621

Laser Physics, Vol. 6, No. 4, 1996, pp. 621–653.

Original Text Copyright © 1996 by Astro, Ltd.English Translation Copyright © 1996 by

åÄàä ç‡Û͇

/Interperiodica Publishing (Russia).

REVIEWS

Mechanisms of Interaction of Electromagnetic Radiation with a Biosystem

S. A. Reshetnyak, V. A. Shcheglov, V. I. Blagodatskikh, P. P. Gariaev, and M. Yu. Maslov

Lebedev Physical Institute, Russian Academy of Sciences, Leninskii pr. 53, Moscow, 117924 Russia

e-mail: misha@sti.ruReceived November 8, 1995

Abstract

—Fine mechanisms of the interaction of weak electromagnetic radiation (EMR) with informationstructures of a living cell—nucleic acids, proteins, and membranes—are considered. Special attention is paidto the mechanisms of action of EMR of the millimeter range. Physicomathematical models of the EMR influ-ence on a biosystem are proposed. Data on nonlinear dynamics of biomacromolecules are presented, in partic-ular, the mechanism of the Davydov soliton for energy transfer in protein molecules is discussed. The processesof soliton formation in polynucleotides and of autosolitons in biosystems are considered. A crucial role ofcoherent effects in the EMR excitation of selected collective modes in biomacromolecules of proteins and DNAis revealed. Several ways of the excitation of levels of collective modes are considered, namely, a set of phasedEMR pulses (an analog of the Veksler method of autophasing), periodic phase modulation of EMR, and ampli-tude–frequency modulation. A common feature of the suggested methods is the possibility of suppressinganharmonicity and, therefore, of the excitation of high quantum levels, which facilitates crossing of a potentialbarrier and transfer of a molecule to a new conformational state. A problem of the interaction of EMR with abiopolymer chain is considered; the resonances are shown to have a collective nature. The influence of EMRon the dynamics of biopolymers with an active site (antenna model) is studied.

CONTENTS

1. FORMULATION OF THE PROBLEM 6221.1. Introduction 622

1.1.1. The application of energy properties of coherent waves in biology and medicine 6221.1.2. Application of coherent resonance radiation of low power in biology and medicine 623

1.2. The influence of low-intensity coherent radiation of millimeter range on biological objects 6251.2.1. Some general considerations 6251.2.2. The main regularities of the action of low-intensity electromagnetic radiation of millimeter range on biological objects 626

1.3. Information aspects associated with the action of resonance electromagnetic radiation of low intensity on bioobjects 6271.3.1. Information aspects associated with the action of radiation on a living organism 6271.3.2. Regulation of functioning in a living organism and generation of coherent waves by its cells 6281.3.3. Mechanisms of generation of coherent waves by cells 6291.3.4. The interrelation of action of coherent low-intensity radiation of EHF, IR, optical, and UV ranges on living organisms 629

1.4. Possible molecular mechanisms of interaction of radiation of the millimeter range with protein and nucleic acid (DNA) macromolecules 6311.4.1. Theoretical models of vibrations in proteins 6311.4.2. Theoretical models of DNA vibrations 6311.4.3. Experimental results 6321.4.4. Some molecular mechanisms of interaction of EHF EMR with protein and DNA macromolecules 632

1.5. Solitons in biomacromolecules 6331.5.1. Introduction 6331.5.2. Davydov soliton mechanism of energy transfer in protein molecules 6341.5.3. Solitons in polynucleotides 6351.5.4. Autosolitons in biological systems 637

2. THE ROLE OF COLLECTIVE AND COHERENT EFFECTS 6382.1. The Role of coherent effects in the EMR excitation of induced collective modes of proteins and DNA 638

2.1.1. Approximation of harmonic oscillator 6382.1.2. Excitation of an anharmonic oscillator by a series of phased pulses 6392.1.3. Periodic phase modulation as a method of cascade excitation 6402.1.4. An approach based on equations of balance type 6422.1.5. Coherent effect of conversion of distribution over the levels of an anharmonic oscillator (parametric excitation) 6432.1.6. Conclusion 645

2.2. Interaction of EMR with biopolymer macromolecules 6452.2.1. Introduction 6452.2.2. Biopolymer chain in a resonance field 646

2.3. Interaction of EMR with biopolymer macromolecules and its influence on the dynamics of the active site (antenna model) 6472.3.1. Introduction 6472.3.2. Antenna model 648

REFERENCES 651

622

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1. FORMULATION OF THE PROBLEM

1.1. Introduction

At present, a tremendous amount of experimentaldata on the action of coherent electromagnetic radiation(EMR) on biological objects of various levels of orga-nization—from macromolecules to a living organ-ism—has been accumulated. Construction of high-power sources of EMR, the possibility of concentration(focusing) of wave beams, synchronization of oscilla-tions of many sources, and, finally, bulk data transferthrough a single channel by means of these waves arethe main advantages that determined successful appli-cation of coherent waves in different areas includingmedicine and biology. Nowadays, sources of coherentEMR of radio (EHF), optical, and UV ranges areapplied in several areas of biology and medicine.

One may consider that the mechanisms of EMRaction on biological objects already formed one of themost interesting interdisciplinary areas of electromag-netobiology.

For a long time, EMR interaction with living organ-isms attracts the attention of scientists and engineers byboth realized and hypothetical, although insufficientlystudied, possibilities of its application in medicine.

According to a general concept put forward byN. Devyatkov, academician of the Russian Academy ofSciences, and his colleagues [1] the areas of applicationof EMR can be rather clearly divided into two groups.The first group uses energy characteristics of coherentwaves, whereas the second group employs informationpotentialities of coherent waves. Obviously, these twogroups may overlap.

1.1.1. The application of energy properties of coher-ent waves in biology and medicine. It is customary todistinguish between ionizing and nonionizing radiationwhen evaluating the effects of interaction with differentobjects at the atomic–molecular level. Normally, elec-tromagnetic oscillations (optical, vacuum-UV, X-ray,and gamma-ray) are classified as ionizing radiation ifthe energy quantum is large enough to cause, forexample, breaking of intermolecular bonds or ionizationof atoms, molecules, and radicals.

1

Electromagneticoscillations with longer wavelengths and low energiesof quanta, including radiation of the millimeter range,are related to nonionizing radiation.

Physiotherapy was one of the first areas of applica-tion of coherent short-wavelength radiation. Heating oftissues was always one of the main (radical) methods ofphysiotherapy. Chemical and biochemical reactions areaccelerated by heating (proportional to an exponentialfactor exp(–

E

A

/

T

), where

E

A

is the activation energyand

T

is the temperature), which determines the physi-ological effect. The advantages of heating by coherent

1

Note that the breaking of bonds of a microobject may be causedby EMR, the quantum of which does not exceed the bindingenergy. This is due to nonlinear effects, in particular, to mul-tiphoton transitions.

waves are associated with the fact that heat releasetakes place mainly in the tissues lying at a certain dis-tance from the surface (diathermy). In this case, wediminish the undesirable heating of surface tissues,which is inevitable in the case when heat sources act onthe surface of a body. The physical mechanism ofdiathermy is not clear yet. However, our studies shedsome light on this problem. Specifically, it becameclear that the so-called heat-shock effects are associ-ated with reversible phase transitions of DNA liquidcrystals in chromosomes during heating and cooling inthe sublethal temperature ranges from 40 to 43

°

C [126].

The nature of EMR interaction with biologicalobjects is determined by both parameters of radiation(frequency or wavelength, propagation speed, coher-ence of oscillations, and polarization) and the physicalproperties of a biological object as a medium in whichan electromagnetic wave propagates (dielectric con-stant, electric conductivity, and nonlinear-dynamicparameters of information biopolymers, as well asparameters depending upon this quantities, namely,wavelength inside the tissue, penetration depth, andreflectance at the air–tissue interface). The decay of theamplitude of a wave during its penetration inside a tis-sue may be characterized by the penetration depth

Λ

,defined as the distance at which the amplitude of oscilla-tions decreases

e

≈

2.72 times. For example, the penetra-tion depth in muscle and skin is

Λ

= 15 cm at

λ

= 10 cm(

ν

≈

3 GHz) and

Λ

= 0.3 mm at

λ

= 8 mm (

ν

≈

37.4 GHz).

The tendency of penetration depth

Λ

to decreasewith the decrease of

λ

is traced while the wavelengthinside the medium is much greater than the size of a cellor cellular organelles (cell nuclei, mitochondria, lysos-omes, ribosomes, Goldgi organ, endoplasm net, etc.).Recall that the mean linear size of a cell is of the orderof 10

–2

μ

m [1], and the length of protein moleculesranges from 4 to 20 nm [3]. However, these estimatesare arbitrary because the protein molecules may consistof many subunits and their sizes may change within awide range, different from 4–20 nm. At very high fre-quencies, the transmissivity of tissues for EMR startsgrowing again. For example, hard X-ray and gamma-ray radiation penetrates through soft tissues virtuallywithout attenuation.

As the penetration depth of waves depends uponthe frequency, the corresponding physiotherapeuticdevices make use of different ranges: 2375–2450, 915,433–460 MHz, etc. [2].

Hyperthermia, a method of destruction of malignanttumors, became a very important step in the energeticapplication of coherent waves. The development of thismethod dates back to 1960–1970s, when it was demon-strated that the viability of tumor cells was strongly sup-pressed already at 40–42

°

C. The increase in the temper-ature influenced tumor cells stronger than the benignones. The destructive action of ionizing radiation andchemotherapeutic drugs on tumor cells is enhanced byoverheating up to a temperature of 43–45

°

C). Therefore,

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nowadays, treatment of tumors uses, as a rule, a combi-nation of hyperthermia and the mentioned factors.Coherent waves localize the heating and accelerate theprocess itself. In this case, both the limiting depth of theheated area and the ultimate possibilities of localizationdepend upon the selected frequency range: the lower thefrequency of oscillations, the deeper the penetration ofradiation. However, the possibility of localization ofheated area is aggravated in this case. In the 1970s, acade-mician N.D. Devyatkov and Dr. E.A. Gel’vich initiatedand supervised pioneering works that resulted in theconstruction of appliances operating in different fre-quency ranges (2450, 915, and 460 MHz) and provid-ing constant temperature during heating with an accu-racy of 0.3

°

C. At present substantial progress is being achieved in

the field of laser application in surgery. A real boom in laser medicine emerged relatively

recently. It is explained, first of all, by the developmentof a series of novel highly effective and reliable lasersand by the development of flexible optical fibers for thedelivery of laser radiation.

Let us enumerate only some of the main branches oflaser medicine: ophthalmology (retina welding, treat-ment of glaucoma); cutting of biotissues (includingbiotissues with a developed net of blood vessels); abla-tion of sclerotic plaques, cutting, layer-by-layer evapora-tion, and coagulation of biotissues; laser dentistry; car-diosurgery (including recanalization of vessels cloggedup by atherosclerotic tissue); laser surgery of cholelith-iasis; laser microsurgery in gynecology; etc.

In practice, different types of gas and solid-statelasers, operating in pulse, cw, and pulse–periodicregimes, are widely applied, including ruby and neody-mium pulse solid-state lasers, CO and CO

2

cw lasers(units that use pulse CO

2

lasers are also available),pulse–periodic copper-vapor lasers, and excimer lasers.

A possibility of the application of pulse lasers onyttrium aluminum garnet with neodymium in laser sur-gery is being studied at present. Unfortunately, the effi-ciency of these lasers is several percent only. The situationis the same with neodymium glass lasers. A relatively lowefficiency of these lasers is associated with the fact thatneodymium ions absorb only a small portion of radia-tion of a pumping lamp, all the rest is converted intoheat. Researchers from the Institute of General Physicsof the Russian Academy of Sciences headed by acade-mician A.M. Prokhorov found the ways of increasingthe efficiency of neodymium lasers. They suggested touse a crystal of gadolinium scandium gallium garnet(GSGG) with Nd and Cr as an active medium. Its effi-ciency is 2–3 times higher than that of widely appliedyttrium aluminum garnet with Nd. A disadvantage ofGSGG lies in its lower, compared to yttrium aluminumgarnet, thermal conductivity. However, GSGG crystalshave several advantages (apart from high efficiency):the rate of the growth of this crystal is 5 times higherand it has a higher radiation stability (the latter was

proved by the scientists from the Institute of NuclearPhysics of the Republic of Uzbekistan). After that, neweffective laser crystals were developed. Their applica-tion seems to be quite promising.

Summarizing we may state that the progress in lasermedicine was achieved due to (1) the high level of stud-ies of interaction of radiation with biotissues, (2) devel-opment of optical fibers with low losses and high radia-tion stability to high-power laser radiation, and (3) con-struction of a variety of original solid-state and gas-discharge pulse–periodic lasers.

1.1.2. Application of coherent resonance radiationof low power in biology and medicine. The seconddirection related to the application of coherent EMR inbiology and medicine includes extensive works on theapplication of sources of low-intensity radiation in radio(EHF), IR, optical, and UV ranges. Interaction of suchradiation with biotissues is not accompanied by anyserious heating. The nature of radiation action in thiscase is not clearly determined yet. To some extent, weclarified the situation by publishing the results of ourstudies on nonlinear dynamic states of biomacromole-cules that depend on EMR modeling these states (see,for example, [127]).

Upon the excitation of oscillations in bioobjectswithin different above-specified frequency ranges, theeffects of radiation action have very much in common.The most substantial (unifying) feature is the manifesta-tion of a sharply resonant response of a bioobject to theaction of coherent radiation within all specified ranges.

Especially extensive studies of sharply resonantaction on bioobjects were carried out by Russian andforeign researchers in the EHF range

2

(30–300 GHz)(see, for example, reviews [4–6], monograph [1], andpapers [10–15] and [20–29]). Conditions under whichthe observation of sharply resonant action of thesewaves on an organism is possible were investigated indetail (see, for example, [7, 8]).

One encounters difficulties when reproducing asharply resonant effect on different biological objects.The analysis of this complicated phenomenon is pre-sented in [16]. This study developed methods allowingone not only to guarantee high reproducibility of thediscussed effects but also to investigate the shape ofresonance curves.

The experimental results clearly demonstrate thedecisive role of a sharply resonant response of livingbioobjects to EHF action in maintaining a homeostasis,which is a combination of adaptability reactions of liv-ing organisms [17–19]. The existence of multiple reso-nances is typical for biomembranes. These resonancesare frequency-shifted relative to each other, and each ofthese resonance corresponds to its own type of biolog-ical action [18, 19]. It is assumed that EHF actionmainly maintains and restores homeostasis, although

2

The range of extremely high frequencies (EHF) from 30 to 300 GHzcorresponds to the range of millimeter waves from 1 to 10 mm [9].

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this approach is too simplified from the point of view ofbiochemistry and physiology. Radiation has virtuallyno influence on the functioning of normal cells that haveno distortions [17]. In case of distortions, EHF action atthe corresponding resonance frequencies speeds uprecovery processes.

The potentialities of medicobiological action of EHFradiation are rather diversified. More than ten yearshave passed since both in our country and abroad EHFmethods were applied for the treatment of ulcers ofstomach and duodenum, trophic ulcers, traumas of softand bone tissues, stenocardia and infarct of myocardium,hypertonia, ophthalmological diseases, osteochondro-pathia of the femur tip of children and youngsters, andburn diseases; for defending hemogenesis from harmfulinfluences; and for regulating and recovery of enzymeactivity of microorganisms [10–12, 20, 30, 31].

We emphasize that the principles of application ofcoherent millimeter waves (

λ

= 1–10 mm) of low inten-sity were tested both on microorganisms and on mam-mals already at the first stages of investigations in thelate 1960s–early 1970s. Academician N.D. Devyatkovconducted general scientific supervision of the studiescarried out in many scientific institutions. From thevery beginning, the studies were aimed at elucidation ofpotentialities of application of these principles in medi-cine and biology. To get an idea of the extent of the works,note that the experiments were carried out with more than10 thousand animals for comprehensive testing.

It is worthwhile to note that the therapeutic action ofcoherent millimeter waves of low intensity is optimal ifcertain resonance frequencies or their combinations usedin experiments correspond to a given type of disease.

As soon as the possibility of application of coherentwaves for the treatment of this or that disease is estab-lished, the introduction of the corresponding methodinto practical medicine occurs. The successful applica-tion of therapeutic methods based on the action of low-intensity waves at the resonance frequencies optimalfor a given disease for the treatment of thousands ofpatients (although the number of working devices is notlarge, because their industrial production has notstarted yet) is the most important general result of theintroduction of these methods into practical medicine.The period of treatment is reduced in comparison withthe periods typical for treatment with medicines,whereas the efficiency of therapy is increased.

Here, it is worthwhile presenting some bright exam-ples of the effective application of EHF methods.

In the recent years, a group headed by ProfessorI.M. Kurochkin and Dr. M.V. Poslavskii carried outfruitful studies and promotion in practical medicine ofEHF methods of treatment of ulcers of stomach andduodenum. As a result, 95% of ulcers were cured with-out any medicine.

The results of the work performed at the PriorovCentral Institute of Traumatology and Orthopedicsshowed that the action of coherent waves of millimeter

range is highly effective in treatment of wounds andtraumas, including infected ones, in specific caseswhen, because of various reasons, neither blood trans-fusion nor antibiotic therapy could be applied.

There are no doubts about the prospects for the appli-cation of the EHF method with sharply resonant radia-tion of low intensity. In this context, we briefly presentthe experimental results illustrating another approach totherapy with the application of low-intensity millimeterradiation (O.V. Betskii, M.B. Golant, N.D. Devyatkov,E.S. Zubenkova, and L.A. Sevost’yanova, 1987). In thisexperiment, the possibility of healing an animal after alethal doze of ionizing radiation when marrow transplan-tation itself could not have any substantial influence onthe recovery process was studied. The resources of anorganism are already exhausted and the direct action ofcoherent waves on an affected organism may onlyspeed up its death. However, such irradiation can beused for the activation of cells of transplanted marrow,thus increasing their activity after transplantation.Indeed, the transplantation of activated marrow made itpossible to cure all the mortally irradiated animals.Their lives elongated approximately 40 times andbecame virtually normal for these animals.

