Long-range resonant interactions in biological systems

0.15 MB·English

by  Jordane Preto

Long-range resonant interactions in biological systems
Jordane Preto∗ and Marco Pettini†
Aix-Marseille University, Campus de Luminy, case 907, CNRS Centre
de Physique Théorique, UMR 7223, 13288 Marseille Cedex 09, France
(Dated: January 26, 2012)
The issue of retarded long-range resonant interactions between two molecules with oscillating
dipole moments is reinvestigated within the framework of classical electrodynamics. By taking
advantageofatheoremincomplexanalysis,wepresentasimplemethodtoworkoutthefrequencies
of the normal modes, which are then used to estimate the interaction potential. The main results
2 thus found are in perfect agreement with several outcomes obtained from quantum computations.
1 Moreover, when applied to a biophysical context, our findings shed new light on Fröhlich’s theory
0 of selective long-range interactions between biomolecules. In particular, at variance with a long-
2 standingbelief,weshowthatsizableresonantlong-rangeinteractionsmayexistonlyiftheinteracting
system is out of thermal equilibrium.
n
a
PACSnumbers: 34.20.Gj,03.50.De,12.20.-m
J

5
2
Introduction—Livingorganismsarewell-knowntosup- interactionisrelevant. Forlargeintermoleculardistances

portmanybiochemicalprocesseswhich,besidesbeingex- r,thepotentialwasgivenatthermalequilibriumas[2,8]
]
h tremely specific, seem to follow a precise time schedule.
p In this context, the motion of the molecules can hardly U(r) 1 1 1 +O 1 (1)
- ∝ r3 ε(ω ) − ε(ω ) r6
o be described on the basis of thermal fluctuations only. (cid:18) + − (cid:19) (cid:18) (cid:19)
i Most studied concepts to this effect refer to electromag- where ε(ω ) is the permittivity of the medium at the
b ±
s. neticshort-range interactions[1],whichleaveslong-range normal frequencies ω± of the interacting system.
c interactions poorly investigated thus far. This is essen- Though Fröhlich’s theory has been well received and
i tially becauseinbiologicalsystemsfreeionsofcellwater developedforalongtimebymanyauthors[9],athorough
s
y tend to screen any electrostatic potential at a distance derivationfromfirstprinciplesofhis mainresultsonres-
h that usually does not exceed a few angstroms. However, onant interactions, as for instance the potential given in
p this screeningprovesgenerallyinefficientfor interactions Eq. (1), is still lacking. In anticipation of advanced ex-
[
involving oscillating electric fields. To this respect, in perimentalinvestigations[7],thepresentpaperisthusin-

1 1972, Fröhlich emphasized [2] that two molecular sys- tendedtobe amorecomprehensiveandanalyticaccount
v tems which would exhibit large oscillating dipole mo- of this theory and to put the mentioned interactions at
7 mentscouldinteractviaalong-range potential,provided theirrightplacewithintheframeworkofelectrodynamic
8
that the oscillation frequencies were roughly similar. forces. Tothatpurpose,wecalculatethenormalfrequen-
1
5 Applied to biological systems, it was suggested that cies of a system of two interacting oscillating dipoles in
1. such resonant forces would have a profound influence on an exact way, including field retardation. This is made
0 the displacement of specific biomolecular entities, and possible by means of a theorem in complex analysis, the
2 thus on the initiation of a particular cascade of chemical Lagrange inversion theorem, that is detailed below. We
1 events. Later on, possible examples of such interactions then show how these results can be used to estimate the
:
v were reported – at the cellular level – between erythro- interaction energy between two atoms on one hand, and
Xi cytes [3]. In parallel, it was shown that the membranes two oscillating dipoles on the other hand. In the former
of these cells have the ability to oscillate at frequencies case, a connection with known QED results is provided
r
and in the latter one the connection with Fröhlich’s the-
a of about 40 GHz [4]. Moreover, since then [5], numer-
ous evidences of electromagnetic long-range interactions oryismade. Inparticular,weprovethattheformforthe
potential(1)ismisleading,suggestingthatlong-range in-
between cells have been identified including yeast cells
teractions exist only if the system is out of equilibrium.
