Water-mediated correlations in DNA-enzyme interactions
WWater-mediated correlations in DNA-enzyme interactions
P. Kurian ∗ National Human Genome Center and Department of Medicine,Howard University College of Medicine, Washington,DC 20059, USA; Computational Physics Laboratory,Howard University, Washington, DC 20059, USA
A. Capolupo
Universit`a degli Studi di Salerno and INFN Gruppo Collegato di Salerno, 84084 Fisciano (Salerno), Italy
T. J. A. Craddock
Departments of Psychology and Neuroscience, Computer Science, and Clinical Immunology,and Clinical Systems Biology Group, Institute for Neuro-Immune Medicine,Nova Southeastern University, Fort Lauderdale, FL 33314, USA
G. Vitiello
Universit`a degli Studi di Salerno and INFN Gruppo Collegato di Salerno, 84084 Fisciano (Salerno), Italy (Dated: July 31, 2018)In this paper we consider dipole-mediated correlations between DNA and enzymes in the context oftheir water environment. Such correlations emerge from electric dipole-dipole interactions betweenaromatic ring structures in DNA and in enzymes. We show that there are matching collectivemodes between DNA and enzyme dipole fields, and that a dynamic time-averaged polarizationvanishes in the water dipole field only if either DNA, enzyme, or both are absent from the sample.This persistent field may serve as the electromagnetic image that, in popular colloquialisms aboutDNA biochemistry, allows enzymes to “scan” or “read” the double helix. Topologically nontrivialconfigurations in the coherent ground state requiring clamplike enzyme behavior on the DNA maystem, ultimately, from spontaneously broken gauge symmetries.
I. INTRODUCTION
In a recent paper [1], the dipole structure of DNA hasbeen studied by examining the molecular dipole compo-nents induced due to London dispersion forces betweenbase pairs. In particular, delocalized electrons in the basepairs of DNA were analyzed and theoretically shown tostimulate dipole formation at the molecular level. Theseinduced dipoles generate collective modes that are suit-ably fine-tuned for enzymatic activity resulting in double-strand breaks in the DNA helix. This class of enzymesrapidly scans the DNA [2] searching for target recognitionsequences, which exhibit a marked palindromic mirrorsymmetry between one DNA strand and its complement.Specific binding of target sequence produces conforma-tional changes in the enzyme and DNA, with charac-teristic release of water and charge-countering ions fromthe DNA-protein interface. Under optimum biologicalconditions, concerted cutting of both strands then oc-curs without producing intermediate single-strand cuts[3–6], which requires, in order to occur, synchronizationof dipole vibrational modes between spatially separatednucleotides and enzymatic molecular subunits. The dipo-lar correlations between DNA and enzyme are requiredin order for these enzymes to cleave DNA in a manner ∗ Corresponding author. E-mail: [email protected]. that preserves the palindromic symmetry of the double-stranded substrates to which they bind.Although the evolution from sequence recognition tocatalysis is perhaps the least understood aspect of the en-zymology, the synchronous long-range correlations overdistance are a hallmark of collective molecular dynam-ics [7, 8]. However, the study of the chemistry ofDNA-enzyme interactions, and perhaps the whole rangeof biomacromolecular interactions, is still fraught withthe lacuna in our understanding between the intrinsi-cally stochastic molecular kinematics and the high effi-ciency and precise targeting of enzymatic catalytic activ-ity [9, 10]. This highly efficient, ultra-precise coordina-tion has been shown to confound the explanatory reachof statistical methods for describing average regularitiesin bulk matter [11]. What is needed is of course notin opposition to the current state of knowledge derivedfrom biochemistry. On the contrary, what we proposeto shed light on is the quantum dynamical basis of themolecular interactions so as to provide a solid physicalfoundation for the biochemistry results. Such inquirydevelops in many respects along the same path of thetraditional study of the molecular dipole and multipoledynamical structure of the electronic quantum confor-mational shells determining the properties of interactingmolecules in chemistry.By following such a line of thought, our aim in thepresent paper is to deepen the understanding of DNA-enzyme interactions by characterizing quantitatively the a r X i v : . [ phy s i c s . b i o - ph ] F e b FIG. 1:
Mediating wave fields or quanta in subatomic and biological physics . (A) Electron-electroncorrelations are mediated by photons in quantum field theory. (B) Analogously, long-range correlations in themolecular water field between DNA and enzymes may be mediated by dipole waves. Note that these are schematicrenderings, neither drawn to scale nor representative of the actual orientations of water molecules.nature of long-range correlations between DNA and en-zyme in water. Water is the matrix of all known livingsystems and constitutes about 65% of the human bodyby mass and about 99% by number of molecules. On theone hand, we exploit the basic feature of any statisticallyoriented analysis, namely the fact that biomolecules arenot isolated from their environment of water molecules.Thus, we have to face the complexity arising from deal-ing with a large number of molecular components andthe quantum vibrational modes of their dipoles, whichbrings us to consider the quantum field theory (QFT)formalism. On the other hand, the generation of collec-tive mode dynamics derived in the framework of spon-taneously broken gauge symmetry theories can explainthe observation of highly ordered patterns characterizingthe catalytic activity, in space and in time (time orderedsequences of steps in the chemical activity). In a rathernatural way we are thus led to adopt in the study ofDNA-enzyme interactions the QFT paradigm of gaugetheories, namely that the interaction between two sys-tems is realized through the exchange between them ofa mediating wave field or quantum, in analogy with thephoton exchanged by two electric charges in quantumelectrodynamics (see Figure 1). Due to the relevant roleplayed by electric dipoles in the molecular interactionsunder study, we center our investigation on a general-ized model of the radiative dipole wave field mediatingthe molecular interactions between DNA and enzyme inwater. This model is discussed in Sections III and IV.The paper is organized as follows. In Section II we dis-cuss and partially review the molecular dipole structureof DNA and enzyme. To connect with previous work, weconsider the case of the type II restriction endonuclease
Eco
RI and, for generality, we similarly analyze the
Taq
DNA polymerase, which is widely used in polymerasechain reaction (PCR) processes for the amplification ofDNA sequences. In Section III we study the collectivedipole dynamics of water molecules in the presence ofthe DNA radiative dipole field. In Section IV we exam-ine how the water dipole wave field interacts with theenzyme radiative dipole field. The water dipole field isidentified as the mediating wave field in the DNA-enzymeinteraction. Concluding remarks are presented in SectionV. The appendices A, B, and C report some details on theenzyme systems, mathematical formalism, and analysisof the system ground states, respectively.
