Research Article On Macroscopic Quantum Phenomena in...

Research ArticleOn Macroscopic Quantum Phenomena in Biomolecules andCells From Levinthal to Hopfield

Dejan RakoviT1 Miroljub DugiT2 Jasmina JekniT-DugiT3 Milenko PlavšiT4

Stevo JaTimovski5 and Jovan ŠetrajIiT67

1 Faculty of Electrical Engineering University of Belgrade 11000 Belgrade Serbia2Department of Physics Faculty of Science University of Kragujevac 34000 Kragujevac Serbia3 Department of Physics Faculty of Science University of Nis 18000 Nis Serbia4 Faculty of Technology and Metallurgy University of Belgrade 11000 Belgrade Serbia5 Academy of Criminalistic and Police Studies 11000 Belgrade Serbia6Department of Physics Faculty of Sciences University of Novi Sad 21000 Novi Sad Vojvodina Serbia7 Academy of Sciences and Arts of the Republic of Srpska 78000 Banja Luka Republic of Srpska Bosnia and Herzegovina

Correspondence should be addressed to Dejan Rakovic rakovicdetfbgacrs

Received 27 February 2014 Accepted 13 May 2014 Published 16 June 2014

Academic Editor K Hun Mok

Copyright copy 2014 Dejan Rakovic et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

In the context of the macroscopic quantum phenomena of the second kind we hereby seek for a solution-in-principle of thelong standing problem of the polymer folding which was considered by Levinthal as (semi)classically intractable To illuminateit we applied quantum-chemical and quantum decoherence approaches to conformational transitions Our analyses imply theexistence of novel macroscopic quantum biomolecular phenomena with biomolecular chain folding in an open environmentconsidered as a subtle interplay between energy and conformation eigenstates of this biomolecule governed by quantum-chemicaland quantum decoherence laws On the other hand within an open biological cell a system of all identical (noninteracting anddynamically noncoupled) biomolecular proteinsmight be considered as corresponding spatial quantum ensemble of these identicalbiomolecular processors providing spatially distributed quantum solution to a single corresponding biomolecular chain foldingwhose density of conformational states might be represented as Hopfield-like quantum-holographic associative neural network too(providing an equivalent global quantum-informational alternative to standard molecular-biology local biochemical approach inbiomolecules and cells and higher hierarchical levels of organism as well)

1 Introduction

On Macroscopic Quantum Phenomena Initially quantummechanics appeared as a theory of microscopic physicalsystems (elementary particles atoms and molecules) andphenomena at small space-time scales typically quantumphenomena are manifested at dimensions smaller than 1 nmand time intervals shorter than 1120583s However from thevery beginning of the quantum mechanical founding thequestion of its universality was raised that is the question ofgeneral validity of the quantum-physical laws formacroscopicphenomena usually treated by the methods of classicalphysics In the history of quantum physics and especially

quantum mechanics this question has been temporarily putaside for very different reasons being considered as a difficultscientific problem The situation is additionally complicatedby the existence of different schools of quantum mechanicsarguing about physical-epistemological status of the so-calledcollapse (reduction) of the wave function In this respect thesituation is not much better today and it can be said freelythat the problem of universal validity of quantum mechanicsis still open [1ndash15] To this end Primas [16] emphasizes thefollowing

ldquoIf we consider quantummechanics as universally valid inthe atomic molecular mesoscopic and engineering domain

Hindawi Publishing CorporationBioMed Research InternationalVolume 2014 Article ID 580491 9 pageshttpdxdoiorg1011552014580491

2 BioMed Research International

then we have to require that a proper mathematical codifica-tion of this theorymust be capable to describe all phenomenaof molecular and engineering science Already rather smallmolecules can have classical properties so that a classicalbehavior is not a characteristic property of large systemsTheexistence of molecular superselection rules and of molecularclassical observables is an empirically well-known fact inchemistry and molecular biology The chirality of somemolecules the knot type of circular DNA-molecules and thetemperature of chemical substances are three rather differentexamples of molecular classical observables Such empiricalfacts can be described in an ad hoc phenomenologicalmanner but it is not so easy to explain these phenomenafrom the first principles of quantummechanics A universallyvalid theory of matter has not only to describe but also toexplain why the chirality of biomolecules (like the L-aminoacids theD-sugars lipids or steroids) is a classical observableThe reality of this breakdown of the superposition principleof traditional quantum mechanics on the molecular level isdramatically demonstrated by the terrible Vontergan tragedywhich caused many severe birth defectsrdquo

Starting from the 1980s mainly in the papers of Leggett[1 2] a new period of investigation of quantum mechanicalphenomena on the macroscopic level began Namely aclarification of the notions and planning of experimentalsituations for observing some physical effects started Thecentral problem in this respect is a notion of macroscopicdifferentiation of the states of quantum system whose quan-tummechanical behavior is explored More precisely Leggettargues that the term macroscopic quantum mechanical effectmust be related tomacroscopically different states that is thesystem states (and observables) that carry macroscopic prop-erties (and behaviors) of the system as a whole These states(ie observables) must carry classical-physical behavior ofthe system as well this poses a task for choosing physicalconditions giving rise to observation of typical quantumeffects related to these states1

Hence different kinds of macroscopic quantum phenom-ena (i) the ones usually explored by the methods of (quan-tum) statistical physics and not related to macroscopicallydifferentiated states (being classified asmacroscopic quantumphenomena of the first kind like solid state phenomena)and (ii) those ones regarding macroscopically different (dif-ferentiated) states (being classified as macroscopic quantumphenomena of the second kind and being interesting to us)Numerous differentmacroscopic quantumphenomena of thesecond kind some of them belonging to the fast developingfield of the quantum computing and information unequiv-ocally sharpen the overall problem of universal validity ofquantum mechanics

In the context of the macroscopic quantum phenomenaof the second kind we shall present a solution-in-principleof the long standing problem of the polymer folding (whichwas considered by Levinthal as (semi)classically intractable[17] as shortly reviewed below)mdashimplying the existence ofnovel macroscopic quantum biomolecular phenomena withfar reaching implications

Levinthal Paradox Revisited Contemporary methods forcalculation of conformational dependent chain propertiesare based on thermodynamic aspect of the problem whichexplores (semi)classically the folding free energy landscapefor protein with several successful attempts to model theseprocesses in silico using molecular dynamics simulationswith full atomic representation of both protein and solvent[18ndash22] producing continuous (semi)classical trajectorieswith the potential to connect static structural snapshotsgenerated from experimental data This is incorporated intothe (semi)classical viewpoint that conformational changes ofproteins due to solvent thermal optical and other influencesof the environment do not occur in a random way (egmovements of gas particles) but fold to their native conforma-tion of deep global minimum in some (semi)classical funnelof low-energy conformations leading toward it [23] Evenin recently reported implementation of quantum annealing(on the programmable superconducting quantum device) forlattice protein folding problems nothing quantum mechan-ical is implied about principles that govern the folding ofprotein chains [24] (rather quantum fluctuations are a toolused for solving the optimization problem of protein foldingconsidered classically intractable [25ndash27])

Hence these (semi)classical calculational methods donot describe properly transitions from one conformationto another which is the kinetic aspect of the problemexploring the conformation change of long flexible chainThis has been illustrated by Levinthal who considered theprobability of folding a proteinmolecule from coiled to nativeconformation [17] Assume 2119899 torsional angles of an 119899-residueprotein each having three stabile rotational states this yields32119899

asymp 10119899 possible conformations for the chain if a protein

can explore new conformations in a random way at the ratethat single bond can rotate it can find approximately 1013conformations per seconds then the time 119905 (s) required fora protein to explore all the conformations available to it is119905 = 10

1198991013 for a rather small protein of 119899 = 100 residues

one obtains 119905 = 1087 s which is immensely more than

the apparent age of the universe (ldquoLevinthal paradoxrdquo) Yetaccording to experiments proteins can fold to their nativeconformation in less than a few seconds [28]

It should be added that (semi)classical kinetic (nonst-ationary) predictions imply the continuous mapconformation change 119896

119894rarr 119896119891which requires a sequence of 119899

local noncommuting successive elementary transformations(local rotations of characteristic time 120591

119900) with the time

necessary for the net transformation much longerthan characteristic time necessary for a local rotation(120591119899sim 119899120591119900≫ 120591119900) and the frequency of corresponding global

transition much lower than the frequency of a local rotation(119891119899

sim 1119899120591119900

sim 119891119900119899 ≪ 119891

119900)mdashstrongly dependent on a

degree of polymerization 119899 (in clear contradistinction withthe experimentally observed poorly dimensionally sensitivedispersion laws of the internal more or less delocalizedquasiparticle excitations in any condensed state quantumsystem electrons phonons and etc [29]) Thus chainfolding based on (semi)classical (nonstationary) predictionscannot be considered kinetically understood the same