A resonance character of biological action wasdetected also in the optical range [32]. Note that thestudy of the shape of resonance curves in this range isaggravated by the difficulty of continuous tuning oflasers within a wide spectral range and needs specialapproaches. Specifically, Karu

et al

. [33] demonstrateda rather narrow dependence of a biological effect on thefrequency with the help of bandpass filters that cut outappropriate bands from the emission spectrum of anincandescent lamp.

Studies [34, 35] are mainly devoted to the emissionof cells in the UV range. The authors also discusssharply resonant irradiation and its relation to the dis-turbances of functioning. The existence of multiple res-onances was stressed. Paper [34], devoted to the role ofUV radiation, describes observations similar to theeffects in the EHF range. It is stressed that any factorsdisturbing cell functioning cause changes of the emis-sion of photons from the cell.

Wide application of helium–neon lasers in medicinegives evidence of the influence of low-intensity coher-ent radiation within the optical range at some resonancefrequencies on the recovery processes in an organism.Radiation of both optical and millimeter ranges atproper resonance frequencies can be successfullyapplied in treatment of several diseases (ulcers of thestomach and duodenum, trophic ulcers, stenocardia, etc.).The similarity of the action of EMR in optical and mil-limeter ranges at appropriate resonance frequencies onbiological tissues is traced even in some details of thephenomenon. In particular, in the course of medicaltreatment of ulcers of stomach and duodenum, rough cic-atrices usually appear, which may cause illness relapses.The application of sources of coherent EHF radiation or

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MECHANISMS OF INTERACTION OF ELECTROMAGNETIC RADIATION 625

lasers of optical range in therapy of these diseases allowsone to get rid of rough cicatrices. The same holds forthe treatment of stenocardia.

The role of coherent waves of the IR range in meta-bolic processes was studied in [32, 36].

In Section 3, we consider the problem of the corre-lation of low-intensity resonant action in the EHF, IR,optical, and UV ranges within the framework of thehypothesis of information–control function of coherentradiation in the course of its interaction with bioobjects[28, 37].

1.2. The Influence of Low-Intensity Coherent Radiation of Millimeter Range on Biological Objects

The problem considered in this section is compre-hensively discussed in monograph [1]. For the reasonsof completeness of presentation, we reproduce hereonly the main points of this analysis.

1.2.1. Some general considerations. The spectrumof EMR ranges from the frequencies of 10

23

Hz (thewavelength is 10

–14

m), typical for

γ

-rays, to the fre-quencies lower than 10 Hz (the wavelength is greaterthan 10 m). The magnitude of a quantum of radiationdetermined the possibility of this or that transformationat the atomic–molecular level in a microobject beinginfluenced by an electromagnetic field. According toSection 1 the entire spectrum may be divided into twoparts: (1) the range of high frequencies, starting from thelow-frequency bands of the visible range (with a fre-quency of 4

×

10

14

Hz and wavelength of 7500 Å), whendissociation is possible, and low-frequency bands of theUV range (

ν

= 7.5

×

10

14

Hz and

λ

= 4000 Å) andhigher, when ionization becomes possible (in thiscase, we consider ionizing radiation); (2) the range oflow frequencies, where no breaks of bonds in an atomor a molecule occur (in this case, we consider nonion-izing magnetic fields).

The second part of the spectrum includes far-IR,microwave, radio, and extremely low frequency ranges.According to the International classification [38], thelow-frequency part may in turn be divided into nar-rower ranges (see Table 1).

For reference, we present the maximum permissiblelevels of radiation according to USSR standards [39](see Table 2).

The idea of applying EMR within the EHF range forthe specific action on biological structures and organ-isms was originally put forward by Soviet scientists(N.D. Devyatkov, M.B. Golant,

et al.

) in 1964–1965.The essence of this concept is as follows. Microwavesare strongly absorbed in the atmosphere of Earth.Therefore, living organisms could not have naturalmechanisms of adaptation to the oscillations of signifi-cant intensity within this range coming from outside.However, they could adapt to similar oscillations oftheir own.

In Section 1, we presented the effect of hyperther-mia as a typical example of the energetic action of radi-ation on an organism, when a useful biological effect isachieved by the transfer of EMR energy into heat.However, such an influence of EMR on an organism ispossible under which the temperature increase is insig-nificant (

�

0.1

°

C) and does not play the main role inbioeffect. In such cases, we usually speak about controlor information action of EMR of low or nonthermalintensity. Numerous studies showed that this is the fea-ture of EMR of the millimeter range (1–10 mm) at alow power density of several tenths of or several milli-watts per square centimeter of the irradiated surface.

The application of millimeter waves in biology andmedicine is highly promising due to several peculiari-ties of coherent interaction of this EMR with bioobjectsat different levels of organization—from biopolymermolecules to the whole body. At the initial steps of theo-retical substantiation, the unique nature of the influenceof EMR of millimeter range on the vital activity oforganisms was inconsistent with standard concepts. Itwas assumed that only incoherent (thermal) oscillationscan be generated in a living organism or in its cells andthat the symptoms of external action are only of nonspe-cific thermal type, and an organism responds to them asto a stress factor. Note that the energy of a quantumwithin this range (at

λ

= 1 mm,

h

ν

= 1.17

×

10

–3

eV) islower than the energy of thermal motion (at

T

= 300 K,

Table 1

Name of the range Frequency range

Extremely low frequencies (natural background)

300 Hz

Very low frequencies 300 Hz–10 kHz

Low frequencies 10 kHz–1 MHz

High frequencies 1–30 MHz

Very high frequencies 30–300 MHz

Ultrahigh frequencies 300 MHz–1 GHz

Microwave range 1–3000 GHz

Table 2

Frequency rangeMaximum permissible level (field strength or

intensity)

Long waves, 30–300 kHz 20 V/m

Medium waves, 0.3–3 MHz 10 V/m

Short waves, 3–30 MHz 4 V/m

Ultrashort waves, 30–300 MHz 2 V/m

Microwaves, 300 MHz–300 GHz (24-hour exposure)

5

μ

W/cm

2

626

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kT

= 2.53

×

10

–2

eV). In Table 3, we present for com-parison the typical values of energies of other types.

It follows from the comparison of energies of differentphysical processes that millimeter radiation may influ-ence substantially the vital activity only in the case ofmultiphoton processes typical for coherent oscillations.

Nevertheless, we should note some of the earlyhypotheses put forward before the implementation ofcorrect experiments and important for understanding ofthe evolution of the concepts of mechanisms of interac-tion of radiation with biological objects.

One of the first mechanisms of generation of coher-ent oscillations by living organisms was proposed byEnglish physicist H. Frölich [40, 41]. One may find fur-ther development of this idea in [15, 42–48].

The essence of the Frölich hypothesis is as follows.Biological systems may have polarization (dipole)oscillations within the frequency range from 100 to1000 Hz (

λ

= 3–0.3 mm). Vital activity in biosystems isaccompanied by energy transport through locally exciteddipole oscillations (biological pumping). A transition ofa system to a metastable state may take place due tononlinear effects of interaction of dipole vibrations anddue to nonlinear coupling of this vibrations with elasticvibrations of DNA, proteins, and membranes. In thistransition, energy is transformed into vibrations of acertain type. Under the action of external EMR, a meta-stable state may turn into a ground state, which givesrise to a “giant dipole”—a specific case of coherentstate in a biological object. Such perturbations of bio-substrates resemble low-temperature condensation ofBose gas (Bose condensation of phonons).

According to another version, a “protein machine”hypothesis of D.S. Chernavskii, Yu.I. Khurgin, andS.E. Shnol’ is accepted [49]. In this case, we assumethat the storage of electromagnetic energy is possible inthe form of mechanical stress of a biomacromolecule,which is also a specific case of a coherent state.

The mentioned models differ mainly by the form ofstorage of energy. All of them are united by an idea thatbiological structures feature a selected degree of free-dom of mechanical nature in which energy can be stored.This selected degree of freedom plays an important func-tional role in biological processes. In particular, thisfeature distinguishes living (thermodynamically non-equilibrium) systems from inorganic ones. The energy ofexternal radiation transforms into the energy of polarmolecules related to rotational degrees of freedom. Polarmolecules of water with a dipole moment of 1.84 D mayserve as such accumulators of energy.

The hypotheses put forward by the authors of [40, 49]are based on the assumption about the existence ofcoherent oscillations in living organisms. However,they do not account for the dependences and regulari-ties characterizing interaction of microwave EMR withliving organisms.

There is also a hypothesis [50] presuming the exist-ence of an unidentified molecule that takes part in theintermediate stages of the development of biochemicalreactions and resides in a triplet state where two non-pair electrons interact with each other through theirmagnetic fields. The molecule has three possible initialstates, each of which corresponds to its own pathway ofchemical reaction. It is assumed that the initial statemay be influenced by EHF pumping, which regulatesthe course of biological processes.

1.2.2. The main regularities of the action of low-intensity electromagnetic radiation of millimeter rangeon biological objects. Let us present in brief the mainregularities of the action of microwave EMR of lowintensity on biosystems based, in particular, on theexperimental data [1].

(1) The change of the power density within widelimits does not influence the reaction of organisms tothe action of EMR as determined by a certain biologicalparameter. Starting from some minimal (threshold)value of the power density and up to its values causingpronounced (exceeding 0.1

°

C) heating of tissue, thebiological effect of microwave EMR remains virtuallythe same [11, 51, 52]. In some cases, the effect relatedto the irradiation of microorganisms and determined bya certain biological parameter does not change with thevariations in the power density up to 10

5

times.

(2) A variation in a certain biological parameter(e.g., of some particular enzyme activity) after theaction of EMR on an organism occurs only within nar-row bands of the acting frequencies that are normally10

3

–10

4

of the medium frequency. This phenomenonwas called a sharply resonant effect of biologicalaction. The number of such bands interchanging withthe bands where no substantial changes of this parame-ter take place may be rather large [11, 51, 53, 54].

(3) The type of sharply resonant biological actiondepends on the frequency of coherent oscillations. At var-ious resonance frequencies detected for one biological

Table 3

Type of energy Typical values of energy, eV

Energy of electron transitions 1–20

Activation energy 0.2

Vibrational energy of molecules 10

–2

–10

–1

Energy of hydrogen bonds 2

×

10

–2

–10

–1

Energy of thermal motion 2.53

×

10

–2

Energy of a quantum for

λ

= 1 mm 1.17

×

10

–3

Energy of rotation of molecules around bonds

10

–3

–10

–4

Energy of Cooper pairs in supercon-ductivity

10

–4

–10

–6

Energy of magnetic ordering 10

–4

–10

–8

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MECHANISMS OF INTERACTION OF ELECTROMAGNETIC RADIATION 627

reaction, the type of changes for another reaction maystrongly differ from that for the first reaction.

(4) Effect of irradiation depends upon the initialstate of the irradiated organisms.

(5) The results of action of microwave EMR may bememorized by organisms for a long time. This happensonly in the case of prolonged (no less than half an hour)or, sometimes, multiple action of EMR [56].

(6) For animals, the biological effect of EMR is notrelated to the direct action of energy falling from theoutside on the surface of a body or on an organ (or asystem) controlling the function modified under theaction of EMR. The distance from the irradiated area tothe corresponding organs or systems may be a hundredor a thousand times longer than the distance at whichthe power density diminishes by an order of magnitudedue to losses in tissues. At the same time, the effective-ness of EMR irradiation is different for different partsof the surface of a body. According to the experimen-tal data [30], the action of EMR may be realized, in par-ticular, in the case of irradiation of acupuncture points,which proved the hypothesis put forward earlier [27].

1.3. Information Aspects Associated with the Action of Resonance Electromagnetic Radiation

of Low Intensity on Bioobjects

1.3.1. Information aspects associated with theaction of radiation on a living organism. A very lowenergy of EMR necessary for substantially affecting thefunctioning of organisms, the specific features of thisphenomenon, and a high reproducibility of theresults—all of this pushed the researchers to a hypoth-esis [27, 28, 57, 58, 126] according to which EMR isnot an accidental factor for organisms. Similar signalsare produced and used for certain purposes by theorganism itself. The external coherent radiation onlyimitates the signals produced by the organism.

The essence of the second hypothesis is as follows.The observed regularities in the action of coherent EHFEMR of nonthermal intensity are explained by the factthat, penetrating inside an organism, this radiation istransformed at certain (resonance) frequencies intoinformation signals controlling and regulating recoveryand adaptation processes in an organism. Both pro-cesses are referred to as an “adaptive growth” [59] orare treated, according to [126], as the wave epigeneticfunctions of chromosome apparatus, cytoskeleton, andextracellular matrix.

Here, we present some of the main facts that providea background for these hypotheses.

(1) The minimal (threshold) power density of theincident radiation necessary for producing a substantialbiological effect is negligibly low in comparison withheat power emitted by the organism itself into space.For human beings and animals, this power density isabout 10

–3

–10

–4 of the heat power [51, 56, 60]. The powerdensity is so low that radiation does not cause even

local heating of tissues by more than 0.1°C. In any case,EMR cannot cause any disturbances in tissues becausethe energy of its quanta is two orders of magnitudelower than the energy of weak hydrogen bonds.

At the same time, the power of external radiation isquite sufficient for the formation of control signals. Inany information system (including technical ones), theenergy of such signals is several orders of magnitudelower than the energy of a system as a whole, which isdetermined by the power of actuators or appliances.

(2) It was noted in Section 2 that the action of EHFradiation, determined by a certain biological parameter,does not depend on its intensity within wide limits.Such a type of the dependence of the action on theintensity of an acting factor is quite natural for informa-tion systems and is due to the specific features of thecontrol regulation process. Apparently, for energeticaction (the effect of which is determined by energy),such a type of dependence has not been observed yet.

(3) Threshold type of the dependence of a biologicaleffect on the intensity of radiation is an imperativecondition for the operation of an information system.If this condition is not met, the operation of the systemcan be disturbed by any external interference or noise.A threshold type of action is possible for energeticaction. However, in this case, a further increase inpower changes biological effect, in contrast to theinformation action.

(4) The type of a biological response of an organismdepends upon the frequency of the acting waves. Eachspecific action is possible only within a narrow fre-quency range. This action may be impossible at differ-ent frequencies or it may be qualitatively and quantita-tively different (including the absence of any action[11, 51, 61]). In other words, the frequencies of oscilla-tions determine the type of action of considered radia-tion on an organism, i.e., the frequency is the carrier ofinformation.

(5) Resonance frequencies related to certain biolog-ical effects are strictly reproducible under reproducibil-ity and control of the experimental conditions [8]. Sucha distinct dependence of action on the numerical valueof the information parameter (which is the frequency ofoscillations in the case at hand) is typical for informa-tion systems designed for the processing of largeamounts of data.

(6) All the above-listed features (2)–(5) are mani-fested only simultaneously. If an effect determined by acertain biological parameter is independent of the EMRpower, then the bioeffect displays a resonance depen-dence on the frequency of oscillations [11, 51, 52, 62].Thus, we deal rather than with a random set of factors,with a deep interrelation of factors which is due to spe-cific features of operation of information systems.

(7) Information basis accounts well for the factthat the changes in living tissues resulting from irradi-ation are absent when the tissues are irradiated after

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the termination of vital activity. Regulating systems donot work in dead tissues [10].

(8) We noted in Section 2 that the action of EMRdepends crucially upon the initial state of an organism.Thus, if in the initial state some function is weakenedseveral times compared to the normal state, then radia-tion may bring it back, whereas the same radiation hasvirtually no influence on the normal state of the organ-ism [61, 63]. An important reason for that is the factthat one of the most important functions of informationsignals in the organism is the maintaining of homeosta-sis, i.e., the ability of biological systems to withstandmutations and to keep relative stability of compositionand properties.

(9) If an organism is large enough, EMR may influ-ence organs rather distant from the irradiated spot,which makes direct energetic effect impossible [62, 67].However, this circumstance seems to be quite naturalwithin the framework of the suggested hypothesis: thesignals periodically amplified due to metabolic energymay spread over large distances along the communica-tion channels within a single information system of aliving organism. The amplification of weak signalsdoes not require great energies and is comparable to theenergetic resources of an organism.

(10) Living organisms under natural conditions donot experience the action of monochromatic EMR ofmillimeter wavelength range because of their absencein the environment. How did all the organisms, frombacteria to a human being, work out during the evolu-tion a specific (depending upon the frequency of oscil-lations) reaction to this radiation? This fact does notcontradict the information hypothesis because, accord-ing to this hypothesis, the efficiency of action of exter-nal coherent radiation is explained by the fact that, pen-etrating inside an organism, they are transformed intosignals similar to information ones that are produced bythe organism itself for the regulation of the process ofrecovery and adaptation to the environmental condi-tions. The existence of the same radiation in the envi-ronment could disturb the information system of theorganism by introducing interference. Therefore, theuse of the control signals that cannot be simulated byradiation from the outside in the inner system is not bio-logically expedient.

(11) It is rather important for substantiating theinformation nature of EMR action on living organismsthat the information content (the ratio of the amount ofthe processed information to the energy consumptionfor this processing) for the millimeter range isextremely high. In particular, for living organisms, thisinformation content substantially exceeds the corre-sponding value for optical and UHF ranges [1].