II. INDUCED DIPOLE NETWORKS IN DNASEQUENCE AND ENZYME SYSTEMS
In this section we present the computation of the col-lective dipole behavior in the DNA and enzyme moleculesby considering the interactions between their constituentaromatic rings. These rings, present in both DNA basepairs and enzyme amino acids as well as a host of otherbiomolecules, contain conjugated planar ring systemswith delocalized π electrons shared across the struc-ture, instead of permanent, alternating single and doublebonds. Benzene is the canonical example of such a ring.This confers on aromatic compounds an unusual stabilityand low reactivity, but also provides an ideal structurefor the formation of electric dipoles, which can interactto produce electromagnetic couplings in biology. A. DNA Sequence
We consider first the linearized DNA sequence, withpolarizability data for its four bases given in Table I. Thesymmetry and regularity of the molecule about its heli-cal axis simplifies the calculation relative to the one forenzyme aromatic networks. By resorting to the resultsof Ref. [1], we observe that in DNA the delocalized elec-trons belong to the planar-stacked base pairs that serveas “ladder rungs” stepping up the longitudinal helix axis.The induced dipole formation is generated by Coulombicinteractions between electron clouds. The helix rungscan be visualized as electronically mobile sleeves vibrat-ing with small perturbations around their fixed positivelycharged core.TABLE I: Polarizabilities for DNA bases [12–14], inunits of 1 au = 1 . × − C m J − . DNA base α xx α yy α zz Adenine (A) 102.5 114.0 49.6Cytosine (C) 78.8 107.1 44.2Guanine (G) 108.7 124.8 51.2Thymine (T) 80.7 101.7 45.9
Consider a molecule of length N nucleotides. TheHamiltonian is H DNA = N − (cid:88) s =0 p s m s + m s (cid:0) ω s,xx x s + ω s,yy y s + ω s,zz z s (cid:1) + V ints , (1)where r s = ( x s , y s , z s ) are the displacement coordinatesbetween each electron cloud and its base-pair core, thecoordinates x s , y s span the transverse plane of each base-pair cloud, and the z s are aligned along the longitudinalaxis. The dipole-dipole interaction terms are given by V ints = 14 π(cid:15) d (cid:20) π s · π s +1 − π s · d )( π s +1 · d ) d (cid:21) , (2)with d = d ˆ z connecting the centers of nearest-neighborbase-pair dipoles π s = Q r s and π s +1 = Q r s +1 .As shown in Table II, ω s,ii are the diagonal elements ofthe angular frequency tensor for each base-pair electronicoscillator and are determined from polarizability data: ω A : T, ii = (cid:115) Q m ( α A, ii + α T, ii ) , (3)and similarly for ω C : G, ii . As shown in Appendix B,Eq. (3) may be derived from the fundamental dipole re-lation π = α · E . The numerical tensor elements α ii account for anisotropies, which have been determinedfrom perturbation theory [12], simulation [13], and ex-periment [14] generally to within five percent agreement. TABLE II: Electronic angular frequencies calculatedfrom polarizability data for DNA base pairs (bp), inunits of 10 radians per second. bp ω xx ω yy ω zz A:T 3.062 2.822 4.242C:G 3.027 2.722 4.244
The mass and charge of an electron are used because sin-gle electrons would be entangled through the base-paircouplings.Due to the twist in the helix about the longitudinalaxis, we must account for cross terms between directionalcomponents of two interacting dipoles. Choosing a sin-gle coordinate frame that corresponds with ( x , y , z )of the 0th base pair, and relating such coordinates tothe ( x s , y s , z s ) ones, the interaction potential for the s thelectronic oscillator can be written as V ints = Q π(cid:15) d [ x s x s +1 cos θ + y s y s +1 cos θ (4)+ ( y s x s +1 − x s y s +1 ) sin θ − z s z s +1 ] , where the orientation of the helix and the twist angle arereflected in the different factors for the quadratic cou-plings.By introducing the normal-mode lowering operator a s,j = (cid:114) m Ω s,j (cid:126) ( r (cid:48) s ) j + i (cid:112) m (cid:126) Ω s,j ( p (cid:48) s ) j , (5)and its conjugate raising operator a † s,j for the s th normalmode of the collective electronic oscillations for the j = xy or j = z potential, with( r (cid:48) s ) j = N − (cid:88) n =0 ( r n ) j exp (cid:18) − πinsN (cid:19) , (6)( p (cid:48) s ) j = N − (cid:88) n =0 ( p n ) j exp (cid:18) − πinsN (cid:19) , the Hamiltonian in Eq. (1) takes the standard diagonal-ized form H DNA = (cid:88) j H j = (cid:88) j N − (cid:88) s =0 (cid:126) Ω s,j (cid:18) a † s,j a s,j + 12 (cid:19) . (7)The eigenstates of H j are given by | ψ s,j (cid:105) = a † s,j | (cid:105) , (8)where s = 0 , , . . . , N − j = xy, z potential.Only the lowest energy states will be considered becausethese modes are the most easily excited.In order to obtain the collective eigenmode frequenciesfor the oscillations, by separating Eq. (1) into energy con-tributions from transverse ( H xy ) and longitudinal ( H z )modes, we may write the symmetric longitudinal poten-tial matrix V z for a four-bp sequence and the diagonalkinetic matrices T j as given in Appendix B. The problemthen consists of solvingdet( V j − Ω s,j T j ) = 0 (9)for the eigenfrequencies Ω s,j .When homogeneity is assumed in the sequence ( m s = m, ω s,zz = ω, γ zs,s +1 = γ ), the longitudinal mode frequen-cies in the four-bp case take on a beautifully simple form:Ω ,z = ω − ϕ γm Ω ,z = ω − ( ϕ − γm Ω ,z = ω + ( ϕ − γm Ω ,z = ω + ϕ γm , (10)where ϕ = (cid:0) √ (cid:1) / Eco
RI DNA recognition sequence(GAATTC) zero-point modes, in units of ε P − O (cid:39) . d . The helixtwist angle θ (cid:39) π/ d (˚A) (cid:126) Ω ,xy / (cid:126) Ω ,xy / (cid:126) Ω ,z / (cid:126) Ω ,z / . . . . . We remark that the energy required in vivo to break asingle phosphodiester bond in the DNA helix backboneis ε P − O (cid:39) .
23 eV [17], which is less than two percentof the energy required to ionize the hydrogen atom, butabout ten times the physiological thermal energy ( k B T ).This value of ε P − O is comparable with the quantum ofbiological energy released during nucleotide triphosphate(e.g., ATP) hydrolysis; it ensures that the bonds of theDNA backbone are not so tight as to be unmodifiable,but remain strong enough to resist thermal degradation.As shown in Table III, at the standard inter-base-pair spacing of 3 . .
5% of2 ε P − O , the energy required for double-strand breakage.It is also remarkable that for Eco
RI, considered in moredetail below, the difference in free energy between thenonspecific and specific complex (i.e., clamping energy)is approximately 2 ε P − O . Thus, enzyme clamping im-parts the quanta of energy necessary to excite the lon-gitudinal mode. Because it is a collective, normal-modeoscillation (where components vibrate in synchrony), thislowest-energy mode along the main axis of the DNA se-quence coherently transports the quanta to break twophosphodiester bonds simultaneously. Parametrizationof this DNA model is briefly discussed in Appendix B. B. Enzyme Systems
We will consider two specific enzymes—
Eco
RI and
Taq polymerase—that represent two broad enzyme classesused in DNA metabolism, namely type II restriction en-donucleases and DNA polymerases, respectively. The for-mer group initiates DNA double-strand breaks with highspecificity and the latter replicates DNA strands withhigh efficiency.We will first consider the enzyme
Eco
RI, shown in Fig-ure 2 and the prototype for a class of enzymes called typeII restriction endonucleases that recognize very specificDNA sequences.
Eco
RI was the first of this class to bediscovered, found in
E.coli bacteria (see Appendix A).It initiates a double-strand break (DSB) in the DNA he-lix after recognizing the sequence GAATTC. The mech-anism of this DSB synchronization has been the subjectof much investigation and was recently proposed to becorrelated through quantum entanglement [1]. One in-teresting characteristic of the
Eco
RI recognition site isthe palindromic symmetry of the sequence. By the rulesof DNA base pairing, the complementary DNA sequenceis CTTAAG, so GAATTC is the mirror image of its com-plement.The other enzyme we consider is the
Taq
DNA poly-merase, shown in Figure 3 and used widely in molecularbiology labs throughout the world for rapid amplifica-tion of DNA sequences. While
Taq does not recognizespecific sequences like
Eco
RI, it employs the use of aninteresting DNA clamp. The DNA clamp fold is an ac-cessory protein that assembles into a multimeric struc-ture completely encircling the DNA double helix as thepolymerase adds nucleotides to the growing strand. Bypreventing dissociation of the enzyme from the templateDNA strand, the clamp dramatically increases the num-ber of nucleotides the polymerase can add to the growingstrand per binding event, up to 1,000-fold. The clamp as-sembles on the DNA at the replication fork, where thedouble helix effectively “unzips” into single strands, and“slides” along the DNA with the advancing polymerase,aided by a layer of water molecules in the central poreof the clamp between the DNA and the protein surface.FIG. 2:
