Complex Systems and Department of Physics,Chung-Yuan Christian University, Chungli 300, Taiwan, Institute for Theoretical Physics, University of Amsterdam,Valckenierstraat 65, 1018 XE Amsterdam, The Netherlands (Dated: November 2, 2018)The thermally induced denaturation of DNA in the presence of attractive solid surface is studied.The two strands of DNA are modeled via two coupled flexible chains without volume interactions. Ifthe two strands are adsorbed on the surface, the denaturation phase-transition disappears. Instead,there is a smooth crossover to a weakly naturated state. Our second conclusion is that even whenthe inter-strand attraction alone is too weak for creating a naturated state at the given temperature,and also the surface-strand attraction alone is too weak for creating an adsorbed state, the combinedeffect of the two attractions can lead to a naturated and adsorbed state.
PACS numbers: 82.35.Gh, 05.90.+m, 82.39.Pj
I. INTRODUCTION
The structure of DNA is the key for understanding its biological functioning, explaining why the physical features ofDNA have attracted attention over the last decades [1–7]. A known component of this structure is the Watson-Crickdouble-strandedness: DNA is composed of two single-strand molecules lined up by relatively weak hydrogen bonds.The double-strand exists for physiological temperatures and is responsible for the stability of the genetic informationstored in DNA. For higher temperatures the double-strand separates into two strands (denaturation). Many processesrelevant for the functioning of DNA—such as transcription and replication of the genetic information and packing ofDNA into chromosomes—proceed via at least partial separation (denaturation) of the two strands due to breaking ofhydrogen bonds [1, 2]. In addition, denaturation is important for a number of technological processes, such as DNAsequence determination and DNA mechanical nano-devices [2].DNA denaturation is driven by changing the temperature or the solvent structure, e.g., the pH factor [1, 2]. Thereare several generations of statistical physics models aiming to describe the physics of denaturation. Early models, basedon the one-dimensional Ising model, focus on the statistics of hydrogen bonds modeling them as two-state variables(open or closed) [4]. More recent models describe a richer physics, in that they try to explore space configurations ofDNA [5, 5–9].Most of the physics literature devoted to DNA denaturation studies this process in isolation from other relevantprocesses involved in DNA functioning [3–9, 11]. However, denaturation is frequently only a component of a largerprocess, such as replication or compactification into a nucleosome, the basic structural unit underlying the chromosome.Here we want to study how another important aspect of DNA physics — adsorption of the double-strand DNAon a surface— influences its denaturation. Surface adsorption of DNA is widely employed in biotechnologies forimmobilization and patterning (drug or gene delivery) of DNA [10, 11]. There are in fact several pertinent situations,where both adsorption and denaturation of DNA are simultaneously at play.1) For DNA at normal conditions (pH= 7 and NaCl concentration of 0 .
15 M) thermal denaturation occurs betweentemperatures 67 C and 110 C (which are the temperatures for A-T and C-G unbinding, respectively) [1, 3, 4]. Thedenaturation temperature can be decreased by increasing the pH factor, i.e., by decreasing the concentration of freeprotons in the solvent, since the negatively charged phosphate groups on each strand are not screened anymore byprotons and strongly repel each other. For the same reason, for the DNA adsorption on a positively charged surface,the increase of the pH will increase the electrostatic attraction to the surface. Thus at certain values of the pH factorand the surface charge, denaturation and adsorption may take place simultaneously.2) Surface adsorption can be realized by the hydrogen-bonding of the negatively charged phosphate residues to anegative surface (e.g., silica surface) [11, 13]. The effect is possible only when the electrostatic repulsion is sufficienctlyscreened by the solvent cations. Thus the same factors (temperature, pH, solvent concentration) that decrease theinter-strand attraction, will weaken the DNA-surface binding [11].3) The binding to hydrophobic surfaces (e.g., aldehyde-derivate glass, or micro-porous membrane) goes via partialdenaturation which exposes the hydrophobic core of the double-helix and leads to the DNA-surface attraction [11].Both naturation and adsorption are simultaneously weakened by increasing the pH [11, 12].4) Human DNA has a total length of 2 m bearing a total charge of 10 electron charge units. This long object iscontained in the cell nucleus with diameter 10 µ m, which is comparable with the persistence length of DNA. Recallthat the persistence length of a polymer is a characteristic length over which the polymer folds freely due to thermalfluctuations. For the double-strand DNA at normal conditions, the persistence length is relatively large and amountsto 50 nm or 100 base-pairs, while the persistence length of the single-stranded DNA is much smaller, about 1-2 nm(i.e., 2-4 base-pairs) [20]. This seems to create a paradoxical situation: not only the large, strongly charged DNAhas to be packed in a very small compartment, but the DNA has to be replicated, repaired, and transcribed. Theproblem is solved by a hierarchical structure: the DNA double-helix is wrapped around positively charged histone(achieving partial charge neutralization), histones condense into nucleosome complex, which in its turn is containedin chromatin, etc . It was recently discovered that packing of DNA into nucleosomes with characteristic size muchsmaller than the persistence length of the DNA chain proceeds via transient denaturation of the double strand [21].Denaturation reduces the persistence length and thereby facilitates the packing process.For all these processes we need to describe the DNA as a double-strand polymer interacting with an attractingsurface. This will be the goal of the present paper.Needless to say that there is an obvious situation, where the double-stranded structure is not relevant for theadsorption. If the two strands are too tightly connected, their separate motion is not resolved. This case is well knownin literature and—due to a large persistence length of a double-strand DNA—can be described via an effectively singlesemi-flexible chain interacting with the surface [18]. These studies complement the classic theory of the flexible chainadsorption, extensively treated in literature [3, 15, 16]. The electrostatic effects of the DNA adsorption, modeledvia a single Gaussian chain, are studied in [22]. Another recent activity couples the Ising-Zimm-Bragg model for thehelix-coil transition with the known theory of flexible chain adsorption on solid surfaces [17]. While interesting fortheir own sake, the results of Ref. [17] do not apply to DNA adsorption-denaturation, since the main assumption ofRef. [17]—that the helical pieces of the polymer interact with the surface much stronger than the coiled ones—doesnot hold for DNA.This paper is organized as follows. In section II we define the model we shall work with. It describes two flexiblechains interacting with each other and with an attracting solid surface. Section II also recalls the known correspon-dence between the equilibrium physics of flexible polymers and quantum mechanics. In its final part this sectiondiscusses limitations of the studied model in including volume interactions. Section III recalls the quantum mechan-ical variational principle which will be the basic tool of our analysis. Section IV shows that if both polymers areadsorbed on the surface, they do not denaturate via a phase-transition. Section V discusses collective scenarios ofbinding, while section VI studies conditions under which the naturated and/or adsorbed state is certainly absent.The next section presents the phase diagram of the model. The last section summarizes our results. Some technicalissues are discussed in appendices. The reader interested in the qualitative message of this work may study section IIfor learning relevant notations and then jump to section VII, which discusses general features of the phase diagram.A short account of the present work has appeared already in Ref. [14]. II. THE MODEL.