(12) The constitution of different living organisms,starting from bacteria up to a human being, the func-tioning of which can be influenced by in the millimeterrange EMR is absolutely different. Irradiation may pro-voke the changes in various functions even for the same

species. For example, in [60] the influence of EMR onthe treatment of different human diseases and the regu-lation of functioning of microorganisms is described.One can hardly imagine that the mechanism of changesis the same for all the diversity of changes observedupon the irradiation of different and similar organisms.At the same time, the considerations mentioned above,which attribute the observed regularities of EMR actionto the information function of this action, give quite anatural answer to this question: the general regularitiesof operation of information systems must be realizedirrespective of the mechanisms implementing the regu-lating signals.

(13) Recently, new data that slightly fall out of theframes of the presented considerations have beenobtained [126]. A notion of a “wave gene” was intro-duced for the most substantial part of “wave metabo-lism” of biosystems where the millimeter range (bothexogenous and endogenous) is only a part of electro-magnetic and acoustic symbol wave processes. Synthe-sis of soliton and holographic matrices performingtransportation and reduction of strategic bioinforma-tion, generated and accepted by organisms, is a partic-ular case of conversion of wave processes into biosym-bolic ones. DNA molecules included into chromo-somes, cytoskeleton, ribosomes, membranes, and thesystem of extracellular matrices provide a substrate forthe development of these processes.

1.3.2. Regulation of functioning in a living organismand generation of coherent waves by its cells. Livingorganisms, even elementary species, such as singlecells, adequately adapt their functioning to continuouschanges of habitat related to the changes both in theorganism itself and in the environment. The number ofpossible changes and the corresponding reactions is tre-mendous. Hence, even a single cell must possess acomplicated system regulating its reactions.

An average size of a cell is about 5 μm. How cansuch a regulating system fit such a small volume andhow does it work? Priority studies carried out in ourcountry during the last 25 years [1, 126] showed thatcells regulate their functioning eliminating disbalancesand adapting to the changing habitat with the help ofcoherent acoustoelectric waves. An alternate electric fieldof these waves, emitted into the surrounding space, allowsinteraction of neighboring cells and cells approachingeach other. Spreading over an organism, they may pro-mote organization of interrelation and regulation of theintracellular processes in the system as a whole.

The wavelength of acoustoelectric waves in anorganism is roughly 106 times shorter than the wave-length of electromagnetic millimeter waves in free space(λ = 1 mm). Thus, the size of a cell is 5 × 103 timeslarger than the wavelength of acoustoelectric waves.Therefore, systems of large electric length may existinside a cell. These systems are able to generate a largenumber of regulating signals and to influence in differ-ent ways the intracellular processes.

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1.3.3. Mechanisms of generation of coherent wavesby cells. The studies showed that the generation ofcoherent waves by cells is a systematic process inwhich cellular membranes, protein molecules, andmetabolism are involved. The frequencies of generatedoscillations are determined by resonances in mem-branes. The energy is fed to the membranes in cyto-plasm and to protein molecules capable of vibrating atthe same resonance frequencies. Protein molecules, inturn, acquire energy of intracellular metabolic pro-cesses. Adjoining are the processes of excimer andexciplex pumping of nitrous nucleotides of DNAaccording to laser principle. In this case, DNA attains ahighly regular liquid-crystal structure and the ability ofemitting an ultraweak coherent photon field within therange from 250 to 800 nm. The role of such a field maybe associated, in particular, with read-out of quasi-holo-graphic gratings of chromosome continuum and therecovery of symbol photon mark-up fields, which orga-nize the spatial–temporal structure of biosystems [126].

In a normal cell, protein molecules are in thermalequilibrium with the medium, and the emission spec-trum of the cell differs slightly from a thermal one, cor-responding to the temperature of the environment. If thenormal functioning is distorted and the shape of the cellis disturbed, the conditions are created for preferredexcitation of cellular membranes at certain resonancefrequencies. At these frequencies, the energy transferfrom protein molecules to membranes is stimulated,which is promoted by the synchronization of the vibra-tions of bound protein molecules by membranes.

Thus generated oscillations reflect the existing mal-functioning, which, in accordance with the La Chatelierprinciple (and its analog in biology—the principle ofconservation of homeostasis), results in the processesbringing the system back to the initial state.

It was revealed that the frequencies of vibrationalprocess excited in a cell in the case of functional disor-ders remain the same until unbalances are removed.Such stability is provided by the formation of substruc-tures consisting of protein conglomerates on the sur-face of the membranes under the action of excitedacoustoelectric waves.

Note that the external irradiation of cells with coher-ent waves may lead to the formation of substructures onthe membranes of healthy cells and induce the corre-sponding generation of coherent waves by cells them-selves. A mechanically stable structure of membranesis not disturbed by the processes related to the genera-tion. Radiation weakly influences the current function-ing of the cells. In the case of the formation on mem-branes of sharp deformations (which corresponds todisturbance of functioning), the processes related to thegeneration of coherent waves at resonance frequenciesresult in relatively fast changes in the shape of mem-branes and easily detectable changes in cell function-ing. The induced substructures gradually disappear as a

result of Brownian movement upon the normalizationof the state of cells.

The studies performed at the Institute of Radioelec-tronics of the USSR Academy of Sciences under thesupervision of academician N.D. Devyatkov showedthat the processes of heating of tissues and substancescaused by coherent waves may influence substantiallytheir functioning, in particular, information processes pro-ceeding under the action of coherent waves. Of especialimportance are the processes related to absorption ofwave energy by water, playing an important role in thelife of an organism. One should keep in mind that pro-tein molecules and DNA polymers may function prop-erly only inside highly organized water surroundings ofintracellular and intercellular medium.

Absorptivity of water depends upon the type ofintermolecular interactions. Therefore, the peculiaritiesof absorption of radiation by water molecules in bio-structures may cause structural heating of water, phasetransitions in biological membranes, etc. Even at lowintensities of radiation of several tenths of or severalmilliwatts per square centimeter, large gradients of theelectric field and related thermal gradients may arise insuch layers. This may stimulate mixing of extracellularliquid, change transport of ions in water and of othersubstances through cellular membranes, and give riseto thermal and acoustic waves. However, these thermalphenomena themselves have no resonance character.

1.3.4. The interrelation of action of coherent low-intensity radiation of EHF, IR, optical, and UV rangeson living organisms. The application of coherent radia-tion of low power of the mentioned ranges in medicineand biology necessitates permanent specification of theconcepts of the processes taking place in an organismunder the action of this radiation. The development ofthe concept of information–control role of radiationwithin all indicated ranges in vital activity is the mostimportant achievement in this way. Such a role is adirect consequence of the fact that the supercomplexsystems of multicellular organisms may operate in astable way only in the presence of a highly developedregulating system using a supply of rather low power(comparable with energetic potentialities of an organ-ism) [37, 127].

The essence of homeostasis (relative stability) pre-sumes that any changes in the organism and in its differ-ent systems, consisting of elements functionally relatedto each other, cannot be isolated. The changes in some ofthe functions must cause the changes in the others, com-pensating to this or that extent for the negative influenceof the former on the functional systems. However, thecompensation is probably incomplete and the action ofeach range has its own peculiarities. The assumption ofthe possibility of correlated excitation of oscillationswithin different frequency ranges seems to be highlyprobable if the most miniature organisms (cells) are con-sidered. The elements of these organisms must take partin implementation of different functions because the

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diversity of the latter is enormous. Radiation of all men-tioned ranges influences primarily the cellular processes.

As the radiation generated by cells at different fre-quencies in different ranges may maintain homeostasisonly in the case of concerted action, it is likely that theonly way providing optimal recovery processes is theuse of correlation of excitation of mentioned waves.3

Under correlated excitation of the oscillations ofdifferent frequency ranges in an organism, the biologi-cal results of action have very much in common:sharply resonant (Δf/f ~ 10–3) biological response of theorganism to the action of coherent radiation of all indi-cated ranges, existence of multiple frequency-shiftedresonances of membranes, correspondence of each res-onance frequency to its own type of biological action,and the absence of influence of radiation on the currentfunctioning of normal cells having no disorders.

At the end of this section, we will point out animportant fact. Since complex cellular systems includenonlinear elements, nonlinear conversion of frequen-cies is inevitable. In particular, Del Giudice et al. [68]considered the change of the dielectric permittivity in astrong electric field as a source of nonlinearity (forexample, the dc component of the field of polarizationfor lipid membranes is of the order of 107 V/m, and theac component of the field is proportional to the dc field).The main genetic molecule (DNA) is another importantnonlinear element of the cell. Such a positively nonlin-ear effect as the formation of solitons is possible in thismolecule. We proved it experimentally within theframes of phenomenon of Fermi, Pasta, and Ulamrecurrence [126]. Holographic associative memory isanother phenomenon of genetic apparatus, which isalso related to a nonlinear effect of phase conjugation.The description of this effect involves the model offour-wave mixing [126].

According to the data presented in [32, 36], the orig-inal resonance frequencies (17910, 18420, 19560,21150, 23970, 25080, 27960, and 28680 Hz) are con-verted in the course of metabolism into a long series ofcombination frequencies which are nothing but the sec-ond and the third harmonics of these frequencies andthe frequencies of oscillations resulting from opticalmixing and parametric generation. Only original fre-quencies are observed at the beginning of a cellularcycle. The spectrum is enriched in the course of meta-bolic processes with the bands of higher frequencies,which can be explained by the increase in the amplitudeof generated oscillations and, consequently, of theamplitude of high-order harmonics. However, the spec-tral bands of higher frequencies related to the describedfrequency transformation do not exceed 9 × 104 GHz(3000 cm–1), i.e., lie within the optical range where thelowest frequency bands are higher than 4 × 105 GHz.They obviously do not reach the range of UV spectrum,

3 This idea does not contradict the earlier hypotheses [37] about thedomination of EHF waves in the system of regulation of cellularprocesses aimed at maintaining and recovery of homeostasis.

starting from 7.5 × 105 GHz. An effective frequencytransformation into these ranges based on the principlesmentioned above is hardly possible.

The generation of coherent waves in the IR rangeseems to be associated with metabolism [32] and the dis-turbance of electric symmetry of cells. However, meta-bolic processes are natural for cell functioning and, thus,they can hardly disturb the homeostasis. The excitation ofcoherent waves of EHF, optical, and UV ranges in cellsresults from the changes inside cells that disturbhomeostasis. We may assume that, under the circum-stances mentioned above, the coherent waves of theIR range, on the one hand, and of EHF, optical, andUV ranges, on the other hand, regulate different pro-cesses. This is the basis for the hypothesis about inter-relation of excitation of coherent waves in differentranges. What is the physical and, so to say, intelligentnature of such regulating logical processes?

The length of the majority of protein moleculesranges from 4 to 20 nm [3]. As the speed of propagationof acoustic waves is several hundred meters per second,the resonance frequencies (the fundamental harmonic)of acoustic vibrations of the majority of these mole-cules belong to the EHF range. It determines [69, 70]the efficiency of participation of protein molecules in asystem process of generation of acoustoelectric wavesin cells. However, we should point out that the same mol-ecules may serve as resonators for electromagnetic oscil-lations of substantially higher frequencies. Thus, in thecase of wave propagation in vacuum, the wavelength invacuum equal to the length of a molecule (1 nm) wouldcorrespond to the resonance frequency of 3 × 1016 Hz.However, if we take into account that an equivalent rel-ative dielectric constant of a molecule is about severalunits and that the linear sizes of the resonators of com-plex shapes are often less than the wavelength byapproximately an order of magnitude [71], then we maydetermine the excited frequencies as roughly 50 timeslower than the indicated value, i.e., 5 × 1014 Hz. Thisvalue is close to the frequency boundary between UVand optical ranges and may account for the excitationof acoustic waves in both ranges. In particular, the exci-tation of acoustic vibrations of protein molecules maybe related to the release of energy during its transfor-mation in the course of metabolism. The same is validfor DNA molecules.

As the electrical charges are inhomogeneously dis-tributed along the peptide bonds, acoustic vibrations ofa protein molecule may give rise to oscillations of bothtypes: acoustoelectric oscillations in the EHF range,which are determined by the total length of a givenmolecule, and oscillations in the UV and opticalranges, which are related to the excitation in a moleculeof electromagnetic vibrations accompanied by the stor-age of energy of elastic deformations at the points ofpronounced bendings of the molecule. As both types ofoscillations are related to the same molecules, thesimultaneous excitation of these oscillations seems to

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be a natural process. One should note that the values ofsynchronously excited resonance frequencies of differ-ent ranges, their ratio, and the ratio of amplitudes ofoscillations at these frequencies depend upon the topol-ogy of protein molecules. Apparently, the optimal typeforms of proteins have been selected in the course ofevolution not only according to optimal enzymaticfunctions of proteins but also with regard to adequateelectromagnetic attributes of metabolism.

One cannot consider the sizes of cells (e.g., along theperimeter of the outer membrane) to be small in compar-ison with the wavelength of radiation in the UV and opti-cal ranges. For cells (the mean diameter is 50 μm), theperimeter of the plasmatic membrane is close to 15 μm,i.e., it is approximately 30 times greater than the wave-length of an electromagnetic wave (even if we considerthe wavelength in vacuum between the UV and opticalranges). That is why the “antenna systems” of cells [18],built up with the help of the waves of the EHF range,for the waves of optical and UV ranges can send a fluxof radiation at significant distances with a rather highefficiency and admissible losses in the power density.There is virtually no evidence of almost complete (as incase of waves of the EHF range) stochastization and theloss of coherence of radiation. It was possibly due tothis reason that, first, A.G. Gurvich [72] and then otherauthors [36, 73] revealed the influence of living organ-isms on each other by means of emitted waves in theUV range (reflection or absorption of these waves dis-turbed communication).

1.4. Possible Molecular Mechanisms of Interaction of Radiation of the Millimeter Range with Protein

and Nucleic Acid (DNA) Macromolecules

The discovery of extremely sharp resonant effects isthe most interesting subject of experiments on the inter-action of low-intensity EHF EMR with biologicalobjects (from “primitive” biomacromolecules to thewhole organism). The elucidation of the molecularnature of the indicated resonance effects is a rathercomplex problem. The current situation is rather con-tradictory. In spite of successful clinical applications ofEHF methods for the treatment of different pathologies(see Section 1), it is still not clear how one can accountfor the pronounced influence of the fields that causelocal heating of tissues by no more than 0.1°C. It is notclear what is the level of reception of EMR and of“purely physical” primary acts of its bioinfluence.However, the universality of the action of weak micro-wave radiation on organisms puts forward a basic ques-tion a to whether the electromagnetic channel of inter-action in the corresponding frequency range is a funda-mental intrinsic way of information and regulationtransfer in living nature.

1.4.1. Theoretical models of vibrations in proteins.In the case of globular proteins, the mechanisms thatassume collective motions of the whole macromoleculeor of its isolated large parts (domains), subunits, etc. are

considered. The calculations of normal modes arebased on X-ray data at the level of atomic resolution.One chooses initial coordinates in conformational spaceof macromolecules and uses different approximations toselect, from the whole matrix of force constants, a partresponsible for the lowest frequency modes with regardto possible nonlinear effects. To introduce the generaltendencies in the development of these studies, we willbriefly discuss some of the main results.

Brooks and Karplus [74] calculated low-frequencyvibrational modes for a small globular protein–pancre-atic trypsin inhibitor (PTI). Vibrational frequencieslower than 120 cm–1 correspond to a model of elasticvibrations of a protein molecule. Modes with frequen-cies lower than 50 cm–1 (λ < 0.2 mm) correspond toanharmonic vibrations.

Chon [75] developed a quasi-continuous model,which presumes that the dominant low-frequency modein a protein molecule is determined by collective fluc-tuations of weak bonds, in particular, of hydrogenbonds and by internal displacement of massive atoms.On the basis of this model, low-frequency vibrationswere calculated in α- and β-structures of proteins. It wasdemonstrated that accordion-type modes in β-sheet andbreathing modes in β-barrels of immunoglobulin C andconcavalin A corresponded to low-frequency modeswith 20–30 cm–1 (λ = 0.5–0.3 mm).

Levy et al. [76] used the quasi-harmonic method forcalculating normal modes. The effective force con-stants are derived from the molecular-dynamic estima-tions of root-mean-square changes of internal virtualcoordinates with regard to harmonic effects. Aminoacid residues of PTI are considered to be interactioncenters. For 168 calculated normal modes, the highestfrequency is 250 cm–1 and the lowest one is 0.32 cm–1

(λ = 3 mm). Thus, many of the predicted vibrations fallwithin the range of interest.

Go et al. [77] used an iteration method allowing oneto extract eigenvectors from the full matrix of secondderivatives of potential energy of the system. A hinge-bending mode (interdomain motion) of lysozyme witha frequency of 3.6 cm–1 (λ = 2.8 mm) was consideredwith the help of this method.

1.4.2. Theoretical models of DNA vibrations. In thecase of DNA, theory also predicts the existence ofvibrations manifesting themselves in the EHF range(the so-called “soft modes”). Here, we note the studycarried out 25 years ago [78]. The model of related twist-ing vibrations of nitrous bases of DNA was considered.The frequencies of these vibrations were estimated to liewithin a range from 10 to 100 cm–1, i.e., λ � 1–0.1 mm.The “solid-state” approach to DNA double helix wasdeveloped in a series of works [79–82]. According tothe assumption used in these studies, long-range collec-tive interactions exist in this helix. The lattice modes inthe phonon spectrum of DNA are considered with the helpof a model of valence and long-range electrostatic forcefield. On the basis of these models, the calculations of

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the frequencies (20.3, 25, and 27 cm–1) of modes activein Raman spectra were calculated [79].

Another model presumes the existence of both longi-tudinal and transverse acoustic waves in the DNA doublehelix. The frequencies of these vibrations are determinedby the properties of DNA and depend upon the length ofthe DNA chain. Changing this length, one may cover awide frequency range, including submillimeter, millime-ter, and centimeter waves (see review in [126]).

Local vibrations of the type of “resonance ondefects” of the DNA structure (end defects and fork-type defects) were also predicted in [81]. These vibra-tions were presumably detected in experiments at lowfrequencies of 600 MHz [83].