Aromatic amino acid network in
Eco RI . Tryptophan (blue), tyrosine (purple), and phenylalanine(green) form induced dipole networks in Eco
RI, shown here bound to its double-stranded DNA substrate, with A:T(yellow) and C:G (orange) base pairs highlighted. Other amino acids (gray) are displayed in the context of theirsecondary structures within the enzyme. Image of
Eco
RI (PDB ID: 1CKQ) at 1.85 ˚A resolution created withPyMOL.Because of the toroidal shape of the assembled multimer,the clamp cannot dissociate from the template strandwithout also dissociating into monomers.We are particularly interested in the polarizability ofthe amino acids, which are the building blocks of proteinand enzyme systems. The most polarizable amino acidsare termed aromatic , and as with DNA we will examinethe effect of induced dipole interactions between theseconstituent ring structures. See Tables IV and V for aro-matic amino acid polarizabilities and electronic angularfrequencies, respectively.TABLE IV: Polarizabilities for aromatic amino acids[18–22], in units of 1 au = 1 . × − C m J − ,with α = (cid:113) α xx + α yy + α zz . For further details, seecomments surrounding Eq. (12). Amino Acid α α xx α yy α zz Trp 153.4 119.5 91.6 29.4Tyr 129.3 89.5 43.0 82.9Phe 118.1 79.0 79.0 38.6
Interestingly, the most aromatic amino acids (tryp- tophan, tyrosine, and phenylalanine) are also the mosthydrophobic. Various proposals have been put forwardarguing the biological significance of such intra-proteinhydrophobic pockets. London force dipoles in such re-gions could couple together and oscillate coherently, thusgenerating a radiative field.TABLE V: Electronic angular frequencies calculatedfrom polarizability data for aromatic amino acids, inunits of 10 radians per second. Amino Acid ω ω xx ω yy ω zz Trp 3.338 3.782 4.320 7.622Tyr 3.635 4.370 6.305 4.541Phe 3.803 4.653 4.653 6.654
In regards to the spacing between aromatic amino acidsin protein, when considering tryptophan, tyrosine, andphenylalanine, the spacings are as close as 5 ˚A in theenzymes of interest, comparable to the inter-bp spacingof DNA (3.4 ˚A). The use of the point-dipole approxima-tion, rather than an extended dipole description, can bequestioned due to these close separations, but this is afirst-order estimate of the contribution of these effects.FIG. 3:
Aromatic amino acid network in
Taq
DNA polymerase . Tryptophan (blue), tyrosine (purple), andphenylalanine (green) form induced dipole networks in
Taq polymerase, shown here bound to a blunt-ended duplexDNA in the polymerase active-site cleft, with A:T (yellow) and C:G (orange) base pairs highlighted. Other aminoacids (gray) are displayed in the context of their secondary structures within the enzyme. Image of
Taq polymerase(PDB ID: 1TAU) at 3.0 ˚A resolution created with PyMOL.Additionally, because we are considering interactions be-tween all aromatic rings, the point-dipole approximationwould indeed hold for the longer distances, which are themajority.As shown in Table V, ω s,ii are the diagonal elements ofthe angular frequency tensor for each amino-acid dipoleand are determined from the polarizability data: ω aa, ii = (cid:115) Q mα aa, ii . (11)The mass and charge of an electron are used becausethe amino acids are approximated as charge-separateddipoles. As shown in Appendix B, Eq. (11), like Eq. (3),may be derived from the fundamental dipole relation π = α · E .Here we use the values and principal directions of thepolarizability tensors based on the experimental and the-oretical calculations for indole [18], phenol [19], and ben-zene [20–22], to represent the polarizability tensors of theamino acids tryptophan, tyrosine, and phenylalanine, re-spectively, because each ring structure defines the aro-maticity of the corresponding amino acid in which it iscontained. Figure 4 illustrates the orientation geome-tries of the principal directions and the molecular centers used in calculating the dipole displacement coordinatesbetween aromatic amino acids in the aromatic coordinateframe. We use only the diagonal elements of the po-larizability tensor α xx , α yy , α zz , and neglect off-diagonalterms. After alignment of the polarizabilities with theorientation of the aromatic amino acids in the proteincoordinate space, we take the magnitude of the averagepolarizability to be α = (cid:113) α (cid:48) xx + α (cid:48) yy + α (cid:48) zz , (12)where the induced dipole direction for each aromatic isdefined by the vector ( α (cid:48) xx , α (cid:48) yy , α (cid:48) zz ) in the protein coor-dinate frame.As we did for the DNA molecule in the first part ofthe present section, we proceed with the Hamiltonian forsuch a collection of N aromatic amino acids: H enz = N − (cid:88) s (cid:54) = t p s m s + m s ω s (cid:0) x s + y s + z s (cid:1) + V ints,t , (13)where the displacement coordinates r s = ( x s , y s , z s ) aredefined as previously done between each electron cloudand its amino acid core. Here ω s are derived from thepolarizabilities α s in Table IV through Eq. (11), andFIG. 4: Orientation geometries and molecular centers of aromatic ring structures in amino acids . (A)Indole ring in tryptophan. (B) Phenol ring in tyrosine. (C) Benzene ring in phenylalanine. The positive z -axis isdirected out of the page.the term V ints,t = π(cid:15) d s,t (cid:104) π s · π t − π s · d s,t )( π t · d s,t ) d s,t (cid:105) de-scribes the pairwise interactions between each amino acidelectric dipole in the enzyme’s aromatic network. Thearomatic networks for Eco
RI and
Taq are shown in Fig-ures 2 and 3 and are determined from crystal structureswith identification codes 1CKQ [23] and 1TAU [24], re-spectively, in the Protein Data Bank [25]. The 1CKQPDB file must be duplicated appropriately to accountfor the symmetric homodimer formed from two identicalprotein subunits. Tryptophan, the most strongly polar-izable amino acid, will generally contribute most to thecollective electronic behavior of each enzyme.We return to solving Eq. (9), but this time in the en-zyme case for larger, denser potential matrices V ints,t . Thislargeness and denseness is due to the sheer number of aro-matic amino acids and the pairwise coupling interactionsbetween them. We simplify this task by recognizing thatEq. (9) is similar to the characteristic equation for matrixeigenvalues: det ( A − λ s I ) = 0 , (14)with V in place of A and λ s = m e Ω s . Numerical pack-ages can then be used to quickly and efficiently solve forthe eigenvalues of V .Collective modes for Eco
RI and
Taq are presented inFigure 5. The average energy for the
Eco
RI modes is1.22 ε P − O and for the Taq modes is 1.20 ε P − O . Note that Taq is a larger enzyme and has more dipolar vibrationalmodes than
Eco
RI (61 vs. 42) because these modes cor-respond to the number of aromatic amino acids in eachenzyme. Though proteins with aromatic amino acids typ-ically absorb in the UV, around 4 − ε P − O (2041, 2226, 2319, and 2412 cm − ). Carboxylicacids ubiquitous in proteins tend to form strongly hy-drogen bonded dimers, which shift water’s O-H stretch-ing frequencies to between 2000-3500 cm − depending onthe strength of the hydrogen bonds [26]. It is interestingand significant to our studies that the enzyme collectivedipole modes displayed in Figure 5 fall within the spec-trum of DNA collective modes presented in Table III anddescribed in full detail in Ref. [1]. The enzyme collectivemodes, like the DNA modes in Eq. (7), are dipole fluc-tuation modes and can be used to construct the creationand annihilation operators (see Eq. (5)) for the radiativeelectromagnetic field quanta (i.e., photons) acting on thewater molecules.We thus arrive at a picture for the case of enzymes sim-ilar from the physical point of view to the one obtainedabove for DNA. The DNA and the enzyme molecules,although sharply different in their chemical structures,are macromolecular systems physically characterized bytheir dipolar structure. According to our quantum elec-trodynamics description, dipole vibrational modes arethe source of radiative electromagnetic (em) dipole fieldsthrough which the interaction between DNA and enzymedevelops. Since they are embedded in the water envi-ronment whose molecules are also characterized by theirmolecular dipoles, DNA and enzyme “talk” to each otherby means of their radiative dipole fields propagating inthe water medium. This long-range communication ismediated by the water dipoles. The chemical activity be-tween DNA and enzyme develops in and is conditioned by (a) Eco
RI (b)
Taq
FIG. 5:
Collective dipole oscillations in aromatic amino acid networks of DNA-interacting enzymes.