When the motion of the single strands is resolved —i.e., when the inter-strand hydrogen bonds are relatively weak,as happens next to denaturation or unzipping transitions—DNA becomes a complex system with different, mutuallybalancing features at play. A realistic model of DNA should take into account the stacking energy between two basepairs and its dependence on the state (open or closed) of these pairs; helical structure of the double-strand; intra-strand and inter-strand volume interactions (e.g., self-avoidance); the pairing energy difference between A-T and C-Gpairs (respectively, 3 k B T and 5 k B T under normal conditions), etc. Such fully realistic models do not seem to exist;there are, however, various models with different degrees of sophistication which are intended to capture at least somefeatures of the double-stranded structure [3, 5, 5–9, 15, 18].The model we shall work with disregards almost all the above complex aspects and focuses on the most basicfeatures of the problem. It consists of two homogeneous flexible chains interacting with each other and coupled to thesurface described as an infinite rigid attractive wall.Consider 2 N coupled classical particles (monomers) with radius-vectors ~r | k and ~r | k ( k = 1 , ..., N ) and potentialenergy Π( ~r α | k ) = N X k =1 ( U ( ~r k ) + X α =1 (cid:20) K (cid:0) ~r α | k − ~r α | k − (cid:1) + V ( ~r α | k ) (cid:21)) , (1)where ~r k ≡ ~r | k − ~r | k , so that | ~r | k − ~r | k | is the distance between two monomers, U is the inter-strand potential and V is the surface-monomer potential. The harmonic interaction with stiffness K (Gaussian chain) between successivemonomers in each strand is responsible for the linear structure of the polymers.The system is embedded in an equilibrium thermal bath at temperature T = 1 /β ( k B = 1). The quadratic kineticenergy of monomers is irrelevant, since it factorizes from the partition function and does not influence the equilibriumprobability distribution: P ( ~r α | k ) = e − β Π( ~r α | k ) Z , (2) Z = Z " Y α =1 , N Y k =1 d ~r α | k e − β Π( ~r α | k ) . (3)This model without adsorbing surface, i.e. V ≡
0, was mentioned in [15] and studied in [5] in the context of DNAdenaturation. When the inter-strand interaction U ( r ) is absent, we get two independent flexible chains interactingwith the solid surface, a well known model for adsorption-desorption phenomenon [3]. Recall that our purpose is instudying these two processes—i.e., surface-polymer interaction and inter-strand attraction—together. When takenseparately, these processes are well studied and well understood.Note that for the considered Gaussian chain model the stiffness parameter K relates to the characteristic persistencelength l p as [3]: K = Tl p . (4) A. Specification of the surface-monomer potential.
We assume that the surface can be represented as an infinite, solid plane at z = 0 (the role of the solid surface canbe played by any body of a smooth shape and the size much larger than the polymer length). Thus for the probabilitydistribution (2) one has [for α = 1 , k = 1 , . . . , N ] P ( ~r α | k ) = P ( x α | k , y α | k , z α | k ) = 0 , for z α | k ≤ . (5)This boundary condition should be imposed as a constraint in (2).The remaining part of the surface-strand interaction is described by a negative (attractive) potential V that dependsonly on the third coordinate: V ( ~r α | k ) = V ( z α | k ). The potential V ( z ) will be assumed to be short-ranged: it is negativefor z → z → ∞ .Let us continue the specification of the potential V ( z ) taking as an example the electrostatic attraction between onenegatively charged DNA strand and a positively charged surface; see, e.g., [16]. We denote by σ the surface chargedensity, q stands for the monomer charge (for DNA the effective monomer charge is roughly q ≃ e , where e is theelectron charge) and ǫ is the dielectric constant of the medium into which the polymer is embedded ( ǫ ≈
80 for waterat room temperature). Now the interaction energy between the surface area d x d y and one monomer reads: qσǫr e − k D r d x d y, (6)where r = p x + y + z is the distance between the surface area d x d y and the monomer, while k D is the inverseDebye screening length. This interaction leads to attraction for opposite charges: qσ <
0. The full expression of theinverse Debye screening length is well-known: k D = s πl B X a n a Z a , l B ≡ e ǫ T , (7)where l B is the Bjerrum length, and where n a and Z a are, respectively, the concentration and valency of ions of thesort a present in the solvent (so that the ion charge is Z a e ). The summation in (7) is taken over all sorts of ionspresent . Under normal conditions the Bjerrum length is ≃ The quantity P a n a Z a is called ionic strength. becomes comparable with the thermal energy T . The Debye length 1 /k D varies between ∼ . ∼ . /k D ∼ µ m.Integrating (6) over x and y from −∞ to + ∞ , we get for the surface-monomer interaction [16]: V ( z ) = 2 πqσǫk D e − k D z . (8)Thus the strength of the potential is πqσǫk D , while the inverse characteristic length is (expectedly) 1 /k D . The potential V ( z ) is short-ranged for all other relevant mechanisms of adsorption (hydrogen-binding, hydrophobic interactions,cation exchange). This means, in particular, that R ∞ d z V ( z ) is finite for all these mechanisms [23].Returning to (8) we note that for a single flexible polymer interacting with the surface the adsorption problem wassolved in Ref. [16] within the Schr¨odinger equation approach to be discussed below in detail; see in particular Eq. (23).The adsorption-desorption phase transition transition temperature found in [16] reads: T c = 8 . π | σq | k l p ǫ , (9)where l p is the persistence length from (4).Let us estimate the Debye length as k − ∼ l p ∼ Z elementary (electron) charges per 1 nm × Z ∼
1, though stronlgy chargedsurfaces achieve Z = 10 −
20. Taking the effective monomer charge one elementary charge (which is a typical value fora single-strand DNA) and recovering the Boltzmann constant, we see that (9) predicts T c of order of room temperature(300 K).When looking at concrete parameters in (9) we should also recall that Eqs. (6–8) account for the surface-monomerelectrostatic interaction, while the monomer-monomer electrostatic interaction within the single polymer is neglected.This is possible when the surface charge σl p at the area l p × l p (where l p is the persistence length of the single strand)is larger than the monomer charge: σl p ≫ | q | . (10)This condition will be satisfied for strongly charged surfaces Z ≈ B. Specification of the monomer-monomer interaction between the two strands.
The inter-strand potential U ( | ~r | k − ~r | k | ) collects the effects of hydrogen-bonding, (partially) stacking, and possibleelectrostatic repulsion. We again assume that it is purely attractive, short-ranged and goes to zero sufficiently fastwhenever the inter-particle distance | ~r | k − ~r | k | goes to infinity. In particular, the short-ranged features implies that R ∞ d rr U ( r ) is finite.Several concrete examples of the inter-strand potential U were studied and favorably compared with denaturationexperiments in [5–7]. For example, Ref. [6] studies the Morse potential U ( r ) = νe − ar ( e − ar − , (11)where ν is the potential strength and a is its characteristic range. Within the Schr¨odinger equation approach [see(23) below] Eq. (11) predicts a second-order denaturation transition at the critical temperature: T c = 16 νa l p . (12)Note that the appearance of the factor a l p in (12) is similar to the that of the factor k l p in (9). Here are thestandard estimates for the parameters in (12): ν ≃ . l p ≃ a l p ≃ T c ∼ C. Effective Schr¨odinger equation.