Note also a model of conformational motility of DNAproposed in [84], where “pendulum” motion of nucle-otides was considered. The frequencies of vibrationsdetermined with the help of this model (10 and 60 cm–1)correspond to submillimeter waves.

The calculations of vibrations of nucleotides (m ≈200mp) of double-stranded DNA carried out on the basisof structural data for different conformations of macro-molecules yield the values ω = 2πν ≈ 1012 Hz, which cor-relates well with the data of Raman spectroscopy.

Our experimental data on the spectroscopy of corre-lation of photons of DNA, presented most completelyin [127], prove the existence of acoustic vibrations withthe frequencies ranging from 100 Hz to several milli-hertz. Typical are the repetitions of different and spe-cific Fourier spectra of DNA dynamics, which is in linewith the assumptions about soliton quasi-spontaneousexcitations of DNA according to the Fermi–Pasta–Ulam return mechanism. Here, we face a problem of anew type of DNA memory and translation of “waveimages” of sequences of nucleotides along DNA chains,which is important for understanding fundamentalmechanisms of precise “piloting” of DNA transposonsin a liquid-crystal multidimensional space of chromo-somes.

1.4.3. Experimental results. The difficulty of exper-imental observation of vibrational states of DNA in theEHF range is explained by (1) the absence of suffi-ciently sensitive spectral apparatus in the range betweenIR and UHF areas and (2) very strong water absorptionin this frequency range. Theoretical models (see Sec-tions 4.1, 4.2) usually neglect the interaction withwater solvent, which results in the shift of resonancefrequencies and broadening of the spectra with theformation of broad bands [80, 86]. Van Zaldt [87] pro-posed a theory of resonance absorption of EHF EMRby solutions of plasmid DNA discovered by Edwardset al. [88]. It is assumed that a hydrate shell of DNA isa complex structure and that the interaction of the firstwater layer with DNA is free of energy losses.

The situation is more clear with Raman spectra ofproteins with different water contents [89–92]. In thecase of globular proteins, low-frequency modes are

detected around 20–30 cm–1. The frequencies dependslightly upon the size and shape of protein globules [89].

Similar Raman studies were carried out with DNA[91, 92]. For the A-form of DNA, the bands wereobserved at 85 and 21 cm–1. With the increase of thewater content in the sample, the low-frequency mode ofthe A-form shifts to 14 cm–1 and that of the B-form“softens” and disappears. This feature allowed theauthors to attribute the indicated mode to an interhelicalvibration.

Didenko et al. [93] used the method of gamma-reso-nance spectroscopy for recording changes in conforma-tions of protein (lyophilized hemoglobin with 57Fe tracer)under the influence of EMR in the range of 40–50 Hz.Very narrow peaks (Δf/f ~ 10–4) of action were detected.

1.4.4. Some molecular mechanisms of interactionof EHF EMR with protein and DNA macromolecules.A major part of works aimed at theoretical consider-ation of effects of biological action of weak EMR,including that of the EHF range, are mainly of the phe-nomenological type. Rather general cooperative, to bemore precise, synergetic models of biosystems are used.Many authors address, in particular, the well-knownFrölich model [90]. However, synergetic approaches tothis problem themselves can hardly clarify the nature ofspecific carriers of “unusual” properties of biosystemsand give an answer to the question about primary mech-anisms of biological action of EMR. For these purposes,a more detailed simulation taking into account quanti-tative parameters of the considered effects is necessary.

Information biopolymers of organisms realize theirfunctions in many directions. Two of them are strategic;these are the levels of substance and field. The substancelevel for nucleic acids and proteins includes matrix andcatalytic functions. The field (or wave) level is under-stood as an ability of DNA and proteins to emit electro-magnetic and acoustic waves with biosignal purposes.

A property of molecular, in particular, conforma-tional, isomerism is inherent in biopolymers. Confor-mation of biomolecules is of great importance for theirfunctioning. Thus, the processes of replication and syn-thesis of proteins are accompanied by transformationsof segments of the DNA double helix from “physiolog-ically normal” B-conformation into A-conformation.Biochemically active states of enzymes correspond tocertain conformations. The transitions between differ-ent conformational states of biomolecules separated bybarriers of the height significantly larger than thermalenergies kT normally require the use of energy ofhydrolysis of ATP, which is a universal source of energyin biosystems.

All the aforesaid is also applicable to fragments ofbiological macromolecules, segments of DNA helix,active sites of enzymatics, etc.

One of the possible primary mechanisms of the influ-ence of EMR on biosystems is the initiation with its helpof the transitions between different isomer states of bio-molecules (as applied to proteins, this idea was sug-

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gested by Frölich). According to this assumption, theenergy of absorbed radiation is spent on crossing the bar-rier separating conformers. The electrical activity of thecorresponding degree of freedom of a biopolymer is nec-essary for the realization of this process.

Functioning of some biomacromolecules, includingenzymes, is determined, to a great extent, by the pro-cesses in their active sites surrounded by biopolymerchains folded up into globules. It is reasonable toassume that resonance interaction with field results inthe excitation of dipole-active vibrations of monomers,which, in turn, induce vibrations of an active site capa-ble of bringing it into another conformational state(antenna effect [94]).

Excitation of dipole oscillations in separate groups(monomers) of a biopolymer may explain the influenceof EMR on the form of a macromolecule (inducedunfolding of a biopolymer). Considering a biopolymerchain as a long thread with bending rigidity a and cur-vature ρ = r–1 (so that the density of free energy forsmall ρ is given by Φ = Φ0 + aρ2/2) and using theexpression W ~ –δPhi/T for the probability of fluctua-tions “twisting” the molecular thread, we obtain the fol-lowing expression for the mean square of scattering fordistant points [96]:

where L is the distance along the thread. Thus, a trans-

formation of a linear form of molecule ~ L2 into a

coil /L2 � 0 is determined by a ratio kT/a. The actionof EMR exciting dipole-active oscillations leads to therepulsion of neighboring dipoles (and, consequently, tothe increase in the rigidity of the chain), which influ-ences the form of a macromolecule.

The efficiency of biological transport systems isprovided, to a great extent, by the polymer properties ofbiomolecules. Resonance interaction of EMR withmonomers of the chain may substantially influencecharge transport, in particular, in transmembrane ionchannels. Thus, here we speak about a possibility ofregulation of charge transport through a membranewith the help of weak EHF EMR at rather high fieldintensities (Em ~ 105 V/cm) on a membrane.

1.5. Solitons in Biomacromolecules

1.5.1. Introduction. The diversity of living organismsis determined by different combinations of the sameatomic groups and compounds. Living organisms com-prise six main chemical elements: nitrogen, oxygen, car-bon, hydrogen, phosphorus, and sulfur. The content ofcalcium, potassium, sodium, magnesium, iron, copper,etc. is substantially smaller. Protein molecules are builtof only 20 amino acids. The main elements of thegenetic apparatus of a cell are formed by four types ofnucleotides (elementary parts of DNA and RNA mole-cules that code a certain portion of genetic informa-

R2

2 a kT⁄( )2LkT a⁄( ) 1– e

LkT 2⁄–,+=

R2

R2

tion). What is the difference between living organismsand inorganic objects? The main differences are as fol-lows. Living organisms are, first of all, open systemswith continuous exchange of energy. Living organismsare intrinsically heterogeneous, they possess a propertyof self-organization, and they are in a state far fromthermodynamic equilibrium. Their evolution is one-directional and results in a state of “death,” because,upon the achievement of complete thermodynamicequilibrium, all the proteins of a living organism disin-tegrate. Such physical notions as entropy, free energy,and other thermodynamic functions are arbitrarilyapplicable to biosystems.

One of the main methods here is the method of sys-tem simulation, where only key processes determiningthe phenomenon under study are taken into account,whereas the secondary processes are ignored. The pos-itive outcome of simulation depends, to a great extent,upon a correct choice of the simplified model and themethods of its description. Below, we consider the non-linear processes playing an important role in bioener-getics. In particular, a process of vibrational energytransfer along a protein molecule is considered on thebasis of a hypothesis of solitons, i.e., waves that satisfynonlinear equations and that spread inside a living tis-sue virtually without losses and changes in their form.

A cell is an elementary structural unit with the sizeof about 10–3 cm. A bacterium is an example of the sim-plest unicellular organism. Inside a surface layer of acell, cytoplasm and nucleus are located (for cells ofhigher animals and plants). Nuclei of cells containchromosomes, carrying genetic information. In addi-tion to the nucleus, some inclusions called organellescan be found inside a cell. These inclusions have theirown membranes, similarly to the external shell of a cell.The most important function of organelles is the supplyof a cell with energy due to synthesis of adenosinetriphosphoric acid (ATP) in the course of hydrolysis(oxygenation of organic compounds). The first studiesof the spatial structure of the cells by means of electronmicroscopes showed that all the inclusions are not free intheir motion inside a cell but occupy certain places dueto the fixation of organelles in a complex 3D structure ofcytoskeleton. It determines the shape of a cell and partic-ipates in the transfer of energy and information.

Cellular membranes consist of molecules of lipids,proteins, and carbohydrates (small amounts). The mainphysicochemical properties of membranes are deter-mined by the molecules of lipids, which, figurativelyspeaking, consist of a tail and a head. The head eithercarries an electric charge or possesses a rather largeelectric dipole moment. The tail of a lipid moleculeconsists of two parallel carbohydrate chains. Since awater molecule has a large dipole moment, the head oflipid is attracted to water molecules, i.e., it is hydro-philic. The tail of a lipid molecule is nonpolar and isrepulsed by water. Thus, it is hydrophobic. At the watersurface, these molecules are arranged in such a way that

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the hydrophilic head is plunged into water and thehydrophobic tail is exposed from it. In aqueous solu-tions, lipid molecules tend to organize closed construc-tions of a bubble type. At the surface, a double layer isformed with the heads of lipids on the outer and innersides. Protein molecules can be found at the exterior orinterior surfaces of the lipid bilayer or they can evenpierce the membrane. The latter proteins may be of twotypes. Stick-like proteins belong to the first type and theglobular proteins belong to the second one. Long alpha-helical structure of fibrous proteins is located outsidethe membrane, whereas the globular structure belongsto the hydrophobic inner part of the membrane. A dou-ble layer of lipid molecules provides positive charge ofthe exterior surface of a living cell and negative chargeof the interior surface. A potential drop appears in thiscase, and the resulting electric field inside the mem-brane may be as high as 102 kV/cm (for the membranethickness of about 5 nm). About 50% of energy pro-duced by a cell is required to maintain its normal func-tioning, potential drop, and difference of ion concentra-tions inside and outside the cell. Complexes of proteinswith lipids perform transportation of substances andions inside a cell thus providing its vital activity. Pro-teins play an important role in a living organism. How-ever, they cannot synthesize themselves. The synthesisof protein molecules is performed by the molecules ofgenetic apparatus: desoxyribonucleic acid (DNA) andribonucleic acid (RNA). Synthesis of one peptide bondbetween amino acids in cells of living organismsrequires 0.14–0.21 eV of energy. However, the disinte-gration of a protein molecule is a rather slow process.For example, the half-life of the proteins of cardiacmuscle equals approximately 30 days. That is why anorganism must gradually replace the disintegrating pro-tein molecules with new ones. Such a state of a systemis classified in physics as a metastable one. A transitionfrom a metastable state to stable thermodynamic equi-librium takes a long time because of a high potentialbarrier to be crossed in the course of a transition.Enzymes help to lower the potential energy barrier andaccelerate this process. A repeated process of polymer-ization results in a formation of long chains of proteinmolecules (polypeptides) with equally spaced peptidegroups. In 1951, Poling and Kory demonstrated that apolypeptide chain may form an ordered right-handhelix. Such helical molecules were called alpha-heli-ces. Transverse hydrogen bonds of a polypeptide chainallow the existence of another elementary configurationsuggested by Poling. This configuration is a zigzagfolding of a polypeptide chain. It is called a beta-sheet.Generally, a polypeptide chain consists of interchang-ing alpha-helical and beta-sheet segments and is foldedto form a globular structure.

1.5.2. Davydov soliton mechanism of energy trans-fer in protein molecules. Vital activity is associatedwith energy consumption and continuous metabolism.Energy in cells is stored due to synthesis of ATP mole-cules from ADP molecules. Thus produced ATP mole-

cules diffuse to the places where the energy is consumedand release stored energy in the presence of enzymes inthe course of back reaction. This energy is utilized byion pumps and for the synthesis of proteins and othermolecules.

Because of large sizes of organic molecules, thesites of energy storage (ATP synthesis) and energy con-sumption (ATP hydrolysis) are separated by the dis-tances significantly larger than interatomic ones. Thus,a question arises about the mechanism of effectiveenergy transfer along large protein molecules. As it wasalready mentioned, ATP hydrolysis results in theenergy release of about 0.5 eV. This energy is sufficientfor effective excitation of only vibrational degrees offreedom of the c = 0 atomic group, which is a constitu-ent of peptide groups of all proteins. However, it is wellknown that, in molecular crystals, the lifetime of vibra-tional excitations is about 10–12 s. Therefore, the trans-fer of vibrational excitation from one peptide group toanother must decay quickly at short distances, and themechanism of effective energy transfer in a cell has noadequate physical interpretation. The question aboutenergy transfer remained open up to a publication byA.S. Davydov and N.I. Kislukha [96]. It was demon-strated therein that waves of collective nature maypropagate in helical protein molecules without energylosses and changes in the shape. Such waves are calledsolitons. Thus, the “crisis” in bioenergetics was over-come. Many studies carried out since that time in manycountries of the world proved the feasibility of effectiveenergy and, possibly, information transfer by solitonsinside a cell and between the cells in biological sys-tems. An extremely high stability of soliton isexplained by the balance of nonlinearity and dispersionof a medium where the soliton was produced. Nonlin-earity of the medium presumes the description of theprocesses on the basis of nonlinear dynamic and kineticequations. Dispersion of the medium is treated as adependence of the phase velocity of wave propagationupon the wavelength. A strong interaction of wavestakes place in nonlinear media. If, in a wave packet con-sisting of a group of monochromatic waves, an energytransfer takes place from the slower waves to the fasterones, such a packet will not spread and will propagateas a single entity like a particle.

Thus, let an intrapeptide vibration amide-1 (with avibrational energy of 0.21 eV) be excited at an end ofalpha-helical protein molecule as a result of ATPhydrolysis. Peptide groups are bound to each other bythe interactions of two types. The interactions of thefirst type are dipole–dipole interactions that are propor-tional to the square of the electric dipole moment (0.35 D)and are inversely proportional to the cube of the dis-tance between peptide groups. The interactions of thesecond type are deformation interactions determinedby displacements of peptide groups from their equilib-rium states in the molecule.

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MECHANISMS OF INTERACTION OF ELECTROMAGNETIC RADIATION 635

The excitation of the first type is characterized by auniform distribution of its probability along the entireprotein molecule without violations of periodicity ofpeptide groups. These excitations are monochromaticwaves of fixed energy, frequency, and phase velocity,and they are called excitons (physical analogy betweena wave and a particle). Excitons determine the mainproperties of light absorption by protein molecules.Note that the efficiency of energy transportation by freeexcitons is low because of dispersion in a medium,which results in spreading of the wave packet of an exci-ton and, consequently, in a strong dispersion of energy.

Deformation interaction plays an important role inenergy transfer along a protein molecule. Since thepeptide groups are bound to each other by weak hydro-gen bonds, their displacements from the equilibriumstate in the molecule are large and the nonlinearitydevelops in interaction of this type to a full extent.

In reality, interactions of these two types develop ina protein molecule simultaneously and influence eachother. The following model was proposed for the solu-tion of this problem [97]. A one-dimensional chain con-sisting of carbonyl oscillators with dipole–dipole inter-action between them was considered. In other words,the interaction between excitons described by means ofquantum mechanics and acoustic phonons, i.e., collec-tive vibrations of oscillators near their equilibriumstates, was taken into account. In the approximation ofa continuous system (when the distance between theoscillators is less than the typical width of a soliton) themodel is described by a set of two coupled differentialequations

(1)

where ψ is the wave function of the exciton, m is itsmass, a is a constant, L is the displacement of oscilla-tors of mass M from equilibrium states, and σ is aparameter of correlation between excitons andphonons.

The solution of this set is a single wave (a bell-shapesoliton) traveling along the chain by means of its defor-mation induced by the wave itself. It is customary tocall this solution the Davydov soliton. Note that freeexcitons do not interact with each other and cannot pro-duce a single wave of a stable form. The interactionbetween excitons takes place through the deformationof the chain due to the displacements of oscillators rel-ative to the equilibrium state. It is due to the deforma-tion of the chains, or phonons, that the interactionbetween excitons occurs. A similar situation is wellknown in the theory of superconductivity. Electrons,the main carriers of current, acquire an attracting force(in spite of the fact that they bear the same electric

iψ∂t∂

------- 12m-------∂2ψ

x2∂

---------– σψ L∂x∂

------, �+ 1,= =

∂2

t2∂

------ U02 ∂2

x2∂

--------–⎝ ⎠⎜ ⎟⎛ ⎞ L∂

x∂------ aσ

M------

x∂∂ ψ 2

,=

charge) due to electron–phonon interaction and form aself-organized state of superconductivity.