Collective normal-mode solutions to networks of aromatic induced dipoles in
Eco
RI (5a) and
Taq (5b) are withinthe energy range of the collective dipole modes of DNA bounded by its relevant protein clamps. These states areprecisely those for which the number operator n s = a † s a s acts in the diagonalized form of Eq. (13), analogous to Eq.(7) for DNA, to produce a zero eigenvalue, thus identifying them with zero-point modes that exist independent ofexternal excitations and arise from the ground state of the enzyme aromatic network. Data for the collective modes(arranged along the abscissa axis according to increasing energy) are presented in units of eV on the ordinate axis.such a physical scenario. In order to study the physicaldynamics providing the environmental conditions underwhich the chemical activity develops, we need to deepenthe understanding of collective dipole modes of liquid wa-ter. This is done in the following section. III. THE WATER DIPOLE FIELD
Water constitutes the environment in which DNA andenzyme macromolecules described in the previous sec-tions are embedded. These macromolecules and the wa-ter molecules are endowed with electrical dipoles, whosequantum vibrational modes characterize their reciprocalinteractions. The picture emerging from the different in-teraction couplings, frequencies, amplitudes, and phasemodulations in different space and time regions of the wa-ter environment provides the em imaging or em field pat-tern of the DNA-water-enzyme interaction. We will focuson the local properties of such a pattern of fields, thusdealing with densities rather than with physical quan-tities integrated over the whole volume, although inte-grations over some relatively small volumes will also beconsidered.We study first the interaction between DNA and water.After that we will introduce the interaction with enzymemacromolecules. We will neglect the static dipole-dipoleinteractions between DNA and water dipoles and focusour attention instead on the radiative quantum electro-magnetic (em) interaction.Consider a system of molecules of liquid water, as-sumed to be homogeneous in a region whose linear size (cid:96) will be elaborated in the course of our discussion (see the comments following Eq. (18)). The magnitude of thewater molecule dipole vector d e is given by | d e | = 2 e d e ,with d e (cid:39) . e denoting the electron charge.In the following discussion we use natural units, where (cid:126) = 1 = c .The system is assumed to be at a non-vanishing tem-perature T . Under such conditions, in the absence ofother actions, the system of water molecules is invariantunder dipole rotations because the molecular dipoles arein general arbitrarily oriented and thus there is no pre-ferred orientation direction. Our aim is to study the ra-diative dipole transitions in the water molecules inducedby interactions with the DNA em field and all molecularwater dipoles.By closely following Ref. [28] in our presentation, wedenote by φ ( x , t ) the complex dipole field collectively de-scribing the N molecule system in the unit volume V sothat integration over the sphere of radius r gives: (cid:90) d Ω | φ ( x , t ) | = N , (15)where d Ω = sin θdθdφ is the element of solid angle andthe polar coordinates are denoted by ( r, θ, φ ). It is conve-nient to introduce the rescaled field χ ( x , t ) = √ N φ ( x , t ).The integration over the unit sphere then gives (cid:90) d Ω | χ ( x , t ) | = 1 . (16)Since the molecule density is assumed to be spatiallyuniform in the unit sphere, the angular variables are theonly relevant ones. This allows to us to expand the field χ ( x , t ) in terms of spherical harmonics in the unit sphere: χ ( x , t ) = (cid:88) l,m α l,m ( t ) Y ml ( θ, φ ) . (17)By setting α l,m ( t ) = 0 for l (cid:54) = 0 ,
1, Eq. (17) reducesto the expansion in the four levels ( l, m ) = (0 ,
0) and(1 , m ) for m = 0 , ±
1. The populations of these levels atthermal equilibrium, in the absence of interaction, followthe Boltzmann distribution and are given by N | α l,m ( t ) | .We will show that the populations of the l = 0 and l = 1levels change due to the presence of DNA and enzyme.The amplitude of α ,m ( t ) does not depend on m since,due to the dipole rotational invariance, there is no pre-ferred direction in the dipole orientation. In other words,the time average of the polarization P n along any direc-tion n must vanish. In order to show that this is indeedthe case, let us introduce the notation α , ( t ) ≡ a ( t ) ≡ A ( t ) e iδ ( t ) ,α ,m ( t ) ≡ A ( t ) e iδ ,m ( t ) e − iω t ≡ a ,m ( t ) e − iω t , (18)where A ( t ), A ( t ), δ ( t ), and δ ,m ( t ) are real quantitiesand ω ≡ /I (in natural units), where I is the averagemoment of inertia of the water molecule. * Note that ω is also the eigenvalue of L / I on the state (1 , m ), with L denoting the total angular momentum for the molecule.In fact, l ( l + 1) / I = 1 /I .Our discussion will be restricted to the resonant radia-tive em modes, i.e., those for which ω ≡ k = 2 π/λ , andwe use the dipole approximation, where exp( i k · x ) ≈ N molecule system. The system linear size (cid:96) is thusconstrained to be (cid:96) ≤ λ = 2 π/ω . We thus see that ω provides a relevant scale for the system. For water,2 π/ω ≈ − ≈ . × − cm (in (cid:126) = 1 = c units), taking I = 1 . × − g · cm as the averageof the moments of inertia presented in the footnote.We now consider in an arbitrary reference frame the z axis and a vector n parallel to it. Use of the explicitexpressions for the spherical harmonics Y = 1 √ π , Y = (cid:114) π cos θ ,Y = − (cid:114) π sin θ e iφ = − [ Y − ] ∗ , (19) * The moment of inertia of water varies within a factor of threedepending on the axis around which it is calculated: 1 . × − g · cm for the axis in the plane of the water molecule withthe origin on the oxygen and orthogonal to the H-O-H angle, i.e.,parallel to the longest dimension of the molecule; 2 . × − g · cm for the axis orthogonal to the plane of the water moleculewith the origin on the oxygen; and 1 . × − g · cm for theaxis in the plane of the water molecule with the origin on theoxygen and bisecting the H-O-H angle (see Ref. [29]). gives P n = (cid:90) d Ω χ ∗ ( x , t )( x · n ) χ ( x , t )= 2 √ A ( t ) A ( t ) cos( ω − ω ) t , (20)where ωt ≡ δ , ( t ) − δ ( t ). Eq. (20) shows indeed thatthe time average of P n is zero, which in turn confirmsthat the three levels (1 , m ) for m = 0 , ± (cid:80) m | α ,m ( t ) | = 3 | a ( t ) | , with a ( t ) ≡ a ,m ( t ) for any m = 0 , ±
1. The normalization condition (16) then canbe written as | α , ( t ) | + (cid:88) m | α ,m ( t ) | = | a ( t ) | + 3 | a ( t ) | = 1 (21)at any time t , so that, by putting Q ≡ | a ( t ) | +3 | a ( t ) | ,we have ∂Q/∂t = 0, which expresses the conservation ofthe total number N of molecules (see Eq. (15) and therescaling adopted for χ ( x , t )). Moreover, the relation ∂∂t | a ( t ) | = − ∂∂t | a ( t ) | (22)shows that, at each time t , the rate of change of the pop-ulation in each of the levels (1 , m ) for m = 0 , ±
1, equallycontributes, in the average, to the rate of change in thepopulation of the level (0 ,
0) (rotational invariance).Eq. (21) shows that the initial conditions at t = 0 canbe set as | a (0) | = cos θ , | a (0) | = 13 sin θ . (23)We exclude the values of θ corresponding to the phys-ically unrealistic cases in which the state (0,0) is com-pletely filled ( θ = nπ , n integer) or completely empty( θ = (2 n + 1) π/
2, n integer). We will see that the lowerbound for the parameter θ is imposed by the dynamicsin a self-consistent way. Note that θ = π/ ,
0) and (1 , m ), as given by the Boltzmann dis-tribution when the temperature T is much higher thanthe energy E ( k ) of the field modes, k B T (cid:29) E ( k ), whichwould be the case in our DNA-enzyme systems.Let u m ( t ) ≡ U ( t ) e iϕ m ( t ) , with U ( t ) and ϕ m ( t ) realquantities, denote the field of the em modes generatedby the DNA molecular dipoles (see Eqs. (5), (7), and thecomments following Eq. (14)) and the ones generated bythe molecular water dipoles. One also finds [28, 30] thatthe amplitude | u m ( t ) | does not depend on m , so that wemay write | u ( t ) | ≡ | u m ( t ) | . Moreover, one can derivethat the phases also do not depend on m ; thus ϕ ≡ ϕ m , δ ( t ) ≡ δ ,m ( t ). We consider the coupling of u ( t ) withwater molecules in the transition (1 , m ) ↔ (0 , u ( t ) are given in Refs. [31, 32]: i ∂χ ( x , t ) ∂t = L I χ ( x , t ) − i (cid:88) k ,r d e √ ρ (cid:114) k (cid:15) r · x )[ u r ( k , t ) e − ikt − u † r ( k , t ) e ikt ] χ ( x , t ) ,i ∂u r ( k , t ) ∂t = id e √ ρ (cid:114) k e ikt (cid:90) d Ω( (cid:15) r · x ) | χ ( x , t ) | , (24)where d e is the magnitude of the electric dipole moment, ρ ≡ NV , and (cid:15) r is the polarization vector of the em mode(for which the transversality condition k · (cid:15) r = 0 is as-sumed to hold). In Eqs. (24) the coupling d e √ ρ is en-hanced by the factor √ N due to the rescaling of the fields.More comments on this point are included later on.Eqs. (C1) - (C3), which are derived from Eqs. (24)and presented in Appendix C, show that the strength ofthe coupling between the em modes and the molecularlevels is given by Ω = 4 ed e (cid:112) N/ (6 ω V ) ω ≡ G ω . Forpure water under standard conditions, the dimensionlessquantity G (cid:39)
13. The coupling Ω scales with √ ρ andthus varies with temperature and pressure. The coupledequations (C1) - (C3) admit the constants of motion Q (see Eq. (21)) and the quantity | u ( t ) | + 2 | a ( t ) | = 23 sin θ . (25)For further details, see Appendix C and Refs. [28, 30, 33].Consistency of Eq. (25) with the initial condition (23)implies that at t = 0 we have | u (0) | = 0 . (26)Since | u ( t ) | ≥