BioMed Research International 3

applies to biomolecular recognition processes based on (semi)classical selective ligand-proteinstarget-receptors keylockinteractions

2 Conformational Transitions inBiomolecules and Cells as MacroscopicQuantum Effects

21 Quantum-Chemical Approach to Conformational Tran-sitions in Biomolecules Within the framework of standardquantum-chemical Hamiltonian (including kinetic energiesand Coulomb interactions of all biomolecular electrons andnuclei) and Born-Oppenheimer adiabatic approximation (ofseparated biomolecular electronic and vibrational degreesof freedom) the (semi)classical problem of many-electronhypersurface 119864

119890(120601(119896)

119890) is replaced by better-defined problem

of two (virtually intersecting) isomeric many-electron hyper-surfaces (hyper-paraboloids) serving as potential hypersur-faces for two vibrational (isomeric) problemsmdashwithin thetheory of nonradiative resonant structural transitions [30]In this approach the conditions for electronic-vibrationalnonradiative resonant transitions between the 119894th and119891thisomeric states are possible only for close states with non-vanishing electronic and vibrational dipole moments andnonvanishing electronic and vibrational overlap integrals (cfFigure 1 and its caption for further explanation)

22 QuantumDecoherence Approach to Conformational Tran-sitions in Biomolecules and Cells From Levinthal to HopfieldQuantum decoherence approach to conformational transi-tions [34ndash42] (cf the Appendix) generally allows repro-duction of both existence and stability of the (stationary)conformations and the short time scales for the quantummechanical processes resulting effectively in (nonstation-ary) conformational transitions under external influences onthe complementary environmental solution This approachmight also be applied to (nonstationary) mismatching-to-matching quantum mechanical conformational transitions inselective ligand-proteinstarget-receptors keylock biomolec-ular recognition processes under external (eg composi-tionalchemical thermaloptical) influences on the cellrsquoscomplementary cytoplasmatic environment [36 37 41 42]

In the context of existence and changes of conformationsof biomolecules it should be particularly pointed out thatbiomolecular operators of Hamiltonian

_119867 (electronic-

vibrational Hamiltonian which includes operators ofkinetic energies and all Coulomb interactions between thebiomolecule electrons and nuclei in the center-of-masscoordinate system) and conformations

_119870 (so-called ldquoreaction

(conformational)rdquo coordinates of the nuclei defining thebiomolecule conformations) do not commute [

_119867

_119870] = 0

Hence quantum-chemical approach described in theprevious section (with simultaneously defined energies andconformations of biomolecules) is essentially (semi)classicaland it is only quantum decoherence that enables appearanceof biomolecular conformational eigenstates (labeled by upperindex 119870 in (1)) from the biomolecular energy eigenstate of

the isolated biomolecule (labeled by upper index 119864 in (1))via nonpotential interaction of the biomolecular quantumsystem (QS) with its quantum environment (QE) whenone of the biomolecular conformational 119870

119896eigenstates is

stochastically selected via quantum decoherence (QD)2

from the biomolecular initial many-electronic energy 119864(119894)

119890

eigenstate of the isolated biomolecule (as only self-Hami-ltonian of the biomolecule was switched-on initially likea proper approximation when interaction with quantumenvironment might be accounted for via potential term of theself-Hamiltonian)3 It should be noted that themost probablebiomolecular conformational eigenstate is the one labeled by119870119894 corresponding to biomolecular initial many-electronic

energy 119864(119894)

119890(with the same index i especially if it corresponds

to biomolecular many-electronic ground state in accordancewith the usually adopted quantum-chemical computationswithin the framework of adiabatic approximation)

Subsequently one of the stochastically QD-selectedbiomolecular conformational 119870

119896eigenstates (119870

119894 in Figure 1)

might be excited by nonstationary external perturbations(photons ) into some resonant electronic-vibrational energyeigenstate (119864(119894)

119890+ 119864(119894)

V = 119864(119891)

119890+ 119864(119891)

V in Figure 1)when self-Hamiltonian of the biomolecule is again a properapproximation (and interaction with quantum environ-ment might be again accounted for via potential termof the self-Hamiltonian) Then in subsequent quantumdeexcitationdecoherence3 there are finally at least two pos-sible biomolecular conformational eigenstates (as depicted inFigure 1) 119870

119894related to biomolecular deexcitation back into

initial many-electronic state 119894 or 119870119891related to biomolecular

deexcitation into final many-electronic state 119891And such fluctuations between eigenstates of energy and

conformation of biomolecules are repeating

10038161003816100381610038161003816Φ119894⟩119864

QS10038161003816100381610038161003816Ψ119894⟩119864

QE

= sum

119895

119888119895

100381610038161003816100381610038161003816Φ119895⟩119870

QS

100381610038161003816100381610038161003816Ψ119895⟩119870

QE

QD997888997888997888997888997888rarr

10038161003816100381610038161003816Φ119896⟩119870

QS10038161003816100381610038161003816Ψ119896⟩119870

QE [997888rarr 120588119870

ΦΨ]+Δ119864exc

997888997888997888997888997888997888997888997888rarr

= sum

119897

1198881015840

119897

10038161003816100381610038161003816Φ119897⟩119864

QS10038161003816100381610038161003816Ψ119897⟩119864

QE

minusΔ119864deecQD997888997888997888997888997888997888997888997888997888997888997888rarr

100381610038161003816100381610038161003816Φ119891⟩119864

QS

100381610038161003816100381610038161003816Ψ119891⟩119864

QE[997888rarr 120588

119864

ΦΨ] = sdot sdot sdot

(1)

and might be observed by applying methods of experimentalmacromolecular biophysics [43]mdashthus becoming a paradigmof macroscopic quantum phenomena of the second kind

So biomolecular chain folding in an open environmentalsolution might be considered as a subtle interplay betweenenergy and conformation eigenstates of a biomolecule gov-erned by local quantum-chemical and quantum decoherencelaws and in this scenario the Levinthalrsquos paradox disappears(cf the Appendix for some aspects of quantum decoherencescenario of conformational transitions also cf footnote 5


Ee

E(i)

E(f)

E(i)e E

(f)e

120601(k)e

(120601 )(120601 ) )

120601(f)e120601(i

(i

)e

(f)

Figure 1 The (semi)classical problem of many-electron hyper-surface 119864

119890(120601(119896)

119890) as a potential energy for adiabatically decoupled

Q1D vibrational and conformational system (with local minima as(semi)classical ldquopositionsrdquo ie many-atomic isomer configurationson many-electronic hypersurface (broken line in the figure))mdashnotadiabatically well-defined when traversing between two adjacentlocal minimamdashis replaced in the framework of theory of nonra-diative resonant transitions [30 31] by better defined problem oftwo (virtually intersecting) isomeric many-electronic hypersurfaces(hyperparaboloids) serving as potential hypersurfaces for two vibra-tional (isomeric) problems (full line in the figure) In this approachby time-dependent external perturbation of the isomer at this veryintersection the conditions for electronic-vibrational nonradiativeresonant transitions between the two isomers (119894 119891) are achievedin the first approximation the matrix element of dipole transitionfrom 119894th to 119891th isomer is given by 120583(119894119891) asymp 120583

(119894119891)

119890119878(119894119891)

V + 120583(119894119891)

V 119878(119894119891)

119890

It is obvious that allowed transitions between isomeric states (119894 119891)

are possible only for close states with nonvanishing electronicand vibrational dipole moments 120583(119894119891)

119890and 120583(119894119891)V and nonvanishing

electronic and vibrational overlap integrals 119878(119894119891)

V and 119878(119894119891)

119890or in

cascade resonant transitions between close intermediate partici-pating isomeric states which might be related to nondissipativepolaronsoliton-like transport [32 33] Also during these resonanttransitions the perturbed biomolecular system is shortly describedby quantum-coherent superposition (120601

(119894)

119890120601(119894)

V plusmn 120601(119891)

119890120601(119891)

V )radic2 beforeits quantum decoherence into final electronic state 120601

(119891)

119890or into

initial electronic state 120601(119894)119890(with subsequent deexcitations into lower

vibrational states)

therein for revealing (semi)classicalmeaning of the harmonic-like vibratingmacromolecule conformations in the vicinity oflocal minimums of many-electron hypersurface)