Depending upon the values of parameters in (1), theDavydov soliton can be considered as a result of disloca-tion motion with no regard to exciton. In this case, set (1)is reduced to the nonlinear sine-Gordon equation [98]:

(2)

On the other hand, the Davydov soliton may betreated as an autolocalized state of an exciton. In thiscase, the deformation of the chain manifests itself as aself-regulation of the exciton, whose motion isdescribed by the well-known nonlinear Schrödingerequation [98]:

(3)

where σ is the parameter of self-action. What does govern the efficiency of energy transfer

by means of Davydov solitons? Firstly, the change froman exciton state to a soliton one gives an energy gaindue to exciton coupling with a deformation grating.The softer the chain and the larger the coupling param-eter σ, the more stable the soliton. Secondly, the speedof propagation of a soliton will always be less than thespeed of sound in the chain because of the interactionof exciton with acoustic vibrations (phonons). It is dueto this circumstance that exciton does not experiencelosses associated with the emission of phonons.Thirdly, according to the Franck–Condon principle [99],the absorption of light in this system of oscillators is notaccompanied by the change of their coordinates at themoment of quantum transition. Thus, light excites avery small number of solitons, and solitons do not emitlight. Solitons can emit light only at the ends of a mol-ecule, which is a result of topological stability of soli-tons. Hence, solitons can appear in relatively shortalpha-helical segments of protein molecules that areparts of majority of enzymes and proteins piercing cel-lular membranes.

1.5.3. Solitons in polynucleotides. The manifesta-tions of nonlinear dynamics in different natural phe-nomena are quite diverse [99]. It is important to knowwhen studying experimentally the biological objectswhich particular aspect of nonlinear dynamics is appro-priate and how it shows up in experiment. In particular,evidence exists [100] that similar solitons can beexcited in polymer nucleic acids. Here also dipole–dipole interaction exists between the chains of carbonyloscillators, and they can be considered as two helicalcomplexes of homopolynucleotide [101] or arbitrarynucleotide sequences [102]. Deformation of the gratingcan be expressed in terms of the change in distancebetween the planes of neighboring residues in a chain.

∂2f

t2∂

-------- A2∂2

f

x2∂

-------- V0 f .sin+=

iψ∂t∂

------- 12m-------∂2ψ

x2∂

--------- 2σψ ψ 2,±–=

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RESHETNYAK et al.

The weak bond in such a carbonyl chain is determinedby the weakness of stacking interaction in the frames ofduplex. In spite of the complexity of the dynamics ofDNA, RNA, and nucleoproteids in vitro and in vivo, itsstudy is urgent because such nonlinear dynamics in itsessence is a new semiotic area generating acoustic andelectromagnetic symbol structures utilized by biosys-tems for their own self-organization [127].

The model of DNA dynamics proposed in [103, 104]is close in its physical meaning to the Davydov solitonmodel. The authors of [103, 104] consider deformationvibrations of phosphodiester O–P–O bond as an exci-ton and the stretching and contraction of Watson–Crickhydrogen bonds as phonons (collective vibrations of agrating). Kashcheev [105, 106] developed the Davydovmodel and applied it to proton transfer along the chainsof peptide hydrogen bonds under their interaction witha system of heavy ions. Classic Davydov solitons areconsidered also with regard to the interaction betweenexcitons and optical and acoustic phonons.

Frölich [107] demonstrated that some biologicalmolecules, in particular, proteins and DNA can be sim-ulated by a chain of oscillating and interacting dipoles.Coherent external action on this system maintains itsnonequilibrium state. One may state that polymers withhigh electric polarizability possess electron–phononinteraction, to be more precise, the interaction betweenelectric and elastic degrees of freedom. If an externalparameter exceeds a certain threshold, the low-fre-quency vibrational mode is excited in a coherent waywith a formation of Bose condensate. One deals with asimilar situation in laser systems that pass over the las-ing threshold. Thus, it is suggested to consider biopoly-mers as structures with laser pumping.

We should note papers [108, 109] among the worksdevoted to DNA solitons describing conformational spe-cific features of a double polynucleotide chain. In theseworks, a double helix is simulated by a chain of masseswith elastic forces. Transverse displacements of massesfrom equilibrium points are also taken into account.The Hamiltonian of this model is presented as

(4)

where m is the mass of a nucleotide particle, L is theconstant of elastic interaction between the neighboringmasses, and U(yi) is the potential energy of transversedisplacement from the equilibrium of the element i.

The Hamilton equations in this case are written as

(5)

Hm2----v i

2 L2--- yi yi 1––( )2

U yi( )+ +i 1=

N

∑ ,=

v i

dyi

dt-------,=

piH∂yi∂

------- mdv i

dt--------⇒– L yi 1+ 2yi– yi 1++( ) U' yi( ),–= =

yiH∂pi∂

-------dyi

dt-------⇒ v i,= =

where pi = mvi . The following system of nonlinear dif-ferential equations follows from (5):

(6)

If a soliton involves a large number of monomers,i.e., its typical size is much larger than the distance abetween the neighboring monomers, then a transfer ispossible to the continuous approximation or continuousdescription. Assuming that

where Δx2 is a physically infinitely small quantity, weobtain

(7)

where C0 = La2/m. Equation (7) is well known in physics of nonlinear

phenomena. For U'(y) ≠ ky, this equation is related tothe class of the Klein–Gordon nonlinear differentialequations [110]. Potential function U(y) is determinedby the anharmonicity range of displacements of mono-mers in the transverse (relative to the chain) direction.It can be seen that, in a particular case when U = U0cosy,this equation is reduced to the sine-Gordon equation.Analysis [108] of equation (7) with a potential contain-ing nonlinear terms revealed the possibility of existenceof nonlinear supersonic traveling waves in a molecularchain in the case of a physically reasonable choice ofthe parameters of the system. Any conformational tran-sition in a monomer unit of a polynucleotide chain isdescribed within the frames of this model by means ofconsidering a two-well potential known as the Lifshitzband. If the ends of the chain are in energetically unfa-vorable state (in case of binding to biologically activemolecules), then, within this model, a two-soliton solu-tion is possible (kink + antikink), i.e., two solitary wavesmoving along a nucleotide chain may exist.

In addition to the works describing these or those fea-tures of biological processes in protein molecules on thebasis of soliton theory, one may also find works doubt-ing the stability of the soliton solutions. Perez [111] con-sidered the interaction of Davydov solitons with grat-ing solitons (grating distortions of large amplitudewithout interaction with excitons). Numerical simula-tion revealed the disappearance of Davydov solitonseven in the case of collisions with grating solitons oflow energy. This effect may be related to the simulationof interactions of neighboring units in a molecularchain with the Lennard–Jones potential. Note that sucha chain is known in physics as the Toda chain [112].

mdyi

2

t2

d-------- L yi 1+ 2yi– yi 1++( ) U' yi( ).–=

dy2

dx2

-------- � 1

Δx------

yi 1+ yi–Δx

--------------------yi yi 1––

Δx-------------------–

Δx a=

yi 1+ 2yi– yi 1++

Δx2

---------------------------------------,=

∂2y

t2∂

-------- C0∂2

y

x2∂

-------- 1m----U' y( ),–=

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MECHANISMS OF INTERACTION OF ELECTROMAGNETIC RADIATION 637

In early DNA models [113–115], a DNA moleculewas considered as a single thread or two equal threadsrigidly bound to each other. The difficulties of explana-tion of experimental data necessitated the considerationof DNA as a duplex of two interacting threads (chains).The authors of [116, 117] suggest describing the mainproperties of the two-chain model with the help of asystem of two coupled sine-Gordon equations:

(8)

Kivshar [118] demonstrated that the solutions sug-gested earlier are special cases of (8) and that these solu-tions are unstable. Moreover, with regard to the fact thatthe system (8) is not integrable, collisions of solitons thatsatisfy this system are inelastic. Hence, the solitons ofsystem (8) can emit transverse phonons and decay.

1.5.4. Autosolitons in biological systems. An impor-tant feature of Davydov solitons and solitary-wavesolutions to the sine-Gordon and Klein–Gordon equa-tions is the fact that all of them are obtained within theframework of the theory of conservative or Hamiltonsystems. These equations are invariant relative to thesubstitution of time –t for t. In other words, Hamiltonsystems are unable to distinguish between “past” and“future,” they are unable to describe the dynamics oftransition to equilibrium state, i.e., they are non-Mark-ovian systems. Indeed, there is no interaction of nucleicacids with their hydrate–ion shell in the theory of con-servative systems. Taking into account the action of theenvironment on a biosystem, one could consider the lat-ter as a highly organized object in a medium with loworganization. The action of the environment on a bio-system determines its transition to thermal equilibrium.Far from the equilibrium state functions of distributionof the parameters may differ strongly from the equilib-rium ones and describe metastable steady dissipativestructures in a sense that the time of transition into theequilibrium state may be rather long due to the existenceof high potential barriers.

Thus, taking into account the action of fluctuating sto-chastic environment on a biosystem requires the applica-tion of nonlinear kinetic equations similar to that used inthe studies of nonequilibrium processes in semiconduc-tors for the description of the dynamics of informationbiopolymers [119, 120]. The construction of nonlinearkinetic equations is the main difficulty in such formula-tion of the problem. However, a substantial simplificationcan be achieved by means of analysis of the first severalmoments of kinetic equations. In physics, equations of gasdynamics and magnetic hydrodynamics may be examplesof such moment equations. These equations follow fromthe Boltzmann equation and are valid within certain timeintervals upon the achievement of the equilibrium state.It was demonstrated in [119, 120] that emergence and

∂2u1

t2∂

----------∂2

u1

x2∂

---------- u1sin– α u1 u2–( ),sin–=

∂2u2

t2∂

----------∂2

u2

x2∂

---------- u2sin– α u1 u2–( ).sin+=

evolution of ordered states (both spatial and temporal) canbe simulated by means of a system of nonlinear diffusionequations:

(9)

where Lij is the matrix of transport coefficients and giare the nonlinear terms taking into account interrelationof the parameters of the biosystem.

An important feature of system (9) is its ability togive rise to traveling solitary waves (autosolitons) as aresponse to an external local disturbance [120, 121].The simplest mechanism of formation of an autosolitoncan be described on the basis of a system of two nonlin-ear diffusion equations. Because biological objects areintrinsically nonequilibrium systems, the approach totheir analysis on the basis of equations (9) seems to bepreferable as compared with that making use of theequations of conservative dynamics.

Solitons and autosolitons are known to be the solu-tions of nonlinear equations. Nowadays, there are nogeneral methods of the construction of their solutions.It is worthwhile to note the method of inverse problem[122, 123] is one of the most well-developed methodsof analysis of a certain class of nonlinear equations.The essence of the method lies in reducing the solutionof a nonlinear equation of a certain type to the solutionof an auxiliary linear integral equation. The simplestmethod of construction of the solution to a nonlinearequation is its presentation in the form of a wave travel-ing at a certain speed v. Let us demonstrate the construc-tion of the solution to, e.g., the sine-Gordon equation:

(10)

Let us look for a solution in the form of a travelingwave: u = u(z), z = x – vt. Then, we arrive at

To reduce the rank of this equation, we make a sub-stitution du/dz = P(u). As a result, we obtain an ordi-nary differential equation with separable variables:

Its solution is given by

(11)

where C is an arbitrary constant. The variables in (11) are also separable and its solu-

tion is presented as an implicit dependence of z on u:

(12)

xi∂t∂

------- ∇ LijΔx j( ) gi x1 … xN, ,( ),–j

∑=

∂2u

t2∂

-------- ∂2u

x2∂

-------- u.sin+=

v2

1–( )d2u

dz2

-------- u.sin=

v2

1–( )PdPdu------- u.sin=

dudz------

2

1 v2

–( )-------------------- u C+cos ,±=

zdu

2 1 v2

–( )1–

ucos C+

----------------------------------------------------- C1.+∫±=

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RESHETNYAK et al.

Taking into account that u is a physical and, conse-quently, positive variable, by an appropriate choice ofthe integration constant, one can build up a solution inthe form of a wave traveling at a speed v of a solitarywave. However, within this scheme, we cannot answerthe question as to whether we have found all the solu-tions to equation (10) or not.

Apparently, such a simplet method can be applied tofind a solution to two coupled nonlinear equations (9)for an autosoliton. The general methods of analysis ofsystem (9) are not developed yet. The method of inte-gration of the system of linear differential equations (9)is well developed. If the system of linear equations (9)is presented in the matrix form,

(13)

its solution can be presented as an expansion in a seriesin eigenvalues and eigenfunctions of self-conjugate

operator that has a positive spectrum of eigenvalues:

(14)

where λi and ψ(i) are the eigenvalues and the eigenvec-

tors of operator determined by the boundary condi-tions and c(i) are the vectors taking into account initialconditions.

Reshetnyak et al. [124, 125] demonstrated that, fortime-independent coefficients Lij, the series (14) isequivalent to the solution in the form of a series in the

powers of operator ,

(15)

where is the operator inverse of the operator and

xe is the solution to the homogeneous equation = 0containing integration constants that generally dependon time.

Applying operator to both sides of (15), we mayverify that expansion (15) is really a solution of equa-tion (14):

The vector xe can be considered as a state that thesystem approaches in the course of transition to ther-modynamic equilibrium, i.e., for t ∞, all the termsin the series, except the first one, disappear. To describethe stage when the system approaches the equilibriumstate, one can consider the first several terms of expan-sion (15). The further the analyzed state from the equi-librium state, the larger the number of terms in (15)

x∂t∂

----- Lx, Lij xi∂∂

Lij x j∂∂

, xx1

�

xN⎝ ⎠⎜ ⎟⎜ ⎟⎜ ⎟⎛ ⎞

,= = =

L

x ci( )ψ i( ) λit–( ), Lψ i( )exp

i 0=

∞

∑ λiψ i( ),–= =

L

E L1–∂ ∂t⁄=

x xe Exe E2xe …,+ + +=

L1–

L

Lxe

L

Lx L xe Exe E2xe …+ + +( )

x∂t∂

-----.= =

should be taken into account. Solution (15) makes a fulluse of the intrinsic property of a biological system—theMarkovian character, the or erasure of memory withtime and, as a consequence, the simplification of itsdescription.

Apparently, the symptotic in time approach to theanalysis of linear equations (13) is applicable to theconstruction to the nonlinear equations (9) provided xe

is considered as a solution of nonlinear homogeneous

equation = g and an iteration scheme is used forapproaching the equilibrium state. Within this scheme,the deviations of the distribution function for theparameters of a biosystem from their equilibrium val-ues can be significant.

Thus, in spite of the complexity of the structure ofbiological objects and diversity of the processes thereinone may state that nowadays sufficient evidence isavailable in the form of experimental data, theoreticalconcepts of functioning of biosystems, simulation ofdifferent processes in biopolymers for adequate com-prehension of fundamental attributes of living systems,namely, synthesis of energy and information and theirtransportation in time and space of organisms.

2. THE ROLE OF COLLECTIVE AND COHERENT EFFECTS

2.1. The Role of Coherent Effects in the EMR Excitation of Induced Collective Modes of Proteins and DNA

In simulation of dynamics of protein and DNA mac-romolecules, the latter are presented as chains of mono-mers involved in local nonlinearly related dipole-activevibrations. Combinations of these vibrations are treatedas the so-called collective modes, which characterizean ensemble as a whole from the energetic point ofview. According to modern concepts, these collective(cooperative) modes play an important role in thedynamics of protein and DNA macromolecules. There-fore, it is interesting to study the resonance action ofEMR on these collective modes.

The considerations below are carried out within theframework of a model of a selected cooperative mode.

2.1.1. Approximation of harmonic oscillator. Let usconsider an elementary system—harmonic oscillatorwith a quantum �ω0 excited by a resonance electric fieldE = E0cosωt. Then, the Hamiltonian is presented as

where is the operator of projection of the dipolemoment of a molecule on the direction of the field,�ω0 is the Hamiltonian of an isolated harmonic oscilla-tor, the matrix with elements εik = (i – 1)δik (i, k = 1, 2, …)corresponds to the operator , and � is the Planck con-stant. It is assumed that the field induces a high-fre-quency polarization only between neighboring pairs ofquantum levels.

Lxe

H �ω0ε E t( )μ,–=

μ

ε

ε

LASER PHYSICS Vol. 6 No. 4 1996

MECHANISMS OF INTERACTION OF ELECTROMAGNETIC RADIATION 639

Using a standard procedure [128], we arrive at theequation for the operator of the induced dipolemoment:

where T2 is the phase relaxation time and square brack-ets correspond to commutation operation. Taking intoaccount the relation between the squares of matrix ele-

ments and the relationship

( is density matrix with elements ρik), we can find theobserved value of the dipole moment

from the equation

(1)

where Ω = μ21E/� is the Rabi frequency. It is easy toderive an equation for operator ,

so that the average number of quanta ε = Sp( ) is gov-erned by

(2)

where T2 is the energy relaxation time.

For small detunings � 1 with � 1,we obtain from (1) and (2) [ε(0) = 0]:

(3)

where Δ = ω0 – ω and .

In a particular case when T2 � t and T2 � T1,the expression for the energy has the same form asthat obtained without taking into account coherenteffects [129],

d2μ

dt2

---------2T2-----dμ

dt------ ω0

2 1

T22

------+⎝ ⎠⎛ ⎞ μ+ +

ω0

�------E t( ) με[ ]μ[ ],=

μk 1+ k,2

kμ212

=

Sp ρ με[ ]μ[ ]( ) 2μ212

=

ρ

p Sp ρμ( ) μm 1+ m, ρm m 1+, μm m 1+, ρm 1+ m,+( )m 1=

∞

∑= =

p2T2----- p ω0

2 1

T22

------+⎝ ⎠⎛ ⎞ p+ + 2μ21ω0Ω ωt,cos=

ε

d εtd

------ εT1-----+

E t( )�ω0--------- dμ

dt------ μ

T2-----+⎝ ⎠

⎛ ⎞ ,=

ρε

ε εT1-----+

E0

�ω0--------- ωt p

pT2-----+⎝ ⎠

⎛ ⎞ ,cos=

Δ ω0⁄ T22ω0 Δ

ε Ω2

2 Δ2T2

2–+( )

---------------------------T1

T2----- 1 e

t T1⁄––( )

Δ2T2τ12( ) 1–

+

Δ2 τ122–

+----------------------------------e

t T1⁄–+

⎩⎨⎧

=

Δ2T2τ12( ) 1–

+( ) Δtcos Δ T21– τ12

1––( ) Δtsin+

Δ2 τ122–

+--------------------------------------------------------------------------------------------------------e

t T2⁄–

⎭⎬⎫

,–

τ121–

T11–

T21–

–=

εT1

2T2--------- Ω2

Δ2T2

2–+

-------------------- 1 et T1⁄–

–( ).=

It is seen that the energy approaches a stationarylevel with time; for small t, we have ε ~ t. In the opposite

limiting case ( and 0), we have from (3)

i.e., the stored energy pulsates with a frequency of opti-cal beats (see [130]). For small t we have ε ~ t2.4

2.1.2. Excitation of an anharmonic oscillator by aseries of phased pulses. A method based on the knownprinciple of phase stability can be used for the excitationof high levels of an anharmonic oscillator by a resonancefield. Veksler [132] was the first to suggest the principleof autophasing for infinite maintaining of a resonancebetween charged particles and a high-frequency field ina cyclotron-type accelerator.