0, it also implies that | a ( t ) | ≤ sin θ and therefore, due to (21), | a ( t ) | ≥ cos θ .Let us highlight a few consequences of our analysis. Inour initial conditions, the value a (0) = 0 has been ex-cluded (see comments following Eq. (23)), on the basisof physical considerations. It is remarkable that the dy-namics self-consistently excludes such a value: a ( t ) = 0appears to be, for any t , the relative maximum for thepotential (see Appendix C), and therefore, for any t , it isan instability point away from which the system (spon-taneously) runs. Consistently, use of the constants ofmotion (21) and (25) shows that | a ( t ) | cannot be zero,since | a ( t ) | = 0 would imply U ( t ) = − cos θ , whichis not possible since U ( t ) is real.Next we observe that, due to the spontaneous sym-metry breaking, the system, moving away from the ini-tial condition values (23) and (26), dynamically tends tothe circle of squared radius γ ( θ ) (Eq. (C7)), wherethe nonvanishing time-independent values A = sin θ and U = sin θ are obtained (see Eqs. (21) and (25)).The bound | a ( t ) | ≥ cos θ discussed after Eq. (25) isdynamically satisfied by | a ( t ) | = γ ( θ ) (see Eq. (C7),which actually implies | a ( t ) | > cos θ ).The discussion presented in Appendix C regarding theem mode u ( t ) shows that the system runs away from the state with u ( t ) = 0 for any t , which is also consistentwith the dynamics of the modes a ( t ) and a ( t ). Thepossibility θ ≤ π/ | u ( t ) | = − (1 /
3) cos 2 θ ≡ v ( θ ), with θ > π/
4, whichis time-independent (see Eq. (C12)). One thus realizesthat a coherent em field pattern (“limit cycle” state) isgenerated as an effect of the spontaneous breakdown ofthe ( U (1)) phase symmetry. This is further discussed inthe next section, where we also discuss the interaction ofthe enzyme dipole field with the collective dipole wavefield. IV. INTERACTION BETWEEN ENZYME ANDDIPOLE WAVE FIELD
In order to study how the enzyme radiative dipole fieldinteracts with the em field pattern described above, letus first discuss a few more consequences emerging fromthe results we have obtained so far. We remark thatEq. (C12) implies the existence of a time-independentamplitude U for the field u ( t ). Inspection of the motionequations (C1)-(C3) shows indeed that˙ U ( t ) = 2Ω A ( t ) A ( t ) cos α ( t ) (27)˙ ϕ ( t ) = 2Ω A ( t ) A ( t ) U ( t ) sin α ( t ) , (28)where α = δ ( t ) − δ ( t ) − ϕ ( t ). We thus see that ˙ U ( t ) = 0,i.e., a time-independent amplitude U exists, if and onlyif the phase-locking relation α = δ ( t ) − δ ( t ) − ϕ ( t ) = π ϕ ( t ) = ˙ δ ( t ) − ˙ δ ( t ) = ω, (30)with ω introduced in Eq. (20) and proportional to Ω whenEq. (29) holds.In such a dynamical regime, the so-called “limit cycle”regime, a central result is that A − A is also time-independent. Indeed, by solving the system of equations(21) and (25) for | a ( t ) | and | a ( t ) | in terms of | u ( t ) | and then subtracting, we obtain A − A = cos θ −
13 sin θ + 2 U (cid:54) = 0 , (31)to be compared with A ( t ) − A ( t ) ≈ a ( t ) and a ( t ) is com-pensated by the change of phase of the em field. In other1words, the gauge invariance of the theory is preserved bythe dynamical emergence of coherence between the mat-ter field (DNA-water-enzyme) and the em field (radiativedipole field from DNA, water, and enzyme).We note here an estimate of the resulting interactionenergy. For 0 < θ < π/
3, the average pulsations (seeAppendix C, especially Eqs. (C11) and (C12)) fall in therange 0 < ν < − , populating bands in the in-frared spectrum [30] and overlapping with the lower endof the 0.1 - 1 eV range mentioned in Section II. This is aremarkable result, considering the rough analytical formof our water model, without an abundance of externalparameters. These wavelengths, which overlap with theenergy scale of collective dipole modes in DNA and en-zyme systems, provide an estimate of the long-range in-teractions mediated by the collective water dipole field,and they are distinct from the more energetic absorptionbands of water due to intramolecular vibrations and elec-tronic transitions. These long-wavelength interactions donot exist in pure water, as evidenced by Eqs. (31) and(35). Water adapts itself to biological agents (e.g., DNAand enzyme) and compensates, through gauge invariance,by phase-locking its electromagnetic field—arising fromwater’s matter field of endowed electric dipoles—to thefield introduced by the biological agents.In summary, as a consequence of the interaction withDNA and the water’s own radiative em field, the com-bined system evolves away from the initial symmetricvacuum to the asymmetric vacuum | u ( t ) | (cid:54) = 0. The co-herent em field pattern that emerges is formally expressedby the phase locking in Eqs. (29) and (30).We consider now the polarization density of the wa-ter environment in the presence of the enzyme radiativeelectric field E , assumed parallel to the z axis, as chosenin Section III. The term H = − d e · E , (32)with d e the electric dipole moment for water, is thenadded to the system energy.By following Ref. [31], it is possible to write the in-teraction Hamiltonian H of Eq. (32) in the form of aJaynes-Cummings-like Hamiltonian for large N : H = (cid:126) √ N γ ( a † S − + aS + ) , (33)where γ is the coupling constant proportional to the ma-trix element of the molecular dipole moment and to theinverse of the volume square root V − / , a † and a are thecreation and annihilation operators discussed in Figure(5) for the electric field E , and S ± denote the creationand annihilation operators of the “dipole wave modes”of the field pattern imaging the DNA-water interaction.Note that the factor √ N multiplying the coupling γ implies that for large N the collective mesoscopic inter-action energy scale is much larger, by the factor √ N ,than the microscopic fluctuations due to individual wa-ter molecules, thus providing a protective gap againstthermalization for the long-range quantum correlations. This fact is well known in studies of superradiance andcooperative robustness for aggregates of two-level sys-tems [34, 35]. Conversely, the time scale is much shorter,by the factor 1 / √ N , than the short-range interactionsamong the molecules. A similar remark applies to theenhancement of the coupling G , also due to the mul-tiplying factor √ N (see the comments after Eq. (23)).In this vein, we observe that in our discussion we havenot considered the losses of the radiative energy fromthe volume V . One can estimate these losses by com-paring lifetimes of the different modes, namely consid-ering the different time scales associated with them,2 π/ω , 2 π/ω , 2 π/ω , and 2 π/ω , where we have put ω ≡ m = 2 √ (cid:113) − (1 /
4) sin θ , in the limit cycleregime θ > π/
4, and ω = Ω √ θ for 0 < θ < π/ N thetime scale of the collective interaction is much shorter bythe factor 1 / √ N than the short-range interactions amongthe water molecules. It then follows that the macroscopicstability of the system is protected against quantum fluc-tuations in the microscopic short-range dynamics. In asimilar way, for sufficiently large N the collective inter-action is protected against thermal fluctuations, whichmay affect the collective process only when k B T ( ∼ .02eV at physiological temperatures) is of the same order orlarger than the energy gap determining the height of theprotection.The interaction H produces mixing between the states Y and Y (see Eq. (17)): Y → Y cos τ + Y sin τ and Y → − Y sin τ + Y cos τ , withtan τ = ω − (cid:112) ω + 4 H H . (34)The polarization P n is now found to be P n = 1 √ A − A ) sin 2 τ + 2 √ A A cos 2 τ cos[( ω − ω H ) t ] , (35)where ω H ≡ (cid:112) ω + 4 H and the limit cycle amplitudes A and A have been used. Time averaging gives the po-larization P n = (1 / √ A − A ) sin 2 τ . P n is non-zero aslong as τ (cid:54) = 0 and provided that the difference ( A − A )is non-vanishing. As we have seen, the former condition( τ (cid:54) = 0) is realized due to the presence of the enzyme elec-tric dipole field, and the latter condition ( A − A (cid:54) = 0) isrealized in the water coherent domains imaging the DNAem radiative field. P n thus carries information on boththe DNA radiative field and the enzyme field, and it isvanishing if one or both fields are absent.We would like to clarify how frequencies in the 0.1 - 1eV range can effectively couple to rotational transitionsof liquid water. Though the rotational transitions of in-dividual water molecules correspond to energies orders ofmagnitude smaller (meV), the point we would like to em-phasize is that the dipole-dipole modes of the aromatic2networks do not couple to the rotational transition ener-gies of individual water molecules but rather to the col-lective polarization modes present in the molecular waterdipole field. The dipole-dipole modes of the aromatic net-works become sources for radiative fields that stimulatethis collective polarization, and thus, because this is afield effect, the electronic polarizability of individual wa-ter molecules is not involved. See, in particular, Eq. (20),which shows the collective polarization as a function ofthe phase shift ( ω − ω ) t between the water dipole fieldand an individual water molecule. Please note that thesepolarization modes have been studied in the formalismof quantum electrodynamics since at least the 1980s [30].Furthermore, the infrared spectrum of liquid water(i.e., the 0.1 - 1 eV range) is dominated by the intense ab-sorption due to fundamental O-H stretching vibrations.Though there is no rotational fine structure in this region,the absorption bands are broader than might be expectedbecause of collective behaviors in the water dipole field.Peak maxima for liquid water are observed at 3450 cm − (0.43 eV), 3615 cm − (0.45 eV), and 1640 cm − (0.20eV), completely within the 0.1 - 1 eV range. Liquid wa-ter also has absorption bands around 5128 cm − (0.64eV), 6896 cm − (0.86 eV), 8333 cm − (1.03 eV), and10300 cm − (1.28 eV).The collective polarization modes we describe areNambu-Goldstone (NG) modes, which are generated inspontaneous symmetry breaking processes. Specifically,in our case, there is a breakdown of rotational dipolesymmetry. In the infinite volume limit, NG modes aremassless; however, as an effect of the system boundariesand defects, NG modes acquire a non-zero mass. TheNG modes with nearly vanishing mass, as dictated bythe Goldstone theorem, are low-energy (low-momentum)modes and thus couple to the longer wavelength rangeof the spectrum mentioned above. Indeed, in their con-densation in the lowest energy state (the ground state),nearly vanishing momentum k implies due to the DeBroglie relation ( k = h/λ ) that low-energy NG modesare quanta associated with long wavelengths ( ∼ V. CONCLUSIONS
