Collective behavior of oscillating electric dipoles
aa r X i v : . [ c ond - m a t . d i s - nn ] N ov Collective behavior of oscillating electric dipoles
Simona Olmi,
1, 2, ∗ Matteo Gori,
3, 4, † Irene Donato, ‡ and Marco Pettini
3, 4, § Institut f¨ur Theoretische Physik, Technische Universit¨atBerlin, Hardenbergstr. 36, 10623 Berlin, Germany CNR - Consiglio Nazionale delle Ricerche - Istitutodei Sistemi Complessi, 50019, Sesto Fiorentino, Italy Aix Marseille Univ, CNRS, CPT, Marseille, France CNRS Centre de Physique Th´eorique UMR7332, 13288 Marseille, France (Dated: November 22, 2017)
Abstract
The present work reports about the dynamics of a collection of randomly distributed, and ran-domly oriented, oscillators in 3D space, coupled by an interaction potential falling as 1 /r , where r stands for the inter-particle distance. This model schematically represents a collection of identicalbiomolecules, coherently vibrating at some common frequency, coupled with a − /r potentialstemming from the electrodynamic interaction between oscillating dipoles. The oscillating dipolemoment of each molecule being a direct consequence of its coherent (collective) vibration. Bychanging the average distance among the molecules, neat and substantial changes in the powerspectrum of the time variation of a collective observable are found. As the average intermolec-ular distance can be varied by changing the concentration of the solvated molecules, and as thecollective variable investigated is proportional to the projection of the total dipole moment of themodel biomolecules on a coordinate plane, we have found a prospective experimental strategy ofspectroscopic kind to check whether the mentioned intermolecular electrodynamic interactions canbe strong enough to be detectable, and thus to be of possible relevance to biology. PACS numbers: 87.10.Mn; 87.15.hg; 87.15.R- ∗ Electronic address: simona.olmi@fi.isc.cnr.it † Electronic address: [email protected] ‡ Electronic address: [email protected] Electronic address: [email protected] . INTRODUCTION Let us quickly summarize what motivates the present work. The starting point is theobservation of the enormous efficiency, rapidity, and robustness against environmental dis-turbances, of the complex network of biochemical reactions in living cells. At the sametime it is hardly conceivable that this pattern of interactions/reactions is driven and reg-ulated only by random encounters between cognate partners [1]. In fact, on the basis ofseveral estimates [2, 3], in many cases the high efficiency that biomolecules display whenmoving toward their specific targets and sites of action can hardly be the result of ther-mal fluctuations (Brownian motion) alone: biochemical players ”need to know” where to goand when. Therefore, in order to accelerate these encounters, selective forces acting at along distance (”long” means possibly up to some hundreds of nanometers) are needed. Inthe physico-chemical conditions typical of the cytoplasm (large value of the static dielectricconstant of water, strong Debye shielding due to high concentrations of freely moving ions)electrostatic forces are ruled out; to the contrary, electrodynamic interactions of sufficientlyhigh frequency can be effective. Quite a long time ago, it was surmised [4] that if each of thecognate partners of a biochemical reaction would undergo a collective vibrational oscillation(involving all the atoms or a large fraction of them in each molecule) at the same or almostthe same frequency, then the associated giant dipole vibrations could excite a sufficientlyintense and resonant (thus selective) electrodynamic attractive interaction [5]. This wouldbe the basic mechanism of molecular recruitment at a distance, beyond all the well-knownshort-range forces (chemical, covalent bonding, H-bonding, Van der Waals). Unfortunately,because of technological limitations, an experimental proof or refutation of this possibilityhas been for a long time and is still sorely lacking. These long range electrodynamic in-teractions are predicted by standard classical electrodynamics, thus they necessarily exist,the point is whether these can attain a sufficient strength to overcome all the dissipationmechanisms that would be activated together with the collective vibration [5]. In our pre-liminary investigations in [6] and [7, 8] we have put forward the idea that an answer tothis conundrum could come from the study of how the diffusion behavior of biomoleculesin solution could change when their concentration is varied (that is, when the average in-termolecular distance is varied) as a consequence of the action of surmised electrodynamicinteractions. The experimental technique envisaged in [7, 8] was Fluorescence Correlation3pectroscopy (FCS), a well established experimental technique [9–11]. In the present paperwe report about a possible alternative/complementary viable experimental procedure for anassessment of the strength - thus of the potential biological relevance - of resonant electro-dynamic intermolecular interactions. The paper is organized as follows: in Section II themodel is defined and discussed, while in Sec. III we report the outcomes of the MolecularDynamics simulations of the chosen model and we comment on the observed phenomenology.Section IV is devoted to some concluding remarks about the results presented throughoutthe present paper.