This method is modified in our case in the followingway. If the frequency of an EMR pulse E = E0cosωtequals the transition frequency of the lowest pair of lev-els 1 and 2 (ω = ω21), then, in the absence of relaxation,the population of each of these levels will change period-ically in time with the Rabi frequency Ω21 = Eμ21/� [128].At time instants t = τ1 determined by Ω21τ1 = (2n + 1)π(n = 0, 1, 2, …), all the particles reside at the upper level.

Assume now that, at the instant τ1, the frequency ofradiation changed in a jumpwise manner and becameequal to the frequency ω32 of the higher lying pair oflevels. The particles “trapped” in resonance by a pulsewill start oscillating again but now between the levels 3and 2 with the frequency Ω32 = E0μ32/�. The choice ofthe phase of the next pulse according to the conditionΩ32τ2 = (2n + 1)π brings a particle even higher, etc.Thus, a series of phased pulses with the frequencies ω21,ω32, ω43, … theoretically allows one to excite high lev-els of an anharmonic oscillator.

Let us determine the portion of the particles that cango to high levels of a collective mode. Suppose thatexternal action consists of a series of sequential pulses.For the kth pulse of the duration τk + 1,k, we have

Within the interval tk – 1 < t < tk, the difference ofpopulations is Wk = ρk + 1,k + 1 – ρk,k, and the sum of pop-ulations Vk = ρk + 1,k + 1 + ρk,k satisfy, correspondingly,the equations

4 The idea of this method was proposed by Prof. A.N. Oraevskii.

T11–

T21–

ε Ω2

2Δ2--------- 1 Δtcos–( ),=

Ek t( ) Ek0 ωk 1+ k, t, kcos 1 2 …,, ,= =

tk 1– t tk, tk< < τi, τ0

i 0=

k

∑ t0 0.= = =

Wk2

T12-------Wk Ωk

2 1T1T2-----------+⎝ ⎠

⎛ ⎞ Wk1

T1T2-----------δ1k+ + + 0,=

Vk1T1-----Vk+

1T1-----δ1k 0, k 1 2 3…,, ,= = =

640

LASER PHYSICS Vol. 6 No. 4 1996

RESHETNYAK et al.

It is assumed that an inequality

is fulfilled. The conditions of coherent phasing of EMR pulses

allow one to determine their durations:

For a portion of particles occupying the kth excitedlevel at the instant of time

we have

In a particular case when T1 = T2 = T and Ωj ≡ Ω , thisexpression is reduced to (nk = 0)

For x = 102, a typical time of population of, e.g., thetenth level is about 0.3T, and the portion of excited mol-ecules at this level is about e–0.25.

2.1.3. Periodic phase modulation as a method ofcascade excitation. Let us consider the second way ofexcitation of particles by EMR allowing one to neutral-ize anharmonicity.

Let us enumerate levels 1, 2, 3, …, N, N + 1. Theenergy of level (N + 1) corresponds to the activation

Wk tk 1–( ) ρkk tk 1–( ), Wk tk 1–( )– 0,= =

Vk tk 1–( ) ρkk tk 1–( ),=

Ωk

E0kμk 1+ k,

�-----------------------,

1T12------- 1

2--- 1

T1----- 1

T2-----+⎝ ⎠

⎛ ⎞ .= =

2 Ωk2 1

T1T2-----------+⎝ ⎠

⎛ ⎞ 1T1----- 1

T2-----+>

Ωk' τk 2 T12Ωk'( )arctan 2πnk, nk+ 1 2 …,, ,= =

Ωk'2

Ωk2 1

4--- 1

T1----- 1

T2-----–⎝ ⎠

⎛ ⎞2

.–=

tk 1– τ j,j 1=

k 1–

∑=

ρkk

21 k–

T1T2Ω212

1 T1T2Ω212

+-------------------------------- 1 τ1 T12⁄–( )exp+[ ]=

× τi 2T1⁄i 2=

k 1–

∑–⎝ ⎠⎜ ⎟⎛ ⎞

exp

× τ j 2T1⁄–( )exp τ j 2T2⁄–( )exp+[ ].j 2=

k 1–

∏

ρkkx

2

1 x2

+-------------- y 2k 3–( )y–( ), k 2≥ ,expcosh=

tk 1– 2T k 1–( )y, x ΩT , y x x.⁄arctan= = =

energy. The frequencies of one-quantum transitions ofan anharmonic oscillator can be presented as

(4)

where ω21 is the frequency of transition 2 1, Δω isthe anharmonic shift, and Δω/ω21 � 1. A typical fea-ture of an anharmonic oscillator under the assumption(4) lies in the fact that the difference of frequencies oftransitions between neighboring pairs of levels does notdepend upon the number of the pair and equals the fre-quency of the anharmonic shift,

An expression for periods is valid with the sameaccuracy:

(5)

Note that the Veksler’s idea is actually based on rela-tionship (5), where Tm + 1,m is treated in [132] as theperiod of the mth cycle of rotation of a charged relativ-istic particle around the magnetic field. Thus, the differ-ence of times of two subsequent cycles remains con-stant irrespective of the energy of a particle, i.e., of m,which is the physical meaning of (5).

Let us consider the vibrations of an anharmonicoscillator in the electric field

(6)

where the carrier frequency ω equals the frequency ofone of the transitions (e.g., for the lowest pair 2 1)and the phase Φ(t) changes periodically in time withthe frequency equal to the frequency of the anharmonicshift Δω . Then, the spectrum of oscillations of the fieldwill evidently contain the frequencies of all one-quan-tum transitions of the cooperative mode. Thus, a peri-odic phase modulation with the frequency of anharmo-nicity allows one to realize a cascade mechanism ofpopulation of the levels of an anharmonic oscillator (6)in the Fourier representation:

(7)

In a particular case of sine modulation, i.e., whenΦ(t) = δsinΔωt, we have for spectral components

cn = Jn(δ),

where Jn is the Bessel function. A process of excitation of an oscillator by coherent

EMR is described by an equation for the density matrix(the Boltzmann equation)

(8)

ωm 1+ m, ω21 m 1–( )Δω m 1 2 … N, , ,=( ),–=

ωm m 1–, ωm 1+ m,– Δω.=

Tm 1+ m, Tm m 1–,– ΔT2πΔω--------.= =

E t( )E0

2----- e

i ωt Φ t( )+( )c.c.+( ),=

E t( )E0

2----- cne

i ω nΔω+( )tc.c.+

n ∞–=

∞

∑⎝ ⎠⎜ ⎟⎛ ⎞

,=

cnΔω2π-------- dt i Φ t( ) nΔω–( )t )[ ].exp

0

2π

∫=

i�ρ∂t∂

------ρ ρ0–

T--------------+⎝ ⎠

⎛ ⎞ Hρ[ ],=

LASER PHYSICS Vol. 6 No. 4 1996

MECHANISMS OF INTERACTION OF ELECTROMAGNETIC RADIATION 641

where the Hamiltonian in the dipole approximation isgiven by

The eigenvalues of Hamiltonian are the energiesof levels of the anharmonic oscillator, is the equilib-rium density matrix, T is the phenomenological time ofrelaxation (for diagonal elements T = T1, for nondiago-nal, T = T2). For simplicity, we assume that only oneresonance spectral component from expansion (7) actson each separate period (m + 1 m):

Then, the following system of equations will corre-spond to Boltzmann equation (8) (we consider onlyone-quantum transitions):

(9)

System (9) is of the order of 2N. It is assumed thatthe equilibrium value of the diagonal element of den-sity matrix ρ11 (the lowest quantum state) is equal tounity and the equilibrium values of other elements areequal to zero.

If we set Em + 1,m(t) = E0(t) in (9), then this systemdescribes, in particular, the excitation of a truncatedharmonic oscillator.

It is convenient to perform further analysis for a renor-malized density matrix. Let us introduce the variables

The quantity ym has a simple physical meaning: this isa portion of particles populating the levels from (m + 1) to(N + 1) (i.e., to the level of the activation energy).

H H0 E t( )μ.–=

H0

ρ0

Em 1+ m, t( )E0

2-----cm iωm 1+ m, t( )exp c.c.+[ ].=

i� ρmmρmm δ1m–

T1-----------------------+⎝ ⎠

⎛ ⎞

= Em 1+ m, t( )Am 1+ m, Em m 1–, t( )Am m 1–, 1 δ1m–( ),–

i� ρm m 1+,ρm m 1+,

T2----------------+⎝ ⎠

⎛ ⎞ �ωm 1+ m, ρm m 1+,–=

– Em 1+ m, t( )μm m 1+, ρm 1+ m 1+, ρmm–( ),

i� ρm 1+ m,ρm 1+ m,

T2----------------+⎝ ⎠

⎛ ⎞ �ωm 1+ m, ρm 1+ m,=

+ Em 1+ m, t( )μm 1+ m, ρm 1+ m 1+, ρmm–( ),

Am 1+ m, μm 1+ m, ρm m 1+, μm m 1+, ρm 1+ m, ,–=

ρmm

m 1=

N 1+

∑ 1, m 1 2 …., ,= =

ym 1 ρ jj t( ), mj 1=

m

∑– 1 2 … N ., , ,= =

It is easy to demonstrate that the system (9) may bereduced to the following system [in calculations, weneglected the second harmonics exp(2iωm + 1,mt)]:

(10)

Here, we restrict ourselves to the case Ωm ≡ Ω0. Let usintroduce collective coordinates

(11)

where

(12)

Then, we obtain a system of uncoupled equations for zk,

(13)

Here, the notations

(14)

are introduced. Solving (13) and substituting the result into (12), we

obtain

(15)

ym1T1----- 1

T2-----+⎝ ⎠

⎛ ⎞ ym1

T1T2-----------ym+ +

= Ωm

2

2------- δ1m ym 1– 2ym– ym 1++ +( ),

m 1 2 … N ym 0( ), , , ym 0( ) 0,= = =

y0 yN 1+ 0, Ωm

E0cmμm 1+ m,

�-----------------------------.= = =

zk2

N 1+------------- ym

mkπN 1+-------------, ksin

m 1=

N

∑ 1 2 … N ,, , ,= =

ym2

N 1+------------- zk

mkπN 1+-------------, msin

k 1=

N

∑ 1 2 … N ., , ,= =

zk1T1----- 1

T2-----+⎝ ⎠

⎛ ⎞ zk νk2 1

T1T2-----------+⎝ ⎠

⎛ ⎞ zk+ +Ω0

2

2------- f k,=

zk 0( ) zk 0( ) 0, k 1 2 … N ., , ,= = =

νk 2Ω0kπ

2 N 1+( )--------------------, f ksin 2

N 1+------------- kπ

2 N 1+( )--------------------sin= =

ym1

N 1+------------- mkπ

N 1+------------- kπ

N 1+-------------

Ω0

ωk'------

⎝ ⎠⎜ ⎟⎛ ⎞ 2

Fk t( ),sinsink 1=

N

∑=

Fk t( ) 1ωk'

ωk

------ t–2---- 1

T1----- 1

T2-----+⎝ ⎠

⎛ ⎞ ωkt φk+( ),cosexp–=

ωk2 νk

2 14--- 1

T1----- 1

T2-----–⎝ ⎠

⎛ ⎞2

; ωk'2

– νk2 1

T1T2-----------;+= =

φkcosωk

ωk'------; k 1 2 … N ., , ,= =

642

LASER PHYSICS Vol. 6 No. 4 1996

RESHETNYAK et al.

Diagonal matrix elements ρmm are determined by therelationships

(16)

For the average number of quanta in an oscillatorwe have

(17)

The physical structure of these relations is evident:formulas (15)–(17) are the superposition of the solutions

of the problem. If the condition ωk > ( )/2 is met,the solutions describe damped oscillations. The latter aredetermined by a coherent mechanism of interaction ofthe field and a medium. The spectrum of pulsations isgiven by the formula for ωk (k = 1, 2, …, N). In a partic-ular case when T1 = T2, the frequencies are ωk = νk.

For t ∞, the amplitudes of pulsations go down,and the solutions approach the stationary level. Usingstandard methods of operational calculus, one can dem-onstrate that the stationary values of the correspondingvariables are

(18)

For strong fields, β = � 1 (α 0), we have

(19)

i.e., the saturation regime is realized where the popula-tion probabilities of levels of anharmonic oscillatorsare equal.

For weak fields (β � 1), the dependences are of thepower type:

ρmm ym 1– ym–=

= 1

N 1+------------- m 1–( )kπ

N 1+------------------------sin mkπ

N 1+-------------sin–

k 1=

N

∑

× kπN 1+-------------

Ω0

ωk'------

⎝ ⎠⎜ ⎟⎛ ⎞ 2

Fk t( ), msin 1 2 … N 1.+, , ,=

ε m 1–( )ρmm

m 1=

N 1+

∑ ym

m 1=

N

∑= =

= 1

N 1+------------- 1 1–( )k

–[ ] kπ2 N 1+( )--------------------

Ω0

ωk'------

⎝ ⎠⎜ ⎟⎛ ⎞ 2

Fk t( ).cosk 1=

N

∑

T11–

t21–

+

ym 0,N 1 m–+( )cosh α

N 1+( )sinh α--------------------------------------------,=

ρmm 0, 2α 2⁄( )sinh

N 1+( )sinh α--------------------------------- N 3/2 m–+( )α,cosh=

m 1 2 … N 1,+, , ,=

ε 12--- Nα 2⁄( )sinh

α 2⁄( ) N 1+( )α 2⁄[ ]coshsinh-------------------------------------------------------------------------,=

αcosh 11

Ω02T1T2

-------------------.+=

Ω02T1T2

ym 0,N 1 m–+

N 1+-----------------------, ρmm 0,

1N 1+-------------, ε N

2----,= = =

ym 0, β 2⁄( )m, ρmm 0, β 2⁄( )m 1–

, ε β 2.⁄≈ ≈ ≈

In a purely coherent regime ( � 1/T1T2), forsmall times t � T2 < T1 (the times of phase relaxationand field are long), the population probabilities of thelevels of oscillator and the average number of quanta init are given by

(20)

where the notations are introduced:

(21)

The first relationship in (20) generalizes the knownrelationship for a trivial two-level medium [128] (exactresonance) to an arbitrary finite number of quantumlevels. It follows from (20) and (21) that the probabili-ties ρmm oscillate near the values (N + 1)–1, and the num-ber of quanta oscillates near the value N/2. Note thatthe spectrum of pulsations of the average number ofquanta stored contains only the frequencies with oddindices.

In the opposite case

(22)

the role of coherent effects is insignificant. The func-tion Fk(t) in (15)–(17) becomes

(23)

In the next section, we demonstrate that relationship(23) can be obtained directly from standard balanceequations for diagonal matrix elements of the densitymatrix. Hence, conditions (22) define the range ofapplicability of the latter equations (for a constant fieldamplitude).

2.1.4. An approach based on equations of balancetype. A standard representation for balance equationsfor population probabilities of the levels of an oscilla-tor ρm = ρmm is

(24)

νk2

Δρmm Umk νkt,cosk 1=

N

∑=

Δε Vk νkt,cosk 1=

N

∑=

Δρmm ρmm t( ) N 1+( ) 1–,–=

Δε ε t( ) N 2,⁄–=

Umk1

N 1+------------- mkπ

N 1+-------------cos m 1–( )kπ

N 1+------------------------cos– ,=

Vk1

2 N 1+( )--------------------- 1–( )k

1–[ ] kπ2 N 1+( )---------------------.coth=

4 T22νk

2T2 T1⁄+( ) � 1, T2 � 1,

Fk t( ) 1 T11–t 1 T1T2νk

2+( )–[ ].exp–=

ρm

ρm δm1–T1

--------------------+

= σmIm ρm 1+ ρm–( ) σm 1– Im 1– ρm ρm 1––( ),–

ρm

m 1=

N 1+

∑ 1, ρm 0( ) δm1.= =

LASER PHYSICS Vol. 6 No. 4 1996

MECHANISMS OF INTERACTION OF ELECTROMAGNETIC RADIATION 643

For the cross sections of induced transitions σm =σm + 1,m (exact resonance) and the spectral intensitiesIm = Im + 1,m, we have

Let us solve the system (24) in general case, when thespectral amplitudes of the field are time-dependent, i.e.,

Introducing, similarly to Section 2.3, variables ym,we obtain from (24) a system of equations:

(25)

Let us consider only the case Ωm = E0μm + 1,m/� ≡ Ω0.Then, the solution of the system (25) is formally deter-mined by (12), and the collective coordinates zk obeythe equations

(26)

Recall that νk and fk are determined in (14). Equa-tions (26) are easily integrable. Substituting zk(t) in (12),we find the dependence of ym on time. This dependencehas the form of (15) where Fk(t) is given by

(27)

The form of presentation for probabilities ρmm andthe average number of quanta of an oscillator ε is simi-lar to (16), (17), but Fk(t) is defined by (27).