We have considered the interaction between DNA andenzyme macromolecules in their water environment. Byresorting to recent results [1], we have discussed the col-lective dipole behavior and the Hamiltonians of DNA andenzyme molecules. We have considered the
Eco
RI restric-tion endonuclease and the
Taq
DNA polymerase, whichare used widely in molecular biology for precise cuttingand rapid amplification of DNA sequences, respectively.We have adopted the QFT paradigm, according to whichany interaction between two systems is mediated by thepropagation of a mediating correlation field or quantum,such as, for example, the photon exchanged between in-teracting electric charges in quantum electrodynamics.In the present study of DNA-enzyme interactions, wehave identified such a correlation field with the collectivedipole wave of the molecular water field. The DNA radia-tive dipole field does indeed trigger the collective dipolewave in the water environment, which in turn coupleswith the enzyme radiative dipole field. By following Ref.[28], we have shown that a nonzero time-independent emamplitude can develop, driving a phase transition, as aneffect of the radiative dipole-dipole interaction. We havethus discussed the transition of the molecular water fieldto the limit cycle regime, where the em DNA dipole fieldand the water dipole field get locked in phase. We haveshown that the specific value of the resulting em dipolefield amplitude controls the boundary conditions of thedynamics and the characteristic time scales of the pro-cess. These time scales turn out to be much shorter thanthe ones associated with thermal noise, providing a pro-tective energy gap for the collective dynamics.While a comparison with the strength of other interac-tions involved—electrostatic, hydrophobic, etc.—is pos-sible, it is not relevant to our study because these inter-actions occur at short range. Water has a unique abilityto shield charged species from each other, so electrostaticinteractions between charges are highly attenuated in wa-ter. The electrostatic force between two charges in solu-tion is inversely proportional to the dielectric constantof the solvent. The dielectric constant of water (80.0) ishuge, over twice that of methanol (33.1) and over five3times that of ammonia (15.5). Water is therefore a goodsolvent for salts because the attractive forces betweencations and anions are minimized. In this respect, it isinteresting to observe that the decoherence time for ion-ion collisions and for interactions with distant ions inaqueous solutions for crystalline binary compounds is ofthe order of 10 − and 10 − of a second, respectively[36]. Such very short decoherence times would make im-possible the formation of salt crystals, which of coursecontradicts common experience, showing that in prac-tice the formation of stable salt crystals does occur andlasts for many orders of magnitude longer than 10 − and10 − of a second. At the crystallization point and lowtemperature, this formation can occur within fractionsof a second to several seconds or even longer times, fromminutes to hours.The way out of the contradiction lies in the fact thatthe binding of the atoms in the crystalline lattice is notthe result of ion-ion collisions or of interactions with dis-tant ions in a quantum mechanical scheme. The atombinding is instead due to long-range correlations me-diated by Nambu-Goldstone modes in a quantum elec-trodynamics framework. In the case of crystal lattices,these modes are indeed the phonons, and they are knownto be extremely important in many areas of solid statephysics [37, 38].It has also long been recognized that hydrogen bond-ing is a dominant mechanism of cohesion in water sys-tems, and some of the failures of previous models haveoriginated in the exclusion of crucially important disper-sion interactions. However, hydrogen bonding is a com-plex phenomenon, which may be decomposed into elec-trostatic attraction, polarization, dispersion, and partialcovalency, etc., even though the relative contributions ofthese components in water remain controversial, depend-ing significantly on definition. The contribution due topartial covalency (often described as charge transfer), forexample, has been particularly contentious. Hydropho-bic effects, furthermore, are a consequence of strong di-rectional interactions between water molecules and aredriven by entropic considerations in bulk water. In con-trast, the phenomenon we describe in this manuscript isa mesoscopic effect that occurs due to the long-range ra-diative correlations of water at biopolymeric interfaces.Several estimates place the strengths of these various con-tributions due to water dimer interactions at the meVscale [39], various orders of magnitude smaller than thecollective dipole modes described in our manuscript, andtherefore too small to couple effectively. Our contention,based on estimates in this paper, is that collective dipolemodes in the 0.1 - 1 eV range may serve a critical role infacilitating interactions between DNA and (endonucleaseor polymerase) enzyme systems.We observe that the system ground state is a co-herent condensate of the massless modes (of Nambu-Goldstone type), which have been identified in our anal-ysis in Appendix C. We have indeed seen that the cylin-drical ( U (1)) phase symmetry gets spontaneously bro- ken. As a final consequence, then, the phase-lockingrelation in Eqs. (29) and (30) is obtained. The con-sistency of such a complex dynamical process in biologywith the gauge invariance of QFT can be understood asfollows: Gauge invariance requires fixing the phase ofthe matter field and implies that a specific gauge func-tion must be selected. This leads to the introductionof the covariant derivative D µ = ∂ µ − igA µ . The vari-ations in the phase of the matter field thus give originto the pure gauge A µ = ∂ µ ϕ . This means that when ϕ ( x , t ) is a regular (continuous and differentiable) func-tion, then E = − ∂ A ∂t + ∇ A = ( − ∂∂t ∇ + ∇ ∂∂t ) ϕ = 0,since time derivative and gradient operator can be inter-changed for regular functions. In such a case, we alsohave B = ∇ × A = ∇ × ∇ ϕ = 0. Therefore, the fields E and B can acquire nonvanishing values in a coherentregion only if ϕ ( x , t ) exhibits a divergence or a topologi-cal singularity within the region [40]. Such an occurrenceis particularly relevant in our analysis, since it accountsfor topologically nontrivial configurations in the coherentground state, thus requiring the “clamping” characteriz-ing a wide swath of restriction endonuclease and DNApolymerase catalytic activity. Eco
RI locates its targetDNA sequence of six base pairs via “facilitated diffu-sion” in a non-specific conformation that is characterizedby interstitial water molecules between the DNA and en-zyme. In this conformation and under optimum condi-tions, these enzymes are able to scan up to 10 base pairsin a single binding event. Upon recognizing its target se-quence, Eco
RI changes to a specific conformation thattightly binds the DNA through exclusion of the intersti-tial water. In the
T aq
DNA clamp, a protein multimericstructure completely encircles the DNA double helix, es-sentially encircling the core of a vortex, which is formednot coincidentally by a layer of water molecules in thecentral pore of the clamp between the DNA and the pro-tein surface. As is well known [41, 42], the dynamicalformation of vortex structures is described by the QFTof spontaneously broken U (1) symmetry theories.These findings may contribute toward addressing anunsolved problem in biology: Why are divalent cationslike Mg so important to enzyme behavior and so pre-cisely controlled for optimum biological functions? Forexample, Eco
RV (a close relative of
Eco
RI) incubated atideal pH cuts both strands of DNA in the synchronized,concerted manner discussed in Ref [1]; in contrast, theenzyme reaction at lower pH involves sequential, inde-pendent cutting of the two strands [4]. The differencein catalysis, which cannot be accounted for by weakenedenzyme-DNA binding, has been traced to the asymmetri-cal binding of Mg to the Eco
RV subunits. Thus we seethat the pH disturbance propagates through the buffersolution, generating a local electromagnetic environmentin which the symmetry of the complex is broken. Sim-ilarly,
Taq is extremely dependent on magnesium, anddetermining the optimum concentration to use is criti-cal to the success of the polymerase chain reaction. Justas complex systems exhibit behavior that cannot be pre-4dicted from the mechanics of microscopic constituents, sobiology has dynamically optimized several parameters toachieve maintenance of long-range correlations. Divalentcations may therefore serve as electromagnetic antennaeto enhance or maintain mediating dipole-wave fields insolution.In conclusion, both the high efficiency and high relia-bility of enzymatic catalytic activity in DNA metabolismrest on the physico-chemical properties of enzymaticmacromolecules, which aid in identifying the water emfield pattern imaging the DNA molecules in their waterenvironment.