[6]. Very recently, supported by experimental evidences
of collective oscillations in biomacromolecules (see Refs. Theory—Let us consider two molecules A and B with
in [7]), possible experimental tests were proposed to as- oscillating dipole moments µ and µ . The equations
A B
sess whether such interactions could be relevant at the of motion can be given in general terms as
biomolecular level [7].
µ¨ +γ µ˙ +ω2µ =ζ E (r ,t)+f (µ ,t)
A A A A A A B A A A
Coming to Fröhlich’s predictions, an intriguing result
of his computations is the possibility of observing long- ( µ¨ +γ µ˙ +ω2µ =ζ E (r ,t)+f (µ ,t),
B B B B B B A B B B
rangeresonantinteractionsevenifthesystemofoscillat- (2)
ing dipoles is at thermal equilibrium. This would occur where ω and γ are the harmonic frequencies and
A,B A,B
when the retardation of the electric field mediating the dampingcoefficientsofthedipoles. Here,theinteraction
2
takesplacethroughtheelectricfieldE (r,t)generated mind that in computing normal frequencies, one drops
A,B
byeachmolecule,locatedatr =r ,whilethecoupling dissipationeffects,onlytherealpartsdenotedbyχ′ will
B,A ii
constants are given by ζ = Q2/m where Q and m be thus considered in what follows.
A A A A A
are the effective charge and mass involved in the dipole SubstitutingintoEq. (3)theexpectedharmonicforms
A, and similarly for ζ . Finally, f and f are func- for µ and Eq. (4), one obtains a system of linear
B A B A,B
tions accounting for possible anharmonic contributions, algebraic equations. The existence of the solutions is
thermal noise as well as possible external excitations. then ensured by the vanishing of the determinant (ω2
ω2)(ω2 ω2) ζ ζ (χ′ (r,ω))2 =0. This indicatesAtw−o
Normalmodes analysis—Apossiblewayofestimating B− − A B ii
possible values for ω > 0 for each i, namely, ω and
the interaction energy of the system (carried out in the i,+
ω verifying
last part of this letter) is to use the normal frequencies i,−
of the associatedharmonic conservative system given by
ω2 +ω2 ω2 ω2 2
ω2 = A B A− B +ζ ζ (χ′ (r,ω ))2.
µ¨A+ωA2µA =ζAEB(rA,t) i,± 2 ±s(cid:18) 2 (cid:19) A B ii i,±
(3) (6)
(µ¨ +ω2µ =ζ E (r ,t).
B B B B A B Atthispoint, itmustbe emphasizedthateachnormal
frequency is strongly dependent on the proximity of the
The normal frequencies are defined as the frequencies
frequencies of the dipoles, as detailed below.
ω such that µ (t) = µ eiωt are solutions of (3). In
A,B A,B
this context, the electric field generated by a harmonic Resonant case—When ω ω ω , equation (6)
A 0 B
≃ ≡
dipole is well-known from classical electrodynamics [11] is readily simplified and at large separations, one can
and one has the following matrix relation expand ω around ω as
i,± 0
EB(rA,t)=χ(r,ω)µBeiωt, (4) ωi,± =ω0+φi,±(r,ωi,±),
√ζ ζ χ′ ζ ζ (χ′ )2 (7)
rweistehntas stihmeilagreneexrpalriezsesdionsufsocrepEtibAi(lritBy,tm).atrHixere(r,eχtarrdeepd- with φi,± ≃± A2ωB0 ii − A B8ω03ii .