II. THE MODELA. Model for the biomolecule
This work aims to study the emergence of collective phenomena in a system of mutuallyinteracting classical electric dipole oscillators out-of-thermal equilibrium. This is intended tobe a little step further in the same direction of [5] where the hypothesis have been explored ofthe possibility that long-range classical electrodynamic interactions can take place in livingmatter at molecular level.As in that case, an oversimplified model for biomolecules has been used, where the level ofaccuracy taken into account is suited to this feasibility study. Each biomolecule has beendescribed as an oscillating electric dipole composed of two material points, each of themwith a mass m and the same absolute value Ze of the electric charge but with oppositesign. The position of the positive and negative charged particles of the i-th biomolecule arerespectively r + ,i and r − ,i . The position of its center of mass of each biomolecule is indicatedby R i while the direction of each dipole is b r i = r + ,i − r − ,i k r + ,i − r − ,i k ; (1)both have been considered to be fixed, so that the charged particle of each biomolecule areconstrained to oscillate along their joining line.Both the constraints on R i and b r i seem to be quite strong with respect to a realistic biologicalmolecular system where particles both diffuse (time dependence R i ) and rotate due to thecollisions with the surrounding water molecules. These assumptions are justified by thecomparison of characteristic time scales for collective giant dipole oscillations of a single4iomolecule with characteristic time scales given by the translational and rotational diffusion(See Appendix A for more details). It follows that the only dynamical variable is themutual distance r i = ( r + ,i − r − ,i ) · b r i between the two centres of charge. The electric dipolemoment is given by p i ( t ) = Zer i ( t ) b r i . Despite its simplicity, this model takes into accountthe main features we are interested in: collective effects on the dynamics of giant dipoleoscillations emerging in a system of biomolecules coupled through electrodynamic long-range interactions.For each dipole representing a biomolecule, an effective potential V ( r i ) = V eff ( k r + ,i − r − ,i k )is supposed to be exerted among material charged points. A stable equilibrium configurationis supposed to be attained for r i = r i,eq such that the effective potential takes the form V eff ( r i ) ≈ mω i ( r i − r i, ) + 14 m ω i Λ ( r i − r i, ) , (2)where the parameter Λ is the characteristic length of the oscillation amplitude for the emer-gence of non-harmonic contributions. So the effective potential of (2) takes in account bothharmonic and non-harmonic contributions in the oscillation of the electric dipole. The non-harmonic contribution has been included for two main reasons: firstly, it accounts for theexchange of energy of the main collective mode with other vibrational normal modes of thebiomolecule; secondly, it has been included in order to prevent instability of the oscillationswhen the electric dipoles, representing biomolecules, are strongly coupled among them. B. Mutual quasi-electrostatic interactions among biomolecules
The physical picture behind the model we intend to analise is an ensemble of oscillatingbiomolecules in watery solutions in presence of freely moving ions. Since the declared interestof this work is to study collective phenomena mediated by long-range interactions amongbiomolecules, we neglect any electrostatic effect due to Debye screening effects. We caneasily make this assumpion as the electrostatic field is exponentially suppressed on a lengthscale of the order of some Angstroms in real biological systems. It follows that, for theintermolecular average distance range we are interested in ( ∼ − ˚ A ), the contributionof electrostatic field is negligible. On the contrary electrodynamics fields are not screened inwatery systems in presence of freely moving ions, as it can be inferred both from theoreticalworks and from dielectric spectroscopic experiments for sufficiently high frequencies ( ω > M Hz ). As mentioned before the expected frequency for the collective oscillation ofa biomolecule is around 0 . − T Hz , thus largely above the upper frequency thresholdfor important screening effects on electrodynamic fields. Collective phenomena are moreprobably expected in systems of resonant oscillators: for such a reason, a system of N identical biomolecules (oscillators) has been considered. Moreover, resonance of electricdipole oscillators, describing biomolecules, has been argued to be a necessary condition