ACKNOWLEDGMENTS
AC and GV acknowledge partial financial supportfrom MIUR and INFN. TJAC would like to acknowledgefinancial support from the Department of Psychologyand Neuroscience and the Institute for Neuro-ImmuneMedicine at Nova Southeastern University (NSU), andwork in conjunction with the NSU President’s FacultyResearch and Development Grant (PFRDG) programPFRDG 335426 (Craddock – PI). PK was supportedin part by the National Center For Advancing Transla-tional Sciences of the National Institutes of Health underAward Number TL1TR001431. The content of researchreported in this publication is solely the responsibilityof the authors and does not necessarily represent theofficial views of the National Institutes of Health. PKwould also like to acknowledge ongoing discussions withG. Dunston, as well as partial financial support from theWhole Genome Science Foundation.
Appendix A
The revolution in genetic sequencing and engineeringhas been made possible through the use of restriction en-donucleases. Originally isolated from bacteria, these en-zymes cut DNA at recognition sequences with high speci-ficity, thereby assuring a consistent final product. Eachendonuclease has been named using a system looselybased on the bacterial genus, species, and strain fromwhich the enzyme is derived: the first identified endonu-clease in the RY13 strain of
Escherichia coli is Eco
RI,and the fifth endonuclease extracted from the same strainis
Eco
RV. The body of literature on the structure andcatalytic mechanisms of these molecular workhorses issubstantial and has been reviewed in multiple instances[2, 43–49].One study [6] revealed that the
Eco
RV DNA-bindingdomains cannot function independently of each other,and that only with asymmetric modifications can an
Eco
RV mutant cleave DNA in a single strand of therecognition site. Also,
Eco
RV mutants are not affectedin ground state binding but rather in the stabilization ofthe transition state, and catalysis is significantly altered compared to binding when the symmetry of the protein-DNA interface is disturbed. Taken together, these datasuggest that an asymmetry in the enzyme is manifestedin the catalytic centers of the two subunits only in thetransition state, and that a nonlocal pathway—in whichsome physical quantity is conserved—may be used forcoordination.The notion that enzyme catalysis can be “substrate-assisted” is not new [44, 50], and previous authors havehypothesized that energy could be transferred from en-zyme clamping to the catalytic transition state [43],though no quantitative mechanisms were proposed. Theidea that enzymes may sequester coherent energy fromDNA zero-point modes for genomic metabolism [1] cer-tainly fills a gap in our understanding of how enzymeclamping on DNA might give rise to energy transport inthe substrate, which then “assists” in synchronization ofthe catalytic process to form double-strand breaks.This process of substrate-assisted catalysis occurs af-ter the
Eco
RI enzyme has bound to its cognate DNAsequence and changed conformation so as to be in thecatalytic transition state. Thus, already having over-come the activation energy barrier, the reaction proceedsspontaneously due to the negative change in Gibbs freeenergy, which is provided here as an estimate for the hy-drolysis of a phosphodiester bond ( ε P − O ). The collectivedipole-dipole oscillation mediates the synchronized catal-ysis of two phosphodiester bonds at about 20 ˚A spatialseparation and ensures that the spontaneous Gibbs freeenergy change is channeled to the appropriate catalyticcenter in a symmetric way. Since the Eco
RI enzyme doesnot use ATP for its catalytic activity, this energy is re-cruited from the collective oscillation after the enzymehas already overcome the activation energy barrier due toconformation change upon target sequence recognition.Developed in 1983 by Kary Mullis and colleagues [51],the polymerase chain reaction (PCR) is now a commonand often indispensable technique used in biological andmedical research, where a single copy or few copies ofDNA must be amplified by several orders of magnitude.The method relies on thermal cycling, which consistsof repeated stages of 1) DNA denaturation (into singlestrands), 2) primer annealing, and 3) DNA replication.This last stage is guided by a heat-tolerant DNA poly-merase called
Taq , which was originally isolated from thethermophilic bacterium
Thermus aquaticus . Taq func-tions optimally between 75 and 80 ◦ C and enzymaticallyassembles a new DNA strand from nucleotides that areadded to the PCR solution. Specially designed primerscomplementary to the target region ensure amplificationof only the desired DNA sequence under consideration.As the target sequence is doubled during each cycle,available substrates in the reaction can become limitingas the PCR progresses, usually between 20 and 40 cycles.5
Appendix B
Using matrix elements for the derivation of the base-pair electronic angular frequencies, by using ( π ) t = Qr t and ( F ) u = k uv r v we write π = α · E = α · F /Q as Qr t = α tu k uv r v /Q , or Q δ tv = α tu k uv , and thus Q α − = k = m ω , (B1)which yields precisely the matrix version of Eq. (3) in themain text.In the case of an infinite helix composed of homoge-neous base pairs, we may transform the infinite sums intoeasily calculable integrals. From the idealized Hamilto-nian H I = 12 M (cid:40) + ∞ (cid:88) n = −∞ p n (cid:41) + 12 M ω (cid:40) + ∞ (cid:88) n = −∞ q n (cid:41) (B2)+ 12 Γ (cid:40) + ∞ (cid:88) n = −∞ ( q n − q n +1 ) (cid:41) , where the uniform M and ω reflect the chain homogene-ity and p n ( t ) , q n ( t ) are the deviation from equilibriumfor the n th component oscillator in a single dimensionwith arbitrary interaction potential Γ, we may use theBessel-Parseval relation [52] and recursion to obtain thediagonalized form for H I : + π(cid:96) (cid:90) − π(cid:96) (cid:96)dk π (cid:26) | P ( k, t ) | M + M (cid:20) ω + 4Γ M sin (cid:18) k(cid:96) (cid:19)(cid:21) | Q ( k, t ) | (cid:27) (B3)where (cid:96) is the unit distance of the oscillator chain andthe new position and momenta coordinates are definedusing Fourier series: Q = + ∞ (cid:88) n = −∞ q n e − ink(cid:96) ,P = + ∞ (cid:88) n = −∞ p n e − ink(cid:96) , Q = + ∞ (cid:88) n = −∞ q n +1 e − ink(cid:96) , ...P = + ∞ (cid:88) n = −∞ p n +1 e − ink(cid:96) , .... (B4)The normal modes of vibration are obtained readilyfrom Eq. (B3): Ω( k ) = (cid:113) ω + M sin (cid:0) k(cid:96) (cid:1) , which ap-plies for any infinite coupled oscillator sequence with ho-mogeneous components in the first Brillouin zone, wherethe wave vector obeys − π(cid:96) ≤ k ≤ π(cid:96) . Transforming co-ordinates by Fourier expansion thus allows us to convertthe idealized Hamiltonian, which was expressed as an in-finite sum of coupled oscillators, into a definite integralover uncoupled collective modes for each physically dis-tinguishable state.Though a helpful comparison, such an analytical so-lution is not rigorously applicable for finite, real DNAsequences. It does not show the complement of modes de-pressed below the homogeneous trapping frequency ω be-cause the generality of the real-valued function Ω would have to be restricted for specified values of Γ. How-ever, numerical analysis will certainly suffice for genomiclength scales.With reference to our computations for DNA sequencesdiscussed in Section II, we note that by separating Eq. (1)into energy contributions from transverse ( H xy ) and lon-gitudinal ( H z ) modes, we may write the symmetric lon-gitudinal potential matrix V z for a four-bp sequence as k ,zz γ z γ z k ,zz γ z γ z k ,zz γ z γ z k ,zz , (B5)where k ,zz = m ω ,zz , etc., and γ zs,s +1 = γ zs +1 ,s = − Q / (2 π(cid:15) d ) denotes the z s z s +1 coefficient fromEq. (4). The symmetric transverse potential matrix V xy is k ,xx γ x γ xy γ x k ,xx γ x γ xy γ xy γ x k ,xx γ x γ xy γ xy γ x k ,xx γ xy γ yx k ,yy γ y γ yx γ yx γ y k ,yy γ y γ yx γ yx γ y k ,yy γ y γ yx γ y k ,yy , (B6)where k ,xx = m ω ,xx , etc., and γ xy = − Q sin θ/ (4 π(cid:15) d ) = − γ xy , etc. The diagonal ki-netic matrices T j consist of the electronic oscillatormasses: T z = diag( m , m , m , m ) , (B7) T xy = diag( m , m , m , m , m , m , m , m ) . Parametrization of our model by the inter-base-pairspacing is shown in Figure 6 for a homogeneous ds-DNA sequence of four base pairs. What becomes quicklyapparent is the rapid convergence of the longitudinalmodes to (cid:126) Ω / ≈ . ε P − O , within less than three timesthe standard observed inter-base-pair spacing, suggest-ing that this parameter has been evolutionarily optimizedto maximize variation in DNA electronic oscillations atphysiologically relevant length scales. Similar behavioris observed for the transverse modes, with a bi-modalconvergence around (cid:126) Ω / ≈ . ε P − O and 4 . ε P − O .Surprisingly, the modes for AAAA converge to thesame values as the triplet codon case (dsAAA, datanot shown) for the zero-point energies over comparablelength scales. The middle harmonics bifurcate quicklyand diverge when the spacing dips below 5 . . (a) dsAAAA longitudinal modes (b) dsAAAA transverse modes FIG. 6:
Zero-point modes of DNA sequence.