Green function) of the electric field – note that the spa-
The implicitness of the equation can then be solved
tialdependenceofχin(4)reducessimplytothedistance
by making use of the Lagrange inversion theorem [13] :
r rA rB because the intermediate medium is sup- let C be a contour in the complex plane surrounding a
≡ | − |
posed to be homogeneous and isotropic. In addition, to point a, and let φ a function analytic inside and on C.
model an aqueous biologicalenvironmentfilled with free If t C is such that the inequality tφ(z) < z a is
moving ions, in what follows we will consider it as a dis- satis∈fiedforallz ontheperimeterofC| ,then| the|eq−uat|ion
persive dielectric medium of complex permittivity ε(ω). ω =a+tφ(ω) in ω has oneroot inside C andany further
At this stage, it should be remarked that the general- function g analytic inside and on C can be expanded as
ized susceptibility of the electric field is analytic every-
a power series in t by the formula
where on the complex plane except on an uncountable
subset where it has a discontinuity. In a homogeneous ∞ tn dn−1 dg
dielectric,thissubsetisgivenby ω C Im(ω ε(ω))= g(ω)=g(a)+ n!dan−1 da{φ(a)}n . (8)
0 while χ reads as [11] { ∈ | nX=1 (cid:20) (cid:21)
} p
Identifying tφ(ω) with φ (r,ω) of equation (7), one
i,±
e±iω√ε(ω)r/c must carefully choose the contour C out of the domain
χ (r,ω)=χ (r,ω)=
11 22 − ε(ω)r3 · ofdiscontinuityofχ′ . Onthe otherhand, sinceeachχ′
ii ii
(and so φ ) is a sum of inverse power laws of r, we can
(5a) i
iω ε(ω)r ω2ε(ω)r2 assume that for large enough separation the inequality
1 ,
∓ pc − c2 ! φthi,a±t(rC,zm)a<y |bze−stωill0|chhoosledns ifnorthzecdloosmeaeinnooufghantaolyωti0cistyo
2e±iω√ε(ω)r/c iω ε(ω)r of χ′ii. Letting g as the identity function and n≤2, one
χ (r,ω)= 1 , (5b) finds after some algebra
33 ε(ω)r3 ∓ pc !
√ζ ζ χ′ (r,ω )
ω (r) ω A B ii 0 +
and χij(r,ω)=0 when i6=j; the ± sign is attributed to i,± ≃ 0± 2ω0
positive or negative values of Im(ω ε(ω)), respectively. (9)
Let us also specify that the diagonapl form of χ is due to ζAζB d χ′ii(r,ω) 2 ,
the choice to set the z axis along r. Finally, it should be 2 dω "(cid:18) ω+ω0 (cid:19) #ω=ω0
stressed that for real values of ω, each χ is a complex
ii
numberwhoseimaginarypartessentiallyaccountsforthe to second order in χ′ , i = 1,2,3. The normal frequen-
ii
dissipation due to the field propagation [12]. Bearing in cies thus found are equal to the resonance frequency ω
0
3
plus a shift due to the (effective) interaction. The first Asacomplementtotheseresults,twoimportantpoints
contribution of each frequency shift is proportional to should be mentioned. First, the above classical compu-
the real part of the susceptibility matrix elements. As tation of normal frequencies does not allow to deduce
detailed below, this term is responsible of the long-range the energy contribution due to virtual photons that ac-
natureoftheresonantinteractionenergyas,accordingto counts for the interaction between the ground states of
(5), each χ-element reads as a polynomial in 1/rα with the atoms. This is confirmed by the fact that this con-
α 3 (the dimension of physical space) : in the limit tribution which is always present whether the system is