It is known, see e.g. [3, 7], that in the thermodynamical limit N ≫ D over which the polymerdensity changes is much larger than the persistence length l p : D ≫ l p . (13)This condition is always satisfied in the vicinity of a second-order phase-transition, where D is naturally large for afixed l p . If condition (13) is satisfied for a short-range potential—see (8, 12) for relevant examples—this potential isnecessarily small for those distances ∼ D , where the flexible polymer is predominantly located [3].For the considered two-strand situation the Schr¨odinger equation reads H Ψ = E Ψ , (14) H ≡ X α =1 [ − ∂ ~r α + V ( z α )] + U ( r ) , (15)where [using also (4)] V ( z ) ≡ Kβ V ( z ) = βl p V ( z ) , U ( r ) ≡ Kβ U ( r ) = βl p U ( r ) . (16)If there is a gap between the lowest two eigenvalues of H , the ground state wave-function Ψ determines the monomerstatistics as n ( ~r , ~r ) = Ψ ( ~r , ~r ) , (17)where n ( ~r , ~r ) is the probability distribution for two neighboring monomers on the strands for the considered trans-lationally invariant system.Recalling the known correspondence between the flexible polymer physics and (stationary) quantum mechanics [3],we can think of ~r , = ( x , , y , , z , ) as the position vectors of two quantum particles representing the strands, while ~r = ~r − ~r is their mutual position.The eigenvalue E is the energy of the quantum pair. It is related to the free energy f N of the system as E = β lf + 3 βl ln 2 πβl . (18)Since the surface is described by an infinite potential wall, we have the following boundary condition for the wavefunction Ψ( ~r , ~r ) = 0 , if z ≤ , or z ≤ . (19)Both V ( z ) and U ( r ) are attractive, V ≤ , U ≤
0, and short-ranged, that is R ∞ d zV ( z ) and R ∞ d rr U ( r ) are finite.When U = 0, the Hamiltonian H reduces to two uncoupled strands (or two uncoupled quantum particles), each onein the potential V ( z ). The corresponding Schr¨odinger equation for the z -coordinate of one strand reads from (15):[ − ∂ z + V ( z )] ψ ( z ) = Eψ ( z ) , ψ ( z = 0) = 0 . (20)It is well-known that if V ( z ) is shallow enough, no bound (negative energy) state exists, while the second-orderbinding transitions corresponds to adsorption of a single flexible polymer [3]. The physical order-parameter for thistransition is the inverse square average distance from the surface, 1 / h z i , which is finite (zero) in the adsorbed(desorbed) state. It is useful to denote by µ the dimensionless coupling constant of V = µ e V such that (for U = 0)the adsorption threshold is µ c, = 1 . (21) In fact, one should be more careful, when defining the boundary condition (19). For the two-particle case it appears to be necessary tofix not only the continuity of Ψ and its value at the wall, as Eq. (19) does, but also the behavior next to the wall: one has to requirethat when z and z go to zero simultaneously, Ψ ∝ z z . Otherwise, there will be (continuous) wave-functions which provide a boundstate for two-particles with an arbitrarily weak V <
U <
0, though the single particleneeds a critical strength of V to get into a bound state. This obviously pathological situation is prevented by the additional boundarycondition Ψ ∝ z z . For the wave-functions we shall consider below this additional boundary condition will be satisfied automatically. Note that the adsorption of a single strand DNA is a part of the renaturation via hybridization [2], a known methodof genetic systematics.For the example (8) the concrete expression for µ reads from (9): µ = 8 . π | σq | T k l p ǫ . (22)Analogously, switching off both V ( z ) and the wall, we shall get a three-dimensional central-symmetric motion inthe potential U ( r ) which again is not bound if U is shallow. This second-order unbinding transition with the orderparameter 1 / h r i , where r is the inter-strand distance, corresponds to thermal denaturation (strand separation) ofthe double-strand polymer [1, 2, 5, 7].The Schr¨odinger equation for the radial motion in the absence of the surface reads from (15) [7][ − ∂ r + U ( r )] χ ( r ) = Eχ ( r ) , χ ( r = 0) = 0 , (23)where χ ( r ) is related to the original wave-function as ψ ( r ) = χ ( r ) r . (24)Note that (23) is again a one-dimensional Schr¨odinger equation, but as compared to the equation (20), Eq. (23)contains an additional factor next to the kinetic-energy term ∂ r . This factor arises due to effective mass; see [23]for more details.Let us write likewise U = λ ˜ U , where λ is the dimensionless naturation strength. We take the naturation thresholdin the bulk to be λ c, = 1 . (25)For the example (11), λ reads from (12): λ = 16 νa l p T . (26)When the wall is included, i.e., condition (19) is imposed, the strands loose in the adsorbed phase part of theirentropy. This is known to lead to a fluctuation induced effective repulsion [19].Let us now remind that the physics of weakly bound quantum particles does not depend on details of bindingpotential [23]. Thus for qualitative understanding of the situation one may employ the delta-shell potential, which iseasily and exactly solvable and has very transparent physical features; see Appendix B.
D. Relevant coordinates.
Let us now return to the basic equation (20). It is convenient to recast this equation in new coordinates: v = 12 ( x + x ) , v = 12 ( y + y ) , (27) x − x = ρ cos ϕ, y − y = ρ sin ϕ, (28)where 0 ≤ ρ, ≤ φ ≤ π, (29)and to re-write the Schr¨odinger equation (14, 15) as − (cid:26) ρ ∂∂ρ ρ ∂∂ρ + 1 ρ ∂∂ϕ + 12 ∂ ∂v + 12 ∂ ∂v + ∂ ∂z + ∂ ∂z (cid:27) Ψ + { V ( z ) + V ( z ) + U ( | ~r − ~r | ) } Ψ = E Ψ . (30)It is seen from (30) that the variables separate, since Ψ( ~r , ~r ) can be written asΨ( ~r , ~r ) = ψ ( ρ, z , z ) ψ ( v ) ψ ( v ) ψ ( ϕ ) , (31)and the lowest energy levels is to be found via the following equation − (cid:26) ρ ∂∂ρ ρ ∂∂ρ + ∂ ∂z + ∂ ∂z (cid:27) Ψ + n V ( z ) + V ( z ) + U ( p ρ + ( z − z ) ) o Ψ = E Ψ . (32)Thus due to the translational invariance along the surface and the invariance under rotations around the z -axis,we are left with three independent coordinates: the projection ρ of the inter-particle distance on the surface, and thedistances z , z between the particles and the surface.Note that within the quantum mechanical setting the problem described by (32) corresponds to a three-bodyproblem, where the role of the third body (with infinite mass) is played by the surface. E. Common action of the surface-strand and inter-strand potentials.
In this and subsequent subsection we shall discuss two possible limitations of the present model.Above we combined together the surface-strand interaction potential V , which was derived separately from studyinginteraction of the surface with one flexible strand, and inter-strand potential U deduced from studying two flexiblestrands without the surface. While this type of combining is widely applied in all areas of statistical physics, itsapplicability needs careful discussions in each concrete case. For instance, it is possible that the presence of adsorbingsurface will directly influence the inter-strand potential. Let us discuss one (perhaps the major) example of that typepertinent for the studied model.It is well-known that the two strands of DNA are negatively charged [1]. For the double-stranded DNA under normalconditions the inter-strand repulsion is screened by positive counterions, so that the hydrogen bonding can overcomethe electrostatic repulsion and create an effective attraction, which is then the main reason of inter-strand binding[1]. Once DNA denaturates and separates into two strands, the counterions are released into the ambient mediumand are clouded around each strand. However, for temperatures not very far from the denaturation temperature thecounterions continue to screen the electrostatic repulsion, so that once the temperature lowers below the denaturationtransition temperature, the two strands reversibly assemble back into the double-strand [1]. We stress that the factthat (partially released) counterions still provide a sufficient screening follows from the existence of the observedreversible renaturation transition.When DNA denaturates in the presence of a positively charged surface the cloud of screening counterions aroundeach strand will tend to rarefy. This will increase the screening length and make the overall inter-strand interactionrepulsive. However, this is possible only for strongly adsorbed strands, where the majority of counterions are within thedirect influence of the surface charge. In the present work we focus on weakly bound strands, where the characteristiclength of the adsorbed layer D is much larger than the persistence length l p (approximately 1nm in normal conditions),which is of the same order of magnitude as the Debye screening length 1 /k D ; see (13). Thus the majority of counterionswill not feel the adsorbing surface, and in this case we do not need to account directly for the influence of the surfaceon the inter-strand potential. For strongly adsorbed DNA strands, i.e., for D ∼ /k D , it can be necessary to coupledirectly the inter-strand potential with the degree of adsorption. F. Self-avoidance and of electrostatic volume interactions.