On the other hand within an open biological cell a sys-temof (noninteracting anddynamically noncoupled)119873

119896pro-

teins identical in their primary chemical structure (and theirbiomolecular targets) might be considered as correspondingglobal spatial quantumensemble of119873

119896identical biomolecular

processors providing a spatially distributed quantum solu-tion to corresponding single local biomolecular chain folding(and key-lock recognition process)mdashwhose time-adaptingdensity of conformational states _

120588119896

119878119896(119905) might be represented

as global cellrsquos Hopfield-like quantum-holographic associativeneural network too [41 42] (cf Figure 2 and its figure captionfor further explanation) We hereby silently assumed ergodichypothesis that is near thermodynamic equilibrium of the119873

119896

proteins in their decoherence-selected (stationary) conforma-tions which is not fulfilled in (nonstationary) conformationaltransitions induced by strong environmental interactions (cfthe Appendix for more details on our decoherence scenario)

ES119896

ΔEk10S119896

ΔEk01S119896 ΔE

k02S119896

ΔEk20S119896

k1119878119896k0119878119896

k2119878119896

|120601k0 ⟩S119896 |120601k2 ⟩S119896 |120601k⟩S119896|120601k1 ⟩S119896

Figure 2 Schematic presentation of the memory attractors inthe (many-electronic) energy-state (119864

119878119896(120601119896)) hypersurface of the

Hopfield-like quantum-holographic memorypropagator of theopen macroscopic quantum (sub)system 119878

119896of cellrsquos particular

spatial quantum ensemble of (noninteracting and dynamicallynoncoupled) 119873

119896chemically identical proteins of 119896th type (and

their corresponding biomolecular targets) [41 42] in Feynmanrsquosrepresentation [44] 119866(119903

2 1199052 1199031 1199051) = sum

119894120601119896119894 (1199032 1199052)120601119896lowast

119894 (1199031 1199051) =

sum119894119860119896119894(1199032 1199052)119860lowast

119896119894(1199031 1199051)119890(119894ℎ)(120572119896119894

(1199032 1199052)minus120572119896119894(1199031 1199051)) It should be pointed

out that quantum decoherence presumably plays a fundamentalrole in biological quantum-holographic neural networks via energy-state hypersurface shape adaptation (in contrast to low-temperatureartificial qubit quantum processors where it must be avoided untilthe very read-out act of quantum computation)mdashwhich impliesthat nature presumably has chosen an elegantroom-temperaturesolution for biological quantum-holographic information process-ing permanently fluctuating between eigenstates of energy andconformation of the proteins of 119896th type (identical in their primarystructure of the amino acids sequence) due to nonstationaryenvironmental perturbations and subsequent decoherence by theenvironment described by time-adapting density of conformationalstates (represented by corresponding depths of the minima in the

figure above)_120588119896

119878119896(119905) =sum

119894119908119896119894(119905)|120601119896119894⟩119878119896⟨120601119896119894 | sum119894119908119896119894(119905) = 1 of cellrsquos

biomolecular open macroscopic quantum (sub)system 119878119896

which might occur far from thermodynamic equilibrium (asis the case in metabolic processes in biological cells [43])

Or to generalize a series of all 119896 intracellular and extra-cellular environmentally driven (compositionallychemicallyor thermallyoptically) local biochemically coupled reactionsmight be equivalently considered as a series of all 119896 cor-responding intracellular and extracellular global Hopfield-like quantum holographically coupled associative neuralnetwork layersmdashproviding an equivalent global quantum-informational alternative to standardmolecular-biology localbiochemical approach in biomolecules and cells (and higherhierarchical levels of organism as well)4

3 Discussion and Conclusion

Biomolecules in a living biological cell are subjected tononequilibrium processes of huge complexity Elaboratequantum mechanical descriptions of such processes are onlya matter of recent considerations [45ndash47] In this regard thephysical methods are a matter of intense current research[48 49] A fully developed quantitatively elaborate quantummechanical background for such biological processes is yet aremote goal


In the context of the macroscopic quantum phenomenaof the second kind we hereby proposed quantum-chemicaland quantum decoherence approaches to biomolecular con-formational transitions which cannot be considered kineti-cally understood based on (semi)classical predictions Ourqualitative proposal has a solid quantum mechanical basis ofwide applicability there are not any particular assumptionson the chemical kind structure or the initial state of themolecule or any assumptions on the chemical kind or on theinitial state of the moleculersquos environment

It seems that our matter-of-principle solution to the longstanding Levinthal paradox offers a natural physical pictureof a number of the important processes with biomoleculesincluding chain folding and biomolecular recognition Thisoffers a basis for some (semi)classical descriptions suchas the recent (also qualitative) proposal of Dill and Chan[23] Actually quantum decoherenceis assumed to provide(quasi)classical behavior of the biomolecules conformationdegrees of freedom which can be further (semi)classicallydescribed to provide more details of the biomoleculersquos confor-mation dynamics in molecular biology and biochemistry

Our model (A2) is stochastic not deterministic andis sensitive to all allowed final conformations Dependingon the details of the physical model (initial state of themolecule the kind and strengths of interactions with thesolventmolecules and the form of the energy landscape etc)there is more than one possible final conformation in the sumequation (A2) that in principle includes the initial conforma-tion Regarding the funnel landscape of Dill and Chan [23]a few scenarios are possible For example if the particle isin a thin local minimum quantum tunneling can cause thehighly semiclassical dynamic that is essentially described byLeggett [1] and qualitatively agrees with Dill and Chan [23]the particle is expected quickly to go down the slope On theother hand for sufficiently deep local minimum the particlecan be trapped (eg in a metastable state) in which case therelated conformation appears in the sum in (A2) If such localminimum is in the vicinity of the absolute minimum andthe related conformations are practically indistinguishablethen our model predicts redefinition of the very conceptof ldquonative state (conformation)rdquo In this case ldquonative staterdquodoes not refer to a single but to a set of close conformationsof the moleculemdashagain in accordance with the qualitativeconsiderations of Dill and Chan [23] (and the referencestherein)

As another virtue of our decoherence model we empha-size existence of a few different mechanisms for the exter-nally induced conformational transitionsThosemechanisms(ldquochannelsrdquo) are defined by local influence on certain subsys-tem without yet influencing the other subsystems (degreesof freedom) of the molecule The subsystems of interestare conformational system vibrational system (vibrationalmodes) the electrons system and the local rotational degreesof freedom of the molecule Realistic transitions can beassumed to be combinations of those local ldquochannelsrdquo forconformational transitions In principle high precision andcontrol of the molecular degrees of freedom can experimen-tally partially distinguish between the different channels Forinstance illuminating the molecule by the microwaves of

the characteristic frequency sim109Hz should influence thelocal rotations in a molecule (with nonnegligible quantumtunneling between the allowed structural rotamers) withoutaffecting the other degrees of freedomwhile the infrared lightof the frequency sim1013Hz should influence the vibrationsin a molecule with possible nonradiative resonant structuralisomeric transitions (like in Figure 1 for transitions withinthe electronic ground state hypersurface) To this end somebasic details on the conformation-transitions mechanismscan be found in [32 33 39] respectively while research isstill in progress Direct influence on the conformation whichis typically considered in the statistical (eg thermal equilib-rium) approach is rather subtle and is often described as a neteffect mainly originating from a change of physicochemicalcharacteristics of the solution (in the manner described byAnfinsen [24]) without resorting to an elaborate model yetsee for example [14 equation (3164)]