r ≤ c/ω (near zone limit), it goes as 1/r3 with the excited or not, was shown to arise purely from vacuum
0
≪ ±
intermolecular distance while it oscillates at longer dis- fluctuations[16]. Secondly, ifwereferto recentquantum
tance (intermediate andfar zone limits) with a 1/r2 or calculations [17], the real photon contribution of the in-
±
even 1/r envelope. teractionenergyinthecaseω =ω wouldbenoticeably
A B
± 6
different from the one suggested in the present paper as
Off-resonance case—On the contrary, when ω ω
A B
≫ wellas in the references mentioned in [15]. This resultis
(or similarly when ω ω ), one can use Eq. (6)
A ≪ B distinguishedbythepresenceof χ 2 insteadof(χ′ )2 in
to approximate ωi,± around ωA,B as ωi,± ωA,B | ii| ii
≃ ± Eq. (10),thatleadstoaspatiallymonotonic–insteadof
ζ ζ (χ′ (r,ω ))2 / 2ω (ω2 ω2) to second order
inAχB′ . Aiigain,i,f±orlarger,Ath,BeLaAgr−angBeinversion theorem oscillating – potential. Nevertheless, deeeper theoretical
ii (cid:2) (cid:3) investigations[18] revealedthat the monotonic potential
can be applied so that tφ of the theorem corresponds to
holds true on time scales much larger than the sponta-
the last term of the equation. To compare the results
neousdecay time ofthe excitedatom(s). Inthe opposite
withthoseoftheresonantcase,itisenoughtoapplyEq.
case,theoscillatingpotentialisobtained,thusrecovering
(8) up to n=1. Thus, the normal frequencies are
the above mentioned quantum-classical correspondence
ζ ζ (χ′ (r,ω ))2 [19].
ω (r) ω A B ii A,B . (10)
i,± ≃ A,B± 2ωA,B(ωA2 −ωB2) Interaction energy (real dipoles)—Contrarily to the
caseoftwoatoms,theinteractionbetweenrealoscillating
At variance with the resonant case, the frequency shifts
dipoles, i.e., molecules whose oscillatingdipole moments
associatedwiththeunperturbedfrequencyω arenow
A,B
proportional to (χ′ )2 at first order. When r c/ω , are not due to electron motions but rather to confor-
ii ≪ A,B mational oscillations, have not been given much atten-
this leads to a short-range 1/r6 contribution while at
very long distances it oscilla±tes decaying as 1/r2. tion in the literature. As mentioned in the Introduction,
± a remarkable exception is given by Fröhlich who tack-
Interactionenergy(atoms)—Nowthatnormalfrequen- led this issue forty years ago in a biophysical context.
cies have been worked out, it is interesting to remark At that time, he emphasized in particular that resonant
thattheenergyshiftsU±(r)=~ i(ωi,±(r)−ωA,B)are long-rangeinteractionsmayoccurbetweentwoharmonic
identicalwiththereal photoncontributionsthatappears dipoles even if none of the normal modes is excited be-
P
intheinteractionenergyoftwotwo-levelatomsinanex- yond thermal equilibrium. However, we show that this
cited state, as it can be found within the framework of statement is actually incorrect.
QED. Since the atoms and the radiation field mediat- To clarify, let us consider the system of dipoles A and
ing the interaction can be considered as an ensemble of Binthermalequilibriumandsupposeasanexamplethat
harmonic coupled oscillators, the correspondence is not ~ω k T so that classicaleffects are dominant[20].