In Hamiltonian (1) we accounted for the surface-strand and inter-strand interaction, but neglected all the volumeinteractions such as self-avoidance and (for charged polymers) electrostatic interaction between various monomers.It is important to note that the volume interactions coming from the intra-chain contributions can be accounted forwithin the present model via renormalizing the persistence length l p ; see (1) and (4) for definitions. As shown in[25] for a single flexible polymer interacting with electrostatically adsorbing surface, the self-avoiding interactions andelectrostatic volume interactions renormalize the persistence length. Provided that the Debye screening length 1 /k D is not very large —a sufficient condition for this is k D l p √ N ≫
1, where N is the number of monomers [25]— bothself-avoiding and electrostatic volume interactions lead to an effective persistence length e l p , which differs from the barepersistence length mainly by the factor N / : e l p ∼ N / l p [25] (the remaining part of renormalization is numericalfactors, which are not essential for the present qualitative discussion) . Once the persistence length is renormalized, In more detail, Ref. [25] considers a continuous polymer model with length L and reports for the square of the effective persistencelength e l p ∼ L / l p . For the present dsicrete model we take naturally L ∝ N . one can still use the flexible polymer coupled to an adsorbing surface [25]. Thus the transition temperatures (9) and(12) are divided by factor N / , where N is the number of monomers. Now for the typical single-strand DNA length N ∼ this renormalization will not make any substantial change in transition temperatures, though it is essentialfor longer polymers, N ≥ . In particular, for such a long polymer the persistence length may increase to an extentthat the condition (13) will be violated.We will see below that for qualitative conclusions of this paper, the precise form of the renormalized persistencelength is not essential, provided that one can still employ the Schr¨odinger equation (20) for describing denaturation anddesorption. The main reason for this is that the renormalization of the persistence length homogeneously renormalizesboth dimensionless couplings λ and µ in (25) and (21), respectively.The above discussion does not account for the inter-chain volume interactions and thus should not create an im-pression that the full volume interactions effect for two coupled chains can be described via a renormalized persistencelength. It is clear that one needs a more specific study of volume interactions for the present model. Since such astudy poses immence analytical problems, it will be concluded at a later time. III. VARIATIONAL PRINCIPLE AND THE EXISTENCE OF THE OVERALL BOUND STATES.
Note that Eqs. (32, 19) follow from a variational principle: δ I{ ψ } = 0 , (33)with I{ ψ } = Z ∞ Z ∞ Z ∞ ρ d ρ d z d z " ( (cid:18) ∂ψ∂ρ (cid:19) + (cid:18) ∂ψ∂z (cid:19) + (cid:18) ∂ψ∂z (cid:19) ) + 12 n V ( z ) + V ( z ) + U ( p ρ + ( z − z ) ) − E o ψ = 0 , (34)where ψ is taken real, since we are interested in bounded (discrete-level) states. We already assumed that ψ is properlynormalized: Z ∞ Z ∞ Z ∞ ρ d ρ d z d z ψ ( z , z , ρ ) = 1 . (35)If either V ( z ) = 0 or U = 0, the criterion for the existence of a bound state is well known, since it reduces to theexistence of a negative energy in the spectrum, or equivalently to the existence of a physically admissible (satisfyingthe proper boundary conditions) wave-function with a negative average energy.The situation is slightly more delicate when the two potentials V and U act together. Let us assume that eitherfor V ( z ) → U ( | ~r − ~r | ) → E { U } < , E { V ( z ) + V ( z ) } = 2 E { V } < , (36)respectively, the corresponding lowest (most negative) energies.Then it suffices to have a normalized wave-function ψ with I{ ψ } < E { U } + 2 E { V } , (37)for at least one overall bound, i.e., adsorbed and naturated, state to exist. IV. ABSENCE OF DENATURATION PHASE-TRANSITION FOR ADSORBED STRANDS.
Let us return to the variational principle (34) and assume that V ( z ) is strong enough to create at least a single(lowest) bound state with energy E { V } <
0. Denote by φ ( z ) the corresponding lowest-energy normalized wavefunction: − φ ′′ ( z ) + V ( z ) φ ( z ) = E { V } φ ( z ) . (38)For the overall problem we shall employ the following variational wave-function: ψ ( ρ, z , z ) = φ ( z ) φ ( z ) ξ ( ρ ) , (39)where ξ ( ρ ) is an unknown, tentatively normalized, viz. Z ∞ d ρρξ ( ρ ) = 1 , (40)wave-function, to be determined from the optimization of (34). Note that in (39) the boundary conditions for thesurface are satisfied via φ ( z ) φ ( z ).Substituting (39) into (34) and varying it over ξ , we get an effective Schr¨odinger equation for ξ ( ρ ): − (cid:26) ρ ∂∂ρ ρ ∂∂ρ (cid:27) ξ + { U eff ( ρ ) − ε } ξ = 0 , (41)where U eff ( ρ ) is an effective potential: U eff ( ρ ) = Z ∞ d z Z ∞ d z φ ( z ) φ ( z ) U ( p ρ + ( z − z ) ) , (42)and where ε is the reduced energy ε = E − E { V } . (43)Two main point about the effective potential (42) is that it is attractive (since so is U ) and goes to zero for ρ → ∞ .The last feature follows from the analogous one of U ( r ) and the fact that φ ( z ) are normalizable. A more explicit formfor U eff can be obtained by assuming that U ( r ) is a delta-shell potential U ( r ) = − λr δ ( r − r ) , (44)with the strength λ > r >
0. The transparent properties of this potential are recalled inAppendix B. The critical binding strength of this potential is λ c, = 1 , (45)as given by (B9). [When comparing Eq. (44) with Eq. (B2), note that the additional factor 2 comes from the reducedmass.]Using (44) we now obtain from (42) after changing variables: U eff ( ρ ) = − λr Z ∞ d v Z v d u φ (cid:18) v + u (cid:19) φ (cid:18) v − u (cid:19) δ (cid:16)p ρ + u − r (cid:17) = − λ θ ( r − ρ ) p r − ρ Z ∞ √ r − ρ d v φ v + p r − ρ ! φ v − p r − ρ ! = − λ θ ( r − ρ ) p r − ρ Z ∞ d v φ (cid:18) v + q r − ρ (cid:19) φ ( v ) . (46)It is now seen explicitly that U eff ( ρ ) is zero for sufficiently large ρ .Note that (41) has the form of two-dimensional Schr¨odinger equation for an effective particle in the attractivepotential U eff ( ρ ). It is well known that any (however weak) attractive potential in two dimensions creates a boundstate [23]. Thus there is a normalizable function ξ ( ρ ) such that ε in (42) is negative. This means that E < E { V } , (47)and, according to our discussion in section III, there is an overall bound (naturated and adsorbed) state provided V ( z ) creates a bound state. In our model a sufficiently attractive surface potential confines fluctuations of the twostrands and prevents the denaturation phase-transition (this however does not mean that the denaturation is absentas a physical process; see below).The physical reason for the existence of an overall bound state for an arbitrary small potential is a peculiar two-dimensional effect: the weakly singular attractive ∝ /ρ potential [19] . Indeed changing in (41) the variables One-dimension in this respect is not much different from the three-dimensional situation. The known statement on the existence ofbound state for any small one-dimensional potential is connected with a different mechanism, that is, with allowing all values of theone-dimensional coordinate (no infinite wall at the origin). The two-dimensional situation is indeed peculiar in this respect. e ξ = ξ √ ρ , (48)we get − ∂ e ξ∂ ρ + (cid:26) U eff ( ρ ) − ρ − ε (cid:27) e ξ = 0 . (49)Eq. (48) implies e ξ (0) = 0 , (50)i.e., the existence of the infinite wall at ρ = 0 for the effectively one-dimensional Eq. (49). It is, however, seen from(49) that there is also an attractive potential 1 / (4 ρ ). It is known that if the strength of such a potential is largerthan 1 / / / (4 ρ ) suffices to create a bound state [19].To illustrate the behavior of U eff for weakly bound state of the potential V ( z ), let us assume that V ( z ) is also adelta-shell potential: V ( z ) = − µ z δ ( z − z ) . (51)Recall that we still have an infinite wall at z = 0 and that for the delta-shell potential the bound state exists for µ > µ c, = 1 , (52)see Appendix B for details. If now µ is close to one, the energy E { V } ≡ − k / φ ( z ) ∝ √ k e − kz , (53)we get U eff ( ρ ) = − λk θ ( r − ρ ) p r − ρ (54)Since for small k , the wave-function ψ ( x ) is almost delocalized, the effective potential U eff ( ρ ) is proportional to k andgoes to zero for k → ν →