Regarding the quantum ensemble prediction of our deco-herence model (DM) (A2) and the resembling Hopfield-like quantum-holographic neural network (HQHNN) bioin-formational framework of the environmentally driven bio-chemical reactions on the level of open biological cell(Figure 2) there are several notes that might be added inproof (i) biochemical reactions involve enzymatic processesand enzymersquos function is the DM conformational-adaptiveone (so fundamentally every single biochemical reactionhas bioinformational structure of HQHNN within the occu-pational basis of enzymersquos conformational states as anindicator of DM enzyme-mediated biochemical reactions)(ii) regarding all biochemical reactions of the particulartype in the cell a higher percentage of the functionallyappropriate enzymes take their native conformation (usu-ally the lowest energy state ) influenced by the prox-imity of the corresponding biomolecular substrate (key-lock enzyme-to-substrate DM conformational adjustments)but not the remaining percentage of those enzymes thatare not yet in close interaction with their biomolecularsubstrates (which then occupy the remaining possible con-formations as well according to DM) (iii) in accordancewith the previous point all biochemical reactions of theparticular type in the cell have bioinformational structureof HQHNN within the occupational basis of the corre-sponding enzymersquos conformational states (iv) taking intoaccount other successive intracellular and extracellular envi-ronmentally driven biochemical reactions that are function-ally interconnected with preceding biochemical reactionsin the cell they can also be successively presented in thebioinformational framework of HQHNNs within the occu-pational bases of conformational states of the correspondingenzymes involved (v) since all these successive biochem-ical reactions are functionally interconnected so are thesuccessive HQHNNs in bioinformational framework withinthe corresponding enzymesrsquo occupational bases (which maybe presented in the form of Hakenrsquos multilevel syner-getic neural network composed of layers of the successiveHQHNNs) and (vi) in such bioinformational frameworkof Hakenrsquos multilevel synergetic neural network each ofthe successive HQHNNs layers representing corresponding


intracellular and extracellular biochemical reactions has aformal Hopfield-like mathematical structure in the formof (nonmorphologicalabstract) ldquoformal neuronsrdquo massivelyinterconnected by ldquoformal connectionsrdquo while the layersof HQHNNs would be mutually quantum holographicallycoupled via their ldquomemory attractorsrdquo (ie their quantum-holographic memory states within the occupational basesof conformational states of the corresponding enzymesinvolved)

Such a generalized bioinformational framework ofHakenrsquos multilevel synergetic neural network representingcorresponding intracellular and extracellular biochemicalreactions is in line with trends of modeling hierarchicalinformation processing in higher cognitive processes [44]andmight also provide possible missing downward causationcontrol mechanism of morphogenesis and psychosomatics[41 42 50ndash52] However it should be noted that conditionsfor the above resemblance between quantum ensembleprediction of our DM and HQHNN frameworks are fulfillednear thermodynamic equilibrium (while predicted verynonstationary conformational transitions induced by strongenvironmental interactions within the decoherence model(cf the Appendix) might occur far from thermodynamicequilibrium)

Appendix

On General Quantum DecoherenceFramework for MacromolecularConformations and Transitions

We assume that the macromoleculersquos environment selects themolecule conformation as the ldquopointer observablerdquo Follow-ing the general phenomenological results and understandingwe assume that decoherence takes place for virtually allkinds of macromolecules and the solvent environment whilethe composite system ldquomacromolecule + environmentrdquo isnot externally disturbed This we call ldquostationary staterdquo forwhich we stipulate occurrence of decoherence that is theenvironment-induced classicality of the molecule conforma-tion Typically the conformation transitions occur due toa severe external influence exerted on the macromoleculardegrees of freedom andor to the molecule environment Ineffect this external influence redefines the macromoleculeenvironment and thereby also the influence exerted by thenew environment to the macromolecule degrees of freedomSuch physical situations which may take some time wedescribe as ldquononstationary staterdquo For the nonstationary stateno particular assumption is made Rather one may expectthat the change in physical characteristics and state of theenvironment would typically violate the conditions assumedfor the occurrence of decoherence

In formal terms the stationary state is defined by theconformation system ldquomixedrdquo state

120588119870

= sum

119894

119908119894

1003816100381610038161003816 119896119894⟩119870 ⟨1198961198941003816100381610038161003816 sum

119894

119908119894= 1 (A1)

where states |119896119894⟩119870

represent the different conformationalstates These (approximately orthogonal) states |119896

119894⟩119870 rep-

resent the preferred (semi)classical ldquopointer basisrdquo statesfor the macromolecule conformation system5 Therefore westipulate the occurrence of decoherence as the fundamen-tal quantum mechanical basis for the phenomenologicallyobserved (semi)classical behavior of the macromoleculesconformation stability

On the other hand as emphasized above the conforma-tional transitions occur due to a severe external influenceThe related nonstationary state is defined by nonvalidity of(A1) for the duration of the external influence Intuitivelyone may say that the external influence redefines the physicalsituation the macromolecule is subjected to In effect thestationary state is disturbed and there is not any semiclassicalconformation state for themacromolecule Formally the con-formation system is in state 1205881015840

119870 which cannot be presented by

(A1)Of course every external influence terminates and leaves

the (redefined) system ldquomolecule + environmentrdquo to relaxthat is to reach another stationary state with the finalconformation state 120588

10158401015840

119870 which is representable in the form

of (A2) The point strongly to be emphasized is that it ishighly unexpected that 120588

119870= 12058810158401015840

119870 That is the final set

of conformations need not be the same as the initial onewhile for the conformations (ie states |119896

119894⟩119870) common for

120588119870

and 12058810158401015840

119870 their statistical weights need not equal each

other 119908119894

= 11990810158401015840

119898 In effect the following transition of the

conformation state occurred

120588119870

= sum

119894

119908119894

1003816100381610038161003816 119896119894⟩119870 ⟨1198961198941003816100381610038161003816

nonstationary997888997888997888997888997888997888997888997888997888997888997888997888rarr 120588

1015840

119870

stationary997888997888997888997888997888997888997888997888997888997888rarr 120588

10158401015840

119870

= sum

119898

11990810158401015840

119898

1003816100381610038161003816 119896119898⟩119870 ⟨1198961198981003816100381610038161003816 sum

119894

119908119894= 1 = sum

119898

11990810158401015840

119898

(A2)

Duration of the whole dynamics presented by (A2) is ofthe order of the time needed for the nonstationary state toterminate (note that decoherence present for both stationarystates the most left hand and the most right hand sidesof (A2) is among the fastest physical processes known todate) So in this scenario the Levinthalrsquos paradox disappearsFurthermore as the concept of ldquotrajectoryrdquo (in configurationspace) is not well-defined quantum mechanically the verybasis of the Levinthalrsquos paradox (ie sampling of trajectoriesin the configuration space) is absent in this quantummechan-ical picture

This general scenario has been analysed [46 47] and afew possible scenarios of the external influence (ie of thenonstationary state) have been distinguished (see Figure 3 fora possible one)

Disclosure

This research was presented at the Fifteenth Annual Confer-ence YUCOMAT2013 HercegNoviMontenegro September2ndash6 2013 httpwwwmrs-serbiaorgrs


K

Decoh decoherence

Eg

E1

DecohDecoh

Dee

xc

V

k1 k2

Exci

tatio

n

k998400 k998400998400

E electrons energy levelK conformationV effective energy for K

Figure 3 The black dot represents one-dimensional ldquoparti-clerdquo (macromolecular electronic-conformational-vibrational sys-tem) which is excited to the upper hypersurface (electronic excitedstate 119864

1) and according to Ehrenfest theorem is descended down

the slope towards the local excited state minimum conformation1198961015840 Then the ldquoparticlerdquo is deexcited to the lower hypersurface

(electronic ground state 119864119892) when according to Ehrenfest theorem

a superposition of the two possible ldquoparticlerdquo states depicted by1198961015840 and 119896

10158401015840 is established with subsequent coherent descent of theldquoparticlerdquo down both slopes of the barrier (which can be thoughtof as the interference of the two paths along the barrier walls)and final decoherence into local ground state minima 119896

1and

1198962(with assumption that environmental influence dominates the

dynamics in the vicinity of the local minima corresponding to theacquired conformations 119896

1and 119896

2mdashwith their statistical weights

being changed as a net effect) For more details see [47]

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

This paper was partly financed by the Serbian Ministry ofScience Technology and Development Project no 171028

Endnotes

1 The Paradigm of the macroscopically differentiatedstates are the Gaussian states of the center of massdegreesof freedom of many-particle system On thecontrary so-called relative coordinates (as observables)neither define macroscopically differentiated states norcarry classical behavior of the system in any knownphysical theory or experimental situation

2 In general the stipulated decoherence-preferred degreesof freedom (biomolecular conformations in our case)are considered to be accessible (directly measurable)and therefore objective for an environmental observer(which is thus a part of the structure he observes)Are there some general rules andor limitations for allpossible bi-partitions QS

119896+ QE

119896in the universe It is

not generally answered in QD theory and still needs

additional fenomenological assumptions [13] So in themanner of quantum chemistry [29 30] it might beplausibly proposed that many-atomic quantum systemsQS119896are fenomenologically limited to the structures

with dynamically coupled electrons considered as iden-tical fermions described by permutationally antisym-metric many-electronic eigenstates which encompassesall existing molecules and electronic condensed stateobjects described by general quantum-chemical elec-tronic self-Hamiltonian in the case of the structureswith dynamically coupled identical bosons they aredescribed by permutationally symmetric many-bosoniceigenstates and corresponding self-Hamiltonian

3 In general only closed composite system QS + QE issubject to the Schrodinger law (although this does nothold true separately for neither QS nor QE as openquantum systems) with Hamiltonian