inorder to activate long range dipole-dipole ( ∼ R − ij ) electrodynamic interactions [5].In our very simple model the force acting on each charge barycentre of the i -th electric dipoledue to the j -th dipole is given by F CED ( r ± ,i ; R j ) = Ze E CED ( r ± ,i ; r j ) . (3)where E CED ( r ; R j ) is the value of the electric field in r generated by the j -th dipole whosecenter is in R j . According to the Classical Electrodynamics (CED), if we assume valid thedipole approximation, i.e. k r − R j k ≫ r j , the expression for the electric field takes the form E CED ( r ; R j ) = Z + ∞ d ω exp h iω (cid:16) t ± p ǫ ( ω ) k r − R j k /c (cid:17)i πǫ ( ω ) k r − R j k × ( [3 b n j ( r )( p j ( ω ) · b n j ( r )) − p j ( ω )] ∓ iω p ǫ ( ω ) k r − R j k c ! + − [ p j ( ω ) − b n j ( r )( p j ( ω ) · b n j ( r ))] ω ǫ ( ω ) k r − R j k c ) . (4)where c is the speed of light, b n j = r − R j / ( k r − R j k ) is direction joining the center of dipole R j to r , p j ( ω ) is the Fourier Transform of the electric dipole moment of the j -th biomoleculein time domain and ǫ ( ω ) is the dielectric constant of the medium.For the range of frequencies we explore ( ω ∼ Ω ≈ T Hz ), the dielectric constant of anelectrolytic aqueous solution can assumed to be real Re ( ǫ ( ω )) ≫ Im ( ǫ ( ω )) and approxima-tively constant ǫ W S (Ω) ≈
3. Moreover both the intermolecular average distance R ij ≈ ˚Aand the characteristic linear dimensions r ≈ λ = 2 πc/ ( ǫω ) ≃ × ˚A. This allows to assume thatthe electromagnetic field has the same value for both centers of charge of each biomolecule,i.e. E CED ( r + ,i ; R j ) = E CED ( r − ,i ; r j ) = E CED ( R i ; R j ), and that any retardation effect canbe neglected, i.e. R ij /λ ≪
1. With these approximations the acceleration of the i -th dipole6s directed along b r i and due to the interaction with the j -th dipole reads as (cid:18) m d r i d t (cid:19) CED = (cid:18) m d r + ,i d t − m d r − ,i d t (cid:19) CED · b r i = 2 Ze X j = i E CED ( R i ; R j ) · b r i == 2 Ze X j = i Z + ∞ d ω exp ( iωt )4 πǫ W S R ij [3( b n ji · b r i )( p j ( ω ) · b n ji ( r )) − p j ( ω ) · b r i ] == 2( Ze ) X j = i Z + ∞ d ω exp ( iωt )4 πǫ W S R ij [3( b n ji · b r i )( b r j · b n ji ) − ( b r j · b r i )] r j ( ω ) == 2( Ze ) X j = i [3( b n ji · b r i )( b r j · b n ji ) − ( b r j · b r i )]4 πǫ W S R ij r j ( t ) = X j = i mω ij ζ ij r j ( t ) , (5)where b n ji = R j − R i R ij is the direction joining the electric dipoles, ω ij = 2 Z i Z j e πǫ WS mR ij (6)is a characteristic frequency describing the strength of the dipole-dipole interactions, ζ ij = [3( b n ji · b r i )( b r j · b n ji ) − ( b r j · b r i )] (7)is a geometrical factor depending of the orientation of the electric dipoles and r j ( ω ) is theFourier Transform of r j ( t ). III. STUDY OF SYNCHRONIZATION IN PRESENCE OF THERMAL BATHAND EXTERNAL SOURCEA. Biological watery environment as thermal bath
This work is inspired by the request for observables in real biological systems at molec-ular level that can detect the presence of long-range electrodynamics interactions amongbiomolecules. As all biomolecules in real biological environment are in watery solution, wehave to take into account the presence of surrounding water molecules. Though recent stud-ies reveal that the water in biological system can have a highly non trivial behaviour withrespect to electrodynamic fields generated by the electric dipole of biomolecules [12–16], inthis article we will assume the surrounding water to play simply the role of a thermal bath.As a consequence of this, the presence of water molecules can be schematized via the intro-duction of a stochastic noise (thermal fluctuations) and a viscous friction term (dissipation)7n the equation of motion for oscillating electric dipoles. In particular friction viscous forcesare due to the aqueous surrounding medium considered as a homogeneous fluid with viscos-ity η w . We assume that the expression of the viscous force is given by Stokes’ Law actingon each barycentre of electric charge (positive and negative) F visc ,i ± = − γ i dr i, ± d t γ i = 6 πη W R i (8)where R i is the hydrodynamic radius of a typical biomolecule ( ∼ A ). From eq.