1. In other words, the trial function (39) does not predict any (overall) bindingfor µ ≤ . (55)Note however that although for µ > ε in (49) is exponentially small for small U eff , i.e., small λ or small k . Recall that this energyis estimated as [23] ε ≃ r exp (cid:20) Z ∞ d ρρU eff ( ρ ) (cid:21) = 2 r e − / ( λkr ) . (56)Thus for a small λ or k we get a very large separation between the strands. In this sense the (incomplete)denaturation phenomenon without the phase transition is present in our model.In summary, the main physical message of this section is that if the two strands are localized near the surface, theoverall DNA molecule does not melt via a phase-transition with increasing the temperature: there is only a smoothcrossover from tightly bound to a (very) weakly bound state. The cause of this effect is that the surface confinesfluctuations of each strand. Mathematically this is expressed by an additional attractive potential − ρ in (49).This result was obtained without taking into account various realistic features of DNA. It is possible that thedenaturation transition in the adsorbed phase will recover upon taking into account some of those neglected features,1e.g., volume interactions between the two strands and within each strand (see [9] for a prediction of such a transitionin a different model of DNA that partially accounts for volume interactions).We nevertheless expect that the obtained result will apply, at least qualitatively, to denaturation-renaturationexperiments, and will be displayed by facilitation of the naturation in the adsorbed phase. We are not aware ofany specific experiment done to check the renaturation-facilitating effect of an attractive surface. There are, however,somewhat related experiments showing that the renaturation rate can significantly increase in the condensed (globular)phase of single-strand DNA [24]. This condensed phase is created by volume (monomer-monomer) interactions. Theeffect was obtained under rather diverse set of conditions, but to our knowledge it did not get any unifying explanation.The analogy with our finding is that in the condensed phase fluctuations of the single strand DNA are also greatlyreduced as compared to coil (free) state. V. COLLECTIVE BINDING.
With the aim to understand the situation when V ( z ) alone does not provide any binding, we take for the variationalfunction ψ ( z , z , ρ ) = φ ( z , z ) ξ ( ρ ) . (57)As compared to (39) we do not require that z and z are factorized, and we are going to optimize over φ ( z , z ). Incontrast, ξ ( ρ ) is a fixed, normalized (see (40) ) known function.Substituting (57) into (34) and varying over φ ( z , z ) we get: − (cid:26) ∂ ∂z + ∂ ∂z (cid:27) φ + { V ( z ) + V ( z ) + V eff ( | z − z | ) − E } φ = 0 , (58)where V eff ( z ) ≡ Z ∞ d ρ ρ ξ ( ρ ) U ( p ρ + z ) . (59)Recall that by the very meaning of the variational approach E provides —for any λ and any normalized function ξ ( ρ )— an upper bound for the real ground state energy. Eq. (58) describes two one-dimensional particles withinter-particle interaction V eff ( | z − z | ) and coupled to an external field V ( z ).For the inter-particle interaction given as in (44), this effective potential V eff ( z ) reads V eff ( z ) = − λ θ ( r − z ) ξ (cid:18)q r − z (cid:19) . (60)We are now going to show that Eq. (58) predicts binding —that is, it predicts E < φ ( z , z )— at the critical point µ = 1 of the potential V ( z ). To this end let us calculate the perturbativecorrection ∆ E introduced by the effective potential V eff . At first glance the application of perturbation theory isproblematic, because we search for a nearly degenerate energy level. However, due to strong delocalization of thecorresponding wave-function, the matrix elements of the perturbing potential V eff appear to be small as well, andapplying perturbation theory is legitimate. This will be also underlined below by a perfectly finite behavior of thesecond order perturbation theory result.Recall that in the first two orders of the perturbation theory we have [23]∆ E ≡ E − E { V ( z ) } = h | V eff | i − Z ∞ d K |h | V eff | nK i| ε K − E { V } , (61)2 E { V } = − k , (62)where h z , z | i = φ ( z ) φ ( z ) is the lowest energy state of the unperturbed system, and where the integration over K involves all excited wave-functions of the unperturbed two-particle system with wave-vector K and energy ε K (allthese wave-functions are in the continuous spectrum). Note that there are three orthogonal families of these states: φ ( z ) e φ ( nz , n ) , ε n = n − k , (63) φ ( z ) e φ ( nz , n ) , ε n = n − k , (64) e φ ( nz , n ) e φ ( nz , n ) , ε n n = n n , (65)2where e φ ( nz, n ) are the corresponding single-particle excited (continuous spectrum) wave-function with the wave-number n . These wave-functions are normalized over the wave-number scale; see Eq. (B14) in Appendix B. This typeof normalization is important for the integration over the wave-number K in (61).The first-order contribution to ∆ E appears to be zero for k → + (i.e., for µ → + ). Indeed, we can use (53) for φ ( z , z ) = φ ( z ) φ ( z ) = 2 k e − k ( z + z ) , (66)to conclude h | H | i = Z ∞ Z ∞ d z d z V eff ( | z − z | ) φ ( z ) φ ( z )= Z ∞ d v Z v d u V eff ( u ) φ (cid:18) v + u (cid:19) φ (cid:18) v − u (cid:19) = 2 k Z ∞ d ve − kv Z v d u V eff ( u ) = O ( k ) . (67)Using (63–66) we shall calculate various matrix elements entering into (61): h | H | n i = Z ∞ Z ∞ d z d z φ ( z ) φ ( z ) V eff ( | z − z | ) φ ( z ) e φ ( nz , n )= √ k k Z ∞ d v Z v d u e − k (3 v + u ) / V eff ( u ) e φ (cid:18) n ( u + v )2 , n (cid:19) , (68) h | H | n , n i = Z ∞ Z ∞ d z d z φ ( z ) φ ( z ) V eff ( | z − z | ) e φ ( nz , n ) e φ ( nz , n )= 2 k Z ∞ d v Z v d u e − kv/ V eff ( u ) e φ (cid:18) n ( u + v )2 , n (cid:19) e φ (cid:18) n ( v − u )2 , n (cid:19) . (69)This results to the following formula for ∆ E ,∆ E = − Z ∞ d n |h | H | n i| n + k − Z ∞ d n d n |h | H | n , n i| n + n + k . (70)Working this out and going to the limit k → + (i.e. µ → + ) we obtain∆ E = − (cid:20)Z ∞ d u V eff ( u ) (cid:21) { Z ∞ d n n (cid:20)Z ∞ d v e − v/ e φ (cid:16) nv , (cid:17)(cid:21) (71)+ Z ∞ d n d n n + n (cid:20)Z ∞ d v e − v e φ (cid:16) n v , (cid:17) e φ (cid:16) n v , (cid:17)(cid:21) } < . (72)This expression for ∆ E is finite in the limit k → (cid:2)R ∞ d u V eff ( u ) (cid:3) . In the limit µ → λ ) we are in the situation whereneither V ( z ) + V ( z ) nor U alone create bound states. Recalling our discussion in section III on the existence ofbound states as reflected in the magnitude of variational energy, we conclude from ∆ E < µ = 1 and for sufficiently small λ . Since the ground state is supposed to be continuous, thevery fact of having a negative energy for µ = 1 and not very large λ implies that a bound state will exist for λ c > λ > , µ c < µ < , (73)where neither of the potentials V and U alone allows binding. Here λ c ≥ E { U } ; recall our discussion in section III. Note that the precise form of ξ ( ρ ) isirrelevant for the argument. This function has to be normalized and such that the effective potential V eff does notbecome large for a sufficiently small λ (and, of course, does not vanish for a finite λ ). For the rest it can be arbitrary.Thus in view of (73) we have found an example of so called Borromean binding , where the involved potentials donot produce bound states separately, but their cumulative effects lead to such a state. It is seen from (57) that thisunusual type of binding is connected with correlations between the z -components of each particles and separatelywith correlations between their x and y components (which enter via ρ ).Note that for three (or more) interacting point-like particles (instead of two particles and a surface) this effect waspredicted in nuclear physics; see, e.g., Ref. [26] for a review.3 VI. NO-BINDING CONDITIONSA. First method.