_119867=

_119867QS +

_119867QE +

_119867int where interaction Hamiltonian

_119867int depends on

observables of both QS and QE However when_119867int

can be reduced to the ldquoexternal fieldrdquo its potential term_119881 can be added to

_119867QS providing new self-Hamiltonian

of the QS dynamically decoupled from the observablesof the QE and then QS can be treated as the closedquantum system This is the case in most situations inquantum chemistry with Schrodingerrsquos equation appliedto the explored closed many-atomic quantum systemwith appropriate boundary conditions and adoptedcomputational approximations (giving rise to station-ary ground and excited electronic-vibrational energyeigenstates of all possible many-atomic isomers corre-sponding to the minimum of the electronic potentialhypersurface depicted in Figure 1 for ground electronicand corresponding excited vibrational energy eigen-states) [29 30] It should be noted that Schrodingerrsquosequation cannot apply to nonstationary excitations andrelaxations of the many-atomic quantum system notonly in between different isomers but also within thesame isomermdashwhen quantumdeexcitationdecoherencemust apply to nonpotential interaction of the openmany-atomic quantum system (nondescribable fully by its self-Hamiltonian) with its quantum environment (generallyfield-related including vacuum) [12 13]

4 The similar Hopfield-like quantum-holographic picturemight also be applied to individual acupuncture system[41 42] (with quantum-like macroscopic resonancesfenomenologically observed in microwave resonancetherapy [50 51] which implies corresponding biparti-tion QSacu + QEacu ie that acupuncture system hasmacroscopic open quantum structure with dynamicallycoupled electrons all along macroscopic network ofacupuncture channels [41 42 50]) This then providesnatural quantum-informational framework for psycho-somatic medicine that is quantum-holographic down-ward coupling of the higher macroscopic quantum levelsof acupuncture system and its projection zones (andpresumably closely related consciousness [10 41 42])with lowermacroscopic quantum cellrsquos biomolecular level


changing the expression of genes (starting from the firstfertilized cell division which initializes differentiation ofthe acupuncture system of nonthreshold electrical GJ-synapses (ldquogap-junctionsrdquo)) [41 42 52]

5 Normalized ldquopointer basisrdquo states 1003816100381610038161003816 119896119894⟩119870 for the macro-molecule conformation system of (A1) are shown to bealmost-classical one-dimensional harmonic oscillationldquocoherent statesrdquo 10038161003816100381610038161003816

120595119902119894(119905)119901119894(119905)

⟩ [40] Then (A1) physicallymeans that each macromolecule in a solution oscillateswith probability 119908

119894along classical harmonic trajectories

(119902119894(119905)119901119894(119905)) of the mean values of position and momen-

tum where the time change of 119902119894(119905) and 119901

119894(119905) is the

classical law for the harmonic oscillator position andmomentum

119902119894(119905) = 119902

0119894cos120596119894119905 + (

1199010119894

119898120596119894

) sin120596119894119905

119901119894(119905) = 119901

0119894cos120596119894119905 minus 119898120596

1198941199020119894sin120596119894119905

(lowast)

in the vicinity of the 119896119894th local minimum (which can

be locally approximated by the harmonic potential cfFigure 3) that is in the vicinity of the 119896

119894th conforma-

tion Bearing in mind that the ldquocoherent statesrdquo do notchange their Gaussian shape (Δ

_119870119894 = const Δ

_119875119894 =

const Δ_119870119894Δ

_119875119894 = ℎ2) in the course of time [53ndash55]

(lowast) has the clear (semi)classical meaning defining theconformations as the harmonic oscillation equilibriumpositions (cf the local minimums 119896

119894in Figure 3) one

obtains the semiclassical vibration of a macromoleculeconformation in the vicinity of a local minimum 119896

119894

References

[1] A J Leggett ldquoMacroscopic quantum systems and the quantumtheory of measurementrdquo Progress of Theoretical Physics no 69supplement pp 80ndash100 1980

[2] A J Leggett and A Garg ldquoQuantum mechanics versus macro-scopic realism is the flux there when nobody looksrdquo PhysicalReview Letters vol 54 no 9 pp 857ndash860 1985

[3] WH Zurek ldquoDecoherence and the transition from quantum toclassicalrdquo Physics Today vol 44 no 10 pp 36ndash44 1991

[4] W H Zurek ldquoDecoherence einselection and the quantumorigins of the classicalrdquo Reviews of Modern Physics vol 75 no3 pp 715ndash765 2003

[5] G C Ghirardi A Rimini and T Weber ldquoUnified dynamics formicroscopic and macroscopic systemsrdquo Physical Review D vol34 no 2 pp 470ndash491 1986

[6] R Penrose ldquoOn gravitys role in quantum state reductionrdquoGeneral Relativity and Gravitation vol 28 no 5 pp 581ndash6001996

[7] J Kofler and C Brukner ldquoClassical world arising out ofquantum physics under the restriction of coarse-grained mea-surementsrdquo Physical Review Letters vol 99 no 18 Article ID180403 2007

[8] J Kofler and C Brukner ldquoConditions for quantum violationof macroscopic realismrdquo Physical Review Letters vol 101 no 9Article ID 090403 2008

[9] D Rakovic and M Dugic ldquoA critical note on the role ofthe quantum mechanical ldquocollapserdquo in quantum modeling ofconsciousnessrdquo Informatica vol 26 no 1 pp 85ndash90 2002

[10] D Rakovic M Dugic andMM Cirkovic ldquoMacroscopic quan-tum effects in biophysics and consciousnessrdquoNeuroQuantologyvol 2 no 4 pp 237ndash262 2004

[11] M Dugic D Rakovic J Jeknic-Dugic andM Arsenijevic ldquoTheghostly quantum worldsrdquo NeuroQuantology vol 10 no 4 pp619ndash628 2012

[12] M Dugic ldquoDecoherence in classical limit of quantummechan-icsrdquo SFIN XVII 2 Institute of Physics Belgrade Serbia 2004(Serbian)

[13] J Jeknic-Dugic M Arsenijevic and M Dugic QuantumStructures A View of the Quantum World LAP LambertSaarbrucken Germany 2013

[14] D Giulini E Joos C Kiefer J Kupsch I-O Stamatescu andHD Zeh Decoherence and the Appearance of a Classical World inQuantumTheory Springer Berlin Germany 1996

[15] V Vedral Decoding Reality The Universe as Quantum Informa-tion Oxford University Press Oxford UK 2010

[16] H Primas ldquoRealism and quantum mechanicsrdquo in LogicMethodology and Philosophy of Science IX D Prawitz BSkyrms and D Westerstahl Eds Elsevier Science BV Ams-terdam The Netherlands 1994

[17] C Levinthal ldquoAre there pathways for protein foldingrdquo Journalde Chimie Physique vol 65 pp 44ndash45 1968

[18] Y Duan and P A Kollman ldquoPathways to a protein folding inter-mediate observed in a 1-microsecond simulation in aqueoussolutionrdquo Science vol 282 no 5389 pp 740ndash744 1998

[19] U Mayor N R Guydosh C M Johnson et al ldquoThe completefolding pathway of a protein from nanoseconds to microsec-ondsrdquo Nature vol 421 no 6925 pp 863ndash867 2003

[20] P L Freddolino F Liu M Gruebele and K Schulten ldquoTen-microsecond molecular dynamics simulation of a fast-foldingWW domainrdquo Biophysical Journal vol 94 no 10 pp L75ndashL772008

[21] D E Shaw PMaragakis K Lindorff-Larsen et al ldquoAtomic-levelcharacterization of the structural dynamics of proteinsrdquo Sciencevol 330 no 6002 pp 341ndash346 2010

[22] G M Seddon and R P Bywater ldquoAccelerated simulation ofunfolding and refolding of a large single chain globular proteinrdquoOpen Biology vol 2 no 7 Article ID 120087 2012

[23] K A Dill and H S Chan ldquoFrom levinthal to pathways tofunnelsrdquo Nature Structural Biology vol 4 no 1 pp 10ndash19 1997

[24] C B Anfinsen ldquoPrinciples that govern the folding of proteinchainsrdquo Science vol 181 no 4096 pp 223ndash230 1973

[25] A Perdomo-Ortiz N Dickson M Drew-Brook G Rose andA Aspuru-Guzik ldquoFinding low-energy conformations of latticeproteinmodels by quantum annealingrdquo Scientific Reports vol 2article 571 2012