(8) itfollows that the acceleration on the dipole length is given by (cid:18) m d r i d t (cid:19) F R = (cid:18) m d d t ( r i, + − r i, − ) (cid:19) F R · b r i = ( F visc ,i + − F visc ,i − ) · b r i = − γ i dr i d t . (9)On the other hand the stochastic forces are due to the collision of water molecules and freelymoving ions on the biomolecules and they correspond to the realization of a thermal bathat temperature T . In particular these forces, acting directly on the charge barycentres ofeach biomolecules, can be described according to the following expression F stoch ,i ± = Ξ i ξ i, ± ( t ) Ξ = p k B T γ i , (10)where ξ i ( t ) represents white noise whose characteristics along each Cartesian component α, β = x, y, z are given by (cid:10) ( ξ ( t ) i, ± ) α (cid:11) t = 0 D ( ξ ( t ) i, ± ) α ( ξ ( t ′ ) j, ± ) β E t = δ ( t − t ′ ) δ ij δ αβ ( δ ++ + δ −− − δ + − − δ − + )(11)The minus sign in the correlation term is due to the constrains we impose for the noise ξ i, + ( t ) = − ξ i − ( t ) , (12)constrains that allows to easily calculate the stochastic force along the dipole direction (cid:18) m d r i d t (cid:19) ST = (cid:0) ξ i, + ( t ) − ξ i, − ( t ) (cid:1) · b r i = 2 ξ i, + ( t ) · b r i = 2Ξ i ξ i ( t ) . (13) B. Exteral forcing to produce out-of-thermal equilibrium conditions
In [5] it has been shown that long-range interactions among biomolecules can be exertedif the system of oscillating dipoles is maintained in out-of-thermal equilibrium. To achievethis goal a forcing term F NE,i ( t ) has been included in the equations of motion for the8lectric dipoles in order to ensure an external injection of energy. The explicit form ofthe force F NE,i ( t ) depends on the specific process that is chosen to inject energy into thesystem. In particular, a possible mechanism that has been used recently in THz spectroscopyexperiments to detect collective giant oscillations in biomolecules, is the injection of energyin vibrational modes through the vibrational decay of the excited fluorochromes attached toeach biomolecules [17]. This process can be represented choosing the following explicit formfor the forcing term F NE,i ( t ) = A NE,i ω pul f pul ( t ; ω pul , φ i ) (14)where f pul is a pulse-like function of the form f pul ( t ; ω pul , φ i ) = 12 π n pul X i =1 a n [1 + cos ( ω pul t + φ i )] n pul a n = 2 n ( n !) (2 n )! . (15)The coefficients in the former equation have been chosen such that the integral of the function f pul over a period T pul = 2 πω − respects the following normalization Z πω pul f pul ( t ; ω pul , φ i )d t = 1 ω pul . (16)With this choice it is clear that A NE,i corresponds to the momentum transferred by thefluorochrome to the protein in a time 2 πω − . The energy losses in vibrational decay canbe estimated to be of the order ∆ E pul = h ∆ ν fluor where ∆ ν fluor is the difference amongfrequencies of absorbed and emitted light by the flourochrome and h is the Planck constant;consequently, if m fluor is the mass of the fluorochrome, the momentum transferred to thebiomolecule can be approximated by∆( m i ˙ r i ) ≈ p h ∆ νm fluor = A NE,i = A NE . (17) C. Equation of motion for the system of oscillating interacting dipoles
The equations of motion that describe the dynamics of the system with mutually oscil-lating dipoles are m d r i d t = − mω ( r i − r i ) − m ω Λ ( r i − r i ) + X j = i mω ij ζ ij r j + − γ d r i d t + 2Ξ ξ ( t ) + F NE,i ( t ) ∀ i = 1 , ..., N (18)9here all the biomolecules are assumed to be identical so that they all have the samecharacteristic frequencies ω i = ω and Λ i = Λ.In order to simplify the discussion we introduce the following scales m = µ e m, t = τω , r i = λx i (19)that substituted in eq.(18) yield tod x i d τ = − ( x i − x i ) − ( x i − x i ) e Λ − Ω frict ,i d x i d τ + N X j = i Ω ij ζ ij x j + e Ψ i e ξ i ( t )++ Ω pul A NE f pul ( τ ; Ω pul , φ i ) ∀ i = 1 , ..., N (20)where˜Λ = Λ λ , Ω ij = ω ij ω , e R i = R i λ , e η W = η W λµω , Ω frict ,i = 6 π e R i e η W ˜ m i , E bath = k B Tµλ ω , e ξ i = ω − / ξ i , ˜Ψ i = π E bath e R i e η W e m i ! / , Ω pul = ω pul ω , E pul = h ∆ ν fluorr µω λ , e m fluor = m fluor µ , A NE = (cid:18) E pul e m fluor e m i (cid:19) / . (21) D. Choice of numerical parameters in eq. (20)
The numerical values of parameters that appear in eq. (20) have been estimated for arealistic biological system. In particular the characteristic fundamental scales for the systemhave been fixed as following: i) the typical mass scale of a biomolecule µ = 1 . × − Kg =1KDa; ii) the characteristic length scale of a biomolecule λ = 10 − m ; iii) the characteristicfrequency of the collective oscillations for a biomolecule ω = 10 s − . Moreover, sincewe are interested in observing self-emergent synchronization, we consider a set of identicalmolecules in order to maximise the probability of observing it; therefore we assume e R i = 1, e m i = 10 and x i ≃ i = 1 , . . . N according to characteristic dimension and masses ofbiomolecules.The parameter that fixes the characteristic length for the emergence of non linear phe-nomenon has been settled to be e Λ ≃ .