Longitudinal (6a) and transverse (6b) zero-point modesparametrized by the inter-base-pair spacing d for the double-stranded DNA sequence AAAA. The abscissae aregiven in ˚A and the ordinates are dimensionless. Notice the divergence of the zero-point modes at the observedequilibrium base-pair spacing d = 3 . ε P − O (cid:39) .
46 eV. DNA sequence is therefore exquisitely structured to channelits own electronic vibrational energy for the emergence and continuity of biological diversity and other life processes.genomic and biological metabolism. This fine-tuning ofDNA architecture for coherent energy transport leads usto postulate that DNA is constructed with a view towardfinding energy in its own vibrations for life processes.We have chosen to examine the so-called “zero-point”modes (on which the number operator N s,j = a † s,j a s,j inEq. (7) acts to produce a zero eigenvalue) because theseare most easily excited by the free energy changes due toenzyme clamping. These zero-point oscillations are col-lective normal modes of the DNA system considered inour model framework. Because the oscillations are nor-mal modes, a four-base-pair sequence will produce fourfrequencies of coherent (phase-synchronized) oscillationin the longitudinal direction, as shown in Eq. (10). Sim-ilarly, a six-(eight-)base-pair sequence will produce six(eight) frequencies of coherent oscillation in the longitu-dinal direction, and so on. The number of frequencies ofcoherent oscillation in the transverse direction is doubledbecause of the coupling between the x and y degrees offreedom due to the helix twist angle. Appendix C
Taking advantage of the rotational symmetry, by us-ing Eq. (17) in Eqs. (24) and writing u ( t ) ≡ u m ( t ) and a ( t ) ≡ a ,m ( t ), we get the set of equations [28, 30]:˙ a ( t ) = 3 Ω u ∗ ( t ) a ( t ) (C1)˙ a ( t ) = − Ω u ( t ) a ( t ) (C2)˙ u ( t ) = 2 Ω a ∗ ( t ) a ( t ) , (C3) with Ω = 4 ed e (cid:112) N/ (6 ω V ) ω ≡ G ω . These equationsare fully consistent with the dipole rotational invarianceexpressed by the zero average polarization, Eq. (20), thenormalization condition (16) (or (21)), and the conser-vation of molecules in Eq. (22). The rate of change ofthe amplitude of the level (0 ,
0) is shown in Eq. (C1)to depend on the coupling between the levels (1 , m ), m = 0 , ± , and the radiative em mode of correspond-ing polarization. Due to rotational invariance, each ofthese couplings contribute in equal measure to the tran-sitions to (0,0). In a similar way, the rate of change ofthe amplitude of each level (1 , m ) is shown in Eq. (C2)to depend on the coupling between the the level (0 , , ↔ (1 , m ) , m = 0 , ±
1, as shown in Eq. (C3). Weremark that, as described by Eq. (C1), each of the levels(1 , m ) may find in the em field, which includes all pos-sible polarizations, the proper mode to couple with, infull respect of the selection rules. Note that use of thecomplex conjugate of Eq. (C2) in (C3) leads to ∂∂t | u ( t ) | = − ∂∂t | a ( t ) | . (C4)By integrating this equation and fixing the integra-tion constants consistently with the initial conditions,Eq. (25) is obtained.We now study in the mean field approximation [53]the ground state (vacuum) of the system for each of themodes a ( t ), a ( t ), and u ( t ). For the mode a ( t ) we find7the equation [28]¨ a ( t ) = − δδa ∗ V [ a ( t ) , a ∗ ( t )] , (C5)where the potential is V [ a ( t ) , a ∗ ( t )] = 2Ω ( | a ( t ) | − γ ( θ )) (C6)and γ ( θ ) ≡ (1 + cos θ ).Denote by a ,R ( t ) and a ,I ( t ) the real and the imag-inary component, respectively, of the a ( t ) field. Thepotential has a relative maximum at a ( t ) = 0 and a(continuum) set of minima on the circle of squared ra-dius γ ( θ ) in the ( a ,R ( t ) , a ,I ( t )) plane, given for any tby | a ( t ) | = 12 (1 + cos θ ) = γ ( θ ) . (C7)The points on the circle transform into each other un-der shifts of the field δ : δ → δ + α (rotations in the( a ,R ( t ) , a ,I ( t )) plane). They represent (infinitely many)possible vacua for the system. The phase symmetry isspontaneously broken when a specific ground state is sin-gled out by fixing the value of the δ field. γ ( θ ) is theorder parameter. One can recognize that there is a quasi-periodic mode with pulsation m = 2Ω (cid:112) (1 + cos θ ) andthat the field δ ( t ) corresponds to a massless mode (theso-called Nambu-Goldstone (NG) field). It is a collectivemode implied by the spontaneous breakdown of symme-try.The motion equation for the the amplitude a ( t ) isfound to be ¨ a ( t ) = − δδa ∗ V [ a ( t ) , a ∗ ( t )] , (C8)where the potential is V [ a ( t ) , a ∗ ( t )] = σ | a ( t ) | − ( | a ( t ) | ) (C9)and σ = 2 Ω (1 + sin θ ). There is a relative minimumat a = 0 and a (continuum) set of maxima on the circleof squared radius | a ( t ) | = 16 (1 + sin θ ) ≡ γ ( θ ) . (C10)For consistency between Eqs. (C5) and (C8), it is ex-cluded that the amplitude A be zero (at the minimumof V [ a ( t ) , a ∗ ( t )]), which would also correspond to thephysically unrealistic situation of the (0 ,
0) level beingcompletely filled, as we already discussed in the com-ments following Eq. (23) in the main text. On the otherhand, we see that the values on the circle of radius γ ( θ )are indeed forbidden for the amplitude A since in thatcase U = − cos θ <
0, which is not acceptable since U is real. Consistently, the conservation law (25) and the realitycondition for U require that | a ( t ) | ≤ sin θ , which isbelow γ ( θ ). Below such a threshold is also the value sin θ taken by A when | a ( t ) | = γ ( θ ). The con-clusion is that the field a ( t ) is a massive field with (real)mass (pulsation) σ = 2 Ω (1 + sin θ ).We consider now the em mode u ( t ). The potential is V [ u ( t ) , u ∗ ( t )] = 3Ω ( | u ( t ) | + 13 cos 2 θ ) (C11)= µ | u ( t ) | + 3 Ω | u ( t ) | + 13 Ω cos θ , where we put µ = 2 Ω cos 2 θ . For θ ≤ π ( µ ≥ V [ u ( t ) , u ∗ ( t )] is a paraboloid potential with cylindri-cal symmetry about an axis orthogonal to the plane( u R ( t ) , u I ( t )), with u R ( t ) and u I ( t ) denoting the real andthe imaginary component, respectively, of the u ( t ) field.The minimum (the ground state) is at u ( t ) = 0, for any t . However, we have seen above that in its time evolutionthe system runs away from u (0) = 0. Thus u ( t ) = 0 isnot acceptable at any t since the system would run awayfrom it at any t and therefore it cannot be a stable groundstate. This means that consistency with the dynamics ofthe modes a ( t ) and a ( t ) excludes the possibility θ ≤ π .Consistency is recovered for θ > π ( µ < u ( t ) = 0, for any t , and a (continuum) setof minima (ground states) on the circle of squared radius | u ( t ) | in the ( u R ( t ) , u I ( t )) plane: | u ( t ) | = −
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