[26] W E Hart and S Istrail ldquoRobust proofs of NP-Hardness forprotein folding general lattices and energy potentialsrdquo Journalof Computational Biology vol 4 no 1 pp 1ndash22 1997

[27] B Berger and T Leighton ldquoProtein folding in the hydrophobic-hydrophilic (HP) model is NP-completerdquo Journal of Computa-tional Biology vol 5 no 1 pp 27ndash40 1998

[28] P Crescenzi D Goldman C Papadimitriou A Piccolboni andM Yannakakis ldquoOn the complexity of protein foldingrdquo Journalof Computational Biology vol 5 no 3 pp 597ndash603 1998

[29] L A Gribov Introduction to Molecular Spectroscopy NaukaMoscow Russia 1976 (Russian)


[30] L A Gribov From Theory of Spectra to Theory of ChemicalTransformations URSS Moscow Russia 2001 (Russian)

[31] D Rakovic M Dugic J Jeknic-Dugic et al ldquoOn some quantumapproaches to biomolecular recognitionrdquo Contemporary Mate-rials vol 1 no 1 pp 80ndash86 2010

[32] G Kekovic D Rakovic and D Davidovic ldquoRelevance ofpolaronsoliton-like transport mechanisms in cascade resonantisomeric transitions of Q1D-molecular chainsrdquo Materials Sci-ence Forum vol 555 pp 119ndash124 2007

[33] G Kekovic D Rakovic and D Davidovic ldquoA new look atthe structural polymer transitions ldquobridging the quantum gaprdquothrough non-radiative processesrdquo in Proceedings of the 9thYugoslav Materials Research Society Conference (YUCOMATrsquo07) pp 10ndash14 Herceg Novi Montenegro September 2007

[34] D Rakovic M Dugic andM Plavsic ldquoThe polymer conforma-tional transitions a quantum decoherence approachrdquoMaterialsScience Forum vol 453-454 pp 521ndash528 2004

[35] M Dugic D Rakovic andM Plavsic ldquoThe polymer conforma-tional stability and transitions a quantum decoherence theoryapproachrdquo in Finely Dispersed Particles Micro- Nano- andAtto-Engineering A Spasic and J-P Hsu Eds chapter 9 CRCPress New York NY USA 2005

[36] D Rakovic M Dugic and M Plavsic ldquoBiopolymer chainfolding and biomolecular recognition a quantum decoherencetheory approachrdquo Materials Science Forum vol 494 pp 513ndash518 2005

[37] D Rakovic M Dugic M Plavsic G Kekovic I Cosic and DDavidovic ldquoQuantum decoherence and quantum-holographicinformation processes from biomolecules to biosystemsrdquoMaterials Science Forum vol 518 pp 485ndash490 2006

[38] J Jeknic M Dugic and D Rakovic ldquoA unified decoherence-based model of microparticles in a solutionrdquo Materials ScienceForum vol 555 pp 405ndash410 2007

[39] J Jeknic-Dugic ldquoThe environmentrsquoinduced-superselectionmodel of the large molecules conformational stability andtransitionsrdquo European Physical Journal D vol 51 no 2 pp193ndash204 2009

[40] J Jeknic-DugicDecoherence model of molecular conformationaltransitions [PhD thesis] Faculty of Science Kragujevac Serbia2010 (Serbian)

[41] D Rakovic Integrative Biophysics Quantum Medicine andQuantum-Holographic Informatics Psychosomatic-CognitiveImplications IASC amp IEPSP Belgrade Serbia 2009

[42] D Rakovic A Skokljev and D Djordjevic Introduction toQuantum-Informational Medicine with Basics of Quantum-Holographic Psychosomatics Acupuncturology and Reflexother-apy ECPD Belgrade Serbia 2010 (Serbian)

[43] M V Volkenshtein Biophysics Mir Moscow Russia 1983[44] M Perus ldquoMulti-level synergetic computation in brainrdquo Non-

linear Phenomena in Complex Systems vol 4 pp 157ndash193 2001[45] L Luo and J Lu ldquoTemperature dependence of protein fold-

ing deduced from quantumtransitionrdquo httparxivorgabs11023748

[46] J Jeknic-Dugic ldquoThe environment-induced-superselectionmodel of the large molecules conformational stability andtransitionsrdquo European Physical Journal D vol 51 no 2 pp193ndash204 2009

[47] J Jeknic-Dugic ldquoProtein folding the optically induced elec-tronic excitation modelrdquo Physica Scripta vol T135 Article ID014031 2009

[48] H P Breuer and F Petruccione The Theory of Open QuantumSystems Clarendon Press Oxford UK 2002

[49] A Rivas and S F HuelgaOpen Quantum Systems An Introduc-tion SpringerBriefs Springer Berlin Germany 2011

[50] S P Sitrsquoko and L N Mkrtchian Introduction to QuantumMedicine Pattern Kiev Ukraine 1994

[51] N D Devyatkov and O Betskii Eds Biological Aspects of LowIntensity Millimetre Waves Seven Plus Moscow Russia 1994

[52] Y Zhang ECIWO Biology and Medicine A New Theory ofConquering Cancer and Completely New Acupuncture TherapyNeimenggu People Press Beijing China 1987

[53] E Schrodinger ldquoDer stetige Ubergang von der Mikro- zurMakromechanikrdquoN1turwissensch1ften vol 14 no 28 pp 664ndash666 1926

[54] R OmnesThe Interpret1tion of Qu1ntumMech1nics PrincetonSeries in Physics Princeton University Press Princeton NJUSA 1994

[55] C Gerry and P L Knight Introductory Qu1ntum OpticsCambridge University Press Cambridge UK 2005

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Anatomy Research International

PeptidesInternational Journal of


Hindawi Publishing Corporation httpwwwhindawicom

International Journal of

Volume 2014

Zoology


Molecular Biology International

GenomicsInternational Journal of


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


BioinformaticsAdvances in

Marine BiologyJournal of



Signal TransductionJournal of


BioMed Research International

Evolutionary BiologyInternational Journal of



Biochemistry Research International

ArchaeaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014


Genetics Research International


Advances in

Virolog y

Hindawi Publishing Corporationhttpwwwhindawicom

Nucleic AcidsJournal of

Volume 2014

Stem CellsInternational



Enzyme Research



Microbiology


then we have to require that a proper mathematical codifica-tion of this theorymust be capable to describe all phenomenaof molecular and engineering science Already rather smallmolecules can have classical properties so that a classicalbehavior is not a characteristic property of large systemsTheexistence of molecular superselection rules and of molecularclassical observables is an empirically well-known fact inchemistry and molecular biology The chirality of somemolecules the knot type of circular DNA-molecules and thetemperature of chemical substances are three rather differentexamples of molecular classical observables Such empiricalfacts can be described in an ad hoc phenomenologicalmanner but it is not so easy to explain these phenomenafrom the first principles of quantummechanics A universallyvalid theory of matter has not only to describe but also toexplain why the chirality of biomolecules (like the L-aminoacids theD-sugars lipids or steroids) is a classical observableThe reality of this breakdown of the superposition principleof traditional quantum mechanics on the molecular level isdramatically demonstrated by the terrible Vontergan tragedywhich caused many severe birth defectsrdquo

Starting from the 1980s mainly in the papers of Leggett[1 2] a new period of investigation of quantum mechanicalphenomena on the macroscopic level began Namely aclarification of the notions and planning of experimentalsituations for observing some physical effects started Thecentral problem in this respect is a notion of macroscopicdifferentiation of the states of quantum system whose quan-tummechanical behavior is explored More precisely Leggettargues that the term macroscopic quantum mechanical effectmust be related tomacroscopically different states that is thesystem states (and observables) that carry macroscopic prop-erties (and behaviors) of the system as a whole These states(ie observables) must carry classical-physical behavior ofthe system as well this poses a task for choosing physicalconditions giving rise to observation of typical quantumeffects related to these states1

Hence different kinds of macroscopic quantum phenom-ena (i) the ones usually explored by the methods of (quan-tum) statistical physics and not related to macroscopicallydifferentiated states (being classified asmacroscopic quantumphenomena of the first kind like solid state phenomena)and (ii) those ones regarding macroscopically different (dif-ferentiated) states (being classified as macroscopic quantumphenomena of the second kind and being interesting to us)Numerous differentmacroscopic quantumphenomena of thesecond kind some of them belonging to the fast developingfield of the quantum computing and information unequiv-ocally sharpen the overall problem of universal validity ofquantum mechanics

In the context of the macroscopic quantum phenomenaof the second kind we shall present a solution-in-principleof the long standing problem of the polymer folding (whichwas considered by Levinthal as (semi)classically intractable[17] as shortly reviewed below)mdashimplying the existence ofnovel macroscopic quantum biomolecular phenomena withfar reaching implications