In a particular case g(t) = 1, (27) coincides withexpression (23), which was obtained earlier for theincoherent regime.

If the spectral amplitudes of the field change period-ically in time, g(t) = cosνt, then, in the absence of col-lisional relaxation (T1 ∞), we obtain

(28)

σm 4πT2ωm 1+ m,μm 1+ m,

2

�c-----------------,=

Imc

8π------

Em 1+ m,2

�ωm 1+ m,---------------------, σmIm

T2

2-----

Em 1+ m, μm 1+ m,

�----------------------------------⎝ ⎠

⎛ ⎞2

.= =

Em 1+ m, Em0g t( ).=

ym1T1-----ym+

Ωm2

2-------T2g

2t( ) δ1m ym 1– 2ym– ym 1++ +( ),=

m 1 2 … N ., , ,=

zk

zk

T1----- 1 νk

2T1T2g

2t( )+[ ]+

Ω02T2

2------------- f kg

2t( ),=

zk 0( ) 0, k 1 2 … N ., , ,= =

Fk t( ) T11–

1 νk2T1T2+( )e

ψk t( )–dt'g

2t'( )e

ψk t'( ),

0

t

∫=

ψk t( ) T11–

t νk2T1T2+( ) g

2t'( ) t'.d

0

t

∫=

Δρmm Umk γ k t( )–( ),expk 1=

N

∑=

Δε Vk γ k t( )–( ),expk 1=

N

∑=

It follows from the second relationship in (28) that,neglecting coherent effects gives the number of vibra-tional quanta stored ε < N/2 because Vk < 0. It meansthat, at any instant, the population probability of thelower half of all levels in the collective mode exceedsthat of the upper half.

In the next section, we will demonstrate that, in thecase of periodic modulation of the amplitude, thecoherent mechanism of interaction can play an impor-tant role even for long time intervals t � T1, T2. In thiscase effects that cannot be described within the frame-work of rate equations arise.

2.1.5. Coherent effect of conversion of distributionover the levels of an anharmonic oscillator (parametricexcitation). It follows from considerations of Section 2.3that, in the case of interaction of an anharmonic oscil-lator with the field whose frequency is periodically mod-ulated in time [see (7)], the maximal possible number ofquanta stored equals N/2. In the coherent regime, this isan average around which the energy of an oscillator ispulsating. If coherent effects are not substantial, thenε N /2 for strong fields (the saturation regime).

An important question arises whether it is possible,in principle, to implement conditions under which thenumber of quanta exceeds N/2. In this case, we con-sider an overexcited state of a quantum oscillator; aninverted distribution over its levels corresponds to sucha state. It is a priori clear that a coherent mechanism ofresonance interaction of the electromagnetic field withan anharmonic oscillator must be a basis for a methodused to realize the corresponding conditions.

Coherent effects involve, in particular, pulsatingchanges in the population probability of levels. In caseof a two-level oscillator, the frequency of pulsationscoincides with the Rabi frequency for a given transi-tion. In a multilevel system, we should analyze thespectrum of pulsations. Assume that the amplitude ofthe field (7) changes periodically in time with the fre-quency ν:

(29)

Then, the Rabi frequencies corresponding to differ-ent transitions of an anharmonic oscillator will be mod-ulated with the same frequency. Selecting properly thefrequency and the modulation depth, one can imple-ment parametric excitation of the oscillator. Here, thesituation is similar to the oscillation of a pendulum witha pulsating point of support (Kapitsa pendulum [131]).A proper choice of modulation frequencies stabilizes anew (dynamic) equilibrium state of a pendulum. An over-excited state is an analog in the case of coherent excitationof an oscillator by a field. In the presence of amplitude

γ k t( )12---νk

2T2t 1 2νtsin

2νt----------------+⎝ ⎠

⎛ ⎞ .=

E t( )E0

2-----g t( ) cne

i ω nΔω+( )tc.c.+

n ∞–=

∞

∑⎝ ⎠⎜ ⎟⎛ ⎞

.=

644

LASER PHYSICS Vol. 6 No. 4 1996

RESHETNYAK et al.

modulation (29), a system of equations similar to (10)is given by

(30)

Note that the change in the field amplitude resultsnot only in the modulation of the Rabi frequency butalso in the modulation of the “friction coefficient.”Such an effect is intrinsically impossible within theframework of the classical approach.

The solution of system (30) is presented in the formof (12) where zk satisfies the equations

(31)

Below, we consider the case T1 = T2 = T. One canshow that the system (31) allows exact solution for anarbitrary function g(t):

(32)

Having determined zk(t), we find ym(t) from (12), sothat for Δρmm and Δε we have

(33)

Let us consider now the case of periodic modulationof the of field amplitude,

g(t) = cosνt.

If the modulation period is shorter than the relax-ation time, Tν � T, then, for time intervals Tν � t � T,averaging of (33) over the period Tν yields

(34)

ym1T1----- 1

T2----- g t( )

g t( )--------–+⎝ ⎠

⎛ ⎞ ym1T1----- 1

T2----- g t( )

g t( )--------–⎝ ⎠

⎛ ⎞ ym+ +

= Ω0

2

2-------g

2t( ) δ1m ym 1– 2ym– ym 1++ +( ).

zk1T1----- 1

T2----- g t( )

g t( )--------–+⎝ ⎠

⎛ ⎞ zk1T1----- 1

T2----- g

g---–⎝ ⎠

⎛ ⎞ zk+ +Ω0

2

2------- f kg

2t( ).=

zk αk 1 Gk t( )–[ ], k 1 2 … N ,, , ,= =

Gk t( ) et T⁄– τk t( )cos=

+1T---e

t T⁄– τk t( ) τk t'( )–[ ]et' T⁄–

cos t',d

0

t

∫

τk t( ) νk g t'( ) t', αkd

0

t

∫ 12--- 2

N 1+-------------

πk2 N 1+( )--------------------.cot= =

Δρmm UmkGk t( ),k 1=

N

∑=

Δε VkGk t( ).k 1=

N

∑=

ρmm⟨ ⟩ 1N 1+------------- Umk

νk

ν-----⎝ ⎠

⎛ ⎞ ,k 1=

N

∑+=

ε⟨ ⟩ N2---- Vk J0

νk

ν-----⎝ ⎠

⎛ ⎞ .k 1=

N

∑+=

In particular, for a two-level system (N = 1) we have

Hence, the probability of population of the upper stateexceeds 1/2 in certain ranges of frequencies of amplitudemodulation where the Bessel function J0 is negative.

For long time intervals (t � T ), functions Gk(t) in(33) and (34) are given by

(35)

Thus a coherent mechanism of interaction deter-mines undamped oscillations of diagonal elements ofthe density matrix for times t longer than the relaxationtime of the system, and the frequencies of pulsationsare aliquot to the modulation frequency ν.

Averaging (35) over the period Tν, we obtain

(36)

where x = (νkT)–1 and i is imaginary unit. In a particularcase x � 1, (33) is reduced to

(37)

It follows from (37) that ⟨ε(∞)⟩ < N/2. If the modulation law is given by the relationship

g(t) = 1 + γcosνt, then, for the functions Gk(t) in (33),one can easily get (Tν = 2π/ν � t � T)

(38)

Thus, the pulsation spectrum consists of combina-tion frequencies

ρ11⟨ ⟩ 12--- 1 J0

Ω0

ν------⎝ ⎠

⎛ ⎞+ , ρ22⟨ ⟩ 12--- 1 J0

Ω0

ν------⎝ ⎠

⎛ ⎞– .= =

Gk t( ) Pk t( )νk

ν----- νtsin⎝ ⎠

⎛ ⎞cos Qk t( )νk

ν----- νtsin⎝ ⎠

⎛ ⎞ ,sin+=

Pk t( )J2n νk ν⁄( )

1 2nνT( )2+

------------------------------ 2nνtcos 2nνT 2νTsin+( ),n ∞–=

∞

∑=

Qk t( ) 2J2n 1+ νk ν⁄( )

1 2n 1+( )νT( )2+

--------------------------------------------n ∞–=

∞

∑=

× 2n 1+( )νt 2n 1+( )νT 2n 1+( )νtcos–sin[ ].

Gk t( )⟨ ⟩Jn νk ν⁄( )

1 nνT( )2+

--------------------------n ∞–=

∞

∑=

= πx

πxsinh-----------------Jix νk ν⁄( )J ix– νk ν⁄( ),

ρmm ∞( )⟨ ⟩ 1N 1+------------- Umk J0

2 νk ν⁄( ),k 1=

N

∑+=

ε ∞( )⟨ ⟩ N2---- Vk J0

2 νk ν⁄( ).k 1=

N

∑+=

Gk t( ) Jn γνk

ν-----⎝ ⎠

⎛ ⎞ νk nν–( )t.cosn ∞–=

∞

∑=

νkn νk nν k 1 … N , n 0 1 2 …, , ,=, ,=( ).±=

LASER PHYSICS Vol. 6 No. 4 1996

MECHANISMS OF INTERACTION OF ELECTROMAGNETIC RADIATION 645

Suppose, that for certain n and k, a condition νk = nνis met, i.e.,

(39)

Then, it follows from (38) that constant componentof the probability ρmm and the number of quanta ε shift.The following values correspond to the dynamic equi-librium state in this case:

(40)

It follows from the second relationship in (40) that,for even k, the number of quanta is = N/2. If

k = 1, 3, 5, … (k ≤ N)

and the parameter of modulation depth γ lies within theranges where the Bessel function Jn(nγ) is negative,then > N/2, and, thus, an overexcited state is realized.

2.1.6. Conclusion. In this section, we consideredimportant methods of excitation of high quantumfields of a collective anharmonic mode using a seriesof phased pulses (analog of the Veksler autophasingmethod), a periodic modulation of the EMR fre-quency, and amplitude–frequency modulation (AFM).Under certain conditions, coherent effects of interactionof the field with an oscillator in all of these cases mayplay an important role. In particular, the AFM methodallows one to employ the coherent mechanism to createan anomalous state of an anharmonic oscillator, whenthe distribution over its levels is inverted as a whole(an overexcited state). A common feature of all pro-posed methods is the possibility of neutralizing anhar-monism and, consequently, of excitation of high quan-tum levels.

2.2. Interaction of EMR with Biopolymer Macromolecules

2.2.1. Introduction. Functioning of biological mac-romolecules (enzymes and other types of proteins,nucleic acids, etc.) is determined, to a great extent, bybiochemical processes in their elements. One of possi-ble primary mechanisms of EMR action on biosystemsis an induction with the help of transitions between dif-ferent isomeric states of biomolecules (as applied toenzymes, this idea was proposed, in particular, byFrölich [40, 42]). The absorbed energy is spent accord-ing to this hypothesis on crossing (complete or partial)of a barrier between the conformers. It is natural thatelectric (or magnetic) activity of the correspondingdegrees of freedom of a biopolymer molecule is neces-sary for realization of such a process. The information

qnk1– ν

Ω0------

2n

------- πk2 N 1+( )--------------------.sin= =

ρmm1

N 1+------------- 1 Jn nγ( ) mkπ

N 1+-------------cos m 1–( )kπ

N 1+------------------------cos+⎝ ⎠

⎛ ⎞+ ,=

ε N2----

12 N 1+( )-------------------- 1–( )k

1–( )Jn nγ( ) kπ2 N 1+( )--------------------cot

2.+=

ε

ε

on protein and nucleic acid spectra may be consideredas a certain substantiation of this method of consider-ation of primary mechanisms as applied to experimentsin the microwave range (see Section 4).

An elementary model illustrating the change of con-formational states of macromolecules by means of EMRcorresponds to the problem of the motion of a particlein a potential well U(x) with two minima separated bya potential barrier. An alternating force F(t) = eE(t)(where e is the charge of a particle and E(t) is thestrength of the electric component of the electromag-netic wave) induces oscillations of a particle initiallylocalized on one of the sides of the potential barrier.The amplitudes of these oscillations become close to thecoordinate of the maximum, so that thermal fluctuationsbecome sufficient for direct crossing of the barrier (dur-ing the time of action of the electromagnetic field).

The applicability of classical (nonquantum) descrip-tion of oscillations of a particle in the EHF range is guar-anteed by a low value of the ratio hν/kT, where hν is theenergy of a quantum and kT is the thermal energy. Forthe frequencies ν ~ 1011 s–1 (ω = 2πν ~ 1012 rad), thisvalue is about 0.01. However, the height of the potentialbarrier exceeds the thermal energy (kT � Umax), whichallows one to avoid direct introduction into considerationof random forces giving rise to thermal fluctuations(insignificant for high-energy vibrations). Thus, the mul-tiparticle nature of the problem is reduced to the intro-duction of friction forces, and we arrive at the equation

(1)

With regard to the anharmonicity of the interactionpotential, equation (1) is reduced to

(2)

In (2), the electric field strength is presented asF(t) = fcosωt. In the linear approximation, we have

(3)

The solution of (3) is trivial:

The first term decays exponentially so that, after arather long period of time, only one term survives. Thisterm corresponds to induced oscillations:

x = Bcos(ωt + φ).

It is not difficult to determine their amplitude andphase:

(5)

m x 2λ x+( ) U x( )∂x∂

--------------+ F t( ).=

x 2λ x ω02x+ +

fm---- ωtcos αx

2– βx

3.–=

x 2λ x ω02x+ +

fm---- ωt.cos=

x Aeλt– ω0

2 λ2t– φ+( )cos B ωt φ+( )cos .+=

Bf

m ω2 ω02

–( )2

4λ2ω2+

--------------------------------------------------------,=

φtan2λω

ω02 ω2

–------------------.=

646

LASER PHYSICS Vol. 6 No. 4 1996

RESHETNYAK et al.

In the case of exact resonance, we have

(6)

Thus, the maximal amplitude of induced oscillations is

(7)

The amplitude of free oscillations of a harmonic

oscillator with the potential energy U(x) = m /2can be expressed through the energy ε as follows: x0 =

. Equating x0 to (7) we obtain

where q = λ/ν0 = 2πλ/ω0 is the logarithmic decrementof damping (the inverse of the oscillator quality factor).

Assuming that ε � 10kT (which is a rather largevalue compared to the energy of thermal oscillationsand, at the same time, is comparable with the energy ofATP hydrolysis, providing conformational transitionsin the course of functioning of biomolecules) and tak-ing the values typical for local vibrations of biomole-cules in the EHF range,

ω0 = 1012 s–1, m = 200mp, e = ,

for E = 1 W/cm (which corresponds to the EHF EMRintensity of 1 mW/cm2), we obtain an estimate q ~3 × 10–7. The corresponding Q factor (about 3 × 106) isseveral orders of magnitude higher than the Q factors(103–104) characteristic of resonances in typical exper-iments on the interaction of low-intensity EHF EMRwith biological objects.

Therefore, the interaction of EMR with isolatedlocal vibrations of biomolecules cannot be a suffi-ciently effective primary mechanism providing experi-mentally observed biological action of EHF fields.

2.2.2. Biopolymer chain in a resonance field. Let usconsider an elementary one-dimensional chain of mono-mers. It is assumed that the interaction with the field ofradiation results in excitation of dipole-active oscilla-tions of monomers that form the chain. The oscillationsof monomers are interrelated, in turn, by means ofdipole–dipole interaction. Note that a model of this typeis used, as a rule, for the study of dynamics of such biom-acromolecules as nucleic acids (DNA).

Assume also that local dipole-active vibrations ofeach monomer characterized by xk coordinates have thesame natural frequency ω0 and damping λ and interactwith each other by means of the electric field of dipolemoments dk = exk. In the considered case of a one-dimensional structure, we can restrict ourselves to theconsideration of the interaction of only neighboringdipoles (the so-called short-range interaction).

Bf

2mλω0-----------------, φ π

2---.==

xmaxeE

2mλω0-----------------.=

ω02x

2

2ε mω2⁄

E 2 2mεω0λ e⁄ q 2mεω0 πe,⁄= =

e

The potential function of a biopolymer chain in thesimplest case is given by

(8)

Let us present the equations of motion for coordi-nates xk under the action of an external isotropic mono-chromatic field as

(9)

where

In the linear approximation, the system of equations isreduced to

(10)Let us introduce collective variables:

(11)

while

(12)

Then, for the variables zm, we obtain a system of uncou-pled equations,

(13)where the frequencies of collective modes are intro-duced,

(14)

and the partial coefficients are defined by

(15)

Uω0xk

2

2-----------

ξx

3-----xk

3

k

∑+k

∑=

+Ω0

4

4------- xk xk 1––( )2

xk xk 1+–( )2+[ ]

k

∑

+ξxx

3------- xk xk 1––( )3

xk xk 1+–( )3+[ ] ….+

k

∑

xk 2λ xk ω02xk+ +

Ω02

2------- xk 1– 2xk– xk 1++( )=

+ f 0 ωtcos Φ xk 1– xk xk 1+, ,( ),+

Φ ξkxk2

– ξxx xk xk 1––( )2xk xk 1+–( )2

+[ ].–=

xk 2λ xk ω02xk+ +

Ω02

2------- xk 1– 2xk– xk 1++( ) f 0 ωt.cos+=

zm2

N 1+------------- xk

mkπN 1+-------------, msin

k 1=

N

∑ 1 2 … N ,, , ,= =

xk2

N 1+------------- zm

mkπN 1+-------------, ksin

m 1=

N

∑ 1 2 … N ., , ,= =

zm λzm νm2

zm+ + sm f 0 ωt, mcos 1 2 … N ,, , ,= =

νm2 ω0

2 Ω02 mπ

2 N 1+( )--------------------sin

2, m+ 1 2 … N ,, , ,= =

sm2

N 1+------------- mkπ

N 1+-------------sin

k 1=

N

∑=

2N 1+-------------

π2---m

π2--- N

N 1+-------------sinsin

π2--- m

N 1+-------------sin

------------------------------------------.=

LASER PHYSICS Vol. 6 No. 4 1996

MECHANISMS OF INTERACTION OF ELECTROMAGNETIC RADIATION 647

Induced oscillations are of the main interest. The solu-tion of (13) is trivial. Let us present it as

The maximal value of zm is achieved for ω =

. Dynamics of induced oscillations ofcoordinates xk is determined according to (12) as

(16)In (16) the resonances at the collective frequencies

(14) are clearly seen. It is evident that resonance actionof an external electromagnetic field exactly at these fre-quencies ensures the highest efficiency.