A,B B
so surprising if we add to this that normal modes and The int≪eraction energy is then given by the difference of
susceptibilities are the same for classical and quantum free energy of the coupled and uncoupled systems, i.e.,
forsocmillaEtqo.rs(.9)Tahruese,qwuhalentoωtAhe≃enωer0gy=shωiBft,sUd±uectoomtphuetiend- UZ((rr))=isFth(re)p−arFti(t∞ion)=fu−ncktBioTnlnof[Zt(hre)/sZy(s∞tem)],whewnhtehree
teraction between two atoms with a common transition dipoles are separated by a distance r. Using Boltzmann
frequency ω0 [14]. Alternatively, in an off-resonance sit- distribution for each normal mode, one can easily show
uation, one finds identical expressions for U± computed that the interaction energy has the following form [8]
from Eq. (10) and the energy shifts of two atoms with
distinct transition frequencies, ωA and ωB respectively ωi,+(r)ωi,−(r)
U(r)=k T ln . (11)
d(wetiathileωdAin≫veωstBigahteiroen) [t1h5r]o.uRgheatdheersreinfetreernescteesdminenatimonoerde B Xi (cid:20) ωAωB (cid:21)
above will identify ζ as ω µ˜ 2 for all i = 1,2,3,
A A| A,i| Taking ωA ω0 ωB, the result exposed by Fröhlich
where µ˜ is the transition dipole moment of atom A ≃ ≡
A,i is obtained by substituting ω (r) and ω (r) for their
th i,± i,−
along the i spatial coordinate (the same for atom B).
implicit form (7). Then, at first order, U reads as
The polarizability of each atom is then simply given by
αA,B(ω)≡ ωA2,ζBA,−Bω2. U(r)= kBT2√ωζ02AζB Xi {χ′ii(r,ωi,+)−χ′ii(r,ωi,−)}.
4
In addition, Fröhlich only considered the limit r respect to the interactions proposed by Fröhlich. This
c/ω together with the condition ε ε′ Re(ε) (no≪n last result could be of utmost relevance for a deeper un-
0
≃ ≡
absorbing medium in the considered frequency range). derstandingofthehighlyorganizedmolecularmachinery
From Eq. (5), one has in this case : in living matter, as emphasized in the Introduction.
Acknowledgments—WewarmlythankH.R.Haakh,P.
kBT√ζAζB 1 1 1 J. Morrison, S. Spagnolo and M. Vittot for useful com-
U(r) σ ,
≃ 2ω2 r3 i ε′(ω ) − ε′(ω ) ments and discussions.
0 i (cid:26) i,+ i,− (cid:27)
X
(12)
with σ = σ = 1 and σ = 2. At this stage, Fröhlich
1 2 3
claimed that the−above 1/r3 form for U may account for
the existence of long-rangeresonantinteractions in ther-
mal equilibrium provided ε′(ω ) = ε′(ω ). However, ∗ [email protected]
i,+ 6 i,− † [email protected]
as already mentioned, this form arises simply since the [1] J. N.Israelachvili, Quart. Rev.Biophys. 6, 341 (1974).
implicitness has been not resolved yet. Hence, by using [2] H. Fröhlich, Phys.Lett. A39, 153 (1972).
the Lagrange inversion theorem and by substituting, for [3] S.Rowlands,L.S.Sewchand,R.E.Lovlin,J.S.Beckand
example, g of equation (8) for 1/ε′, one get immediately E. G. Enns, Phys. Lett. A82, 436 (1981); S. Rowlands,
L. S. Sewchand, and E. G. Enns, Phys. Lett. A87, 256
that the terms in curly brackets in Eq. (12) vanish at
(1982).
first order. As a result, expansion to second order shows
[4] H. Fröhlich, Phys.Lett. A110, 480 (1985).
that U is actually proportionalto 1/r6. To compute the
[5] See the review paper : M. Cifra, J. Z. Fields and A.
completeformforU,onethenneedstocomebacktoEq. Farhadi, Prog. Biophys. Mol. Biol. 105, 223 (2011).
(11). Using the explicit form of the normal frequencies [6] H. Pohl, J. Biol. Phys. 8, 45 (1980).
derived above (9), one easily obtains [7] J. Preto, E. Floriani, I. Nardecchia, P. Ferrier and M.
Pettini, Phys.Rev.E (2012), in press; arXiv/1201.2607.
3k Tζ ζ 1 dln[ε′(ω )] [8] H.Fröhlich,Adv.Electron.ElectronPhys.53,85(1980).