Levinthal Paradox Revisited Contemporary methods forcalculation of conformational dependent chain propertiesare based on thermodynamic aspect of the problem whichexplores (semi)classically the folding free energy landscapefor protein with several successful attempts to model theseprocesses in silico using molecular dynamics simulationswith full atomic representation of both protein and solvent[18ndash22] producing continuous (semi)classical trajectorieswith the potential to connect static structural snapshotsgenerated from experimental data This is incorporated intothe (semi)classical viewpoint that conformational changes ofproteins due to solvent thermal optical and other influencesof the environment do not occur in a random way (egmovements of gas particles) but fold to their native conforma-tion of deep global minimum in some (semi)classical funnelof low-energy conformations leading toward it [23] Evenin recently reported implementation of quantum annealing(on the programmable superconducting quantum device) forlattice protein folding problems nothing quantum mechan-ical is implied about principles that govern the folding ofprotein chains [24] (rather quantum fluctuations are a toolused for solving the optimization problem of protein foldingconsidered classically intractable [25ndash27])

Hence these (semi)classical calculational methods donot describe properly transitions from one conformationto another which is the kinetic aspect of the problemexploring the conformation change of long flexible chainThis has been illustrated by Levinthal who considered theprobability of folding a proteinmolecule from coiled to nativeconformation [17] Assume 2119899 torsional angles of an 119899-residueprotein each having three stabile rotational states this yields32119899

asymp 10119899 possible conformations for the chain if a protein

can explore new conformations in a random way at the ratethat single bond can rotate it can find approximately 1013conformations per seconds then the time 119905 (s) required fora protein to explore all the conformations available to it is119905 = 10

1198991013 for a rather small protein of 119899 = 100 residues

one obtains 119905 = 1087 s which is immensely more than

the apparent age of the universe (ldquoLevinthal paradoxrdquo) Yetaccording to experiments proteins can fold to their nativeconformation in less than a few seconds [28]

It should be added that (semi)classical kinetic (nonst-ationary) predictions imply the continuous mapconformation change 119896

119894rarr 119896119891which requires a sequence of 119899

local noncommuting successive elementary transformations(local rotations of characteristic time 120591

119900) with the time

necessary for the net transformation much longerthan characteristic time necessary for a local rotation(120591119899sim 119899120591119900≫ 120591119900) and the frequency of corresponding global

transition much lower than the frequency of a local rotation(119891119899

sim 1119899120591119900

sim 119891119900119899 ≪ 119891

119900)mdashstrongly dependent on a

degree of polymerization 119899 (in clear contradistinction withthe experimentally observed poorly dimensionally sensitivedispersion laws of the internal more or less delocalizedquasiparticle excitations in any condensed state quantumsystem electrons phonons and etc [29]) Thus chainfolding based on (semi)classical (nonstationary) predictionscannot be considered kinetically understood the same





119890(120601(119896)









_119867

_119870] = 0






119890


energy 119864(119894)





119894 in Figure 1)


119890+ 119864(119894)

V = 119864(119891)

119890+ 119864(119891)






10038161003816100381610038161003816Φ119894⟩119864

QS10038161003816100381610038161003816Ψ119894⟩119864

QE

= sum

119895

119888119895

100381610038161003816100381610038161003816Φ119895⟩119870

QS

100381610038161003816100381610038161003816Ψ119895⟩119870

QE

QD997888997888997888997888997888rarr

10038161003816100381610038161003816Φ119896⟩119870

QS10038161003816100381610038161003816Ψ119896⟩119870

QE [997888rarr 120588119870

ΦΨ]+Δ119864exc

997888997888997888997888997888997888997888997888rarr

= sum

119897

1198881015840

119897

10038161003816100381610038161003816Φ119897⟩119864

QS10038161003816100381610038161003816Ψ119897⟩119864

QE


100381610038161003816100381610038161003816Φ119891⟩119864

QS

100381610038161003816100381610038161003816Ψ119891⟩119864

QE[997888rarr 120588

119864


(1)




Ee

E(i)

E(f)

E(i)e E

(f)e

120601(k)e

(120601 )(120601 ) )

120601(f)e120601(i

(i

)e

(f)


119890(120601(119896)



(119894119891)

119890119878(119894119891)

V + 120583(119894119891)

V 119878(119894119891)

119890





V and 119878(119894119891)

119890or in


(119894)

119890120601(119894)

V plusmn 120601(119891)

119890120601(119891)


(119891)

119890or into


vibrational states)



119896pro-




120588119896



119896


ES119896

ΔEk10S119896

ΔEk01S119896 ΔE

k02S119896

ΔEk20S119896

k1119878119896k0119878119896

k2119878119896

|120601k0 ⟩S119896 |120601k2 ⟩S119896 |120601k⟩S119896|120601k1 ⟩S119896








2 1199052 1199031 1199051) = sum

119894120601119896119894 (1199032 1199052)120601119896lowast

119894 (1199031 1199051) =

sum119894119860119896119894(1199032 1199052)119860lowast

119896119894(1199031 1199051)119890(119894ℎ)(120572119896119894




119878119896(119905) =sum

















Appendix




120588119870

= sum

119894

119908119894

1003816100381610038161003816 119896119894⟩119870 ⟨1198961198941003816100381610038161003816 sum

119894

119908119894= 1 (A1)

where states |119896119894⟩119870


119894⟩119870 rep-






10158401015840



119870= 12058810158401015840



119894⟩119870) common for

120588119870

and 12058810158401015840


other 119908119894

= 11990810158401015840



120588119870

= sum

119894

119908119894

1003816100381610038161003816 119896119894⟩119870 ⟨1198961198941003816100381610038161003816


1015840

119870

stationary997888997888997888997888997888997888997888997888997888997888rarr 120588

10158401015840

119870

= sum

119898

11990810158401015840

119898

1003816100381610038161003816 119896119898⟩119870 ⟨1198961198981003816100381610038161003816 sum

119894

119908119894= 1 = sum

119898

11990810158401015840

119898

(A2)



Disclosure



K

Decoh decoherence

Eg

E1

DecohDecoh

Dee

xc

V

k1 k2

Exci

tatio

n

k998400 k998400998400








1and



1and 119896





Acknowledgment


Endnotes



119896+ QE






_119867=

_119867QS +

_119867QE +











120595119902119894(119905)119901119894(119905)





119894(119905) is the


119902119894(119905) = 119902

0119894cos120596119894119905 + (

1199010119894

119898120596119894

) sin120596119894119905

119901119894(119905) = 119901

0119894cos120596119894119905 minus 119898120596

1198941199020119894sin120596119894119905

(lowast)



119894th conforma-


_119870119894 = const Δ

_119875119894 =

const Δ_119870119894Δ





119894

References
































































Volume 2014

Zoology






















Advances in

Virolog y



Volume 2014




Enzyme Research



Microbiology





119890(120601(119896)









_119867

_119870] = 0






119890


energy 119864(119894)





119894 in Figure 1)


119890+ 119864(119894)

V = 119864(119891)

119890+ 119864(119891)






10038161003816100381610038161003816Φ119894⟩119864

QS10038161003816100381610038161003816Ψ119894⟩119864

QE

= sum

119895

119888119895

100381610038161003816100381610038161003816Φ119895⟩119870

QS

100381610038161003816100381610038161003816Ψ119895⟩119870

QE

QD997888997888997888997888997888rarr

10038161003816100381610038161003816Φ119896⟩119870

QS10038161003816100381610038161003816Ψ119896⟩119870

QE [997888rarr 120588119870

ΦΨ]+Δ119864exc

997888997888997888997888997888997888997888997888rarr

= sum

119897

1198881015840

119897

10038161003816100381610038161003816Φ119897⟩119864

QS10038161003816100381610038161003816Ψ119897⟩119864

QE


100381610038161003816100381610038161003816Φ119891⟩119864

QS

100381610038161003816100381610038161003816Ψ119891⟩119864

QE[997888rarr 120588

119864


(1)




Ee

E(i)

E(f)

E(i)e E

(f)e

120601(k)e

(120601 )(120601 ) )

120601(f)e120601(i

(i

)e

(f)


119890(120601(119896)



(119894119891)

119890119878(119894119891)

V + 120583(119894119891)

V 119878(119894119891)

119890





V and 119878(119894119891)

119890or in


(119894)

119890120601(119894)

V plusmn 120601(119891)

119890120601(119891)


(119891)

119890or into


vibrational states)