It follows from (15) that variables sm equal zero foreven m. Therefore, the number of resonances is r = [N/2],i.e., it is twice less than the number of oscillators in thesystem. This circumstance is directly related to thesymmetry of the problem.

It is interesting to determine the distances betweenthe frequencies of the collective modes:

(17)

It follows from (17) that, for m close to unity, thespectrum becomes more concentrated, near m ≈ N/2 itis rarefied, and for m ≈ N it is concentrated again.Hence, the most pronounced resonances can beexpected for m close to N/2.

Assume that the external signal is modulated in itsamplitude,

(18)

where 2γ is the modulation depth and Ω is the modula-tion frequency. Then the expansion in (18) yields

Thus, amplitude modulation provides additionalpotentialities of implementing resonance action onbiomacromolecules:

(19)

Thus, is the case of amplitude modulation of theexternal field, the number of possible resonance real-izations increases (in comparison with a purely mono-

zm

sm f 0

ω2 νm2

–( )2

λ2ω2+

------------------------------------------------ ωt φm+( )cos ,=

φmtan2λω

ω2 νm2

–------------------.=

νm2 λ2

2⁄–

xk2

N 1+------------- f 0

smmkπN 1+-------------sin

ω2 νm2

–( )2

λ2ω2+

------------------------------------------------ ωt φm+( ).cosk 1=

N

∑=

νm 2+2 νm

2– Ω0

2 m 2+( )π2 N 1+( )----------------------sin mπ

2 N 1+( )--------------------sin

2–=

= Ω02 π

N 1+------------- π m 1+( )

N 1+--------------------sin .sin

f f 0 1 2γ Ωtcos+( ) ωt,cos=

f f 0 γ ωtcos γ ω Ω–( )t ω Ω+( )tcos+cos( )+[ ].=

νm ω

, standard situation ω Ω

, Stokes resonances– ω Ω

, anti-Stokes resonances.+

=

chromatic field) up to 3[

N

/2], where

N

is the number ofmonomers.

If the solution (16) to a linear system is assumed tobe a zeroth-order approximation, it is not difficult tofind the first approximation taking anharmonicity intoaccount [see also (9)]. For this purpose, one has to sub-stitute the solution (16) into the right-hand side of (9)and integrate the resulting equation. This procedure isnot difficult. However, the final results are rather cum-bersome, and we do not present them here.

Note an important fact that the nonlinearity in theright-hand side of (9) is a quadratic nonlinearity. Con-sequently, the right-hand side involves a constant term,which is independent of time, and the terms propor-tional to the second harmonic. Therefore, new reso-nance potentialities emerge,

ν

m

= 2

ω

,

associated with the second harmonic. The latter widensthe potentialities of active resonance action on biopoly-mer systems. One should keep in mind that the reso-nances at the second harmonic may be realized due toboth anharmonicity of the monomers themselves andthe cubic nonlinearity in the potential of interactionbetween the monomers of a macromolecule.

Finally, for the most interesting case of amplitudemodulation (18), it can be easily demonstrated that asubstantial broadening of resonances takes place:

(20)

With regard to (19) and (20), the number of possi-ble realizations of resonance excitation is equal to(3 + 5)[

N

/2] = 4[

N

]. Thus, the above consideration demonstrates that the

resonances in the interaction of EMR with biopolymerchains are of collective nature. The number of reso-nances in the interaction of a monochromatic field witha macromolecule is twice less than the number of oscil-lators (the spectrum is not equidistant). Amplitudemodulation increases the number of possible resonancerealizations due to Stokes and anti-Stokes frequencies.The number of resonance realizations increases sub-stantially when nonlinearity is taken into account, inparticular, a two-quantum resonance becomes possible.

2.3. Interaction of EMR with Biopolymer Macromolecules and Its Influence on the Dynamics of

the Active Site (Antenna Model)

2.3.1. Introduction. As it was mentioned several

times, functioning of several biological macromole-cules (in particular, enzymes) and other biologicalcompounds is determined, to a great extent, by the pro-cesses in the active sites surrounded by biopolymer

νm

Ω2ω2ω Ω±2 ω Ω±( ).

=

648

LASER PHYSICS

Vol. 6

No. 4

1996

RESHETNYAK

et al

.

chains folded into globules. Based on such a notion ofthe structure of a biomolecule, one may assume that theinteraction with the field of radiation results in the exci-tation of dipole-active vibrations of monomers consti-tuting the chain which, in turn, induces oscillations inthe active site (antenna model). Thus excited oscilla-tions may cause biomacromolecule transition intoanother conformational state.

5

This concept is completely adequate to several

biomacromolecules, e.g., chlorophyll, hemoglobin,myoglobin, etc. These macromolecules have two com-mon structural properties: (1) an ion is situated in theirgeometrical centers (a magnesium ion in chlorophylland iron ion in hemoglobin); (2) the ion is arranged atthe center of symmetry of a tetrapyrrole ring(pseudoplane structure).

Recall that chlorophyll (along with the multitude ofits derivatives) is a pigment playing an important role inphotosynthesis, hemoglobin is a well-known protein ofblood that enters into erythrocytes and that is a carrierof oxygen, and myoglobin is a muscle protein (ofsperm whales) that is an analog of hemoglobin in bio-logical function (it is an oxygen carrier).

Note also that the third class of biomacromoleculescorresponding to the antenna model is based on thechemistry of macrocycle compounds.

Pointing out three classes of biomolecular systems,we specify once again the notion of an antenna model.An external energy (in particular, associated with reso-nance interaction of EHF EMR with bioobjects or withbiochemical reactions, etc.) is transferred to the periph-ery of the system, i.e., to an ensemble of subunits(which are not necessarily identical in their structure).An associating center (a metal ion in the case underconsideration) actively interacts via biochemical bondswith peripheral acceptors that receive encoded energy.Thus, the former acquires the information, whichcauses biological action. Reaction ability of biomacro-molecules substantially depends on the level of excita-tion of central subunits.

Swiss scientist L. Ruzhechka was the first to per-form in 1926 the synthesis of macrocyclic compounds.He obtained cyclic ketones with a 10–30-link ring afterdry vacuum distillation of thorium salts of dicarboxylicacids. Nowadays, synthesized and studied are suchmacrocyclic compounds as crown ethers, cryptands,annulens, cyclophans, catenans, and rotaxans.

Macroheterocyclic compounds, such as crownethers, cryptands, and their heteroanalogs are of specialinterest in the chemistry of macrocycles. The study ofthese substances is closely related to the discovery ofnatural macroheterocycles, namely, cyclic peptides, dep-sipeptides, and depsides. Natural macroheterocycliccompounds are able to form stable complexes with ions

5

Antenna effect is known to be realized in photosynthetic systems(a photosynthetic subunit includes a light-collecting antenna anda photoreaction center coupled with this antenna).

of alkaline and alkaline-earth metals and to transportions through natural and artificial membranes. Theywere called membrane-active complexes or ionophores.

In 1955, valinomycin was extracted from fungousculture. This antibiotic is a 36-link cyclododecadep-side. Soon a remarkably high selectivity of valinomy-cin in the complex formation with certain metal ionswas discovered. This compound binds potassium ionsten thousand times more effectively than sodium ions.According to the X-ray data on the complex salt of vali-nomycin with potassium ion, the latter is located in thecentral hollow of the ring and interacts with carbonyloxygen of the ester group. Other similar natural macro-heterocycles were discovered: valinomycin analogs,nonactins, enniatins, and other ionophores. Complex-ons (in particular, crown ethers) with ions of alkalinemetals (lithium, sodium, potassium, rubidium, andcesium); with ions of alkaline-earth metals (magne-sium, calcium, strontium, and barium); and with cop-per, silver, mercury, lanthanum, thallium, lead, cobalt,and nickel ions are synthesized already.

2.3.2. Antenna model. As an elementary model thatillustrates the antenna effect, we consider a one-dimen-sional closed (circular) chain of monomers. An activesite is located at the center of the ring. It is coupled withmonomers of the chain by means of dipole–dipoleinteraction.

Let us denote the coordinate displacements ofmonomers as

x

1

, …,

x

N

and the displacements of theactive site as

y

. For the potential function, we have

(1)

The first two terms in (1) correspond to the oscilla-tions of monomers (the second term takes anharmonic-ity into account), and the last two terms correspond tothe bonds between monomers. The other terms corre-spond to the coupling of monomers with the active site.Equations of motion are given by

(2)

U x1 … xN y, , ,( )12---ωx

2xk

2 13---ξxxk

3+

k 1=

N

∑=

+12---ωy

2y

2 13---ξyy

3 12---ωxy

2y xk–( )2 1

3---ξxy y xk–( )3

+k 1=

N

∑+ +

12---ωxx

2xk xk 1––( )2

xk xk 1+–( )2+[ ]

k 1=

N

∑+

+ξxx

3------- xk xk 1––( )3

xk xk 1+–( )3+[ ] xk f 0 ωt.cos

k 1=

N

∑–k 1=

N

∑

xk λ xk+U xk( )∂

xk∂---------------- f t( ),+–=

y λ y+U∂y∂

-------,–=

LASER PHYSICS Vol. 6 No. 4 1996

MECHANISMS OF INTERACTION OF ELECTROMAGNETIC RADIATION 649

where f(t) = f0cosωt is an external monochromaticforce acting only on monomers and λ is the dampingcoefficient introduced phenomenologically (for sim-plicity, this coefficient is assumed to be the same forboth monomers and the active site). With regard to (1),equation (2) is reduced to

(3)

or

(4)

Let us introduce a common coordinate for theensemble of monomers,

(5)

Then, the system of equations (4) in the linearapproximation is

(6)

where

N is the number of monomers.

xk λ xk+ ωx2xk– ξxxk

2– ωxx

22xk xk 1–– xk 1+–( )–=

+ ωxy2

y xk–( ) ξxy y xk–( )2

f t( )+ +

+ ξxx xk xk 1––( )2xk xk 1+–( )2

+[ ],

y λ y+ ωy2y– ξyy

2–=

– ωxy2

y xk–( ) ξxy y xk–( )2,

k 1=

N

∑–k 1=

N

∑

xk λ xk ωx2 ωxy

2+( )xk ωxy

2y–+ +

= ωxx2

xk 1– 2xk– xk 1++( ) ξxxk2

– ξxy y xk–( )2

+

+ f t( ) ξxx xk xk 1––( )2xk xk 1+–( )2

+[ ],+

y λ y ωy2

Nωxy2

+( )y ωxy2

xk

k 1=

N

∑–+ +

= ξyy2

– ξxy y xk–( )2.

k 1=

N

∑–

x xk.k 1=

N

∑=

xk λ xk ω12xk ω0

2y–+ +

= Ω0

2

2------- xk 1– 2xk– xk 1++( ) ξxxk

2– f t( ),+

y λ y ω22y ω0

2x–+ + 0,=

ω12 ωx

2 ωxy2

,+=

ω22 ωy

2Nωxy

2,+=

ω02 ωxy

2,=

Ω02

2ωxx2

.=

With regard to (5), we have

(7.1)

(7.2)

It follows from (7.2) that

(8)

Substitution of (8) into (7.1) yields

(9)

The corresponding characteristic equation obtainedafter the substitution of y ~ ekt into a homogeneousequation is

(10)

Denoting zk = k2 + λk, we obtain

so that

(11)

It is assumed below that the following inequalitiesare fulfilled:

(12)

The first inequality corresponds to a weak couplingof monomers with the active site, and the second onecorresponds to weak damping of monomer oscillators.

For eigenvalues, we have

(13)

where the collective frequencies are introduced:

(14)

We consider the induced oscillations (for an exter-nal force f0cosωt):

y = Acosωt + Bsinωt. (15)

x λ x ω12x Nω0

2y–+ + Nf t( ),=

y λ y ω22y ω0

2x–+ + 0.=

x1

ω02

------ y λ y ω22y+ +( ).=

y4( )

2λy3( ) ω1

2 ω22 λ2

+ +( )y2( )

+ +

+ λ ω12 ω2

2+( )y

1( ) ω12ω2

2Nω0

4–( )y+ Nω0

4f t( ).=

k2 λk ω1

2+ +( ) k

2 λk ω22

+ +( ) Nω04.=

z2 ω1

2 ω22

+( )z ω12ω2

2Nω0

4–+ + 0,=

z1 2,12--- ω1

2 ω22

+( ) 14--- ω1

2 ω22

+( )2

ω12ω2

2– Nω0

4+ .±–=

ω02 ω1ω2

N------------, λ ω1

2 ω22

+ .< <

k1 2,λ2---– i Ω1

2 λ2

4-----– , k3 4,± λ

2---– i Ω2

2 λ2

4-----– ,±= =

Ω112--- ω1

2 ω22

+( ) 14--- ω1

2 ω22

–( )2

Nω04

+1/2

+⎩ ⎭⎨ ⎬⎧ ⎫

1/2

,=

Ω212--- ω1

2 ω22

+( ) 14--- ω1

2 ω22

–( )2

Nω04

+1/2

–⎩ ⎭⎨ ⎬⎧ ⎫

1/2

.=

650

LASER PHYSICS Vol. 6 No. 4 1996

RESHETNYAK et al.

Substituting (15) into (9) and equalizing the corre-sponding coefficients of cosωt and sinωt yield a systemof algebraic equations:

where

Finally, we obtain

where

After simple but cumbersome transformations, weobtain the expression for the induced oscillations of theactive site:

(16)

It follows from (16) that the maximal amplitude ofthe induced oscillations of the active site is achievedunder the conditions of collective resonance:

either ω = Ω1

or ω = Ω2.

In any of these cases, the amplitude of the inducedoscillations is given by

(17)

It follows from (17) that the maximal effect of reso-nance excitation of the active site is achieved for a largenumber of peripheral “antenna” subunits, high coeffi-cients of coupling of the active site with monomers,minimal damping coefficient, and minimal detuning ofthe collective modes.

It is not difficult to determine the “choreography”(dynamics of the induced oscillations) of separatemonomers. According to (6), the equation for the kthmonomer is

(18)

A ω4 α2ω2

– α0+( ) B 2λω3 α1ω–( )– F0=

A 2λω3 α1ω–( ) B ω4 α2ω2– α0+( )+ 0,=

α0 ω12ω2

2Nω0

4, α1+ λ ω1

2 ω22

+( ),= =

α2 ω12 ω2

2 λ2, F0+ + Nω0

2f 0.= =

yF0

p2

q2

+--------------------- ωt φ+( )cos ,=

p ω2 ω12

–( ) ω2 ω22

–( ) ω2λ2– Nω0

4,+=

q λω 2ω2 ω12

– ω22

–( ),=

φtanqp---.=

y Nω02

f 0 ωt φ+( ) ω2 Ω12

–( )[2

ω2 Ω22

–( )2

cos=

ω2λ2 ω2λ2 ω2 Ω12

–( )2

ω2 Ω22

–( )2

+ +[ ] ]+1 2/–

.

y0

Nω02

f 0

ωλ ω2λ2 Ω22 Ω1

2–( )

2+

---------------------------------------------------------.=

xk 2λ xk ω12xk+ +

= 12---Ω

0

2

xk 1– 2xk– xk 1++( ) ω02y f t( ).+ +

Introducing collective coordinates

and applying the methods of linear algebra (alreadyused in Section 1), we obtain for the induced oscilla-tions of monomers:

(19)

where

and y0 is determined from (16). Thus, within the framework of the antenna model,

the maximal effect of external action of a monochro-matic field f = f0cosωt is achieved under the conditionsof collective resonance:

Ω1 = ω, Ω2 = ω .

Repeating the considerations of Section 2, we canmake the following conclusions:

(1) Amplitude modulation of an external signal pro-vides additional potentialities of resonance action onbiomacromolecules at the frequencies

(2) Nonlinearity of quadratic coupling for a mono-chromatic signal gives rise to an additional resonance atthe second harmonic,

Ω1,2 = 2ω .

(3) Taking into account nonlinearity of amplitudemodulation, we obtain additional series of resonances,

Thus, collective effects play a rather important role inthe interaction of resonance EMR with biomacromole-cules with incorporated active sites. The properties ofradiation itself determine extensive potentialities of

zm2

N 1+------------- xk

mkπN 1+-------------,sin

k 1=

N

∑=

xk2

N 1+-------------

smmkπN 1+-------------sin

ω2 νm2

–( )2

λ2ω2+

------------------------------------------------m 1=

N

∑=

× f 0 ωt δm1+( )cos y0 ωt δm2+( )cos+[ ],

νm2 ω1

2 Ω02 mπ

2 N 1+( )--------------------,sin+=

sm

mπ( ) mπNN 1+-------------sinsin

mπ2 N 1+( )--------------------sin

------------------------------------------,=

Ω1 2,

ωω Ω–

ω Ω.+

=

Ω1 2,

Ω2ω2ω Ω±2 ω Ω±( ).

=

LASER PHYSICS Vol. 6 No. 4 1996

MECHANISMS OF INTERACTION OF ELECTROMAGNETIC RADIATION 651

active control of the dynamics of a biomacromolecule asa whole and, consequently, of biochemical processes thatinvolve such molecules. Hence, the required biologicalaction is implemented either directly or indirectly.

This work was supported by the Russian Foundationfor Basic Research, project no. 96-02-18855-a.

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