Here we shall consider certain lower bounds on the sought ground state energy. Although these bounds are basicallyalgebraic, they are non-trivial, and they allow to find out under which conditions both the adsorption and naturationare absent. In this way we complement the study of the previous section. We employ —with necessary modificationsand elaborations for our situation— the method suggested in [27].Note from (19) that the presence of the infinite wall can be modeled via the boundary condition at the plane z = 0:Ψ( ~r , ~r ) = 0 , if z = 0 , or z = 0 . (74)Though the physical content of the problem demands that Ψ( ~r , ~r ) is also zero for z < z <
0, we can formallyrequire only (74) and continue the potential V ( z ) to z < V ( − z ) = V ( | z | ) . (75)The ground state energy of the new problem defined with help of (74, 75) will be obviously equal to the ground stateof the original problem.Let us now introduce a fictive particle with the mass M and the radius vector ~r = ( x , y , z ) . (76)Now Eq. (75) is generalized to the corresponding translation-invariant interaction with the fictive particle: V ( | z k − z | ) , k = 1 , . (77)It is again obvious that upon taking the limit M → ∞ , the motion of the fictive particle will completely freeze, ~r will reduce to a constant which can be taken equal to zero.Thus the three-particle (two real particles plus the fictive one) Schr¨odinger equation reads analogously to (14, 15) (cid:26) − M ∂ ∂ ~r − ∂ ∂ ~r − ∂ ∂ ~r + V ( | z − z | ) + V ( | z − z | ) + U ( | ~r − ~r | ) − E ( M ) (cid:27) Ψ = 0 , (78)the correct two-particle energy being recovered in the limit M → ∞ .Note that the boundary conditions (74) are modified as wellΨ = 0 , if z = z , or z = z . (79)It is seen that the Hamiltonian in (78) is invariant with respect to simultaneous shift of all three radius vectors ~r k ( k = 1 , ,
3) by some vector. Since we consider a finite-particle quantum system, symmetry of the Hamiltonianimplies the symmetry of the corresponding ground-state wave-function. Thus we deduce for this functionΨ = Ψ( ~r − ~r , ~r − ~r , ~r − ~r ) , (80)which implies (cid:26) ∂∂ ~r + ∂∂ ~r + ∂∂ ~r (cid:27) Ψ = 0 . (81)We shall now decompose the Hamiltonian in (78) such that (81) is employed and that the separate sectors of theproblem —i.e., surface-particle and inter-particle interaction— are made transparent: H ≡ − M ∂ ∂ ~r − ∂ ∂ ~r − ∂ ∂ ~r + V ( | z − z | ) + V ( | z − z | ) + U ( | ~r − ~r | ) , (82)= H + H + H + H , (83) H ≡ − (cid:18) ∂∂ ~r + ∂∂ ~r + ∂∂ ~r (cid:19) (cid:18) a ∂∂ ~r + b ∂∂ ~r + b ∂∂ ~r (cid:19) (84) H ≡ − c (cid:18)
11 + x ∂∂ ~r − x x ∂∂ ~r (cid:19) + V ( | z − z | ) (85) H ≡ − c (cid:18)
11 + x ∂∂ ~r − x x ∂∂ ~r (cid:19) + V ( | z − z | ) (86) H ≡ − d (cid:18) ∂∂ ~r − ∂∂ ~r (cid:19) + U ( | ~r − ~r | ) . (87)4The coefficients a, b, c and d are read off directly from (82–87): a = − x (1 + 2 x ) , (88) c = (1 + x ) (1 + 2 x ) , (89) b = d = 2 x (1 + x )(1 + 2 x ) , (90)where the limit M → ∞ has already been taken. Here x is a free parameter; the boundaries of its change are to bedetermined below.Let us now take average of the Hamiltonian H with the ground-state wave function Ψ. The term h Ψ | H | Ψ i iszero due to (81). We shall now establish when the remaining terms in h Ψ | H | Ψ i are certainly positive, that is whenbound—i.e., naturated or adsorbed— states are certainly absent.Changing the variables as ~ξ = (1 + 2 x ) ~r + ~r , ~ξ = x~r + ~r , (91)one reduces H to a form H = − c ∂ ∂~ξ + V ( | ξ z − ξ z | ) , (92)where ξ z and ξ z are the third components of the vectors ~ξ and ~ξ , respectively. The constant factor 2 ξ z willobviously not change the binding conditions. Recalling boundary conditions (79) we see that h Ψ | H | Ψ i is certainlypositive for µ ≤ c, (93)where µ is the coupling constant of V , such that H with c = 1 has the binding threshold µ = 1 [compare with (21,52)]. Obviously, h Ψ | H | Ψ i is positive under the same condition (93).As for h Ψ | H | Ψ i we change the variables as ~r = ~r − ~r , ~R = ~r + ~r , (94)to see that H takes the form H = − dm ∂ ∂~r + U ( r ) . (95)Thus, h Ψ | H | Ψ i is certainly positive for λ ≤ d, (96)where λ is the coupling constant of U , such that H with 2 d = 1 has the critical binding threshold λ = 1 [comparewith (25,45)].Let us now recall that we employed c and 2 d as inverse effective masses which should be positive; thus we shouldrestrict ourselves to the situations x ≥ x < −
1, as seen from (89, 90). As inspection shows, the relevant no-binding condition is produced for x changingfrom zero to plus infinity, i.e., for the branch (97).Thus, under conditions (93, 96), where the limit M → ∞ is being taken, the overall bound states are certainlyabsent.5 B. Second method.