85. The temperature of the system has been settledat T = 300 K and consequently for our choices E bath = 2 . × − , while water viscosity is η W ≃ . × − Pa · s and e η W = 0 .
56 yielding to Ω frict ,i = Ω frict = 1 .
05. With our choice offree parameters of the system, the strength of thermal noise results e Ψ ≃ . × − .10he frequencies associated to the electrodynamic interactions Ω ij can be expressed interms of adimensionalized units Ω ij = 1 ω e πǫ W S µλ Z e m e R ij (22)where e R ij is the mutual distance among the centers of the dipoles expressed in unit of λ and e m is the mass of a molecule expressed in adimensionalized units. In the performedsimulations the position of each dipole representing a biomolecule is assigned in a cube boxof unitary side, i.e. the components of the vector position of the center of each dipole havecoordinates e R i = ( x i , y i , z i ), with x i , y i , z i ∈ [0 , N / h e d i ], where N is the total number ofdipoles and h e d i is the average intermolecular distance in λ units. As a reference case in oursimulations the parameters have been chosen to be e m = 10, Z i = 1000, while the averageintermolecular distance h e d i = λ h e d i = 1 . × ˚ A = 1 . × − m . The reason for choosing sucha large value of Z is justified under the hypothesis that the surrounding water moleculesparticipate to the effective dipole of each biomolecule and enhance it. Therefore for theconsidered choice of parameters Ω ij ∼ . × − . Finally, in order to consider differentcases with stronger interactions (corresponding to shorter average intermolecular distances,for instance) the coupling term is multiplied by a factor K > E pul can be estimated assuming that the energy injection on eachbiomolecule is due to the vibrational decay of a fluorescent dye. It is realistic [17] to considera difference between the absorbed and emitted frequency of the order of ∆ ν fluor ≃ × s − and e m fluor ≃ . A NE ≃ . × − . The characteristic frequency for the energytransfer Ω pul is one of the most delicate parameters to be settled. As this term in principleaccounts for the continuous injection of energy into the system, but the release must be donewithout perturbing too much the oscillating behavior, we can assume that Ω i ≫ Ω pul ≃ − . IV. NUMERICAL RESULTS
The reported analyses have been done using a single system size (N=50) and random ini-tial conditions both for positions and velocities. However, similar results have been obtainedfor N=100, 200 (not shown). The collective evolution of the population and in particular11he level of coherence is usually characterized in terms of the macroscopic field ρ ( t ) = r ( t ) e i Φ( t ) = 1 N N X j =1 e iθ j ( t ) , (23)where the modulus r is an order parameter for the synchronization transition being one( O ( N − / )) for synchronous (asynchronous) states, while Φ is the phase of the macroscopicindicator [18]. However, in our case, the molecules are pivoted to the center of mass and can-not rotate: the effective degree of freedom of these objects consists in an elongation/shrinkagealong the direction identified by the mutual distance between the two centers of charges.Therefore it is not possible to describe the movement of the dipole in terms of an oscillatorrotating along the unit-circle via the identification of a time-dependent phase. The solutionthat we have adopted is to calculate the phase of the single molecule by using the inversionformulas sin θ i = x i − x i p v i + ( x i − x i ) , cos θ i = v i p v i + ( x i − x i ) (24)to associate a phase θ i ∈ [ − π, π ] according to θ i = arcsin(sin θ i ) if cos θ i ≥ π − arcsin(sin θ i ) if sin θ i > ∧ cos θ i < − π − arcsin(sin θ i ) if sin θ i < ∧ cos θ i < . (25)However the calculation of the order parameter r does not lead to the identification ofemergent (phase) synchronization in the system; in particular r does not show any depen-dence on the coupling constant (see Fig. 1(a)), as we would expect when the moleculesare interacting with increasing strength. In addition to this, the emergence of a collectivebehavior is not identifiable in a straightforward manner neither looking at the order param-eter usually employed to identify the emergence of 2-clusters ( r ( t ) = | N N X j =1 e i θ j ( t ) | ), norat the distribution of positions and velocities of the molecules (see Fig. 1, panels (b)-(m)).Looking at the phase space ( x, v ) it does not emerge a clear separation in synchronizedclusters among the dipoles and also the probability distributions of positions and velocitiesare simply Boltzmann-distributed, as we expect from a set of indepent oscillators subjectedto a single asymmetric well potential in absence of coupling. Only for very strong coupling(K=50) we can observe the emergence of a secondary small cluster in the phase space ( x, v )(see Fig. 1(i)) that leads to a modification of the probability distribution of the positions,12hat is no more simply Boltzmann-distributed, and to an increasing of the average value of r ( t ) (see Fig. 1 panels (l) and (b) respectively). r Time r K=0K=5K=50 (a)(b) v P(x) -4 0 400.250.5