B A B 0
U(r)=−ω4[ε′(ω )]2r6 1+ω0 dω , (13) [9] J.PokornýandT.M.Wu,BiophysicalAspects ofCoher-
0 0 (cid:26) 0 (cid:27) ence and Biological Order, (Springer,Berlin, 1998).
[10] H. Fröhlich, Int.J. Quantum Chem. 2, 641 (1968).
where the ability of the potential to be attractive or
[11] J. D. Jackson, Classical Electrodynamics (John Wiley &
repulsive depends on the derivativeof ε′ atω0, represen- Sons, Inc., Sixth Printing, 1967), p. 268-273.
tative of the dispersive properties of the medium. [12] L. D. Landau and E. M. Lifshitz, Statistical Physics,
In the end, it goes without saying that the long-range (Pergamon Press, NY,1980).
contributionsthatappearintheexpressionofthenormal [13] E.T.WhittakerandG.N.Watson,ACourse ofModern
Analysis, (Cambridge University Press, Fourth Edition
frequenciesatresonance,Eq. (9),canceleachothersince
1927), p.131-133.
the energy of both modes is given by Boltzmann distri-
[14] This is the well-known condition for exchange degen-
bution; this happens independently of considering retar-
eracy for which U are associated with the approxi-
dationeffects. Byinference,long-rangeresonantinterac- mate eigenfunction±s |ψ i= 1 [|eA,gBi∓|gA,eBi] re-
tionsbetweentwodipoles(biomolecules)mayoccuronly spectively, where |eA,gB±i or |√g2A,eBi refer to the states
beyondthermalequilibriumprovidedthatthe excitation where just one atom A or B is excited in the absence of
of one normal mode is statistically “favored” compared interaction. See for example: M. J. Stephen, J. Chem.
to the other. In this case, anharmonicity of the dipoles, Phys. 40, 669 (1964); A.D.McLachlan, Molecular Phys.
8, 409 (1964), in thecase of two identical atoms.
as mentioned in commenting Eqs. (2) would allow en-
[15] Here,U areassociatedwiththeapproximateeigenfunc-
ergyredistributionamongnormalmodes whereasenergy ±
tions |ψ+i = |eA,gBi or |ψ i = |gA,eBi. See for exam-
supply could be essential to maintain a high degree of −
ple: R. R. McLone and E. A. Power, Proc. Roy. Soc.
excitation despite energy losses. In a biological context A286, 573 (1965); G.-I. Kweon and N. M. Lawandy,
this energy supply may be attributed to environmental Phys.Rev.A47,4513 (1993), erratum: Phys.Rev.A49,
metabolic activity. This scenario was depicted by Fröh- 2205 (1994).
lich in general terms. He showed that a set of coupled [16] E. A. Power and T. Thirunamachandran, Phys. Rev.
A48, 4761 (1993).
normal modes can undergo a condensation phenomenon
[17] Y. Sherkunov,Phys.Rev. A72,052703 (2005).
characterizedby the emerging of the mode of lowest fre-
[18] H. R. Haakh, J. Schiefele, C. Henkel, Proceedings of
quency containing, in the average, nearly all the energy
QFExt (2011); arXiv/1111.3748.
supply [10]. Here such a process would result in an ef- [19] Despite numerous efforts, it seems that the mentioned
fectiveattractive potentialwhoseamplitudeisdependent monotonic potentialis not derivablefrom classical argu-
on the “stored” energy [9] (of course, the above given re- ments similar to those exposed here when dissipation is
marksstillapplyinthequantumcasewhen~ω k T). involved phenomenologically as it is thecase when writ-
0 B
≫ ing equations (2).
To conclude, it is worth noting that retardations effects
[20] Note that at physiological temperature, i.e., T ∼ 35°C,
atlargerbringaboutinteractionswitha1/rdependence
this condition is fulfilled for a wide range of frequencies
[last terms in Eqs. (5)], i.e., of much longer range with as kBT/~≃2.1013 s−1.