Let us now turn to another, simpler way of deriving no-binding regions. For some range of parameters the presentmethod will have a priority over the considered one, and then by combining the two methods we shall get an extendedno-binding region. We return to the very original quantum Hamiltonian in (15) and write it as − α (cid:18) ∂∂ ~r + ∂∂ ~r (cid:19) (98) − α (cid:18) ∂∂ ~r − ∂∂ ~r (cid:19) + U ( | ~r − ~r | ) (99) − − α ∂ ∂ ~r + V ( ~r ) (100) − − α ∂ ∂ ~r + V ( ~r ) , (101)where 0 ≤ α ≤ . (102)The term in (98) is seen to be always positive; for the term (99) we change the variables as in (94), to get that itis always positive for λ ≤ α, (103)while the terms in (100, 101) are both positive under µ ≤ − α. (104)Here λ and µ have the same meaning as in (93, 96). Thus no binding is possible if (103) and (104) are satisfiedsimultaneously. C. Convexity argument.
As for the last ingredient of our construction, we note that the coupling constants µ and λ enter into Hamiltonian H ( µ, λ ) in the linear way, and that the following convexity feature is valid for the ground state as a function of µ and λ : min [ H ( νµ + (1 − ν ) µ , νλ + (1 − ν ) λ ) ] (105)= min [ νH ( µ , λ ) + (1 − ν ) H ( µ , λ )] ≥ ν min [ H ( µ , λ )] + (1 − ν ) min [ H ( µ , λ )] . (106)In other words, if in the phase diagram the binding—i.e., naturation or adsorption— is prohibited at points ( µ , λ )and ( µ , λ ) —that is min [ H ( µ , λ )] ≥ H ( µ , λ )] ≥
0— then there is no binding on the whole lineconnecting those two points, because from (105, 106) one has min [ H ( νµ + (1 − ν ) µ , νλ + (1 − ν ) λ ) ] ≥ x and α , respectively— and complete it to a convex figure ensuring that for every two points belongingto (93, 96, 103, 104) the line joining them is also considered as binding-prohibited. The result is presented in Fig. 1.It is seen that there is the critical strength µ c = 0 .
25— which is necessary for binding.The latter value of µ is special for the following reason: for λ → ∞ , i.e., when the inter-particle attraction is toostrong, the two particles are tightly connected to each other. The mass of the composite particle is two times larger,and (at the same time) the potential acting on it is two times larger. This leads to the adsorption threshold µ = 0 . VII. PHASE DIAGRAM.
We are now prepared to present in Fig. 1 the qualitative phase diagram of the model. The axes of the phasediagram are λ and µ . The dimensionless parameter λ enters into the inter-strand interaction energy λ e U , such that6 Μ Λ DDND NA cba
FIG. 1: Schematic phase diagram for the inter-strand coupling λ versus the strand-surface coupling µ . The bold lines confinethree thermodynamical phases. ND : Naturation and desorption. NA : Naturation and adsorption. DD : Desorption anddenaturation. The critical naturation strength in the bulk is λ c = 1, for single strand adsorption it is µ c = 1. The followingsubregions are confined by normal lines. a : Domain described by the no-binding condition of section VI. b (bounded by thebold DD - NA line and two straight segments): Borromean naturation and adsorption. c : Adsorption and naturation due toovercritical coupling to the surface. without the adsorbing surface the naturated phase of the two strands exists only for λ ≥
1; see sections II B and II Cfor details. In this phase the two strands are localized next to each other and their fluctuations are correlated. Thetypical form of λ for the considered short-range potentials is λ = cνa l p T , (107)where c is a numerical prefactor, ν is the strength of the inter-strand potential (i.e., the modulus of its minimal value), l p is the persistence length, and T is temperature (recall that Boltzmann’s constant is unity, k B = 1); see sections II Bfor mode details. In particular, recall that for the Morse potential discussed around Eq. (26) the concrete formula for λ reads λ = νa l p T , where ν ≃ . l p ≃ a l p ≃ T we get that λ ∼ µ enters the strand-surface attractive potential as µ e V , such that theadsorbed phase of one single strand (that is without inter-strand interaction) exists for µ ≥
1; see sections II A andII C for details. Note that µ has the same qualitative form (107), where ν is the strength of the surface-strandpotential. Recall that for the electrostatic surface-monomer attraction the concrete expression for µ is discussed in(22): µ = . π | σq | T k l p ǫ ,. where k − is the Debye screening length ( k − = 0 . q is the monomercharge (around one electron charge for a single-strand DNA), l p is the persistence legth (around 1nm for a single-strandDNA), and finally σ is the charge density of the surface. Strongly charged surfaces have typically 1 −
10 electroncharges per 1 nm . At room temperatures µ ∼ ND , NA and DD refer, respectively, to thenaturated-desorbed, naturated-adsorbed, and denaturated-desorbed phases. The meaning of these term should beself-explanatory, e.g., in the ND phase the two strands are localized next to each other, but they are far from thesurface.First of all we see that there is no adsorbed and denaturated phase: as we have shown already in section IV, evensmall (but generic) inter-strand (inter-particle) attraction suffices to create a naturated state, provided that the twostrands (particles) are adsorbed. Thus the rectangular region c in Fig. 1, which belongs to the naturated and adsorbedphase NA , refers to conditions where the overall binding is due to sufficiently strong attraction to the surface.The curved line going from ( µ = 1 , λ = 0) to ( µ = 0 . , λ = 1) in Fig. 1 confines region a , where no overall binding(i.e., no-denaturation and no-adsorption) is possible according to the lower bounds obtained in the previous section.The region b , confined by two straight normal lines and the thick curve, refers to the the collective binding situation.It is seen that this region lies below both adsorption and denaturation thresholds. While we do not know the preciseposition of the thick curve confining the region b , we proved its existence in section V.Finally, the line separating NA (naturated-adsorbed) phase from ND (naturated-desorbed) phase extends mono-tonically to µ = 0 .
25 for λ → ∞ . Please note that the monotonicity of this line is conjectured. Still this conjectureis, to our opinion, quite likely to be correct.7 VIII. SUMMARY.
The main purpose of this paper was in studying DNA denaturation in the presence of an adsorbing plane surface.As we argued in the introduction, there are several relevant situations when the two processes, adsorption anddenaturation, are encountered together. Taking into account the importance of these processes in the physics of DNA,as well as for DNA-based technologies, it is important to understand how specifically adsorption and denaturationinteract with each other.Our two basic findings can be summarized as follows. First we saw that provided the two strands of DNA are (evenweakly) adsorbed on the surface, there is no denaturation phase transition. There is only a smooth crossover fromthe naturated state to a (very) weakly bound state. Second we have shown that when the inter-strand attractionalone and the surface-strand attraction alone are too weak to create naturated and adsorbed state, respectively, theircombined effect (“Borromean binding”) can create such a naturated and adsorbed state.The results were displayed on a simple model of two coupled homopolymers (strands) interacting with the planesurface. The volume interaction within each homopolymer can be accounted for via renormalizing the persistencelength; see section II F. Many realistic features of DNA are thereby put aside; see the beginning of section II. Weplan to investigate some of them elsewhere. Another interesting subject is to study the DNA adsorption on a curvedsurface [28, 29].We, nevertheless, hope that the basic qualitative aspects of the presented problem are caught adequately, and thatthe presented results increase our understanding of DNA physics.