P(v) x -404 0 5 10 x v Figure 1: Synchronization properties of the system. Order parameters r (a), r (b) as a functionof time for different coupling constants. Panels (c),(f), (i): snapshots of the velocities of the singledipoles as a function of their positions for K=0 (c), K=5 (f), K=50 (i). Panels (d), (g), (l):probability distribution of the positions of the dipoles for different coupling constants. The panelsrefer to K=0 (d), K=5 (g), K=50 (l). Panels (e), (h), (m): probability distribution of the velocitiesof the dipoles for different coupling constants. The panels refer to K=0 (e), K=5 (h), K=50 (m).The parameters values used for these simulations are: Ω i = 1, x i = 5, Ω frict,i = 0 .
105 (for every i = 1 , . . . , N ), Ω pul = 0 . A NE = 1 . Therefore, in order to investigate the emergence of a collective behavior due to the inter-actions among the molecules we consider the variable P ( t ) = vuut N X i =1 { [( x i ( t ) − x i ) sin β i cos φ i ] + [( x i ( t ) − x i ) sin β i sin φ i ] + [( x i ( t ) − x i ) cos β i ] } (26)which represents the ensemble average of the projection of the dipole position in the cartesiancoordinates system X, Y, Z . The biomolecule in our model is identified via the intermolecularmutual distance between the two centers of charges measured along the radial x directionand we need to express this variable in cartesian coordinates. In other words, each termunder the square root represents the component of the dipole position along one of thedirections X, Y, Z , thanks to the respective projection angle β i of each molecule’s radius to13he Z-axis and φ i of the projection of x i in the XY plane to the X-axis. These angles aregenerated together with the initial conditions and do not vary in time.Due to the fact that the system is not deterministic and a white noise source is presentinto the differential equations, we have developed a method similar to the second-orderRunge-Kutta one for solving numerically ordinary differential equations. In particular wehave implemented the Heun method [19] in the Runge-Kutta algorithm as suggested in [20],and we have used an integration time step 0.002 to perform the simulations. In addition tothis, in order to compare the results for different coupling constant values and for differentstrengths of the thermal noise, we implemented a low-pass filter to analyse the power spectra.This filter relies on the differentiation properties of the Fourier transform; in particular,since the Fourier transform of a generic function f is related to the Fourier transform ofits derivative via the relationship F (cid:20) ∂f ( x ) ∂x j (cid:21) = 2 πiν j b f ( ν ), it is possible to filter the low-frequency components of the spectrum just using the Fourier transform of the derivative.Therefore we calculated the power spectrum of dP/d x to investigate the role played bythe interactions among the dipoles to enhance a collective motion. While in absence ofinteractions (K=0), the system shows a single pronounced peak at frequency ≈ . ± . K > ≈ . ± . K >
20 is related to the fact that power spectra become richer and richer for higher couplingand secondary peaks arise. One of these secondary peaks (the main one) emerging at biggercoupling constant is also reported in Fig. 3 (panels (a), (b)), and it is termed “Third Peak”.Finally, if we analyse in more details the behavior of the first peak, related to the emergentcollective motion, as a function of the coupling contant, it is possible to identify two differentscales, once the figure is plotted in log-log scale (Fig. 3(c)). In particular, the different scalespresent for low coupling constant (
K <
5) and for sufficiently strong coupling (
K > ν ν (a) st Peak2 nd Peak (b)(c) (d) P o w e r S p ec t r u m ν ν (e) rd Peak3 rd Peak (f)(g) (h) P o w e r S p ec t r u m Figure 2: Investigation of the emergence of a collective behavior as a characteristic peak in thepower spectrum. Panels (a)-(h): Power spectrum of dP/d x for different values of the couplingconstant K and for thermal noise strength ˜Ψ i = 0 .