119896pro-




120588119896



119896


ES119896

ΔEk10S119896

ΔEk01S119896 ΔE

k02S119896

ΔEk20S119896

k1119878119896k0119878119896

k2119878119896

|120601k0 ⟩S119896 |120601k2 ⟩S119896 |120601k⟩S119896|120601k1 ⟩S119896








2 1199052 1199031 1199051) = sum

119894120601119896119894 (1199032 1199052)120601119896lowast

119894 (1199031 1199051) =

sum119894119860119896119894(1199032 1199052)119860lowast

119896119894(1199031 1199051)119890(119894ℎ)(120572119896119894




119878119896(119905) =sum

















Appendix




120588119870

= sum

119894

119908119894

1003816100381610038161003816 119896119894⟩119870 ⟨1198961198941003816100381610038161003816 sum

119894

119908119894= 1 (A1)

where states |119896119894⟩119870


119894⟩119870 rep-






10158401015840



119870= 12058810158401015840



119894⟩119870) common for

120588119870

and 12058810158401015840


other 119908119894

= 11990810158401015840



120588119870

= sum

119894

119908119894

1003816100381610038161003816 119896119894⟩119870 ⟨1198961198941003816100381610038161003816


1015840

119870

stationary997888997888997888997888997888997888997888997888997888997888rarr 120588

10158401015840

119870

= sum

119898

11990810158401015840

119898

1003816100381610038161003816 119896119898⟩119870 ⟨1198961198981003816100381610038161003816 sum

119894

119908119894= 1 = sum

119898

11990810158401015840

119898

(A2)



Disclosure



K

Decoh decoherence

Eg

E1

DecohDecoh

Dee

xc

V

k1 k2

Exci

tatio

n

k998400 k998400998400








1and



1and 119896





Acknowledgment


Endnotes



119896+ QE






_119867=

_119867QS +

_119867QE +











120595119902119894(119905)119901119894(119905)





119894(119905) is the


119902119894(119905) = 119902

0119894cos120596119894119905 + (

1199010119894

119898120596119894

) sin120596119894119905

119901119894(119905) = 119901

0119894cos120596119894119905 minus 119898120596

1198941199020119894sin120596119894119905

(lowast)



119894th conforma-


_119870119894 = const Δ

_119875119894 =

const Δ_119870119894Δ





119894

References
































































Volume 2014

Zoology






















Advances in

Virolog y



Volume 2014




Enzyme Research



Microbiology


Ee

E(i)

E(f)

E(i)e E

(f)e

120601(k)e

(120601 )(120601 ) )

120601(f)e120601(i

(i

)e

(f)


119890(120601(119896)



(119894119891)

119890119878(119894119891)

V + 120583(119894119891)

V 119878(119894119891)

119890





V and 119878(119894119891)

119890or in


(119894)

119890120601(119894)

V plusmn 120601(119891)

119890120601(119891)


(119891)

119890or into


vibrational states)



119896pro-




120588119896



119896


ES119896

ΔEk10S119896

ΔEk01S119896 ΔE

k02S119896

ΔEk20S119896

k1119878119896k0119878119896

k2119878119896

|120601k0 ⟩S119896 |120601k2 ⟩S119896 |120601k⟩S119896|120601k1 ⟩S119896








2 1199052 1199031 1199051) = sum

119894120601119896119894 (1199032 1199052)120601119896lowast

119894 (1199031 1199051) =

sum119894119860119896119894(1199032 1199052)119860lowast

119896119894(1199031 1199051)119890(119894ℎ)(120572119896119894




119878119896(119905) =sum

















Appendix




120588119870

= sum

119894

119908119894

1003816100381610038161003816 119896119894⟩119870 ⟨1198961198941003816100381610038161003816 sum

119894

119908119894= 1 (A1)

where states |119896119894⟩119870


119894⟩119870 rep-






10158401015840



119870= 12058810158401015840



119894⟩119870) common for

120588119870

and 12058810158401015840


other 119908119894

= 11990810158401015840



120588119870

= sum

119894

119908119894

1003816100381610038161003816 119896119894⟩119870 ⟨1198961198941003816100381610038161003816


1015840

119870

stationary997888997888997888997888997888997888997888997888997888997888rarr 120588

10158401015840

119870

= sum

119898

11990810158401015840

119898

1003816100381610038161003816 119896119898⟩119870 ⟨1198961198981003816100381610038161003816 sum

119894

119908119894= 1 = sum

119898

11990810158401015840

119898

(A2)



Disclosure



K

Decoh decoherence

Eg

E1

DecohDecoh

Dee

xc

V

k1 k2

Exci

tatio

n

k998400 k998400998400








1and



1and 119896





Acknowledgment


Endnotes



119896+ QE






_119867=

_119867QS +

_119867QE +











120595119902119894(119905)119901119894(119905)





119894(119905) is the


119902119894(119905) = 119902

0119894cos120596119894119905 + (

1199010119894

119898120596119894

) sin120596119894119905

119901119894(119905) = 119901

0119894cos120596119894119905 minus 119898120596

1198941199020119894sin120596119894119905

(lowast)



119894th conforma-


_119870119894 = const Δ

_119875119894 =

const Δ_119870119894Δ





119894

References
































































Volume 2014

Zoology






















Advances in

Virolog y



Volume 2014




Enzyme Research



Microbiology











Appendix




120588119870

= sum

119894

119908119894

1003816100381610038161003816 119896119894⟩119870 ⟨1198961198941003816100381610038161003816 sum

119894

119908119894= 1 (A1)

where states |119896119894⟩119870


119894⟩119870 rep-






10158401015840



119870= 12058810158401015840



119894⟩119870) common for

120588119870

and 12058810158401015840


other 119908119894

= 11990810158401015840



120588119870

= sum

119894

119908119894

1003816100381610038161003816 119896119894⟩119870 ⟨1198961198941003816100381610038161003816


1015840

119870

stationary997888997888997888997888997888997888997888997888997888997888rarr 120588

10158401015840

119870

= sum

119898

11990810158401015840

119898

1003816100381610038161003816 119896119898⟩119870 ⟨1198961198981003816100381610038161003816 sum

119894

119908119894= 1 = sum

119898

11990810158401015840

119898

(A2)



Disclosure



K

Decoh decoherence

Eg

E1

DecohDecoh

Dee

xc

V

k1 k2

Exci

tatio

n

k998400 k998400998400








1and



1and 119896





Acknowledgment


Endnotes



119896+ QE






_119867=

_119867QS +

_119867QE +











120595119902119894(119905)119901119894(119905)





119894(119905) is the


119902119894(119905) = 119902

0119894cos120596119894119905 + (

1199010119894

119898120596119894

) sin120596119894119905

119901119894(119905) = 119901

0119894cos120596119894119905 minus 119898120596

1198941199020119894sin120596119894119905

(lowast)



119894th conforma-


_119870119894 = const Δ

_119875119894 =

const Δ_119870119894Δ





119894

References
































































Volume 2014

Zoology






















Advances in

Virolog y



Volume 2014




Enzyme Research



Microbiology


K

Decoh decoherence

Eg

E1

DecohDecoh

Dee

xc

V

k1 k2

Exci

tatio

n

k998400 k998400998400








1and



1and 119896





Acknowledgment


Endnotes



119896+ QE






_119867=

_119867QS +

_119867QE +











120595119902119894(119905)119901119894(119905)





119894(119905) is the


119902119894(119905) = 119902

0119894cos120596119894119905 + (

1199010119894

119898120596119894

) sin120596119894119905

119901119894(119905) = 119901

0119894cos120596119894119905 minus 119898120596

1198941199020119894sin120596119894119905

(lowast)



119894th conforma-


_119870119894 = const Δ

_119875119894 =

const Δ_119870119894Δ





119894

References
































































Volume 2014

Zoology






















Advances in

Virolog y



Volume 2014




Enzyme Research



Microbiology




120595119902119894(119905)119901119894(119905)





119894(119905) is the


119902119894(119905) = 119902

0119894cos120596119894119905 + (

1199010119894

119898120596119894

) sin120596119894119905

119901119894(119905) = 119901

0119894cos120596119894119905 minus 119898120596

1198941199020119894sin120596119894119905

(lowast)



119894th conforma-


_119870119894 = const Δ

_119875119894 =

const Δ_119870119894Δ





119894

References
































































Volume 2014

Zoology






















Advances in

Virolog y



Volume 2014




Enzyme Research



Microbiology



































Volume 2014

Zoology






















Advances in

Virolog y



Volume 2014




Enzyme Research



Microbiology

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