Acknowledgments
The authors thank the unknown referees for their constructive remarks.This work was supported by the National Science Council of the Republic of China (Taiwan) under Grant No. NSC95-2112-M 001-008, National Center of Theoretical Sciences in Taiwan, and Academia Sinica (Taiwan) under GrantNo. AS-95-TP-A07.A.E. A. was supported by Volkswagenstiftung and by CRDF Grant No. ARP2-2647-YE-05. [1] R.R. Sinden,
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Imposing the periodic boundary conditions, the partition function (3) can be written as Z = Tr T N ≡ Z d ~r d ~r T N ( ~r , ~r ; ~r , ~r ) , (A1)where T is the transfer operator parametrized with two continuous indices: T ( ~r , ~r ; ~r ′ , ~r ′ ) = exp (cid:20) − β ( U ( ~r − ~r ) + V ( ~r ) + V ( ~r )) − βK ~r − ~r ′ ) − βK ~r − ~r ′ ) (cid:21) . (A2)Thus in the thermodynamic limit N → ∞ : Z = Λ N , (A3)where Λ is the largest eigenvalue of T .For simplicity reasons, the subsequent discussion will be done in terms of a transfer matrix, which depends on atwo scalar variables z ′ and z . The extension to the more general case (A2) is straightforward.Write the eigenvalue equation for the right eigenvector as Z d z ′ e − β V ( z ) − βK ( z − z ′ ) ψ ( z ′ ) = e − βf ψ ( z ) , (A4)where e − βf and ψ are, respectively, eigenvalue and eigenvector. It is seen from (A3) that N f is the free energy of themodel in the thermodynamic limit N ≫ βKD ≫ , (A5)where D is the characteristic length of ψ ( z ). Since ψ ( z ) is the density of monomers, we see that D quantifies thethickness of the adsorbed layer. Recalling (4) we can write condition (A5) as D ≫ l p , (A6)i.e., the thickness is much larger than the persistence length.Under condition (A5) the dominant part of the integration in (A4) is z ≈ z ′ . With this in mind we expand ψ ( z ′ )in (A4) as ψ ( z ′ ) = ψ ( z ) + ( z − z ′ ) ψ ′ ( z ) + ( z − z ′ ) ψ ′′ ( z ) + ..., (A7)and substitute this expansion into (A4). The outcome is √ π √ Kβ e − β V ( z ) (cid:18) βK d d z (cid:19) ψ ( z ) = e − βf ψ ( z ) . (A8)The corrections to this equation are of order O ( K β D ) = O ( l p D ).Eq. (A8) can be re-written as12 βK d d z ψ ( z ) = h e β ( V ( z ) − e f ) − i ψ ( z ) , e f ≡ f + T πKβ . (A9)For weakly-bound states |V ( z ) − e f | ≪ , (A10)for those z , where | ψ ( z ) | is sufficiently far from zero. Thus in (A9) we can expand e β ( V ( z ) − e f ) − ≃ β ( V ( z ) − e f ) (A11)and get the Schr¨odinger equation: (cid:18) −
12 d d z + β K V ( z ) (cid:19) ψ ( z ) = (cid:18) β Kf + βK πβK (cid:19) ψ ( z ) ≡ Eψ ( z ) . (A12)The ground-state energy E of this Schr¨odinger equation relates to the free energy f of the original polymer problem.For weakly-bound states E is negative and close to zero.0 APPENDIX B: SOLUTION OF SCHR ¨ODINGER EQUATION WITH THE DELTA-SHELL POTENTIAL.1. Discrete spectrum.
Here we outline bound-state solutions of a one-dimensional Schr¨odinger equation − m ψ ′′ ( x ) + ( V ( x ) − E ) ψ ( x ) = 0 (B1)with the attractive delta-shell potential V ( x ) = − µ mx δ ( x − x ) , (B2)and with the infinite wall at x = 0: ψ (0) = 0 . (B3)Here λ > x is the radius of attraction. m is the particle mass.In contrast to the main text, here we do not put m = 1.Due to boundary condition (B3) the considered problem is equivalent to the corresponding three-dimensionalSchr¨odinger problem with centrally-symmetric potential.Let us rewrite (B1) as ψ ′′ ( x ) − k ψ ( x ) = − µx δ ( x − x ) ψ, k ≡ p m | E | ≥ . (B4)For x = x (B4) is a free wave-equation. Its solution for x < x and x > x are found from the boundary conditions ψ ( x = 0) = 0 and ψ ( x → ∞ ) = 0, respectively. Thus the overall solution is obtained as ψ ( x ) = N − / sinh( kx < ) e − kx > , (B5) x < ≡ min ( x, x ) , x > ≡ max ( x, x ) , (B6)where N is the normalization constant determined via R ∞ d x ψ ( x ) = 1, N = e − kx k [sinh(2 kx ) − kx ] + sinh ( kx )2 k e − kx . (B7)Substituting (B5) into (B4) we get an equation for the energy of the single bound-state: kx = µ sinh( kx ) e − kx . (B8)The critical strength of the potential is seen to be µ = 1 , (B9)because for small kx Eq. (B8) gives kx = µ − µ . (B10)Note the following form of ψ ( x ) for small values of kx : ψ ( x ) = √ k x < x e − k ( x > − x ) + O ( k ) . (B11)For large values of λ the bound state energy increases as2 kx = µ. (B12) We should like to clarify the physical meaning of studying the delta-shell potential (B2). First of all it should be clear that the weak-potential condition (A11) does not (formally) hold for the strongly singular potential (B2). Thus the transition from the transfer-matrixequation to the Schr¨odinger equation is formally not legitimate. Nevertheless, there is a clear reason for studying the potential (B2) inthe context of polymer physics, since it is known that the physics of weakly-bound quantum particles in a short-range binding potentialdoes not depend on details of this potential [23]. So once the conditions for going from the transfer-matrix equation to the Schr¨odingerequation are satisfied for some short-range potential, one can employ the singular potential (B2) for modeling features of weakly-boundparticles in that potential. This is in fact the standard idea of using singular potentials.
2. Continuous spectrum.
For studying the continuous (positive-energy) spectrum of Eq. (B1), we re-write it as e ψ ′′ ( x ) + n e ψ ( x ) = − µx δ ( x − x ) e ψ, n ≡ √ mE ≥ . (B13)The solution is found as e ψ ( nx, n ) = √ √ π sin( n x ) sin( nx < ) sin( nx > + δ ( n ) ) (B14)where δ ( n ) is the phase-shift to be determined below, and where e N is the normalization constant determined viaorthogonalization on the n -scale: Z ∞ d x e ψ ( n x, n ) e ψ ( n x, n ′ ) = δ ( n − n ′ ) . (B15)This normalization can be checked via the large- x behavior of e ψ ( x, n ) [23]. Note that for e ψ ( nx, n ) there are twotypes of dependence on the wave-number n : as a prefactor for the argument and as a parameter entering into thenormalization and the phase-shift.For the phase-shift δ ( n ) we get: n x sin δ ( n ) = µ sin( n x ) sin( n x + δ ( n ) ) , (B16)that for n → µ − n x δ ( n ) ) . (B17)Thus δ (0) = 0 , (B18)and for e ψ ( nx,
0) we have e ψ ( nx,
0) = √ x < √ π x sin( nx > ) . (B19) APPENDIX C
While the finiteness of the integral (71) is obvious (because the integral R ∞ d n/ (1 + n ) is already convergent), theconvergence of the integral in (72) is less trivial. Estimating from (B14, B19) e φ ( nx,
0) = r π sin( nx ) , (C1)we get for the integral in (72):4 π Z ∞ d n d n + n + n n n [( n − n ) + 2( n + n ) + 1] = 14 π Z π Z ∞ d α n d n + n n sin α [ n cos α + 2 n + 1] (C2)= 14 π Z π Z ∞ d α d n + n n sin α [ n cos α + 2 n + 1] . (C3)The integral over n in (C3) is convergent and produces an integrable logarithmic singularity ∼ ln cos αα