46. The black curve represents, in each panel, thepower spectrum of the system without coupling (K=0). The other curves shown are, respectively,for K = 1 (a); K = 2 (b); K = 5 (c), K = 10 (d); K = 21 (e); K = 31 (f); K = 41 (g); K = 50(h). Other parameters as in Fig. 1.
20 40 100001e+05 P ea k H e i gh t K F r e qu e n c y V a l u e (a)(b) K P ea k H e i gh t y = A x b (c) Figure 3: Dependence of the system’s characteristic frequencies on the coupling constant. Panels(a), (b): Peak height (a) and frequency value (b) of the first three main peaks that characterize thedynamics of the system. Panel (c): Fitting of the dependence of the peak height on the couplingconstant. Fitting values are A = 6188 , ± . b = 0 . ± .
03. For all the panels the black dottedcurve represents the first peak, the red diamonds curve represents the second peak and the squaregreen curve represents the third peak. Parameters as in Fig. 1.
If we now investigate the response of the system under the effect of the thermal noisestrength, we obtain a stochastic resonance effect [21]: the signal at low frequency ( ≈ . ± .
09) can be boosted by adding white noise to the signal, which contains a wide spectrumof frequencies. The frequencies in the white noise spectrum corresponding to the originalsignal’s frequencies resonate with each other, thus amplifying the original signal (i.e. thesignal at low frequency) while not amplifying the rest of the white noise. Furthermore thesignal-to-noise ratio is increased, while the added white noise is filtered out thanks to theband-pass filter that we have implemented calculating the power spectrum of dP/d x . Inparticular the low frequency peak, that corresponds in our case to the collective motion,is more visible for thermal noise strength ˜Ψ = 0 .
03, to which corresponds a maximum inthe peak high (see Fig.4 panels (a),(b)). This peak is depressed for higher temperature andless likely to be revealed. On the other hand the peak at high frequency ( ≈ . ± . ν P o w e r S p ec t r u m Ψ =0.02 Ψ =0.03 Ψ =0.06 Ψ =0.2 Ψ =0.4 Ψ =0.6 Ψ =0.8 st Peak 2 nd Peak (a) P ea k H e i gh t Ψ F r e qu e n c y V a l u e First PeakSecond Peak(b)(c)
Figure 4: Response of the system under the effect of the thermal noise strength. Panel (a): Powerspectrum of dP/d x for different values of the thermal noise strength and for coupling constantK=5. Panels (b), (c): Peak height (b) and frequency value (c) of the first two main peaks thatcharacterize the dynamics of the system. Parameters as in Fig. 1. The values of the differentthermal noise strengths reported in the caption of panel (a) and the axix label in panel (c) mustbe intended as ˜Ψ: the ˜ has been suppressed in the figure for the sake of simplicity. V. DISCUSSION
Let us now comment about the physical meaning, and about the prospective relevance,of the results described in the previous Sections. The present work was motivated by theneed of finding an experimental strategy complementary to the diffusion based one alreadysuggested in [6–8] - to detect an intermolecular long range electrodynamic interactions, ifany. The background scientific framework is the following. By pumping energy in thebiomolecules of a watery solution, that is by keeping these molecules warmer than thesolvent (out-of-thermal equilibrium), when the input energy rate exceeds a threshold value,then all, or almost all, the excess energy (that is, energy input minus energy losses dueto dissipation) is channeled into the vibrational mode of the lowest frequency. In otherwords, the shape of the entire molecule is periodically deformed resulting in a “breathing”movement [17]. In so doing the biomolecules behave as microscopic antennas that absorb theelectromagnetic radiation tuned at their “breathing” (collective) oscillation frequency. But17ntennas at the same time absorb and re-emit electromagnetic radiation, thus, accordingto a theoretical prediction, these antennas (biomolecules) can attractively interact at alarge distance through their oscillating near-fields, and through the emitted electromagneticradiation, if these oscillations are resonant, that is, take place at the same frequency [17]. Thestill open question is whether these electrodynamic interactions can be strong enough to beexperimentally detectable, and ultimately relevant to biology. In our schematic modeling of awatery solution of biomolecules, we have then assumed that the above mentioned collectivevibrations of each individual molecule are present so that they interact with a potentialfalling as 1 /r with the intermolecular distance r . By adopting physically reasonable valuesfor the molecular parameters entering the equations of motion of the molecular dipoles, wehave numerically investigated the effect of varying the mutual dipole-dipole electrodynamicinteractions by changing the parameter K . The novel phenomenon observed and reported inthe preceding Section is the appearance of a spectral signature of an intermolecular collectivephenomenon which manifests itself with an increasing evidence when the parameter K israised. Physically, this suggests that the stepping up of supposedly activated electrodynamicintermolecular interactions could be, in principle, spectroscopically detected by varying theconcentration of the soluted biomolecules. This latter fact, of course, entails the variation ofthe average interparticle distance h d i according to the relation h d i = C − / , where C is theconcentration of the solution. And varying C would be a practical way of experimentallyvarying the parameter K . In order to detect the emergence of a collective behavior due tothe interactions among the molecules we considered the variable P ( t ) in Eq.(26) representingthe ensemble average of the projection of the dipole positions in the cartesian coordinatessystem. Strictly speaking, this is not yet directly spectroscopically measurable, but it istightly related with the overall dipole moment of the solution that could be more directlyspectroscopically accessible. However, this is a technical detail which will be more thoroughlyaddressed while designing a specific experiment. For the moment being, the results reportedin the present work outline a very promising strategy - complementary to the diffusionbased one - to reach a proof of concept, or a refutation, of the possible relevance of longrange electrodynamic intermolecular interactions to our understanding of the biochemicalmachinery at work in living matter. 18 ppendix A: Discussion about characteristic time scales on the system The characteristic frequency for giant dipole oscillations has been conjectured to liein a range between 0 . −
10 THz so that the characteristic time for the oscillations is τ osc ∼ ω − ≃ − − − s . Recent experiments seem to provide a first evidence of theexistence of collective biomolecule oscillations in this range of frequency in out-of-thermalequilibrium conditions. The characteristic time scale associated with translational diffusionof biomolecules can be estimated by τ trs ≈ δR D trs (A1)where δR is the tolerance in defining center of mass position of two biomolecules and D trs isthe self-diffusion coefficient of a biomolecule. We are interested in studing collective phenom-ena emerging due to long-range interactions in “diluted” system, meaning that the averageintermolecular distance h R i ≈ − m is much larger then the characteristic molecular lineardimension scale λ bio & − m of biomolecules: this allows to consider δR ≈ λ bio . UsingEinstein’s formula for Brownian self-diffusion coefficient D trs = k B T / (6 πη W λ bio ) in eq. (A1)we obtain τ trs ≃ πλ bio k B T & × − s ≈ τ osc . (A2)This makes plausible the hypothesis that the center of mass of each biomolecule can beconsidered as a parameter and not a dynamical variable. Analogously, the characteristictime for biomolecules rotational diffusion has been estimated using τ rot ≈ D − rot = (cid:18) k Bol T πη W λ bio (cid:19) − & × − s ≈ τ osc . (A3)It follows that also diffusive rotation can be neglected on time scales characteristics for giantdipole oscillations and the orientation of dipole can be assumed to be initially fixed.19 cknowledgments The authors acknowledge the financial support of the Future and Emerging Technologies(FET) Program within the Seventh Framework Program (FP7) for Research of the Euro-pean Commission, under the FET-Proactive TOPDRIM Grant No. FP7-ICT-318121. S.O. thanks Stefano Lepri for useful discussions and suggestions and she acknowledges theDeutsche Forschungsgemeinschaft via Project A1 in the framework of SFB 910.
Author Contributions
S.O. performed the numerical simulations. M. G. elaborated the dynamical model. M.G. and S. O. prepared the manuscript. All the authors developed the theoretical methodsand reviewed the manuscript. As team leader, M.P. supervised all the aspects of the work.
Additional Information
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